With the many problems that p-values have, and the temptation to "bless" research when the p-value falls below an arbitrary threshold such as 0.05 or 0.005, researchers using p-values should at least be fully aware of what they are getting. They need to know exactly what a p-value means and what are the assumptions required for it to have that meaning. ♦ A p-value is the probability of getting, in another study, a test statistic that is more extreme than the one obtained in your study if a series of assumptions hold. It is strictly a probability about data, not a probability about a hypothesis or about the effect of a variable. ♦ The study must be capable of being repeated infinitely often, or one must play a mind game in which this is so. ♦ The repeated studies have data generated by exactly the same data model as the original study with one exception: the null hypothesis is forced to be exactly true. [And we don't know how to count negative treatment benefit.] ♦ To do the repetitions, you must know the exact design and sampling plan that was in effect for your study. [It's not even clear that one would want to use the same sample size in repeating a study.] ♦ You must know all the investigators' intensions for testing, including intended timing and frequency of looks at the data. ♦ You must know the exact stopping rule for the study, or to pretend that the actual final sample size was magical and would be chosen again and again. Note: all of these design features must be the ones actually used in conducting the study, not those in the original study plan. If calculations are based on the original study plan, no deviations from that plan during the study are allowed. ♦ The study repetitions used to compute the p-value must be executed using exactly this sampling plan, stopping rule, data look schedule, and investigator intentions.

In my opinion, null hypothesis testing and p-values have done significant harm to science. The purpose of this note is to catalog the many problems caused by p-values. As readers post new problems in their comments, more will be incorporated into the list, so this is a work in progress.

The American Statistical Association has done a great service by issuing its Statement on Statistical Significance and P-values. Now it’s time to act. To create the needed motivation to change, we need to fully describe the depth of the problem.

It is important to note that no statistical paradigm is perfect. Statisticians should choose paradigms that solve the greatest number of real problems and have the fewest number of faults. This is why I believe that the Bayesian and likelihood paradigms should replace frequentist inference.

Consider an assertion such as “the coin is fair”, “treatment A yields the same blood pressure as treatment B”, “B yields lower blood pressure than A”, or “B lowers blood pressure at least 5mmHg before A.” Consider also a compound assertion such as “A lowers blood pressure by at least 3mmHg and does not raise the risk of stroke.”

A. Problems With Conditioning

p-values condition on what is unknown (the assertion of interest; H~0~) and do not condition on what is known (the data). This conditioning does not respect the flow of time and information; p-values are backward probabilities.

B. Indirectness

Because of A above, p-values provide only indirect evidence and are problematic as evidence metrics. They are sometimes monotonically related to the evidence (e.g., when the prior distribution is flat) we need but are not properly calibrated for decision making. p-values are used to bring indirect evidence against an assertion but cannot bring evidence in favor of the assertion. As detailed here, the idea of proof by contradiction is a stretch when working with probabilities, so trying to quantify evidence for an assertion by bringing evidence against its complement is on shaky ground. Because of A, p-values are difficult to interpret and very few non-statisticians get it right. The best article on misinterpretations I’ve found is here.

C. Problem Defining the Event Whose Probability is Computed

In the continuous data case, the probability of getting a result as extreme as that observed with our sample is zero, so the p-value is the probability of getting a result more extreme than that observed. Is this the correct point of reference? How does more extreme get defined if there are sequential analyses and multiple endpoints or subgroups? For sequential analyses do we consider planned analyses are analyses intended to be run even if they were not?

D. Problems Actually Computing p-values

In some discrete data cases, e.g., comparing two proportions, there is tremendous disagreement among statisticians about how p-values should be calculated. In a famous 2x2 table from an ECMO adaptive clinical trial, 13 p-values have been computed from the same data, ranging from 0.001 to 1.0. And many statisticians do not realize that Fisher’s so-called “exact” test is not very accurate in many cases. Outside of binomial, exponential, and normal (with equal variance) and a few other cases, p-values are actually very difficult to compute exactly, and many p-values computed by statisticians are of unknown accuracy (e.g., in logistic regression and mixed effects models). The more non-quadratic the log likelihood function the more problematic this becomes in many cases. One can compute (sometimes requiring simulation) the type-I error of many multi-stage procedures, but actually computing a p-value that can be taken out of context can be quite difficult and sometimes impossible. One example: one can control the false discovery probability (incorrectly usually referred to as a rate), and ad hoc modifications of nominal p-values have been proposed, but these are not necessarily in line with the real definition of a p-value.

