# Toward Evidence-Based Medical Statistics. 1: The *P* Value Fallacy

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## Abstract

*P*values and hypothesis tests, is widely perceived as a mathematically coherent approach to inference. There is little appreciation in the medical community that the methodology is an amalgam of incompatible elements, whose utility for scientific inference has been the subject of intense debate among statisticians for almost 70 years. This article introduces some of the key elements of that debate and traces the appeal and adverse impact of this methodology to the

*P*value fallacy, the mistaken idea that a single number can capture both the long-run outcomes of an experiment and the evidential meaning of a single result. This argument is made as a prelude to the suggestion that another measure of evidence should be used—the Bayes factor, which properly separates issues of long-run behavior from evidential strength and allows the integration of background knowledge with statistical findings.

*frequentist statistics,*which might be described as error-based. I explicate the logical fallacy at the heart of this system and the reason that it maintains such a tenacious hold on the minds of investigators, policymakers, and journal editors. In the second article (3), I present an evidence-based approach derived from Bayesian statistical methods, an alternative perspective that has been one of the most active areas of biostatistical development during the past 20 years. Bayesian methods have started to make inroads into medical journals;

*Annals,*for example, has included a section on Bayesian data interpretation in its Information for Authors section since 1 July 1997.

*Bayes factor*(which in its simplest form is also called a

*likelihood ratio*) instead of the

*P*value can facilitate the integration of statistical summaries and biological knowledge and lead to a better understanding of the role of scientific judgment in the interpretation of medical research.

## An Example of the Problem

*P*= 0.06 (4). The discussion section began, “ … hydrocortisone treatment was associated with an improvement in symptoms … This is the first such study … to demonstrate improvement with a drug treatment of [the chronic fatigue syndrome]” (4).

*P*= 0.06. This is a natural consequence of a statistical method that has almost eliminated our ability to distinguish between statistical results and scientific conclusions. We will see how this is a natural outgrowth of the “

*P*value fallacy.”

## Philosophical Preliminaries

*P*value fallacy, we must consider the basic elements of reasoning. The process that we use to link underlying knowledge to the observed world is called

*inferential reasoning,*of which there are two logical types:

*deductive inference*and

*inductive inference*. In deductive inference, we start with a given hypothesis (a statement about how nature works) and predict what we should see if that hypothesis were true. Deduction is objective in the sense that the predictions about what we will see are always true if the hypotheses are true. Its problem is that we cannot use it to expand our knowledge beyond what is in the hypotheses.

*problem of induction*(5-7).

*top*). Much harder is the inductive art of differential diagnosis: specifying the likelihood of different diseases on the basis of a patient's signs, symptoms, and laboratory results. The deductions are more certain and “objective” but less useful than the inductions.

*bottom*). But once we observe a particular outcome, as in the result of a clinical trial, it is not easy to answer the more important inductive question, “How likely is it that the treatments are equivalent?”

*Bayes theorem;*it was not divulged until 1762, 20 years after his death (11). Figure 2 shows Bayes theorem in words.

## Conventional (Frequentist) Statistical Inference

*P*value, proposed by R.A. Fisher in the 1920s (13), and a method for choosing between hypotheses, called a hypothesis test, developed in the early 1930s by the mathematical statisticians Jerzy Neyman and Egon Pearson (14). These two methods were incompatible but have become so intertwined that they are mistakenly regarded as part of a single, coherent approach to statistical inference (6, 15, 16).

### The *P* Value

*P*value is defined as the probability, under the assumption of no effect or no difference (the

*null hypothesis*), of obtaining a result equal to or more extreme than what was actually observed (Figure 3). Fisher proposed it as an informal index to be used as a measure of discrepancy between the data and the null hypothesis. It was not part of a formal inferential method. Fisher suggested that it be used as part of the fluid, non-quantifiable process of drawing conclusions from observations, a process that included combining the

*P*value in some unspecified way with background information (17).

*P*value (18-21). Most researchers and readers think that a

*P*value of 0.05 means that the null hypothesis has a probability of only 5%. In my experience teaching many academic physicians, when physicians are presented with a single-sentence summary of a study that produced a surprising result with

*P*= 0.05, the overwhelming majority will confidently state that there is a 95% or greater chance that the null hypothesis is incorrect. This is an understandable but categorically wrong interpretation because the

*P*value is calculated on the assumption that the null hypothesis is true. It cannot, therefore, be a direct measure of the probability that the null hypothesis is false. This logical error reinforces the mistaken notion that the data alone can tell us the probability that a hypothesis is true. Innumerable authors have tried to correct this misunderstanding (18, 20). Diamond and Forrester (19) reanalyzed several large clinical trials, and Brophy and Joseph (22) revisited the GUSTO (Global Use of Streptokinase and tPA for Occluded Coronary Arteries) trial to show that the final probability of no effect, which can be calculated only with Bayesian methods, can differ greatly from the

*P*value. However, serious as that issue is, this article will focus on the subtler and more vexing problems created by using the

*P*value as it was originally intended: as a measure of inductive evidence.

