null hypothesis rejected at the 0.05 level

P-Value And Statistical Significance: What It Is & Why It Matters

Saul McLeod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul McLeod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

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The p-value in statistics quantifies the evidence against a null hypothesis. A low p-value suggests data is inconsistent with the null, potentially favoring an alternative hypothesis. Common significance thresholds are 0.05 or 0.01.

Hypothesis testing

When you perform a statistical test, a p-value helps you determine the significance of your results in relation to the null hypothesis.

The null hypothesis (H0) states no relationship exists between the two variables being studied (one variable does not affect the other). It states the results are due to chance and are not significant in supporting the idea being investigated. Thus, the null hypothesis assumes that whatever you try to prove did not happen.

The alternative hypothesis (Ha or H1) is the one you would believe if the null hypothesis is concluded to be untrue.

The alternative hypothesis states that the independent variable affected the dependent variable, and the results are significant in supporting the theory being investigated (i.e., the results are not due to random chance).

What a p-value tells you

A p-value, or probability value, is a number describing how likely it is that your data would have occurred by random chance (i.e., that the null hypothesis is true).

The level of statistical significance is often expressed as a p-value between 0 and 1.

The smaller the p -value, the less likely the results occurred by random chance, and the stronger the evidence that you should reject the null hypothesis.

Remember, a p-value doesn’t tell you if the null hypothesis is true or false. It just tells you how likely you’d see the data you observed (or more extreme data) if the null hypothesis was true. It’s a piece of evidence, not a definitive proof.

Example: Test Statistic and p-Value

Suppose you’re conducting a study to determine whether a new drug has an effect on pain relief compared to a placebo. If the new drug has no impact, your test statistic will be close to the one predicted by the null hypothesis (no difference between the drug and placebo groups), and the resulting p-value will be close to 1. It may not be precisely 1 because real-world variations may exist. Conversely, if the new drug indeed reduces pain significantly, your test statistic will diverge further from what’s expected under the null hypothesis, and the p-value will decrease. The p-value will never reach zero because there’s always a slim possibility, though highly improbable, that the observed results occurred by random chance.

P-value interpretation

The significance level (alpha) is a set probability threshold (often 0.05), while the p-value is the probability you calculate based on your study or analysis.

A p-value less than or equal to your significance level (typically ≤ 0.05) is statistically significant.

A p-value less than or equal to a predetermined significance level (often 0.05 or 0.01) indicates a statistically significant result, meaning the observed data provide strong evidence against the null hypothesis.

This suggests the effect under study likely represents a real relationship rather than just random chance.

For instance, if you set α = 0.05, you would reject the null hypothesis if your p -value ≤ 0.05. 

It indicates strong evidence against the null hypothesis, as there is less than a 5% probability the null is correct (and the results are random).

Therefore, we reject the null hypothesis and accept the alternative hypothesis.

Example: Statistical Significance

Upon analyzing the pain relief effects of the new drug compared to the placebo, the computed p-value is less than 0.01, which falls well below the predetermined alpha value of 0.05. Consequently, you conclude that there is a statistically significant difference in pain relief between the new drug and the placebo.

What does a p-value of 0.001 mean?

A p-value of 0.001 is highly statistically significant beyond the commonly used 0.05 threshold. It indicates strong evidence of a real effect or difference, rather than just random variation.

Specifically, a p-value of 0.001 means there is only a 0.1% chance of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is correct.

Such a small p-value provides strong evidence against the null hypothesis, leading to rejecting the null in favor of the alternative hypothesis.

A p-value more than the significance level (typically p > 0.05) is not statistically significant and indicates strong evidence for the null hypothesis.

This means we retain the null hypothesis and reject the alternative hypothesis. You should note that you cannot accept the null hypothesis; we can only reject it or fail to reject it.

Note : when the p-value is above your threshold of significance,  it does not mean that there is a 95% probability that the alternative hypothesis is true.

One-Tailed Test

Two-Tailed Test

How do you calculate the p-value ?

Most statistical software packages like R, SPSS, and others automatically calculate your p-value. This is the easiest and most common way.

Online resources and tables are available to estimate the p-value based on your test statistic and degrees of freedom.

These tables help you understand how often you would expect to see your test statistic under the null hypothesis.

Understanding the Statistical Test:

Different statistical tests are designed to answer specific research questions or hypotheses. Each test has its own underlying assumptions and characteristics.

For example, you might use a t-test to compare means, a chi-squared test for categorical data, or a correlation test to measure the strength of a relationship between variables.

Be aware that the number of independent variables you include in your analysis can influence the magnitude of the test statistic needed to produce the same p-value.

This factor is particularly important to consider when comparing results across different analyses.

Example: Choosing a Statistical Test

If you’re comparing the effectiveness of just two different drugs in pain relief, a two-sample t-test is a suitable choice for comparing these two groups. However, when you’re examining the impact of three or more drugs, it’s more appropriate to employ an Analysis of Variance ( ANOVA) . Utilizing multiple pairwise comparisons in such cases can lead to artificially low p-values and an overestimation of the significance of differences between the drug groups.

How to report

A statistically significant result cannot prove that a research hypothesis is correct (which implies 100% certainty).

Instead, we may state our results “provide support for” or “give evidence for” our research hypothesis (as there is still a slight probability that the results occurred by chance and the null hypothesis was correct – e.g., less than 5%).

Example: Reporting the results

In our comparison of the pain relief effects of the new drug and the placebo, we observed that participants in the drug group experienced a significant reduction in pain ( M = 3.5; SD = 0.8) compared to those in the placebo group ( M = 5.2; SD  = 0.7), resulting in an average difference of 1.7 points on the pain scale (t(98) = -9.36; p < 0.001).

The 6th edition of the APA style manual (American Psychological Association, 2010) states the following on the topic of reporting p-values:

“When reporting p values, report exact p values (e.g., p = .031) to two or three decimal places. However, report p values less than .001 as p < .001.

The tradition of reporting p values in the form p < .10, p < .05, p < .01, and so forth, was appropriate in a time when only limited tables of critical values were available.” (p. 114)

• Do not use 0 before the decimal point for the statistical value p as it cannot equal 1. In other words, write p = .001 instead of p = 0.001.
• Please pay attention to issues of italics ( p is always italicized) and spacing (either side of the = sign).
• p = .000 (as outputted by some statistical packages such as SPSS) is impossible and should be written as p < .001.
• The opposite of significant is “nonsignificant,” not “insignificant.”

Why is the p -value not enough?

A lower p-value  is sometimes interpreted as meaning there is a stronger relationship between two variables.

However, statistical significance means that it is unlikely that the null hypothesis is true (less than 5%).

To understand the strength of the difference between the two groups (control vs. experimental) a researcher needs to calculate the effect size .

When do you reject the null hypothesis?

In statistical hypothesis testing, you reject the null hypothesis when the p-value is less than or equal to the significance level (α) you set before conducting your test. The significance level is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10.

Remember, rejecting the null hypothesis doesn’t prove the alternative hypothesis; it just suggests that the alternative hypothesis may be plausible given the observed data.

The p -value is conditional upon the null hypothesis being true but is unrelated to the truth or falsity of the alternative hypothesis.

What does p-value of 0.05 mean?

If your p-value is less than or equal to 0.05 (the significance level), you would conclude that your result is statistically significant. This means the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis.

Are all p-values below 0.05 considered statistically significant?

No, not all p-values below 0.05 are considered statistically significant. The threshold of 0.05 is commonly used, but it’s just a convention. Statistical significance depends on factors like the study design, sample size, and the magnitude of the observed effect.

A p-value below 0.05 means there is evidence against the null hypothesis, suggesting a real effect. However, it’s essential to consider the context and other factors when interpreting results.

Researchers also look at effect size and confidence intervals to determine the practical significance and reliability of findings.

How does sample size affect the interpretation of p-values?

Sample size can impact the interpretation of p-values. A larger sample size provides more reliable and precise estimates of the population, leading to narrower confidence intervals.

With a larger sample, even small differences between groups or effects can become statistically significant, yielding lower p-values. In contrast, smaller sample sizes may not have enough statistical power to detect smaller effects, resulting in higher p-values.

Therefore, a larger sample size increases the chances of finding statistically significant results when there is a genuine effect, making the findings more trustworthy and robust.

Can a non-significant p-value indicate that there is no effect or difference in the data?

No, a non-significant p-value does not necessarily indicate that there is no effect or difference in the data. It means that the observed data do not provide strong enough evidence to reject the null hypothesis.

There could still be a real effect or difference, but it might be smaller or more variable than the study was able to detect.

Other factors like sample size, study design, and measurement precision can influence the p-value. It’s important to consider the entire body of evidence and not rely solely on p-values when interpreting research findings.

Can P values be exactly zero?

While a p-value can be extremely small, it cannot technically be absolute zero. When a p-value is reported as p = 0.000, the actual p-value is too small for the software to display. This is often interpreted as strong evidence against the null hypothesis. For p values less than 0.001, report as p < .001

Further Information

• P Value Calculator From T Score
• P-Value Calculator For Chi-Square
• P-values and significance tests (Kahn Academy)
• Hypothesis testing and p-values (Kahn Academy)
• Wasserstein, R. L., Schirm, A. L., & Lazar, N. A. (2019). Moving to a world beyond “ p “< 0.05”.
• Criticism of using the “ p “< 0.05”.
• Publication manual of the American Psychological Association
• Statistics for Psychology Book Download

Bland, J. M., & Altman, D. G. (1994). One and two sided tests of significance: Authors’ reply.  BMJ: British Medical Journal ,  309 (6958), 874.

Goodman, S. N., & Royall, R. (1988). Evidence and scientific research.  American Journal of Public Health ,  78 (12), 1568-1574.

Goodman, S. (2008, July). A dirty dozen: twelve p-value misconceptions . In  Seminars in hematology  (Vol. 45, No. 3, pp. 135-140). WB Saunders.

Lang, J. M., Rothman, K. J., & Cann, C. I. (1998). That confounded P-value.  Epidemiology (Cambridge, Mass.) ,  9 (1), 7-8.

Hypothesis Testing (cont...)

Hypothesis testing, the null and alternative hypothesis.

In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis). So, with respect to our teaching example, the null and alternative hypothesis will reflect statements about all statistics students on graduate management courses.

The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen ( hint: it usually states that something equals zero). For example, the two different teaching methods did not result in different exam performances (i.e., zero difference). Another example might be that there is no relationship between anxiety and athletic performance (i.e., the slope is zero). The alternative hypothesis states the opposite and is usually the hypothesis you are trying to prove (e.g., the two different teaching methods did result in different exam performances). Initially, you can state these hypotheses in more general terms (e.g., using terms like "effect", "relationship", etc.), as shown below for the teaching methods example:

Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the mean exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things. As such, we can state:

Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.

Significance levels

The level of statistical significance is often expressed as the so-called p -value . Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p -value) of observing your sample results (or more extreme) given that the null hypothesis is true . Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?

So, you might get a p -value such as 0.03 (i.e., p = .03). This means that there is a 3% chance of finding a difference as large as (or larger than) the one in your study given that the null hypothesis is true. However, you want to know whether this is "statistically significant". Typically, if there was a 5% or less chance (5 times in 100 or less) that the difference in the mean exam performance between the two teaching methods (or whatever statistic you are using) is as different as observed given the null hypothesis is true, you would reject the null hypothesis and accept the alternative hypothesis. Alternately, if the chance was greater than 5% (5 times in 100 or more), you would fail to reject the null hypothesis and would not accept the alternative hypothesis. As such, in this example where p = .03, we would reject the null hypothesis and accept the alternative hypothesis. We reject it because at a significance level of 0.03 (i.e., less than a 5% chance), the result we obtained could happen too frequently for us to be confident that it was the two teaching methods that had an effect on exam performance.

Whilst there is relatively little justification why a significance level of 0.05 is used rather than 0.01 or 0.10, for example, it is widely used in academic research. However, if you want to be particularly confident in your results, you can set a more stringent level of 0.01 (a 1% chance or less; 1 in 100 chance or less).

One- and two-tailed predictions

When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:

The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted direction of this effect? Let's use our example to highlight these two points.

Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.

Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been:

In other words, we simply take out the word "positive", which implies the direction of our effect. In our example, making a two-tailed prediction may seem strange. After all, it would be logical to expect that "extra" tuition (going to seminar classes as well as lectures) would either have a positive effect on students' performance or no effect at all, but certainly not a negative effect. However, this is just our opinion (and hope) and certainly does not mean that we will get the effect we expect. Generally speaking, making a one-tail prediction (i.e., and testing for it this way) is frowned upon as it usually reflects the hope of a researcher rather than any certainty that it will happen. Notable exceptions to this rule are when there is only one possible way in which a change could occur. This can happen, for example, when biological activity/presence in measured. That is, a protein might be "dormant" and the stimulus you are using can only possibly "wake it up" (i.e., it cannot possibly reduce the activity of a "dormant" protein). In addition, for some statistical tests, one-tailed tests are not possible.

Rejecting or failing to reject the null hypothesis

Let's return finally to the question of whether we reject or fail to reject the null hypothesis.

If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.

Rejecting the Null Hypothesis Using Confidence Intervals

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In an introductory statistics class, there are three main topics that are taught: descriptive statistics and data visualizations, probability and sampling distributions, and statistical inference. Within statistical inference, there are two key methods of statistical inference that are taught, viz. confidence intervals and hypothesis testing . While these two methods are always taught when learning data science and related fields, it is rare that the relationship between these two methods is properly elucidated.

In this article, we’ll begin by defining and describing each method of statistical inference in turn and along the way, state what statistical inference is, and perhaps more importantly, what it isn’t. Then we’ll describe the relationship between the two. While it is typically the case that confidence intervals are taught before hypothesis testing when learning statistics, we’ll begin with the latter since it will allow us to define statistical significance.

Hypothesis Tests

The purpose of a hypothesis test is to answer whether random chance might be responsible for an observed effect. Hypothesis tests use sample statistics to test a hypothesis about population parameters. The null hypothesis, H 0 , is a statement that represents the assumed status quo regarding a variable or variables and it is always about a population characteristic. Some of the ways the null hypothesis is typically glossed are: the population variable is equal to a particular value or there is no difference between the population variables . For example:

• H 0 : μ = 61 in (The mean height of the population of American men is 69 inches)
• H 0 : p 1 -p 2 = 0 (The difference in the population proportions of women who prefer football over baseball and the population proportion of men who prefer football over baseball is 0.)

Note that the null hypothesis always has the equal sign.