E. The Multiplicity Mess

Frequentist statistics does not have a recipe or blueprint leading to a unique solution for multiplicity problems, so when many p-values are computed, the way they are penalized for multiple comparisons results in endless arguments. A Bonferroni multiplicity adjustment is consistent with a Bayesian prior distribution specifying that the probability that all null hypotheses are true is a constant no matter how many hypotheses are tested. By contrast, Bayesian inference reflects the facts that P(A ∪ B) ≥ max(P(A), P(B)) and P(A ∩ B) ≤ min(P(A), P(B)) when A and B are assertions about a true effect. There remains controversy over the choice of 1-tailed vs. 2-tailed tests. The 2-tailed test can be thought of as a multiplicity penalty for being potentially excited about either a positive effect or a negative effect of a treatment. But few researchers want to bring evidence that a treatment harms patients; a pharmaceutical company would not seek a licensing claim of harm. So when one computes the probability of obtaining an effect larger than that observed if there is no true effect, why do we too often ignore the sign of the effect and compute the (2-tailed) p-value? Because it is a very difficult problem to compute p-values when the assertion is compound, researchers using frequentist methods do not attempt to provide simultaneous evidence regarding such assertions and instead rely on ad hoc multiplicity adjustments. Because of A1, statistical testing with multiple looks at the data, e.g., in sequential data monitoring, is ad hoc and complex. Scientific flexibility is discouraged. The p-value for an early data look must be adjusted for future looks. The p-value at the final data look must be adjusted for the earlier inconsequential looks. Unblinded sample size re-estimation is another case in point. If the sample size is expanded to gain more information, there is a multiplicity problem and some of the methods commonly used to analyze the final data effectively discount the first wave of subjects. How can that make any scientific sense? Most practitioners of frequentist inference do not understand that multiplicity comes from chances you give data to be extreme, not from chances you give true effects to be present.

F. Problems With Non-Trivial Hypotheses

It is difficult to test non-point hypotheses such as “drug A is similar to drug B”. There is no straightforward way to test compound hypotheses coming from logical unions and intersections.

G. Inability to Incorporate Context and Other Information

Because extraordinary claims require extraordinary evidence, there is a serious problem with the p-value’s inability to incorporate context or prior evidence. A Bayesian analysis of the existence of ESP would no doubt start with a very skeptical prior that would require extraordinary data to overcome, but the bar for getting a “significant” p-value is fairly low. Frequentist inference has a greater risk for getting the direction of an effect wrong (see here for more). p-values are unable to incorporate outside evidence. As a converse to 1, strong prior beliefs are unable to be handled by p-values, and in some cases the results in a lack of progress. Nate Silver in The Signal and the Noise beautifully details how the conclusion that cigarette smoking causes lung cancer was greatly delayed (with a large negative effect on public health) because scientists (especially Fisher) were caught up in the frequentist way of thinking, dictating that only randomized trial data would yield a valid p-value for testing cause and effect. A Bayesian prior that was very strongly against the belief that smoking was causal is obliterated by the incredibly strong observational data. Only by incorporating prior skepticism could one make a strong conclusion with non-randomized data in the smoking-lung cancer debate. p-values require subjective input from the producer of the data rather than from the consumer of the data.

H. Problems Interpreting and Acting on “Positive” Findings

With a large enough sample, a trivial effect can cause an impressively small p-value (statistical significance ≠ clinical significance). Statisticians and subject matter researchers (especially the latter) sought a “seal of approval” for their research by naming a cutoff on what should be considered “statistically significant”, and a cutoff of p=0.05 is most commonly used. Any time there is a threshold there is a motive to game the system, and gaming (p-hacking) is rampant. Hypotheses are exchanged if the original H~0~ is not rejected, subjects are excluded, and because statistical analysis plans are not pre-specified as required in clinical trials and regulatory activities, researchers and their all-too-accommodating statisticians play with the analysis until something “significant” emerges. When the p-value is small, researchers act as though the point estimate of the effect is a population value. When the p-value is small, researchers believe that their conceptual framework has been validated.

I. Problems Interpreting and Acting on “Negative” Findings

Because of B2, large p-values are uninformative and do not assist the researcher in decision making (Fisher said that a large p-value means “get more data”).

J. Distortion of Scientific Conclusions

Greenwald, Gonzalez, Harris, and Guthrie’s paper Effect sizes and p values: What should be reported and what should be replicated? nicely describes subtle distortions in the scientific research process caused by the usage of null hypotheses:

One of the more important varieties of prejudince against the null hypothesis ... comes about as a consequence of researchers much more identifying their own theoretical predictions with rejections (rather than with acceptances) of the null hypothesis. The consequence is an ego involvement with rejection of the null hypothesis that often leads researchers to interpret null hypothesis rejections as valid confirmations of their theoretical beliefs while interpreting nonrejections as uninformative and possibly the result of flawed mehods.

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