*P*value (23, 24). Perhaps the most powerful criticism was that it was a measure of evidence that did not take into account the size of the observed effect. A small effect in a study with large sample size can have the same

*P*value as a large effect in a small study. This criticism is the foundation for today's emphasis on confidence intervals rather than

*P*values (25-28). Ironically, the

*P*value was effectively immortalized by a method designed to supplant it: the hypothesis testing approach of Neyman and Pearson.

### Hypothesis Tests

*P*value as an incomplete answer to the problem of developing an inferential method without Bayes theorem. In their hypothesis test, one poses

*two*hypotheses about nature: a null hypothesis (usually a statement that there is a null effect) and an alternative hypothesis, which is usually the opposite of the null hypothesis (for example, that there is a nonzero effect). The outcome of a hypothesis test was to be a behavior, not an inference: to reject one hypothesis and accept the other, solely on the basis of the data. This puts the researcher at risk for two types of errors—behaving as though two therapies differ when they are actually the same (also known as a

*false-positive result,*a

*type I error,*or an α

*error*[Figure 3]) or concluding that they are the same when in fact they differ (also known as a

*false-negative result,*a

*type II error,*or a β

*error*).

no test based upon a theory of probability can by itself provide any valuable evidence of the truth or falsehood of a hypothesis.

But we may look at the purpose of tests from another viewpoint. Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behaviour with regard to them, in following which we insure that, in the long run of experience, we shall not often be wrong.

*P*value,” much to the dismay of Fisher, Neyman, Pearson, and many experts on statistical inference who followed.

## The *P* Value “Solution”

*P*value seem to solve an insoluble problem? It did so in part by appearing to be a measure of evidence in a single experiment that did not violate the long-run logic of the hypothesis test. Figure 3 shows how similar the

*P*value and the α value (the false-positive error rate) appear. Both are tail-area probabilities under the null hypothesis. The tail area corresponding to the false-positive error rate (α) of the hypothesis test is fixed before the experiment begins (almost always at 0.05), whereas the

*P*value tail area starts from a point determined by the data. Their superficial similarity makes it easy to conclude that the

*P*value is a special kind of false-positive error rate, specific to the data in hand. In addition, using Fisher's logic that the

*P*value measured how severely the null hypothesis was contradicted by the data (that is, it could serve as a measure of evidence against the null hypothesis), we have an index that does double duty. It seems to be a Neyman-Pearson data-specific, false-positive error rate and a Fisher measure of evidence against the null hypothesis (6, 15, 17).

*P*value and the false-positive error rate is made (30):

The statement “P < 0.01” indicates that the discrepancy between the sample mean and the null hypothesis mean is significant even if such a conservative significance level as 1 percent is adopted. The statement “P = 0.006” indicates that the result is significant at any level up to 0.6 percent.

## The *P* Value Fallacy

*P*value can play both of these roles is based on a fallacy: that an event can be viewed simultaneously both from a long-run and a short-run perspective. In the long-run perspective, which is error-based and deductive, we group the observed result together with other outcomes that might have occurred in hypothetical repetitions of the experiment. In the “short run” perspective, which is evidential and inductive, we try to evaluate the meaning of the observed result from a single experiment. If we could combine these perspectives, it would mean that inductive ends (drawing scientific conclusions) could be served with purely deductive methods (objective probability calculations).

*P*value of 0.11, whereas one who planned to stop as soon as treatment B was preferred (up to a maximum of six patients) would calculate a

*P*value of 0.03 (Appendix). We have the same patients, the same treatments, and the same outcomes but two very different

*P*values (which might produce different conclusions), which differ only because the experimenters have different mental pictures of what the results could be if the experiment were repeated. A confidence interval would show this same behavior.

*P*values (Appendix).

*P*value fallacy is that a result cannot at the same time be an anonymous (interchangeable) member of a group of results (the long-run view) and an identifiable (unique) member (the short-run view) (6, 15, 31). In my second article in this issue, we will see that if we stick to the short-run perspective when measuring evidence, identical data produce identical evidence regardless of the experimenters' intentions.