The alternative hypothesis, denoted either H 1 or H a , is the statement that is opposed to the null hypothesis (e.g., the population variable is not equal to a particular value  or there is a difference between the population variables ):

• H 1 : μ > 61 im (The mean height of the population of American men is greater than 69 inches.)
• H 1 : p 1 -p 2 ≠ 0 (The difference in the population proportions of women who prefer football over baseball and the population proportion of men who prefer football over baseball is not 0.)

The alternative hypothesis is typically the claim that the researcher hopes to show and it always contains the strict inequality symbols (‘<’ left-sided or left-tailed, ‘≠’ two-sided or two-tailed, and ‘>’ right-sided or right-tailed).

When carrying out a test of H 0 vs. H 1 , the null hypothesis H 0 will be rejected in favor of the alternative hypothesis only if the sample provides convincing evidence that H 0 is false. As such, a statistical hypothesis test is only capable of demonstrating strong support for the alternative hypothesis by rejecting the null hypothesis.

When the null hypothesis is not rejected, it does not mean that there is strong support for the null hypothesis (since it was assumed to be true); rather, only that there is not convincing evidence against the null hypothesis. As such, we never use the phrase “accept the null hypothesis.”

In the classical method of performing hypothesis testing, one would have to find what is called the test statistic and use a table to find the corresponding probability. Happily, due to the advancement of technology, one can use Python (as is done in the Flatiron’s Data Science Bootcamp ) and get the required value directly using a Python library like stats models . This is the p-value , which is short for the probability value.

The p-value is a measure of inconsistency between the hypothesized value for a population characteristic and the observed sample. The p -value is the probability, under the assumption the null hypothesis is true, of obtaining a test statistic value that is a measure of inconsistency between the null hypothesis and the data. If the p -value is less than or equal to the probability of the Type I error, then we can reject the null hypothesis and we have sufficient evidence to support the alternative hypothesis.

Typically the probability of a Type I error ɑ, more commonly known as the level of significance , is set to be 0.05, but it is often prudent to have it set to values less than that such as 0.01 or 0.001. Thus, if p -value ≤ ɑ, then we reject the null hypothesis and we interpret this as saying there is a statistically significant difference between the sample and the population. So if the p -value=0.03 ≤ 0.05 = ɑ, then we would reject the null hypothesis and so have statistical significance, whereas if p -value=0.08 ≥ 0.05 = ɑ, then we would fail to reject the null hypothesis and there would not be statistical significance.

Confidence Intervals

The other primary form of statistical inference are confidence intervals. While hypothesis tests are concerned with testing a claim, the purpose of a confidence interval is to estimate an unknown population characteristic. A confidence interval is an interval of plausible values for a population characteristic. They are constructed so that we have a chosen level of confidence that the actual value of the population characteristic will be between the upper and lower endpoints of the open interval.

The structure of an individual confidence interval is the sample estimate of the variable of interest margin of error. The margin of error is the product of a multiplier value and the standard error, s.e., which is based on the standard deviation and the sample size. The multiplier is where the probability, of level of confidence, is introduced into the formula.

The confidence level is the success rate of the method used to construct a confidence interval. A confidence interval estimating the proportion of American men who state they are an avid fan of the NFL could be (0.40, 0.60) with a 95% level of confidence. The level of confidence is not the probability that that population characteristic is in the confidence interval, but rather refers to the method that is used to construct the confidence interval.

For example, a 95% confidence interval would be interpreted as if one constructed 100 confidence intervals, then 95 of them would contain the true population characteristic. 

Errors and Power

A Type I error, or a false positive, is the error of finding a difference that is not there, so it is the probability of incorrectly rejecting a true null hypothesis is ɑ, where ɑ is the level of significance. It follows that the probability of correctly failing to reject a true null hypothesis is the complement of it, viz. 1 – ɑ. For a particular hypothesis test, if ɑ = 0.05, then its complement would be 0.95 or 95%.

While we are not going to expand on these ideas, we note the following two related probabilities. A Type II error, or false negative, is the probability of failing to reject a false null hypothesis where the probability of a type II error is β and the power is the probability of correctly rejecting a false null hypothesis where power = 1 – β. In common statistical practice, one typically only speaks of the level of significance and the power.

The following table summarizes these ideas , where the column headers refer to what is actually the case, but is unknown. (If the truth or falsity of the null value was truly known, we wouldn’t have to do statistics.)

Hypothesis Tests and Confidence Intervals

Since hypothesis tests and confidence intervals are both methods of statistical inference, then it is reasonable to wonder if they are equivalent in some way. The answer is yes, which means that we can perform hypothesis testing using confidence intervals.

Returning to the example where we have an estimate of the proportion of American men that are avid fans of the NFL, we had (0.40, 0.60) at a 95% confidence level. As a hypothesis test, we could have the alternative hypothesis as H 1 ≠ 0.51. Since the null value of 0.51 lies within the confidence interval, then we would fail to reject the null hypothesis at ɑ = 0.05.

On the other hand, if H 1 ≠ 0.61, then since 0.61 is not in the confidence interval we can reject the null hypothesis at ɑ = 0.05. Note that the confidence level of 95% and the level of significance at ɑ = 0.05 = 5%  are complements, which is the “H o is True” column in the above table.

In general, one can reject the null hypothesis given a null value and a confidence interval for a two-sided test if the null value is not in the confidence interval where the confidence level and level of significance are complements. For one-sided tests, one can still perform a hypothesis test with the confidence level and null value. Not only is there an added layer of complexity for this equivalence, it is the best practice to perform two-sided hypothesis tests since one is not prejudicing the direction of the alternative.

In this discussion of hypothesis testing and confidence intervals, we not only understand when these two methods of statistical inference can be equivalent, but now have a deeper understanding of statistical significance itself and therefore, statistical inference.

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Understanding Hypothesis Tests: Significance Levels (Alpha) and P values in Statistics

Topics: Hypothesis Testing , Statistics

What do significance levels and P values mean in hypothesis tests? What is statistical significance anyway? In this post, I’ll continue to focus on concepts and graphs to help you gain a more intuitive understanding of how hypothesis tests work in statistics.

To bring it to life, I’ll add the significance level and P value to the graph in my previous post in order to perform a graphical version of the 1 sample t-test. It’s easier to understand when you can see what statistical significance truly means!

Here’s where we left off in my last post . We want to determine whether our sample mean (330.6) indicates that this year's average energy cost is significantly different from last year’s average energy cost of $260.

The probability distribution plot above shows the distribution of sample means we’d obtain under the assumption that the null hypothesis is true (population mean = 260) and we repeatedly drew a large number of random samples.

I left you with a question: where do we draw the line for statistical significance on the graph? Now we'll add in the significance level and the P value, which are the decision-making tools we'll need.

We'll use these tools to test the following hypotheses:

• Null hypothesis: The population mean equals the hypothesized mean (260).
• Alternative hypothesis: The population mean differs from the hypothesized mean (260).

What Is the Significance Level (Alpha)?

The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

These types of definitions can be hard to understand because of their technical nature. A picture makes the concepts much easier to comprehend!

The significance level determines how far out from the null hypothesis value we'll draw that line on the graph. To graph a significance level of 0.05, we need to shade the 5% of the distribution that is furthest away from the null hypothesis.

In the graph above, the two shaded areas are equidistant from the null hypothesis value and each area has a probability of 0.025, for a total of 0.05. In statistics, we call these shaded areas the critical region for a two-tailed test. If the population mean is 260, we’d expect to obtain a sample mean that falls in the critical region 5% of the time. The critical region defines how far away our sample statistic must be from the null hypothesis value before we can say it is unusual enough to reject the null hypothesis.

Our sample mean (330.6) falls within the critical region, which indicates it is statistically significant at the 0.05 level.

We can also see if it is statistically significant using the other common significance level of 0.01.

The two shaded areas each have a probability of 0.005, which adds up to a total probability of 0.01. This time our sample mean does not fall within the critical region and we fail to reject the null hypothesis. This comparison shows why you need to choose your significance level before you begin your study. It protects you from choosing a significance level because it conveniently gives you significant results!

Thanks to the graph, we were able to determine that our results are statistically significant at the 0.05 level without using a P value. However, when you use the numeric output produced by statistical software , you’ll need to compare the P value to your significance level to make this determination.

Ready for a demo of Minitab Statistical Software? Just ask! 

What Are P values?

P-values are the probability of obtaining an effect at least as extreme as the one in your sample data, assuming the truth of the null hypothesis.

This definition of P values, while technically correct, is a bit convoluted. It’s easier to understand with a graph!

To graph the P value for our example data set, we need to determine the distance between the sample mean and the null hypothesis value (330.6 - 260 = 70.6). Next, we can graph the probability of obtaining a sample mean that is at least as extreme in both tails of the distribution (260 +/- 70.6).

In the graph above, the two shaded areas each have a probability of 0.01556, for a total probability 0.03112. This probability represents the likelihood of obtaining a sample mean that is at least as extreme as our sample mean in both tails of the distribution if the population mean is 260. That’s our P value!

When a P value is less than or equal to the significance level, you reject the null hypothesis. If we take the P value for our example and compare it to the common significance levels, it matches the previous graphical results. The P value of 0.03112 is statistically significant at an alpha level of 0.05, but not at the 0.01 level.

If we stick to a significance level of 0.05, we can conclude that the average energy cost for the population is greater than 260.

A common mistake is to interpret the P-value as the probability that the null hypothesis is true. To understand why this interpretation is incorrect, please read my blog post  How to Correctly Interpret P Values .

Discussion about Statistically Significant Results

A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. A test result is statistically significant when the sample statistic is unusual enough relative to the null hypothesis that we can reject the null hypothesis for the entire population. “Unusual enough” in a hypothesis test is defined by:

• The assumption that the null hypothesis is true—the graphs are centered on the null hypothesis value.
• The significance level—how far out do we draw the line for the critical region?
• Our sample statistic—does it fall in the critical region?

Keep in mind that there is no magic significance level that distinguishes between the studies that have a true effect and those that don’t with 100% accuracy. The common alpha values of 0.05 and 0.01 are simply based on tradition. For a significance level of 0.05, expect to obtain sample means in the critical region 5% of the time when the null hypothesis is true . In these cases, you won’t know that the null hypothesis is true but you’ll reject it because the sample mean falls in the critical region. That’s why the significance level is also referred to as an error rate!

This type of error doesn’t imply that the experimenter did anything wrong or require any other unusual explanation. The graphs show that when the null hypothesis is true, it is possible to obtain these unusual sample means for no reason other than random sampling error. It’s just luck of the draw.

Significance levels and P values are important tools that help you quantify and control this type of error in a hypothesis test. Using these tools to decide when to reject the null hypothesis increases your chance of making the correct decision.

If you like this post, you might want to read the other posts in this series that use the same graphical framework:

• Previous: Why We Need to Use Hypothesis Tests
• Next: Confidence Intervals and Confidence Levels

If you'd like to see how I made these graphs, please read: How to Create a Graphical Version of the 1-sample t-Test .

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How to Find the Cutoff Point for Rejecting a Null Hypothesis

Statistics for dummies.

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In statistics, if you want to draw conclusions about a null hypothesis H 0 (reject or fail to reject) based on a p- value, you need to set a predetermined cutoff point where only those p -values less than or equal to the cutoff will result in rejecting H 0 .

While 0.05 is a very popular cutoff value for rejecting H 0 , cutoff points and resulting decisions can vary — some people use stricter cutoffs, such as 0.01, requiring more evidence before rejecting H 0 , and others may have less strict cutoffs, such as 0.10, requiring less evidence.

If H 0 is rejected (that is, the p -value is less than or equal to the predetermined significance level), the researcher can say she’s found a statistically significant result. A result is statistically significant if it’s too unlikely to have occurred by chance assuming H 0 is true. If you get a statistically significant result, you have enough evidence to reject the claim, H 0 , and conclude that something different or new is in effect (that is, H a ).

The significance level can be thought of as the highest possible p- value that would reject H 0 and declare the results statistically significant. Following are the general rules for making a decision about H 0 based on a p- value:

If the p- value is less than or equal to your significance level, then it meets your requirements for having enough evidence against H 0 ; you reject H 0 .

If the p- value is greater than your significance level, your data failed to show evidence beyond a reasonable doubt; you fail to reject H 0 .

However, if you plan to make decisions about H 0 by comparing the p- value to your significance level, you must decide on your significance level ahead of time. It wouldn’t be fair to change your cutoff point after you’ve got a sneak peak at what’s happening in the data.

You may be wondering whether it’s okay to say “Accept H 0 ” instead of “Fail to reject H 0 .” The answer is a big no. In a hypothesis test, you are not trying to show whether or not H 0 is true (which accept implies) — indeed, if you knew whether H 0 was true, you wouldn’t be doing the hypothesis test in the first place. You’re trying to show whether you have enough evidence to say H 0 is false, based on your data. Either you have enough evidence to say it’s false (in which case you reject H 0 ) or you don’t have enough evidence to say it’s false (in which case you fail to reject H 0 ).

These guidelines help you make a decision (reject or fail to reject H 0 ) based on a p- value when your significance level is 0.05:

If the p- value is less than 0.01 (very small), the results are considered highly statistically significant — reject H 0 .

If the p- value is between 0.05 and 0.01 (but not super-close to 0.05), the results are considered statistically significant — reject H 0 .

If the p- value is really close to 0.05 (like 0.051 or 0.049), the results should be considered marginally significant — the decision could go either way.

If the p- value is greater than (but not super-close to) 0.05, the results are considered non-significant — you fail to reject H 0 .

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Statistics By Jim

Making statistics intuitive

How Hypothesis Tests Work: Significance Levels (Alpha) and P values

By Jim Frost 45 Comments

Hypothesis testing is a vital process in inferential statistics where the goal is to use sample data to draw conclusions about an entire population . In the testing process, you use significance levels and p-values to determine whether the test results are statistically significant.

You hear about results being statistically significant all of the time. But, what do significance levels, P values, and statistical significance actually represent? Why do we even need to use hypothesis tests in statistics?

In this post, I answer all of these questions. I use graphs and concepts to explain how hypothesis tests function in order to provide a more intuitive explanation. This helps you move on to understanding your statistical results.

Hypothesis Test Example Scenario

To start, I’ll demonstrate why we need to use hypothesis tests using an example.

A researcher is studying fuel expenditures for families and wants to determine if the monthly cost has changed since last year when the average was $260 per month. The researcher draws a random sample of 25 families and enters their monthly costs for this year into statistical software. You can download the CSV data file: FuelsCosts . Below are the descriptive statistics for this year.

We’ll build on this example to answer the research question and show how hypothesis tests work.

Descriptive Statistics Alone Won’t Answer the Question

The researcher collected a random sample and found that this year’s sample mean (330.6) is greater than last year’s mean (260). Why perform a hypothesis test at all? We can see that this year’s mean is higher by $70! Isn’t that different?

Regrettably, the situation isn’t as clear as you might think because we’re analyzing a sample instead of the full population. There are huge benefits when working with samples because it is usually impossible to collect data from an entire population. However, the tradeoff for working with a manageable sample is that we need to account for sample error.

The sampling error is the gap between the sample statistic and the population parameter. For our example, the sample statistic is the sample mean, which is 330.6. The population parameter is μ, or mu, which is the average of the entire population. Unfortunately, the value of the population parameter is not only unknown but usually unknowable. Learn more about Sampling Error .

We obtained a sample mean of 330.6. However, it’s conceivable that, due to sampling error, the mean of the population might be only 260. If the researcher drew another random sample, the next sample mean might be closer to 260. It’s impossible to assess this possibility by looking at only the sample mean. Hypothesis testing is a form of inferential statistics that allows us to draw conclusions about an entire population based on a representative sample. We need to use a hypothesis test to determine the likelihood of obtaining our sample mean if the population mean is 260.

Background information : The Difference between Descriptive and Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

A Sampling Distribution Determines Whether Our Sample Mean is Unlikely

It is very unlikely for any sample mean to equal the population mean because of sample error. In our case, the sample mean of 330.6 is almost definitely not equal to the population mean for fuel expenditures.

If we could obtain a substantial number of random samples and calculate the sample mean for each sample, we’d observe a broad spectrum of sample means. We’d even be able to graph the distribution of sample means from this process.

This type of distribution is called a sampling distribution. You obtain a sampling distribution by drawing many random samples of the same size from the same population. Why the heck would we do this?

Because sampling distributions allow you to determine the likelihood of obtaining your sample statistic and they’re crucial for performing hypothesis tests.

Luckily, we don’t need to go to the trouble of collecting numerous random samples! We can estimate the sampling distribution using the t-distribution, our sample size, and the variability in our sample.

We want to find out if the average fuel expenditure this year (330.6) is different from last year (260). To answer this question, we’ll graph the sampling distribution based on the assumption that the mean fuel cost for the entire population has not changed and is still 260. In statistics, we call this lack of effect, or no change, the null hypothesis . We use the null hypothesis value as the basis of comparison for our observed sample value.

Sampling distributions and t-distributions are types of probability distributions.

Related posts : Sampling Distributions and Understanding Probability Distributions

Graphing our Sample Mean in the Context of the Sampling Distribution

The graph below shows which sample means are more likely and less likely if the population mean is 260. We can place our sample mean in this distribution. This larger context helps us see how unlikely our sample mean is if the null hypothesis is true (μ = 260).

The graph displays the estimated distribution of sample means. The most likely values are near 260 because the plot assumes that this is the true population mean. However, given random sampling error, it would not be surprising to observe sample means ranging from 167 to 352. If the population mean is still 260, our observed sample mean (330.6) isn’t the most likely value, but it’s not completely implausible either.

The Role of Hypothesis Tests

The sampling distribution shows us that we are relatively unlikely to obtain a sample of 330.6 if the population mean is 260. Is our sample mean so unlikely that we can reject the notion that the population mean is 260?

In statistics, we call this rejecting the null hypothesis. If we reject the null for our example, the difference between the sample mean (330.6) and 260 is statistically significant. In other words, the sample data favor the hypothesis that the population average does not equal 260.

However, look at the sampling distribution chart again. Notice that there is no special location on the curve where you can definitively draw this conclusion. There is only a consistent decrease in the likelihood of observing sample means that are farther from the null hypothesis value. Where do we decide a sample mean is far away enough?

To answer this question, we’ll need more tools—hypothesis tests! The hypothesis testing procedure quantifies the unusualness of our sample with a probability and then compares it to an evidentiary standard. This process allows you to make an objective decision about the strength of the evidence.

We’re going to add the tools we need to make this decision to the graph—significance levels and p-values!

These tools allow us to test these two hypotheses:

• Null hypothesis: The population mean equals the null hypothesis mean (260).
• Alternative hypothesis: The population mean does not equal the null hypothesis mean (260).

Related post : Hypothesis Testing Overview

What are Significance Levels (Alpha)?

A significance level, also known as alpha or α, is an evidentiary standard that a researcher sets before the study. It defines how strongly the sample evidence must contradict the null hypothesis before you can reject the null hypothesis for the entire population. The strength of the evidence is defined by the probability of rejecting a null hypothesis that is true. In other words, it is the probability that you say there is an effect when there is no effect.

For instance, a significance level of 0.05 signifies a 5% risk of deciding that an effect exists when it does not exist.

Lower significance levels require stronger sample evidence to be able to reject the null hypothesis. For example, to be statistically significant at the 0.01 significance level requires more substantial evidence than the 0.05 significance level. However, there is a tradeoff in hypothesis tests. Lower significance levels also reduce the power of a hypothesis test to detect a difference that does exist.

The technical nature of these types of questions can make your head spin. A picture can bring these ideas to life!

To learn a more conceptual approach to significance levels, see my post about Understanding Significance Levels .

Graphing Significance Levels as Critical Regions

On the probability distribution plot, the significance level defines how far the sample value must be from the null value before we can reject the null. The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.

To represent a significance level of 0.05, I’ll shade 5% of the distribution furthest from the null value.

The two shaded regions in the graph are equidistant from the central value of the null hypothesis. Each region has a probability of 0.025, which sums to our desired total of 0.05. These shaded areas are called the critical region for a two-tailed hypothesis test.

The critical region defines sample values that are improbable enough to warrant rejecting the null hypothesis. If the null hypothesis is correct and the population mean is 260, random samples (n=25) from this population have means that fall in the critical region 5% of the time.

Our sample mean is statistically significant at the 0.05 level because it falls in the critical region.

Related posts : One-Tailed and Two-Tailed Tests Explained , What Are Critical Values? , and T-distribution Table of Critical Values

Comparing Significance Levels

Let’s redo this hypothesis test using the other common significance level of 0.01 to see how it compares.

This time the sum of the two shaded regions equals our new significance level of 0.01. The mean of our sample does not fall within with the critical region. Consequently, we fail to reject the null hypothesis. We have the same exact sample data, the same difference between the sample mean and the null hypothesis value, but a different test result.

What happened? By specifying a lower significance level, we set a higher bar for the sample evidence. As the graph shows, lower significance levels move the critical regions further away from the null value. Consequently, lower significance levels require more extreme sample means to be statistically significant.

You must set the significance level before conducting a study. You don’t want the temptation of choosing a level after the study that yields significant results. The only reason I compared the two significance levels was to illustrate the effects and explain the differing results.

The graphical version of the 1-sample t-test we created allows us to determine statistical significance without assessing the P value. Typically, you need to compare the P value to the significance level to make this determination.

Related post : Step-by-Step Instructions for How to Do t-Tests in Excel

What Are P values?

P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.

This tortuous, technical definition for P values can make your head spin. Let’s graph it!

First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.

The total probability of the two shaded regions is 0.03112. If the null hypothesis value (260) is true and you drew many random samples, you’d expect sample means to fall in the shaded regions about 3.1% of the time. In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true. That’s the P value!

Learn more about How to Find the P Value .

Using P values and Significance Levels Together

If your P value is less than or equal to your alpha level, reject the null hypothesis.

The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01. Again, in practice, you pick one significance level before the experiment and stick with it!

Using the significance level of 0.05, the sample effect is statistically significant. Our data support the alternative hypothesis, which states that the population mean doesn’t equal 260. We can conclude that mean fuel expenditures have increased since last year.

P values are very frequently misinterpreted as the probability of rejecting a null hypothesis that is actually true. This interpretation is wrong! To understand why, please read my post: How to Interpret P-values Correctly .

Discussion about Statistically Significant Results

Hypothesis tests determine whether your sample data provide sufficient evidence to reject the null hypothesis for the entire population. To perform this test, the procedure compares your sample statistic to the null value and determines whether it is sufficiently rare. “Sufficiently rare” is defined in a hypothesis test by:

• Assuming that the null hypothesis is true—the graphs center on the null value.
• The significance (alpha) level—how far out from the null value is the critical region?
• The sample statistic—is it within the critical region?

There is no special significance level that correctly determines which studies have real population effects 100% of the time. The traditional significance levels of 0.05 and 0.01 are attempts to manage the tradeoff between having a low probability of rejecting a true null hypothesis and having adequate power to detect an effect if one actually exists.

The significance level is the rate at which you incorrectly reject null hypotheses that are actually true ( type I error ). For example, for all studies that use a significance level of 0.05 and the null hypothesis is correct, you can expect 5% of them to have sample statistics that fall in the critical region. When this error occurs, you aren’t aware that the null hypothesis is correct, but you’ll reject it because the p-value is less than 0.05.

This error does not indicate that the researcher made a mistake. As the graphs show, you can observe extreme sample statistics due to sample error alone. It’s the luck of the draw!

Related posts : Statistical Significance: Definition & Meaning and Types of Errors in Hypothesis Testing

Hypothesis tests are crucial when you want to use sample data to make conclusions about a population because these tests account for sample error. Using significance levels and P values to determine when to reject the null hypothesis improves the probability that you will draw the correct conclusion.

Keep in mind that statistical significance doesn’t necessarily mean that the effect is important in a practical, real-world sense. For more information, read my post about Practical vs. Statistical Significance .

If you like this post, read the companion post: How Hypothesis Tests Work: Confidence Intervals and Confidence Levels .

You can also read my other posts that describe how other tests work:

• How t-Tests Work
• How the F-test works in ANOVA
• How Chi-Squared Tests of Independence Work

To see an alternative approach to traditional hypothesis testing that does not use probability distributions and test statistics, learn about bootstrapping in statistics !

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Reader Interactions

December 11, 2022 at 10:56 am

For very easy concept about level of significance & p-value 1.Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 5% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 5% (may be 4% or 3% or 2% even less, it is p-value) it means his results are significant. Otherwise he has error > 5% (may be 6% or 7% or 8% even more, it is p-value) it means his results are non-significant. 2. Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 1% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 1% (may be 0.9% or 0.8% or 0.7% even less, it is p-value) it means his results are significant. Otherwise he has error > 1% (may be 1.1% or 1.5% or 2% even more, it is p-value) it means his results are non-significant. p-value is significant or not mainly dependent upon the level of significance.

December 11, 2022 at 7:50 pm

I think that approach helps explain how to determine statistical significance–is the p-value less than or equal to the significance level. However, it doesn’t really explain what statistical significance means. I find that comparing the p-value to the significance level is the easy part. Knowing what it means and how to choose your significance level is the harder part!

December 3, 2022 at 5:54 pm

What would you say to someone who believes that a p-value higher than the level of significance (alpha) means the null hypothesis has been proven? Should you support that statement or deny it?

December 3, 2022 at 10:18 pm

Hi Emmanuel,

When the p-value is greater than the significance level, you fail to reject the null hypothesis . That is different than proving it. To learn why and what it means, click the link to read a post that I’ve written that will answer your question!

April 19, 2021 at 12:27 am

Thank you so much Sir

April 18, 2021 at 2:37 pm

Hi sir, your blogs are much more helpful for clearing the concepts of statistics, as a researcher I find them much more useful. I have some quarries:

1. In many research papers I have seen authors using the statement ” means or values are statically at par at p = 0.05″ when they do some pair wise comparison between the treatments (a kind of post hoc) using some value of CD (critical difference) or we can say LSD which is calculated using alpha not using p. So with this article I think this should be alpha =0.05 or 5%, not p = 0.05 earlier I thought p and alpha are same. p it self is compared with alpha 0.05. Correct me if I am wrong.

2. When we can draw a conclusion using critical value based on critical values (CV) which is based on alpha values in different tests (e.g. in F test CV is at F (0.05, t-1, error df) when alpha is 0.05 which is table value of F and is compared with F calculated for drawing the conclusion); then why we go for p values, and draw a conclusion based on p values, even many online software do not give p value, they just mention CD (LSD)

3. can you please help me in interpreting interaction in two factor analysis (Factor A X Factor b) in Anova.

Thank You so much!

(Commenting again as I have not seen my comment in comment list; don’t know why)

April 18, 2021 at 10:57 pm

Hi Himanshu,

I manually approve comments so there will be some time lag involved before they show up.

Regarding your first question, yes, you’re correct. Test results are significant at particular significance levels or alpha. They should not use p to define the significance level. You’re also correct in that you compare p to alpha.

Critical values are a different (but related) approach for determining significance. It was more common before computer analysis took off because it reduced the calculations. Using this approach in its simplest form, you only know whether a result is significant or not at the given alpha. You just determine whether the test statistic falls within a critical region to determine statistical significance or not significant. However, it is ok to supplement this type of result with the actual p-value. Knowing the precise p-value provides additional information that significant/not significant does not provide. The critical value and p-value approaches will always agree too. For more information about why the exact p-value is useful, read my post about Five Tips for Interpreting P-values .

Finally, I’ve written about two-way ANOVA in my post, How to do Two-Way ANOVA in Excel . Additionally, I write about it in my Hypothesis Testing ebook .

January 28, 2021 at 3:12 pm

Thank you for your answer, Jim, I really appreciate it. I’m taking a Coursera stats course and online learning without being able to ask questions of a real teacher is not my forte!

You’re right, I don’t think I’m ready for that calculation! However, I think I’m struggling with something far more basic, perhaps even the interpretation of the t-table? I’m just not sure how you came up with the p-value as .03112, with the 24 degrees of freedom. When I pull up a t-table and look at the 24-degrees of freedom row, I’m not sure how any of those numbers correspond with your answer? Either the single tail of 0.01556 or the combined of 0.03112. What am I not getting? (which, frankly, could be a lot!!) Again, thank you SO much for your time.

January 28, 2021 at 11:19 pm

Ah ok, I see! First, let me point you to several posts I’ve written about t-values and the t-distribution. I don’t cover those in this post because I wanted to present a simplified version that just uses the data in its regular units. The basic idea is that the hypothesis tests actually convert all your raw data down into one value for a test statistic, such as the t-value. And then it uses that test statistic to determine whether your results are statistically significant. To be significant, the t-value must exceed a critical value, which is what you lookup in the table. Although, nowadays you’d typically let your software just tell you.

So, read the following two posts, which covers several aspects of t-values and distributions. And then if you have more questions after that, you can post them. But, you’ll have a lot more information about them and probably some of your questions will be answered! T-values T-distributions

January 27, 2021 at 3:10 pm

Jim, just found your website and really appreciate your thoughtful, thorough way of explaining things. I feel very dumb, but I’m struggling with p-values and was hoping you could help me.

Here’s the section that’s getting me confused:

“First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.

** I’m good up to this point. Draw the picture, do the subtraction, shade the regions. BUT, I’m not sure how to figure out the area of the shaded region — even with a T-table. When I look at the T-table on 24 df, I’m not sure what to do with those numbers, as none of them seem to correspond in any way to what I’m looking at in the problem. In the end, I have no idea how you calculated each shaded area being 0.01556.

I feel like there’s a (very simple) step that everyone else knows how to do, but for some reason I’m missing it.

Again, dumb question, but I’d love your help clarifying that.

thank you, Sara

January 27, 2021 at 9:51 pm

That’s not a dumb question at all. I actually don’t show or explain the calculations for figuring out the area. The reason for that is the same reason why students never calculate the critical t-values for their test, instead you look them up in tables or use statistical software. The common reason for all that is because calculating these values is extremely complicated! It’s best to let software do that for you or, when looking critical values, use the tables!

The principal though is that percentage of the area under the curve equals the probability that values will fall within that range.

And then, for this example, you’d need to figure out the area under the curve for particular ranges!

January 15, 2021 at 10:57 am

HI Jim, I have a question related to Hypothesis test.. in Medical imaging, there are different way to measure signal intensity (from a tumor lesion for example). I tested for the same 100 patients 4 different ways to measure tumor captation to a injected dose. So for the 100 patients, i got 4 linear regression (relation between injected dose and measured quantity at tumor sites) = so an output of 4 equations Condition A output = -0,034308 + 0,0006602*input Condition B output = 0,0117631 + 0,0005425*input Condition C output = 0,0087871 + 0,0005563*input Condition D output = 0,001911 + 0,0006255*input

My question : i want to compare the 4 methods to find the best one (compared to others) : do Hypothesis test good to me… and if Yes, i do not find test to perform it. Can you suggest me a software. I uselly used JMP for my stats… but open to other softwares…

THank for your time G

November 16, 2020 at 5:42 am

Thank you very much for writing about this topic!

Your explanation made more sense to me about: Why we reject Null Hypothesis when p value < significance level

Kind greetings, Jalal

September 25, 2020 at 1:04 pm

Hi Jim, Your explanations are so helpful! Thank you. I wondered about your first graph. I see that the mean of the graph is 260 from the null hypothesis, and it looks like the standard deviation of the graph is about 31. Where did you get 31 from? Thank you

September 25, 2020 at 4:08 pm

Hi Michelle,

That is a great question. Very observant. And it gets to how these tests work. The hypothesis test that I’m illustrating here is the one-sample t-test. And this graph illustrates the sampling distribution for the t-test. T-tests use the t-distribution to determine the sampling distribution. For the t-distribution, you need to specify the degrees of freedom, which entirely defines the distribution (i.e., it’s the only parameter). For 1-sample t-tests, the degrees of freedom equal the number of observations minus 1. This dataset has 25 observations. Hence, the 24 DF you see in the graph.

Unlike the normal distribution, there is no standard deviation parameter. Instead, the degrees of freedom determines the spread of the curve. Typically, with t-tests, you’ll see results discussed in terms of t-values, both for your sample and for defining the critical regions. However, for this introductory example, I’ve converted the t-values into the raw data units (t-value * SE mean).

So, the standard deviation you’re seeing in the graph is a result of the spread of the underlying t-distribution that has 24 degrees of freedom and then applying the conversion from t-values to raw values.

September 10, 2020 at 8:19 am

Your blog is incredible.

I am having difficulty understanding why the phrase ‘as extreme as’ is required in the definition of p-value (“P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.”)

Why can’t P-Values simply be defined as “The probability of sample observation if the null hypothesis is correct?”

In your other blog titled ‘Interpreting P values’ you have explained p-values as “P-values indicate the believability of the devil’s advocate case that the null hypothesis is correct given the sample data”. I understand (or accept) this explanation. How does one move from this definition to one that contains the phrase ‘as extreme as’?

September 11, 2020 at 5:05 pm

Thanks so much for your kind words! I’m glad that my website has been helpful!

The key to understanding the “at least as extreme” wording lies in the probability plots for p-values. Using probability plots for continuous data, you can calculate probabilities, but only for ranges of values. I discuss this in my post about understanding probability distributions . In a nutshell, we need a range of values for these probabilities because the probabilities are derived from the area under a distribution curve. A single value just produces a line on these graphs rather than an area. Those ranges are the shaded regions in the probability plots. For p-values, the range corresponds to the “at least as extreme” wording. That’s where it comes from. We need a range to calculate a probability. We can’t use the single value of the observed effect because it doesn’t produce an area under the curve.

I hope that helps! I think this is a particularly confusing part of understanding p-values that most people don’t understand.

August 7, 2020 at 5:45 pm

Hi Jim, thanks for the post.

Could you please clarify the following excerpt from ‘Graphing Significance Levels as Critical Regions’:

“The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.”

I’m not sure if I understood this correctly. If the sample value fall in one of the shaded regions, doesn’t mean that the null hypothesis can be rejected, hence that is not correct?

August 7, 2020 at 10:23 pm

Think of it this way. There are two basic reasons for why a sample value could fall in a critical region:

• The null hypothesis is correct and random chance caused the sample value to be unusual.
• The null hypothesis is not correct.

You don’t know which one is true. Remember, just because you reject the null hypothesis it doesn’t mean the null is false. However, by using hypothesis tests to determine statistical significance, you control the chances of #1 occurring. The rate at which #1 occurs equals your significance level. On the hand, you don’t know the probability of the sample value falling in a critical region if the alternative hypothesis is correct (#2). It depends on the precise distribution for the alternative hypothesis and you usually don’t know that, which is why you’re testing the hypotheses in the first place!

I hope I answered the question you were asking. If not, feel free to ask follow up questions. Also, this ties into how to interpret p-values . It’s not exactly straightforward. Click the link to learn more.

June 4, 2020 at 6:17 am

Hi Jim, thank you very much for your answer. You helped me a lot!

June 3, 2020 at 5:23 pm

Hi, Thanks for this post. I’ve been learning a lot with you. My question is regarding to lack of fit. The p-value of my lack of fit is really low, making my lack of fit significant, meaning my model does not fit well. Is my case a “false negative”? given that my pure error is really low, making the computation of the lack of fit low. So it means my model is good. Below I show some information, that I hope helps to clarify my question.

SumSq DF MeanSq F pValue ________ __ ________ ______ __________

Total 1246.5 18 69.25 Model 1241.7 6 206.94 514.43 9.3841e-14 . Linear 1196.6 3 398.87 991.53 1.2318e-14 . Nonlinear 45.046 3 15.015 37.326 2.3092e-06 Residual 4.8274 12 0.40228 . Lack of fit 4.7388 7 0.67698 38.238 0.0004787 . Pure error 0.088521 5 0.017704

June 3, 2020 at 7:53 pm

As you say, a low p-value for a lack of fit test indicates that the model doesn’t fit your data adequately. This is a positive result for the test, which means it can’t be a “false negative.” At best, it could be a false positive, meaning that your data actually fit model well despite the low p-value.

I’d recommend graphing the residuals and looking for patterns . There is probably a relationship between variables that you’re not modeling correctly, such as curvature or interaction effects. There’s no way to diagnose the specific nature of the lack-of-fit problem by using the statistical output. You’ll need the graphs.

If there are no patterns in the residual plots, then your lack-of-fit results might be a false positive.

I hope this helps!

May 30, 2020 at 6:23 am

First of all, I have to say there are not many resources that explain a complicated topic in an easier manner.

My question is, how do we arrive at “if p value is less than alpha, we reject the null hypothesis.”

Is this covered in a separate article I could read?

Thanks Shekhar

May 25, 2020 at 12:21 pm

Hi Jim, terrific website, blog, and after this I’m ordering your book. One of my biggest challenges is nomenclature, definitions, context, and formulating the hypotheses. Here’s one I want to double-be-sure I understand: From above you write: ” These tools allow us to test these two hypotheses:

Null hypothesis: The population mean equals the null hypothesis mean (260). Alternative hypothesis: The population mean does not equal the null hypothesis mean (260). ” I keep thinking that 260 is the population mean mu, the underlying population (that we never really know exactly) and that the Null Hypothesis is comparing mu to x-bar (the sample mean of the 25 families randomly sampled w mean = sample mean = x-bar = 330.6).

So is the following incorrect, and if so, why? Null hypothesis: The population mean mu=260 equals the null hypothesis mean x-bar (330.6). Alternative hypothesis: The population mean mu=269 does not equal the null hypothesis mean x-bar (330.6).

And my thinking is that usually the formulation of null and alternative hypotheses is “test value” = “mu current of underlying population”, whereas I read the formulation on the webpage above to be the reverse.

Any comments appreciated. Many Thanks,

May 26, 2020 at 8:56 pm

The null hypothesis states that population value equals the null value. Now, I know that’s not particularly helpful! But, the null value varies based on test and context. So, in this example, we’re setting the null value aa $260, which was the mean from the previous year. So, our null hypothesis states:

Null: the population mean (mu) = 260. Alternative: the population mean ≠ 260.

These hypothesis statements are about the population parameter. For this type of one-sample analysis, the target or reference value you specify is the null hypothesis value. Additionally, you don’t include the sample estimate in these statements, which is the X-bar portion you tacked on at the end. It’s strictly about the value of the population parameter you’re testing. You don’t know the value of the underlying distribution. However, given the mutually exclusive nature of the null and alternative hypothesis, you know one or the other is correct. The null states that mu equals 260 while the alternative states that it doesn’t equal 260. The data help you decide, which brings us to . . .

However, the procedure does compare our sample data to the null hypothesis value, which is how it determines how strong our evidence is against the null hypothesis.

I hope I answered your question. If not, please let me know!

May 8, 2020 at 6:00 pm

Really using the interpretation “In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”, our head seems to tie a knot. However, doing the reverse interpretation, it is much more intuitive and easier. That is, we will observe the sample effect of at least 70.6 in about 96.9% of the time, if the null is false (that is, our hypothesis is true).

May 8, 2020 at 7:25 pm

Your phrasing really isn’t any simpler. And it has the additional misfortune of being incorrect.

What you’re essentially doing is creating a one-sided confidence interval by using the p-value from a two-sided test. That’s incorrect in two ways.

• Don’t mix and match one-sided and two-sided test results.
• Confidence levels are determine by the significance level, not p-values.

So, what you need is a two-sided 95% CI (1-alpha). You could then state the results are statistically significant and you have 95% confidence that the population effect is between X and Y. If you want a lower bound as you propose, then you’ll need to use a one-sided hypothesis test with a 95% Lower Bound. That’ll give you a different value for the lower bound than the one you use.

I like confidence intervals. As I write elsewhere, I think they’re easier to understand and provide more information than a binary test result. But, you need to use them correctly!

One other point. When you are talking about p-values, it’s always under the assumption that the null hypothesis is correct. You *never* state anything about the p-value in relation to the null being false (i.e. alternative is true). But, if you want to use the type of phrasing you suggest, use it in the context of CIs and incorporate the points I cover above.

February 10, 2020 at 11:13 am

Muchas gracias profesor por compartir sus conocimientos. Un saliud especial desde Colombia.

August 6, 2019 at 11:46 pm

i found this really helpful . also can you help me out ?

I’m a little confused Can you tell me if level of significance and pvalue are comparable or not and if they are what does it mean if pvalue < LS . Do we reject the null hypothesis or do we accept the null hypothesis ?

August 7, 2019 at 12:49 am

Hi Divyanshu,

Yes, you compare the p-value to the significance level. When the p-value is less than the significance level (alpha), your results are statistically significant and you reject the null hypothesis.

I’d suggest re-reading the “Using P values and Significance Levels Together” section near the end of this post more closely. That describes the process. The next section describes what it all means.

July 1, 2019 at 4:19 am

sure.. I will use only in my class rooms that too offline with due credits to your orginal page. I will encourage my students to visit your blog . I have purchased your eBook on Regressions….immensely useful.

July 1, 2019 at 9:52 am

Hi Narasimha, that sounds perfect. Thanks for buying my ebook as well. I’m thrilled to hear that you’ve found it to be helpful!

June 28, 2019 at 6:22 am

I have benefited a lot by your writings….Can I share the same with my students in the classroom?

June 30, 2019 at 8:44 pm

Hi Narasimha,

Yes, you can certainly share with your students. Please attribute my original page. And please don’t copy whole sections of my posts onto another webpage as that can be bad with Google! Thanks!

February 11, 2019 at 7:46 pm

Hello, great site and my apologies if the answer to the following question exists already.

I’ve always wondered why we put the sampling distribution about the null hypothesis rather than simply leave it about the observed mean. I can see mathematically we are measuring the same distance from the null and basically can draw the same conclusions.

For example we take a sample (say 50 people) we gather an observation (mean wage) estimate the standard error in that observation and so can build a sampling distribution about the observed mean. That sampling distribution contains a confidence interval, where say, i am 95% confident the true mean lies (i.e. in repeated sampling the true mean would reside within this interval 95% of the time).

When i use this for a hyp-test, am i right in saying that we place the sampling dist over the reference level simply because it’s mathematically equivalent and it just seems easier to gauge how far the observation is from 0 via t-stats or its likelihood via p-values?

It seems more natural to me to look at it the other way around. leave the sampling distribution on the observed value, and then look where the null sits…if it’s too far left or right then it is unlikely the true population parameter is what we believed it to be, because if the null were true it would only occur ~ 5% of the time in repeated samples…so perhaps we need to change our opinion.

Can i interpret a hyp-test that way? Or do i have a misconception?

February 12, 2019 at 8:25 pm

The short answer is that, yes, you can draw the interval around the sample mean instead. And, that is, in fact, how you construct confidence intervals. The distance around the null hypothesis for hypothesis tests and the distance around the sample for confidence intervals are the same distance, which is why the results will always agree as long as you use corresponding alpha levels and confidence levels (e.g., alpha 0.05 with a 95% confidence level). I write about how this works in a post about confidence intervals .

I prefer confidence intervals for a number of reasons. They’ll indicate whether you have significant results if they exclude the null value and they indicate the precision of the effect size estimate. Corresponding with what you’re saying, it’s easier to gauge how far a confidence interval is from the null value (often zero) whereas a p-value doesn’t provide that information. See Practical versus Statistical Significance .

So, you don’t have any misconception at all! Just refer to it as a confidence interval rather than a hypothesis test, but, of course, they are very closely related.

January 9, 2019 at 10:37 pm

Hi Jim, Nice Article.. I have a question… I read the Central limit theorem article before this article…

Coming to this article, During almost every hypothesis test, we draw a normal distribution curve assuming there is a sampling distribution (and then we go for test statistic, p value etc…). Do we draw a normal distribution curve for hypo tests because of the central limit theorem…

Thanks in advance, Surya

January 10, 2019 at 1:57 am

These distributions are actually the t-distribution which are different from the normal distribution. T-distributions only have one parameter–the degrees of freedom. As the DF of increases, the t-distribution tightens up. Around 25 degrees of freedom, the t-distribution approximates the normal distribution. Depending on the type of t-test, this corresponds to a sample size of 26 or 27. Similarly, the sampling distribution of the means also approximate the normal distribution at around these sample sizes. With a large enough sample size, both the t-distribution and the sample distribution converge to a normal distribution regardless (largely) of the underlying population distribution. So, yes, the central limit theorem plays a strong role in this.

It’s more accurate to say that central limit theorem causes the sampling distribution of the means to converge on the same distribution that the t-test uses, which allows you to assume that the test produces valid results. But, technically, the t-test is based on the t-distribution.

Problems can occur if the underlying distribution is non-normal and you have a small sample size. In that case, the sampling distribution of the means won’t approximate the t-distribution that the t-test uses. However, the test results will assume that it does and produce results based on that–which is why it causes problems!

November 19, 2018 at 9:15 am

Dear Jim! Thank you very much for your explanation. I need your help to understand my data. I have two samples (about 300 observations) with biased distributions. I did the ttest and obtained the p-value, which is quite small. Can I draw the conclusion that the effect size is small even when the distribution of my data is not normal? Thank you

November 19, 2018 at 9:34 am

Hi Tetyana,

First, when you say that your p-value is small and that you want to “draw the conclusion that the effect size is small,” I assume that you mean statistically significant. When the p-value is low, the null hypothesis must go! In other words, you reject the null and conclude that there is a statistically significant effect–not a small effect.

Now, back to the question at hand! Yes, When you have a sufficiently large sample-size, t-tests are robust to departures from normality. For a 2-sample t-test, you should have at least 15 samples per group, which you exceed by quite a bit. So, yes, you can reliably conclude that your results are statistically significant!

You can thank the central limit theorem! 🙂

September 10, 2018 at 12:18 am

Hello Jim, I am very sorry; I have very elementary of knowledge of stats. So, would you please explain how you got a p- value of 0.03112 in the above calculation/t-test? By looking at a chart? Would you also explain how you got the information that “you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”?

July 6, 2018 at 7:02 am

A quick question regarding your use of two-tailed critical regions in the article above: why? I mean, what is a real-world scenario that would warrant a two-tailed test of any kind (z, t, etc.)? And if there are none, why keep using the two-tailed scenario as an example, instead of the one-tailed which is both more intuitive and applicable to most if not all practical situations. Just curious, as one person attempting to educate people on stats to another (my take on the one vs. two-tailed tests can be seen here: http://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ )

Thanks, Georgi

July 6, 2018 at 12:05 pm

There’s the appropriate time and place for both one-tailed and two-tailed tests. I plan to write a post on this issue specifically, so I’ll keep my comments here brief.

So much of statistics is context sensitive. People often want concrete rules for how to do things in statistics but that’s often hard to provide because the answer depends on the context, goals, etc. The question of whether to use a one-tailed or two-tailed test falls firmly in this category of it depends.

I did read the article you wrote. I’ll say that I can see how in the context of A/B testing specifically there might be a propensity to use one-tailed tests. You only care about improvements. There’s probably not too much downside in only caring about one direction. In fact, in a post where I compare different tests and different options , I suggest using a one-tailed test for a similar type of casing involving defects. So, I’m onboard with the idea of using one-tailed tests when they’re appropriate. However, I do think that two-tailed tests should be considered the default choice and that you need good reasons to move to a one-tailed test. Again, your A/B testing area might supply those reasons on a regular basis, but I can’t make that a blanket statement for all research areas.

I think your article mischaracterizes some of the pros and cons of both types of tests. Just a couple of for instances. In a two-tailed test, you don’t have to take the same action regardless of which direction the results are significant (example below). And, yes, you can determine the direction of the effect in a two-tailed test. You simply look at the estimated effect. Is it positive or negative?

On the other hand, I do agree that one-tailed tests don’t increase the overall Type I error. However, there is a big caveat for that. In a two-tailed test, the Type I error rate is evenly split in both tails. For a one-tailed test, the overall Type I error rate does not change, but the Type I errors are redistributed so they all occur in the direction that you are interested in rather than being split between the positive and negative directions. In other words, you’ll have twice as many Type I errors in the specific direction that you’re interested in. That’s not good.

My big concerns with one-tailed tests are that it makes it easier to obtain the results that you want to obtain. And, all of the Type I errors (false positives) are in that direction too. It’s just not a good combination.

To answer your question about when you might want to use two-tailed tests, there are plenty of reasons. For one, you might want to avoid the situation I describe above. Additionally, in a lot of scientific research, the researchers truly are interested in detecting effects in either direction for the sake of science. Even in cases with a practical application, you might want to learn about effects in either direction.

For example, I was involved in a research study that looked at the effects of an exercise intervention on bone density. The idea was that it might be a good way to prevent osteoporosis. I used a two-tailed test. Obviously, we’re hoping that there was positive effect. However, we’d be very interested in knowing whether there was a negative effect too. And, this illustrates how you can have different actions based on both directions. If there was a positive effect, you can recommend that as a good approach and try to promote its use. If there’s a negative effect, you’d issue a warning to not do that intervention. You have the potential for learning both what is good and what is bad. The extra false-positives would’ve cause problems because we’d think that there’d be health benefits for participants when those benefits don’t actually exist. Also, if we had performed only a one-tailed test and didn’t obtain significant results, we’d learn that it wasn’t a positive effect, but we would not know whether it was actually detrimental or not.

Here’s when I’d say it’s OK to use a one-tailed test. Consider a one-tailed test when you’re in situation where you truly only need to know whether an effect exists in one direction, and the extra Type I errors in that direction are an acceptable risk (false positives don’t cause problems), and there’s no benefit in determining whether an effect exists in the other direction. Those conditions really restrict when one-tailed tests are the best choice. Again, those restrictions might not be relevant for your specific field, but as for the usage of statistics as a whole, they’re absolutely crucial to consider.

On the other hand, according to this article, two-tailed tests might be important in A/B testing !

March 30, 2018 at 5:29 am

Dear Sir, please confirm if there is an inadvertent mistake in interpretation as, “We can conclude that mean fuel expenditures have increased since last year.” Our null hypothesis is =260. If found significant, it implies two possibilities – both increase and decrease. Please let us know if we are mistaken here. Many Thanks!

March 30, 2018 at 9:59 am

Hi Khalid, the null hypothesis as it is defined for this test represents the mean monthly expenditure for the previous year (260). The mean expenditure for the current year is 330.6 whereas it was 260 for the previous year. Consequently, the mean has increased from 260 to 330.7 over the course of a year. The p-value indicates that this increase is statistically significant. This finding does not suggest both an increase and a decrease–just an increase. Keep in mind that a significant result prompts us to reject the null hypothesis. So, we reject the null that the mean equals 260.

Let’s explore the other possible findings to be sure that this makes sense. Suppose the sample mean had been closer to 260 and the p-value was greater than the significance level, those results would indicate that the results were not statistically significant. The conclusion that we’d draw is that we have insufficient evidence to conclude that mean fuel expenditures have changed since the previous year.

If the sample mean was less than the null hypothesis (260) and if the p-value is statistically significant, we’d concluded that mean fuel expenditures have decreased and that this decrease is statistically significant.

When you interpret the results, you have to be sure to understand what the null hypothesis represents. In this case, it represents the mean monthly expenditure for the previous year and we’re comparing this year’s mean to it–hence our sample suggests an increase.

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Null hypothesis significance testing: a short tutorial

Cyril pernet.

1 Centre for Clinical Brain Sciences (CCBS), Neuroimaging Sciences, The University of Edinburgh, Edinburgh, UK

Version Changes

Revised. amendments from version 2.

This v3 includes minor changes that reflect the 3rd reviewers' comments - in particular the theoretical vs. practical difference between Fisher and Neyman-Pearson. Additional information and reference is also included regarding the interpretation of p-value for low powered studies.

Peer Review Summary

Although thoroughly criticized, null hypothesis significance testing (NHST) remains the statistical method of choice used to provide evidence for an effect, in biological, biomedical and social sciences. In this short tutorial, I first summarize the concepts behind the method, distinguishing test of significance (Fisher) and test of acceptance (Newman-Pearson) and point to common interpretation errors regarding the p-value. I then present the related concepts of confidence intervals and again point to common interpretation errors. Finally, I discuss what should be reported in which context. The goal is to clarify concepts to avoid interpretation errors and propose reporting practices.

The Null Hypothesis Significance Testing framework

NHST is a method of statistical inference by which an experimental factor is tested against a hypothesis of no effect or no relationship based on a given observation. The method is a combination of the concepts of significance testing developed by Fisher in 1925 and of acceptance based on critical rejection regions developed by Neyman & Pearson in 1928 . In the following I am first presenting each approach, highlighting the key differences and common misconceptions that result from their combination into the NHST framework (for a more mathematical comparison, along with the Bayesian method, see Christensen, 2005 ). I next present the related concept of confidence intervals. I finish by discussing practical aspects in using NHST and reporting practice.

Fisher, significance testing, and the p-value

The method developed by ( Fisher, 1934 ; Fisher, 1955 ; Fisher, 1959 ) allows to compute the probability of observing a result at least as extreme as a test statistic (e.g. t value), assuming the null hypothesis of no effect is true. This probability or p-value reflects (1) the conditional probability of achieving the observed outcome or larger: p(Obs≥t|H0), and (2) is therefore a cumulative probability rather than a point estimate. It is equal to the area under the null probability distribution curve from the observed test statistic to the tail of the null distribution ( Turkheimer et al. , 2004 ). The approach proposed is of ‘proof by contradiction’ ( Christensen, 2005 ), we pose the null model and test if data conform to it.

In practice, it is recommended to set a level of significance (a theoretical p-value) that acts as a reference point to identify significant results, that is to identify results that differ from the null-hypothesis of no effect. Fisher recommended using p=0.05 to judge whether an effect is significant or not as it is roughly two standard deviations away from the mean for the normal distribution ( Fisher, 1934 page 45: ‘The value for which p=.05, or 1 in 20, is 1.96 or nearly 2; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not’). A key aspect of Fishers’ theory is that only the null-hypothesis is tested, and therefore p-values are meant to be used in a graded manner to decide whether the evidence is worth additional investigation and/or replication ( Fisher, 1971 page 13: ‘it is open to the experimenter to be more or less exacting in respect of the smallness of the probability he would require […]’ and ‘no isolated experiment, however significant in itself, can suffice for the experimental demonstration of any natural phenomenon’). How small the level of significance is, is thus left to researchers.

What is not a p-value? Common mistakes

The p-value is not an indication of the strength or magnitude of an effect . Any interpretation of the p-value in relation to the effect under study (strength, reliability, probability) is wrong, since p-values are conditioned on H0. In addition, while p-values are randomly distributed (if all the assumptions of the test are met) when there is no effect, their distribution depends of both the population effect size and the number of participants, making impossible to infer strength of effect from them.

Similarly, 1-p is not the probability to replicate an effect . Often, a small value of p is considered to mean a strong likelihood of getting the same results on another try, but again this cannot be obtained because the p-value is not informative on the effect itself ( Miller, 2009 ). Because the p-value depends on the number of subjects, it can only be used in high powered studies to interpret results. In low powered studies (typically small number of subjects), the p-value has a large variance across repeated samples, making it unreliable to estimate replication ( Halsey et al. , 2015 ).

A (small) p-value is not an indication favouring a given hypothesis . Because a low p-value only indicates a misfit of the null hypothesis to the data, it cannot be taken as evidence in favour of a specific alternative hypothesis more than any other possible alternatives such as measurement error and selection bias ( Gelman, 2013 ). Some authors have even argued that the more (a priori) implausible the alternative hypothesis, the greater the chance that a finding is a false alarm ( Krzywinski & Altman, 2013 ; Nuzzo, 2014 ).

The p-value is not the probability of the null hypothesis p(H0), of being true, ( Krzywinski & Altman, 2013 ). This common misconception arises from a confusion between the probability of an observation given the null p(Obs≥t|H0) and the probability of the null given an observation p(H0|Obs≥t) that is then taken as an indication for p(H0) (see Nickerson, 2000 ).

Neyman-Pearson, hypothesis testing, and the α-value

Neyman & Pearson (1933) proposed a framework of statistical inference for applied decision making and quality control. In such framework, two hypotheses are proposed: the null hypothesis of no effect and the alternative hypothesis of an effect, along with a control of the long run probabilities of making errors. The first key concept in this approach, is the establishment of an alternative hypothesis along with an a priori effect size. This differs markedly from Fisher who proposed a general approach for scientific inference conditioned on the null hypothesis only. The second key concept is the control of error rates . Neyman & Pearson (1928) introduced the notion of critical intervals, therefore dichotomizing the space of possible observations into correct vs. incorrect zones. This dichotomization allows distinguishing correct results (rejecting H0 when there is an effect and not rejecting H0 when there is no effect) from errors (rejecting H0 when there is no effect, the type I error, and not rejecting H0 when there is an effect, the type II error). In this context, alpha is the probability of committing a Type I error in the long run. Alternatively, Beta is the probability of committing a Type II error in the long run.

The (theoretical) difference in terms of hypothesis testing between Fisher and Neyman-Pearson is illustrated on Figure 1 . In the 1 st case, we choose a level of significance for observed data of 5%, and compute the p-value. If the p-value is below the level of significance, it is used to reject H0. In the 2 nd case, we set a critical interval based on the a priori effect size and error rates. If an observed statistic value is below and above the critical values (the bounds of the confidence region), it is deemed significantly different from H0. In the NHST framework, the level of significance is (in practice) assimilated to the alpha level, which appears as a simple decision rule: if the p-value is less or equal to alpha, the null is rejected. It is however a common mistake to assimilate these two concepts. The level of significance set for a given sample is not the same as the frequency of acceptance alpha found on repeated sampling because alpha (a point estimate) is meant to reflect the long run probability whilst the p-value (a cumulative estimate) reflects the current probability ( Fisher, 1955 ; Hubbard & Bayarri, 2003 ).

The figure was prepared with G-power for a one-sided one-sample t-test, with a sample size of 32 subjects, an effect size of 0.45, and error rates alpha=0.049 and beta=0.80. In Fisher’s procedure, only the nil-hypothesis is posed, and the observed p-value is compared to an a priori level of significance. If the observed p-value is below this level (here p=0.05), one rejects H0. In Neyman-Pearson’s procedure, the null and alternative hypotheses are specified along with an a priori level of acceptance. If the observed statistical value is outside the critical region (here [-∞ +1.69]), one rejects H0.

Acceptance or rejection of H0?

The acceptance level α can also be viewed as the maximum probability that a test statistic falls into the rejection region when the null hypothesis is true ( Johnson, 2013 ). Therefore, one can only reject the null hypothesis if the test statistics falls into the critical region(s), or fail to reject this hypothesis. In the latter case, all we can say is that no significant effect was observed, but one cannot conclude that the null hypothesis is true. This is another common mistake in using NHST: there is a profound difference between accepting the null hypothesis and simply failing to reject it ( Killeen, 2005 ). By failing to reject, we simply continue to assume that H0 is true, which implies that one cannot argue against a theory from a non-significant result (absence of evidence is not evidence of absence). To accept the null hypothesis, tests of equivalence ( Walker & Nowacki, 2011 ) or Bayesian approaches ( Dienes, 2014 ; Kruschke, 2011 ) must be used.

Confidence intervals

Confidence intervals (CI) are builds that fail to cover the true value at a rate of alpha, the Type I error rate ( Morey & Rouder, 2011 ) and therefore indicate if observed values can be rejected by a (two tailed) test with a given alpha. CI have been advocated as alternatives to p-values because (i) they allow judging the statistical significance and (ii) provide estimates of effect size. Assuming the CI (a)symmetry and width are correct (but see Wilcox, 2012 ), they also give some indication about the likelihood that a similar value can be observed in future studies. For future studies of the same sample size, 95% CI give about 83% chance of replication success ( Cumming & Maillardet, 2006 ). If sample sizes however differ between studies, CI do not however warranty any a priori coverage.

Although CI provide more information, they are not less subject to interpretation errors (see Savalei & Dunn, 2015 for a review). The most common mistake is to interpret CI as the probability that a parameter (e.g. the population mean) will fall in that interval X% of the time. The correct interpretation is that, for repeated measurements with the same sample sizes, taken from the same population, X% of times the CI obtained will contain the true parameter value ( Tan & Tan, 2010 ). The alpha value has the same interpretation as testing against H0, i.e. we accept that 1-alpha CI are wrong in alpha percent of the times in the long run. This implies that CI do not allow to make strong statements about the parameter of interest (e.g. the mean difference) or about H1 ( Hoekstra et al. , 2014 ). To make a statement about the probability of a parameter of interest (e.g. the probability of the mean), Bayesian intervals must be used.

The (correct) use of NHST

NHST has always been criticized, and yet is still used every day in scientific reports ( Nickerson, 2000 ). One question to ask oneself is what is the goal of a scientific experiment at hand? If the goal is to establish a discrepancy with the null hypothesis and/or establish a pattern of order, because both requires ruling out equivalence, then NHST is a good tool ( Frick, 1996 ; Walker & Nowacki, 2011 ). If the goal is to test the presence of an effect and/or establish some quantitative values related to an effect, then NHST is not the method of choice since testing is conditioned on H0.

While a Bayesian analysis is suited to estimate that the probability that a hypothesis is correct, like NHST, it does not prove a theory on itself, but adds its plausibility ( Lindley, 2000 ). No matter what testing procedure is used and how strong results are, ( Fisher, 1959 p13) reminds us that ‘ […] no isolated experiment, however significant in itself, can suffice for the experimental demonstration of any natural phenomenon'. Similarly, the recent statement of the American Statistical Association ( Wasserstein & Lazar, 2016 ) makes it clear that conclusions should be based on the researchers understanding of the problem in context, along with all summary data and tests, and that no single value (being p-values, Bayesian factor or else) can be used support or invalidate a theory.

What to report and how?

Considering that quantitative reports will always have more information content than binary (significant or not) reports, we can always argue that raw and/or normalized effect size, confidence intervals, or Bayes factor must be reported. Reporting everything can however hinder the communication of the main result(s), and we should aim at giving only the information needed, at least in the core of a manuscript. Here I propose to adopt optimal reporting in the result section to keep the message clear, but have detailed supplementary material. When the hypothesis is about the presence/absence or order of an effect, and providing that a study has sufficient power, NHST is appropriate and it is sufficient to report in the text the actual p-value since it conveys the information needed to rule out equivalence. When the hypothesis and/or the discussion involve some quantitative value, and because p-values do not inform on the effect, it is essential to report on effect sizes ( Lakens, 2013 ), preferably accompanied with confidence or credible intervals. The reasoning is simply that one cannot predict and/or discuss quantities without accounting for variability. For the reader to understand and fully appreciate the results, nothing else is needed.

Because science progress is obtained by cumulating evidence ( Rosenthal, 1991 ), scientists should also consider the secondary use of the data. With today’s electronic articles, there are no reasons for not including all of derived data: mean, standard deviations, effect size, CI, Bayes factor should always be included as supplementary tables (or even better also share raw data). It is also essential to report the context in which tests were performed – that is to report all of the tests performed (all t, F, p values) because of the increase type one error rate due to selective reporting (multiple comparisons and p-hacking problems - Ioannidis, 2005 ). Providing all of this information allows (i) other researchers to directly and effectively compare their results in quantitative terms (replication of effects beyond significance, Open Science Collaboration, 2015 ), (ii) to compute power to future studies ( Lakens & Evers, 2014 ), and (iii) to aggregate results for meta-analyses whilst minimizing publication bias ( van Assen et al. , 2014 ).

[version 3; referees: 1 approved

Funding Statement

The author(s) declared that no grants were involved in supporting this work.

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Referee response for version 3

Dorothy vera margaret bishop.

1 Department of Experimental Psychology, University of Oxford, Oxford, UK

I can see from the history of this paper that the author has already been very responsive to reviewer comments, and that the process of revising has now been quite protracted.

That makes me reluctant to suggest much more, but I do see potential here for making the paper more impactful. So my overall view is that, once a few typos are fixed (see below), this could be published as is, but I think there is an issue with the potential readership and that further revision could overcome this.

I suspect my take on this is rather different from other reviewers, as I do not regard myself as a statistics expert, though I am on the more quantitative end of the continuum of psychologists and I try to keep up to date. I think I am quite close to the target readership , insofar as I am someone who was taught about statistics ages ago and uses stats a lot, but never got adequate training in the kinds of topic covered by this paper. The fact that I am aware of controversies around the interpretation of confidence intervals etc is simply because I follow some discussions of this on social media. I am therefore very interested to have a clear account of these issues.

This paper contains helpful information for someone in this position, but it is not always clear, and I felt the relevance of some of the content was uncertain. So here are some recommendations:

• As one previous reviewer noted, it’s questionable that there is a need for a tutorial introduction, and the limited length of this article does not lend itself to a full explanation. So it might be better to just focus on explaining as clearly as possible the problems people have had in interpreting key concepts. I think a title that made it clear this was the content would be more appealing than the current one.
• P 3, col 1, para 3, last sentence. Although statisticians always emphasise the arbitrary nature of p < .05, we all know that in practice authors who use other values are likely to have their analyses queried. I wondered whether it would be useful here to note that in some disciplines different cutoffs are traditional, e.g. particle physics. Or you could cite David Colquhoun’s paper in which he recommends using p < .001 ( http://rsos.royalsocietypublishing.org/content/1/3/140216) - just to be clear that the traditional p < .05 has been challenged.

What I can’t work out is how you would explain the alpha from Neyman-Pearson in the same way (though I can see from Figure 1 that with N-P you could test an alternative hypothesis, such as the idea that the coin would be heads 75% of the time).

‘By failing to reject, we simply continue to assume that H0 is true, which implies that one cannot….’ have ‘In failing to reject, we do not assume that H0 is true; one cannot argue against a theory from a non-significant result.’

I felt most readers would be interested to read about tests of equivalence and Bayesian approaches, but many would be unfamiliar with these and might like to see an example of how they work in practice – if space permitted.

• Confidence intervals: I simply could not understand the first sentence – I wondered what was meant by ‘builds’ here. I understand about difficulties in comparing CI across studies when sample sizes differ, but I did not find the last sentence on p 4 easy to understand.
• P 5: The sentence starting: ‘The alpha value has the same interpretation’ was also hard to understand, especially the term ‘1-alpha CI’. Here too I felt some concrete illustration might be helpful to the reader. And again, I also found the reference to Bayesian intervals tantalising – I think many readers won’t know how to compute these and something like a figure comparing a traditional CI with a Bayesian interval and giving a source for those who want to read on would be very helpful. The reference to ‘credible intervals’ in the penultimate paragraph is very unclear and needs a supporting reference – most readers will not be familiar with this concept.

P 3, col 1, para 2, line 2; “allows us to compute”

P 3, col 2, para 2, ‘probability of replicating’

P 3, col 2, para 2, line 4 ‘informative about’

P 3, col 2, para 4, line 2 delete ‘of’

P 3, col 2, para 5, line 9 – ‘conditioned’ is either wrong or too technical here: would ‘based’ be acceptable as alternative wording

P 3, col 2, para 5, line 13 ‘This dichotomisation allows one to distinguish’

P 3, col 2, para 5, last sentence, delete ‘Alternatively’.

P 3, col 2, last para line 2 ‘first’

P 4, col 2, para 2, last sentence is hard to understand; not sure if this is better: ‘If sample sizes differ between studies, the distribution of CIs cannot be specified a priori’

P 5, col 1, para 2, ‘a pattern of order’ – I did not understand what was meant by this

P 5, col 1, para 2, last sentence unclear: possible rewording: “If the goal is to test the size of an effect then NHST is not the method of choice, since testing can only reject the null hypothesis.’ (??)

P 5, col 1, para 3, line 1 delete ‘that’

P 5, col 1, para 3, line 3 ‘on’ -> ‘by’

P 5, col 2, para 1, line 4 , rather than ‘Here I propose to adopt’ I suggest ‘I recommend adopting’

P 5, col 2, para 1, line 13 ‘with’ -> ‘by’

P 5, col 2, para 1 – recommend deleting last sentence

P 5, col 2, para 2, line 2 ‘consider’ -> ‘anticipate’

P 5, col 2, para 2, delete ‘should always be included’

P 5, col 2, para 2, ‘type one’ -> ‘Type I’

I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

The University of Edinburgh, UK

I wondered about changing the focus slightly and modifying the title to reflect this to say something like: Null hypothesis significance testing: a guide to commonly misunderstood concepts and recommendations for good practice

Thank you for the suggestion – you indeed saw the intention behind the ‘tutorial’ style of the paper.

• P 3, col 1, para 3, last sentence. Although statisticians always emphasise the arbitrary nature of p < .05, we all know that in practice authors who use other values are likely to have their analyses queried. I wondered whether it would be useful here to note that in some disciplines different cutoffs are traditional, e.g. particle physics. Or you could cite David Colquhoun’s paper in which he recommends using p < .001 ( http://rsos.royalsocietypublishing.org/content/1/3/140216)  - just to be clear that the traditional p < .05 has been challenged.

I have added a sentence on this citing Colquhoun 2014 and the new Benjamin 2017 on using .005.

I agree that this point is always hard to appreciate, especially because it seems like in practice it makes little difference. I added a paragraph but using reaction times rather than a coin toss – thanks for the suggestion.

Added an example based on new table 1, following figure 1 – giving CI, equivalence tests and Bayes Factor (with refs to easy to use tools)

Changed builds to constructs (this simply means they are something we build) and added that the implication that probability coverage is not warranty when sample size change, is that we cannot compare CI.

I changed ‘ i.e. we accept that 1-alpha CI are wrong in alpha percent of the times in the long run’ to ‘, ‘e.g. a 95% CI is wrong in 5% of the times in the long run (i.e. if we repeat the experiment many times).’ – for Bayesian intervals I simply re-cited Morey & Rouder, 2011.

It is not the CI cannot be specified, it’s that the interval is not predictive of anything anymore! I changed it to ‘If sample sizes, however, differ between studies, there is no warranty that a CI from one study will be true at the rate alpha in a different study, which implies that CI cannot be compared across studies at this is rarely the same sample sizes’

I added (i.e. establish that A > B) – we test that conditions are ordered, but without further specification of the probability of that effect nor its size

Yes it works – thx

P 5, col 2, para 2, ‘type one’ -> ‘Type I’ 

Typos fixed, and suggestions accepted – thanks for that.

Stephen J. Senn

1 Luxembourg Institute of Health, Strassen, L-1445, Luxembourg

The revisions are OK for me, and I have changed my status to Approved.

I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Referee response for version 2

On the whole I think that this article is reasonable, my main reservation being that I have my doubts on whether the literature needs yet another tutorial on this subject.

A further reservation I have is that the author, following others, stresses what in my mind is a relatively unimportant distinction between the Fisherian and Neyman-Pearson (NP) approaches. The distinction stressed by many is that the NP approach leads to a dichotomy accept/reject based on probabilities established in advance, whereas the Fisherian approach uses tail area probabilities calculated from the observed statistic. I see this as being unimportant and not even true. Unless one considers that the person carrying out a hypothesis test (original tester) is mandated to come to a conclusion on behalf of all scientific posterity, then one must accept that any remote scientist can come to his or her conclusion depending on the personal type I error favoured. To operate the results of an NP test carried out by the original tester, the remote scientist then needs to know the p-value. The type I error rate is then compared to this to come to a personal accept or reject decision (1). In fact Lehmann (2), who was an important developer of and proponent of the NP system, describes exactly this approach as being good practice. (See Testing Statistical Hypotheses, 2nd edition P70). Thus using tail-area probabilities calculated from the observed statistics does not constitute an operational difference between the two systems.

A more important distinction between the Fisherian and NP systems is that the former does not use alternative hypotheses(3). Fisher's opinion was that the null hypothesis was more primitive than the test statistic but that the test statistic was more primitive than the alternative hypothesis. Thus, alternative hypotheses could not be used to justify choice of test statistic. Only experience could do that.

Further distinctions between the NP and Fisherian approach are to do with conditioning and whether a null hypothesis can ever be accepted.

I have one minor quibble about terminology. As far as I can see, the author uses the usual term 'null hypothesis' and the eccentric term 'nil hypothesis' interchangeably. It would be simpler if the latter were abandoned.

Referee response for version 1

Marcel alm van assen.

1 Department of Methodology and Statistics, Tilburgh University, Tilburg, Netherlands

Null hypothesis significance testing (NHST) is a difficult topic, with misunderstandings arising easily. Many texts, including basic statistics books, deal with the topic, and attempt to explain it to students and anyone else interested. I would refer to a good basic text book, for a detailed explanation of NHST, or to a specialized article when wishing an explaining the background of NHST. So, what is the added value of a new text on NHST? In any case, the added value should be described at the start of this text. Moreover, the topic is so delicate and difficult that errors, misinterpretations, and disagreements are easy. I attempted to show this by giving comments to many sentences in the text.

Abstract: “null hypothesis significance testing is the statistical method of choice in biological, biomedical and social sciences to investigate if an effect is likely”. No, NHST is the method to test the hypothesis of no effect.

Intro: “Null hypothesis significance testing (NHST) is a method of statistical inference by which an observation is tested against a hypothesis of no effect or no relationship.” What is an ‘observation’? NHST is difficult to describe in one sentence, particularly here. I would skip this sentence entirely, here.

Section on Fisher; also explain the one-tailed test.

Section on Fisher; p(Obs|H0) does not reflect the verbal definition (the ‘or more extreme’ part).

Section on Fisher; use a reference and citation to Fisher’s interpretation of the p-value

Section on Fisher; “This was however only intended to be used as an indication that there is something in the data that deserves further investigation. The reason for this is that only H0 is tested whilst the effect under study is not itself being investigated.” First sentence, can you give a reference? Many people say a lot about Fisher’s intentions, but the good man is dead and cannot reply… Second sentence is a bit awkward, because the effect is investigated in a way, by testing the H0.

Section on p-value; Layout and structure can be improved greatly, by first again stating what the p-value is, and then statement by statement, what it is not, using separate lines for each statement. Consider adding that the p-value is randomly distributed under H0 (if all the assumptions of the test are met), and that under H1 the p-value is a function of population effect size and N; the larger each is, the smaller the p-value generally is.

Skip the sentence “If there is no effect, we should replicate the absence of effect with a probability equal to 1-p”. Not insightful, and you did not discuss the concept ‘replicate’ (and do not need to).

Skip the sentence “The total probability of false positives can also be obtained by aggregating results ( Ioannidis, 2005 ).” Not strongly related to p-values, and introduces unnecessary concepts ‘false positives’ (perhaps later useful) and ‘aggregation’.

Consider deleting; “If there is an effect however, the probability to replicate is a function of the (unknown) population effect size with no good way to know this from a single experiment ( Killeen, 2005 ).”

The following sentence; “ Finally, a (small) p-value  is not an indication favouring a hypothesis . A low p-value indicates a misfit of the null hypothesis to the data and cannot be taken as evidence in favour of a specific alternative hypothesis more than any other possible alternatives such as measurement error and selection bias ( Gelman, 2013 ).” is surely not mainstream thinking about NHST; I would surely delete that sentence. In NHST, a p-value is used for testing the H0. Why did you not yet discuss significance level? Yes, before discussing what is not a p-value, I would explain NHST (i.e., what it is and how it is used). 

Also the next sentence “The more (a priori) implausible the alternative hypothesis, the greater the chance that a finding is a false alarm ( Krzywinski & Altman, 2013 ;  Nuzzo, 2014 ).“ is not fully clear to me. This is a Bayesian statement. In NHST, no likelihoods are attributed to hypotheses; the reasoning is “IF H0 is true, then…”.

Last sentence: “As  Nickerson (2000)  puts it ‘theory corroboration requires the testing of multiple predictions because the chance of getting statistically significant results for the wrong reasons in any given case is high’.” What is relation of this sentence to the contents of this section, precisely?

Next section: “For instance, we can estimate that the probability of a given F value to be in the critical interval [+2 +∞] is less than 5%” This depends on the degrees of freedom.

“When there is no effect (H0 is true), the erroneous rejection of H0 is known as type I error and is equal to the p-value.” Strange sentence. The Type I error is the probability of erroneously rejecting the H0 (so, when it is true). The p-value is … well, you explained it before; it surely does not equal the Type I error.

Consider adding a figure explaining the distinction between Fisher’s logic and that of Neyman and Pearson.

“When the test statistics falls outside the critical region(s)” What is outside?

“There is a profound difference between accepting the null hypothesis and simply failing to reject it ( Killeen, 2005 )” I agree with you, but perhaps you may add that some statisticians simply define “accept H0’” as obtaining a p-value larger than the significance level. Did you already discuss the significance level, and it’s mostly used values?

“To accept or reject equally the null hypothesis, Bayesian approaches ( Dienes, 2014 ;  Kruschke, 2011 ) or confidence intervals must be used.” Is ‘reject equally’ appropriate English? Also using Cis, one cannot accept the H0.

Do you start discussing alpha only in the context of Cis?

“CI also indicates the precision of the estimate of effect size, but unless using a percentile bootstrap approach, they require assumptions about distributions which can lead to serious biases in particular regarding the symmetry and width of the intervals ( Wilcox, 2012 ).” Too difficult, using new concepts. Consider deleting.

“Assuming the CI (a)symmetry and width are correct, this gives some indication about the likelihood that a similar value can be observed in future studies, with 95% CI giving about 83% chance of replication success ( Lakens & Evers, 2014 ).” This statement is, in general, completely false. It very much depends on the sample sizes of both studies. If the replication study has a much, much, much larger N, then the probability that the original CI will contain the effect size of the replication approaches (1-alpha)*100%. If the original study has a much, much, much larger N, then the probability that the original Ci will contain the effect size of the replication study approaches 0%.

“Finally, contrary to p-values, CI can be used to accept H0. Typically, if a CI includes 0, we cannot reject H0. If a critical null region is specified rather than a single point estimate, for instance [-2 +2] and the CI is included within the critical null region, then H0 can be accepted. Importantly, the critical region must be specified a priori and cannot be determined from the data themselves.” No. H0 cannot be accepted with Cis.

“The (posterior) probability of an effect can however not be obtained using a frequentist framework.” Frequentist framework? You did not discuss that, yet.

“X% of times the CI obtained will contain the same parameter value”. The same? True, you mean?

“e.g. X% of the times the CI contains the same mean” I do not understand; which mean?

“The alpha value has the same interpretation as when using H0, i.e. we accept that 1-alpha CI are wrong in alpha percent of the times. “ What do you mean, CI are wrong? Consider rephrasing.

“To make a statement about the probability of a parameter of interest, likelihood intervals (maximum likelihood) and credibility intervals (Bayes) are better suited.” ML gives the likelihood of the data given the parameter, not the other way around.

“Many of the disagreements are not on the method itself but on its use.” Bayesians may disagree.

“If the goal is to establish the likelihood of an effect and/or establish a pattern of order, because both requires ruling out equivalence, then NHST is a good tool ( Frick, 1996 )” NHST does not provide evidence on the likelihood of an effect.

“If the goal is to establish some quantitative values, then NHST is not the method of choice.” P-values are also quantitative… this is not a precise sentence. And NHST may be used in combination with effect size estimation (this is even recommended by, e.g., the American Psychological Association (APA)).

“Because results are conditioned on H0, NHST cannot be used to establish beliefs.” It can reinforce some beliefs, e.g., if H0 or any other hypothesis, is true.

“To estimate the probability of a hypothesis, a Bayesian analysis is a better alternative.” It is the only alternative?

“Note however that even when a specific quantitative prediction from a hypothesis is shown to be true (typically testing H1 using Bayes), it does not prove the hypothesis itself, it only adds to its plausibility.” How can we show something is true?

I do not agree on the contents of the last section on ‘minimal reporting’. I prefer ‘optimal reporting’ instead, i.e., the reporting the information that is essential to the interpretation of the result, to any ready, which may have other goals than the writer of the article. This reporting includes, for sure, an estimate of effect size, and preferably a confidence interval, which is in line with recommendations of the APA.

I have read this submission. I believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

The idea of this short review was to point to common interpretation errors (stressing again and again that we are under H0) being in using p-values or CI, and also proposing reporting practices to avoid bias. This is now stated at the end of abstract.

Regarding text books, it is clear that many fail to clearly distinguish Fisher/Pearson/NHST, see Glinet et al (2012) J. Exp Education 71, 83-92. If you have 1 or 2 in mind that you know to be good, I’m happy to include them.

I agree – yet people use it to investigate (not test) if an effect is likely. The issue here is wording. What about adding this distinction at the end of the sentence?: ‘null hypothesis significance testing is the statistical method of choice in biological, biomedical and social sciences used to investigate if an effect is likely, even though it actually tests for the hypothesis of no effect’.

I think a definition is needed, as it offers a starting point. What about the following: ‘NHST is a method of statistical inference by which an experimental factor is tested against a hypothesis of no effect or no relationship based on a given observation’

The section on Fisher has been modified (more or less) as suggested: (1) avoiding talking about one or two tailed tests (2) updating for p(Obs≥t|H0) and (3) referring to Fisher more explicitly (ie pages from articles and book) ; I cannot tell his intentions but these quotes leave little space to alternative interpretations.

The reasoning here is as you state yourself, part 1: ‘a p-value is used for testing the H0; and part 2: ‘no likelihoods are attributed to hypotheses’ it follows we cannot favour a hypothesis. It might seems contentious but this is the case that all we can is to reject the null – how could we favour a specific alternative hypothesis from there? This is explored further down the manuscript (and I now point to that) – note that we do not need to be Bayesian to favour a specific H1, all I’m saying is this cannot be attained with a p-value.

The point was to emphasise that a p value is not there to tell us a given H1 is true and can only be achieved through multiple predictions and experiments. I deleted it for clarity.

This sentence has been removed

Indeed, you are right and I have modified the text accordingly. When there is no effect (H0 is true), the erroneous rejection of H0 is known as type 1 error. Importantly, the type 1 error rate, or alpha value is determined a priori. It is a common mistake but the level of significance (for a given sample) is not the same as the frequency of acceptance alpha found on repeated sampling (Fisher, 1955).

A figure is now presented – with levels of acceptance, critical region, level of significance and p-value.

I should have clarified further here – as I was having in mind tests of equivalence. To clarify, I simply states now: ‘To accept the null hypothesis, tests of equivalence or Bayesian approaches must be used.’

It is now presented in the paragraph before.

Yes, you are right, I completely overlooked this problem. The corrected sentence (with more accurate ref) is now “Assuming the CI (a)symmetry and width are correct, this gives some indication about the likelihood that a similar value can be observed in future studies. For future studies of the same sample size, 95% CI giving about 83% chance of replication success (Cumming and Mallardet, 2006). If sample sizes differ between studies, CI do not however warranty any a priori coverage”.

Again, I had in mind equivalence testing, but in both cases you are right we can only reject and I therefore removed that sentence.

Yes, p-values must be interpreted in context with effect size, but this is not what people do. The point here is to be pragmatic, does and don’t. The sentence was changed.

Not for testing, but for probability, I am not aware of anything else.

Cumulative evidence is, in my opinion, the only way to show it. Even in hard science like physics multiple experiments. In the recent CERN study on finding Higgs bosons, 2 different and complementary experiments ran in parallel – and the cumulative evidence was taken as a proof of the true existence of Higgs bosons.

Daniel Lakens

1 School of Innovation Sciences, Eindhoven University of Technology, Eindhoven, Netherlands

I appreciate the author's attempt to write a short tutorial on NHST. Many people don't know how to use it, so attempts to educate people are always worthwhile. However, I don't think the current article reaches it's aim. For one, I think it might be practically impossible to explain a lot in such an ultra short paper - every section would require more than 2 pages to explain, and there are many sections. Furthermore, there are some excellent overviews, which, although more extensive, are also much clearer (e.g., Nickerson, 2000 ). Finally, I found many statements to be unclear, and perhaps even incorrect (noted below). Because there is nothing worse than creating more confusion on such a topic, I have extremely high standards before I think such a short primer should be indexed. I note some examples of unclear or incorrect statements below. I'm sorry I can't make a more positive recommendation.

“investigate if an effect is likely” – ambiguous statement. I think you mean, whether the observed DATA is probable, assuming there is no effect?

The Fisher (1959) reference is not correct – Fischer developed his method much earlier.

“This p-value thus reflects the conditional probability of achieving the observed outcome or larger, p(Obs|H0)” – please add 'assuming the null-hypothesis is true'.

“p(Obs|H0)” – explain this notation for novices.

“Following Fisher, the smaller the p-value, the greater the likelihood that the null hypothesis is false.”  This is wrong, and any statement about this needs to be much more precise. I would suggest direct quotes.

“there is something in the data that deserves further investigation” –unclear sentence.

“The reason for this” – unclear what ‘this’ refers to.

“ not the probability of the null hypothesis of being true, p(H0)” – second of can be removed?

“Any interpretation of the p-value in relation to the effect under study (strength, reliability, probability) is indeed

wrong, since the p-value is conditioned on H0”  - incorrect. A big problem is that it depends on the sample size, and that the probability of a theory depends on the prior.

“If there is no effect, we should replicate the absence of effect with a probability equal to 1-p.” I don’t understand this, but I think it is incorrect.

“The total probability of false positives can also be obtained by aggregating results (Ioannidis, 2005).” Unclear, and probably incorrect.

“By failing to reject, we simply continue to assume that H0 is true, which implies that one cannot, from a nonsignificant result, argue against a theory” – according to which theory? From a NP perspective, you can ACT as if the theory is false.

“(Lakens & Evers, 2014”) – we are not the original source, which should be cited instead.

“ Typically, if a CI includes 0, we cannot reject H0.”  - when would this not be the case? This assumes a CI of 1-alpha.

“If a critical null region is specified rather than a single point estimate, for instance [-2 +2] and the CI is included within the critical null region, then H0 can be accepted.” – you mean practically, or formally? I’m pretty sure only the former.

The section on ‘The (correct) use of NHST’ seems to conclude only Bayesian statistics should be used. I don’t really agree.

“ we can always argue that effect size, power, etc. must be reported.” – which power? Post-hoc power? Surely not? Other types are unknown. So what do you mean?

The recommendation on what to report remains vague, and it is unclear why what should be reported.

This sentence was changed, following as well the other reviewer, to ‘null hypothesis significance testing is the statistical method of choice in biological, biomedical and social sciences to investigate if an effect is likely, even though it actually tests whether the observed data are probable, assuming there is no effect’

Changed, refers to Fisher 1925

I changed a little the sentence structure, which should make explicit that this is the condition probability.

This has been changed to ‘[…] to decide whether the evidence is worth additional investigation and/or replication (Fisher, 1971 p13)’

my mistake – the sentence structure is now ‘ not the probability of the null hypothesis p(H0), of being true,’ ; hope this makes more sense (and this way refers back to p(Obs>t|H0)

Fair enough – my point was to stress the fact that p value and effect size or H1 have very little in common, but yes that the part in common has to do with sample size. I left the conditioning on H0 but also point out the dependency on sample size.

The whole paragraph was changed to reflect a more philosophical take on scientific induction/reasoning. I hope this is clearer.

Changed to refer to equivalence testing

I rewrote this, as to show frequentist analysis can be used  - I’m trying to sell Bayes more than any other approach.

I’m arguing we should report it all, that’s why there is no exhausting list – I can if needed.

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9.3 - the p-value approach, example 9-4 section  .

Up until now, we have used the critical region approach in conducting our hypothesis tests. Now, let's take a look at an example in which we use what is called the P -value approach .

Among patients with lung cancer, usually, 90% or more die within three years. As a result of new forms of treatment, it is felt that this rate has been reduced. In a recent study of n = 150 lung cancer patients, y = 128 died within three years. Is there sufficient evidence at the \(\alpha = 0.05\) level, say, to conclude that the death rate due to lung cancer has been reduced?

The sample proportion is:

\(\hat{p}=\dfrac{128}{150}=0.853\)

The null and alternative hypotheses are:

\(H_0 \colon p = 0.90\) and \(H_A \colon p < 0.90\)

The test statistic is, therefore:

\(Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{0.853-0.90}{\sqrt{\dfrac{0.90(0.10)}{150}}}=-1.92\)

And, the rejection region is:

Since the test statistic Z = −1.92 < −1.645, we reject the null hypothesis. There is sufficient evidence at the \(\alpha = 0.05\) level to conclude that the rate has been reduced.

Example 9-4 (continued) Section  

What if we set the significance level \(\alpha\) = P (Type I Error) to 0.01? Is there still sufficient evidence to conclude that the death rate due to lung cancer has been reduced?

In this case, with \(\alpha = 0.01\), the rejection region is Z ≤ −2.33. That is, we reject if the test statistic falls in the rejection region defined by Z ≤ −2.33:

Because the test statistic Z = −1.92 > −2.33, we do not reject the null hypothesis. There is insufficient evidence at the \(\alpha = 0.01\) level to conclude that the rate has been reduced.

In the first part of this example, we rejected the null hypothesis when \(\alpha = 0.05\). And, in the second part of this example, we failed to reject the null hypothesis when \(\alpha = 0.01\). There must be some level of \(\alpha\), then, in which we cross the threshold from rejecting to not rejecting the null hypothesis. What is the smallest \(\alpha \text{ -level}\) that would still cause us to reject the null hypothesis?

We would, of course, reject any time the critical value was smaller than our test statistic −1.92:

That is, we would reject if the critical value were −1.645, −1.83, and −1.92. But, we wouldn't reject if the critical value were −1.93. The \(\alpha \text{ -level}\) associated with the test statistic −1.92 is called the P -value . It is the smallest \(\alpha \text{ -level}\) that would lead to rejection. In this case, the P -value is:

P ( Z < −1.92) = 0.0274

So far, all of the examples we've considered have involved a one-tailed hypothesis test in which the alternative hypothesis involved either a less than (<) or a greater than (>) sign. What happens if we weren't sure of the direction in which the proportion could deviate from the hypothesized null value? That is, what if the alternative hypothesis involved a not-equal sign (≠)? Let's take a look at an example.

What if we wanted to perform a " two-tailed " test? That is, what if we wanted to test:

\(H_0 \colon p = 0.90\) versus \(H_A \colon p \ne 0.90\)

at the \(\alpha = 0.05\) level?

Let's first consider the critical value approach . If we allow for the possibility that the sample proportion could either prove to be too large or too small, then we need to specify a threshold value, that is, a critical value, in each tail of the distribution. In this case, we divide the " significance level " \(\alpha\) by 2 to get \(\alpha/2\):

That is, our rejection rule is that we should reject the null hypothesis \(H_0 \text{ if } Z ≥ 1.96\) or we should reject the null hypothesis \(H_0 \text{ if } Z ≤ −1.96\). Alternatively, we can write that we should reject the null hypothesis \(H_0 \text{ if } |Z| ≥ 1.96\). Because our test statistic is −1.92, we just barely fail to reject the null hypothesis, because 1.92 < 1.96. In this case, we would say that there is insufficient evidence at the \(\alpha = 0.05\) level to conclude that the sample proportion differs significantly from 0.90.

Now for the P -value approach . Again, needing to allow for the possibility that the sample proportion is either too large or too small, we multiply the P -value we obtain for the one-tailed test by 2:

That is, the P -value is:

\(P=P(|Z|\geq 1.92)=P(Z>1.92 \text{ or } Z<-1.92)=2 \times 0.0274=0.055\)

Because the P -value 0.055 is (just barely) greater than the significance level \(\alpha = 0.05\), we barely fail to reject the null hypothesis. Again, we would say that there is insufficient evidence at the \(\alpha = 0.05\) level to conclude that the sample proportion differs significantly from 0.90.

Let's close this example by formalizing the definition of a P -value, as well as summarizing the P -value approach to conducting a hypothesis test.

The P -value is the smallest significance level \(\alpha\) that leads us to reject the null hypothesis.

Alternatively (and the way I prefer to think of P -values), the P -value is the probability that we'd observe a more extreme statistic than we did if the null hypothesis were true.

If the P -value is small, that is, if \(P ≤ \alpha\), then we reject the null hypothesis \(H_0\).

Note! Section  

By the way, to test \(H_0 \colon p = p_0\), some statisticians will use the test statistic:

\(Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}}\)

rather than the one we've been using:

\(Z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}\)

One advantage of doing so is that the interpretation of the confidence interval — does it contain \(p_0\)? — is always consistent with the hypothesis test decision, as illustrated here:

For the sake of ease, let:

\(se(\hat{p})=\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\)

Two-tailed test. In this case, the critical region approach tells us to reject the null hypothesis \(H_0 \colon p = p_0\) against the alternative hypothesis \(H_A \colon p \ne p_0\):

if \(Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \geq z_{\alpha/2}\) or if \(Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \leq -z_{\alpha/2}\)

which is equivalent to rejecting the null hypothesis:

if \(\hat{p}-p_0 \geq z_{\alpha/2}se(\hat{p})\) or if \(\hat{p}-p_0 \leq -z_{\alpha/2}se(\hat{p})\)

if \(p_0 \geq \hat{p}+z_{\alpha/2}se(\hat{p})\) or if \(p_0 \leq \hat{p}-z_{\alpha/2}se(\hat{p})\)

That's the same as saying that we should reject the null hypothesis \(H_0 \text{ if } p_0\) is not in the \(\left(1-\alpha\right)100\%\) confidence interval!

Left-tailed test. In this case, the critical region approach tells us to reject the null hypothesis \(H_0 \colon p = p_0\) against the alternative hypothesis \(H_A \colon p < p_0\):

if \(Z=\dfrac{\hat{p}-p_0}{se(\hat{p})} \leq -z_{\alpha}\)

if \(\hat{p}-p_0 \leq -z_{\alpha}se(\hat{p})\)

if \(p_0 \geq \hat{p}+z_{\alpha}se(\hat{p})\)

That's the same as saying that we should reject the null hypothesis \(H_0 \text{ if } p_0\) is not in the upper \(\left(1-\alpha\right)100\%\) confidence interval:

\((0,\hat{p}+z_{\alpha}se(\hat{p}))\)

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what does For a significance threshold of 0.05, if the null hypothesis is true mean

Although it's a simple question and I know what the null hypothesis means, I couldn't understand this sentence. For a significance threshold of 0.05, if the null hypothesis is true, then the probability of correctly failing to reject the null is 1 – 0.05 = 0.95. Can someone please explain?

• hypothesis-testing

3 Answers 3

If the null is true it means that, for example, there is no difference between the distribution of population X and population Y.

Imagine you take a sample from X and a sample from Y and you do a t-test for difference between the two. If you repeat this sampling and testing many times, you will find that the t-test returns p > 0.05 in 95% of the cases. I.e. the null has not being rejected 95% of the time and this the "correct" outcome.

This simulation in R may help. Hopefully, you can understand it even if you don't use R. We compare a sample from population x and y 1000 times, apply a test and check how many times we get p < 0.05:

We can also show that if the null is true, p-values follow a uniform distribution:

( correctly failing to reject is a bit contrived, I think.)

Frequentist statistics (what you’re doing when you use a p-value) puts controls on how often you will make a fool out of yourself and assert something incorrect.

Traditionally, people have considered it a major error to assert that a null hypothesis is false when it isn’t. Setting $\alpha=0.05$ means that there is only a $1$ in $20$ chance of making such an error when the null is true. $^{\dagger}$

$$ P(\text{ reject }\vert \text{ }H_0\text{ is true })=0.05 $$

So you make mistakes of this sort fairly infrequently.

If you like thinking of sensitivity and specificity, the test has $95\%$ specificity (and then power is sensitivity).

$^{\dagger}$ This assumes that the hypothesis test has proper calibration (t-test does) and that the assumptions are met. More exotic hypothesis tests might not have calibration as good as the t-test. If you get into such tests, you will have to make a judgment call about if it is acceptable to have a $6\%$ or $8\%$ chance of making such a false rejection.

A p-value provides the probability of receiving a particular test statistic or a more extreme one given all of the p-value's assumptions hold true. One of these assumptions is that the null hypothesis (often "of no effect") is indeed true. Given all assumptions are met, including that the null hypothesis is true, and an infinite number of trials, a result with p ≤ .05 is expected to happen only in 5% (.05) of trials (as also visible in the uniform p-value distribution of @dariober's plot).

If you set a significant threshold/level of .05, you consider every result/test statistic with a p-value < .05 as "significant" and reject the null hypothesis. Now if in a significance test the null hypothesis is in fact true and you receive p < .05, you'd incorrectly reject the null hypothesis in these cases. And since by definition the probabilities of disjunct events sum to 1, you correctly do not reject the null hypothesis in 1 - 0.05 = 0.95 = 95% of trials/samples (in the long run, i.e. given an infinite number of repetitions).

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4.2 Two-tailed tests

Hypotheses that have an equal (=) or not equal (≠) supposition (sign) in the statement are called non-directional hypotheses . In non-directional hypotheses, the researcher is interested in whether there is a statistically significant difference or relationship between two or more variables, but does not have any specific expectation about which group or variable will be higher or lower. For example, a non-directional hypothesis might be: ‘There is a difference in the preference for brand X between male and female consumers.’ In this hypothesis, the researcher is interested in whether there is a statistically significant difference in the preference for brand X between male and female consumers, but does not have a specific prediction about which gender will have a higher preference. The researcher may conduct a survey or experiment to collect data on the brand preference of male and female consumers and then use statistical analysis to determine whether there is a significant difference between the two groups.

Non-directional hypotheses are also known as two-tailed hypotheses. The term ‘two-tailed’ comes from the fact that the statistical test used to evaluate the hypothesis is based on the assumption that the difference or relationship could occur in either direction, resulting in two ‘tails’ in the probability distribution. Using the coffee foam example (from Activity 1), you have the following set of hypotheses:

H 0 : µ = 1cm foam

H a : µ ≠ 1cm foam

In this case, the researcher can reject the null hypothesis for the mean value that is either ‘much higher’ or ‘much lower’ than 1 cm foam. This is called a two-tailed test because the rejection region includes outcomes from both the upper and lower tails of the sample distribution when determining a decision rule. To give an illustration, if you set alpha level (α) equal to 0.05, that would give you a 95% confidence level. Then, you would reject the null hypothesis for obtained values of z < 1.96 and z > 1.96 (you will look at how to calculate z-scores later in the course).

This can be plotted on a graph as shown in Figure 7.

A symmetrical graph reminiscent of a bell. The x-axis is labelled ‘z-score’ and the y-axis is labelled ‘probability density’. The x-axis increases in increments of 1 from -2 to 2.

The top of the bell-shaped curve is labelled ‘Foam height = 1cm’. The graph circles the rejection regions of the null hypothesis on both sides of the bell curve. Within these circles are two areas shaded orange: beneath the curve from -2 downwards which is labelled z < -1.96 and α = 0.025; and beneath the curve from 2 upwards which is labelled z > 1.96 and α = 0.025.

In a two-tailed hypothesis test, the null hypothesis assumes that there is no significant difference or relationship between the two groups or variables, and the alternative hypothesis suggests that there is a significant difference or relationship, but does not specify the direction of the difference or relationship.

When performing a two-tailed test, you need to determine the level of significance, which is denoted by alpha (α). The value of alpha, in this case, is 0.05. To perform a two-tailed test at a significance level of 0.05, you need to divide alpha by 2, giving a significance level of 0.025 for each distribution tail (0.05/2 = 0.025). This is done because the two-tailed test is looking for significance in either tail of the distribution. If the calculated test statistic falls in the rejection region of either tail of the distribution, then the null hypothesis is rejected and the alternative hypothesis is accepted. In this case, the researcher can conclude that there is a significant difference or relationship between the two groups or variables.

Assuming that the population follows a normal distribution, the tail located below the critical value of z = –1.96 (in a later section, you will discuss how this value was determined) and the tail above the critical value of z = +1.96 each represent a proportion of 0.025. These tails are referred to as the lower and upper tails, respectively, and they correspond to the extreme values of the distribution that are far from the central part of the bell curve. These critical values are used in a two-tailed hypothesis test to determine whether to reject or fail to reject the null hypothesis. The null hypothesis represents the default assumption that there is no significant difference between the observed data and what would be expected under a specific condition.

If the calculated test statistic falls within the critical values, then the null hypothesis cannot be rejected at the 0.05 level of significance. However, if the calculated test statistic falls outside the critical values (orange-coloured areas in Figure 7), then the null hypothesis can be rejected in favour of the alternative hypothesis, suggesting that there is evidence of a significant difference between the observed data and what would be expected under the specified condition.