*P*value is grounded in this fundamental problem. The

*multiple comparisons debate*is whether a comparison should be considered part of a group of all comparisons made (that is, as an anonymous member) or separately (as an identifiable member) (32-35). The controversy over how to cite a

*P*value when a study is stopped because of a large treatment effect is about whether we consider the result alone or as part of all results that might have arisen from such monitoring (36-39). In a trial of extracorporeal membrane oxygenation in infants, a multitude of

*P*values were derived from the same data (40). This problem also has implications for the design of experiments. Because frequentist inference requires the “long run” to be unambiguous, frequentist designs need to be rigid (for example, requiring fixed sample sizes and prespecified stopping rules), features that many regard as requirements of science rather than as artifacts of a particular inferential philosophy.

*P*value, in trying to serve two roles, serves neither one well. This is seen by examining the statement that “a result with

*P*= 0.05 is in a group of outcomes that has a 5% chance of occurring under the null hypothesis.” Although that is literally the case, we know that the result is not just

*in*that group (that is, anonymous); we know where it is, and we know that it is the most probable member (that is, it is identifiable). It is

*in*that group in the same way that a student who ranks 10 out of 100 is

*in*the top 10% of the class, or one who ranks 20th is

*in*the top 20% (15). Although literally true, these statements are deceptive because they suggest that a student could be anywhere in a top fraction when we know he or she is at the lowest level of that top group. This same property is part of what makes the

*P*value an inappropriate measure of evidence against the null hypothesis. As will be explored in some depth in the second article, the evidential strength of a result with a

*P*value of 0.05 is actually much weaker than the number 0.05 suggests.

*P*value fallacy were limited to the realm of statistics, it would be a mere technical footnote, hardly worth an extended exposition. But like a single gene whose abnormality can disrupt the functioning of a complex organism, the

*P*value fallacy allowed the creation of a method that amplified the fallacy into a conceptual error that has profoundly influenced how we think about the process of science and the nature of scientific truth.

## Creation of a Combined Method

*P*value and the subtlety of the fallacy that it embodied enabled the combination of the hypothesis test and

*P*value approaches. This combination method is characterized by setting the type I error rate (almost always 5%) and power (almost always ≥ 80%) before the experiment, then calculating a

*P*value and rejecting the null hypothesis if the

*P*value is less than the preset type I error rate.

*P*value) with the null hypothesis within the context of a method that controls the chances of errors. The key word here is

*probability*, because a probability has an absoluteness that overwhelms caveats that it is not a probability of truth or that it should not be used mechanically. Such features as biological plausibility, the cogency of the theory being tested, and the strength of previous results all become mere side issues of unclear relevance. None of these change the probability, and the probability does not need them for interpretation. Thus, we have an objective inference calculus that manufactures conclusions seemingly without paying Neyman and Pearson's price (that is, that it not be used to draw conclusions from individual studies) and without Fisher's flexibility (that is, that background knowledge be incorporated).

*P*value is identified as equivalent to the chance of a false-positive error. In a tutorial on statistics for surgeons, under the unwittingly revealing subheading of “Errors in statistical inference,” we are told that “Type I error is incurred if H

_{o}[the null hypothesis] is falsely rejected, and the probability of this corresponds to the familiar P-value” (41).

## Implications for Interpretation of Medical Research

What used to be called judgment is now called prejudice, and what used to be called prejudice is now called a null hypothesis … it is dangerous nonsense (dressed up as the “scientific method”) and will cause much trouble before it is widely appreciated as such (69).

*P*= 0.06” and “treatment was associated with improvement in symptoms.” That bridge consists of everything that the authors put into the latter part of their discussion: the magnitude of the change (small), the failure to change other end points, the absence of supporting studies, and the weak support for the proposed biological mechanism. Ideally, all of this other information should have been combined with the modest statistical evidence for the main end point to generate a conclusion about the likely presence or absence of a true hydrocortisone effect. The authors did recommend against the use of the treatment, primarily because the risk for adrenal suppression could outweigh the small beneficial effect, but the claim for the benefit of hydrocortisone remained.

*P*value seemed to play almost no role. The initial conclusion was phrased no differently than if the

*P*value had been less than 0.001. This omission is the legacy of the hypothesis test component of the combined method of inference. The authors (and journal) are to be lauded for not hewing rigidly to hypothesis test logic, which would dismiss the

*P*value of 0.06 as nonsignificant, but if one does not use the hypothesis test framework, conclusions must incorporate the graded nature of the evidence. Unfortunately, even Fisher could offer little guidance on how the size of a

*P*value should affect a conclusion, and neither has anyone else. In contrast, we will see in the second article how Bayes factors offer a natural way to incorporate different grades of evidence into the formation of conclusions.

*P*values, a practice whose incoherence is most apparent when the “significance” verdict is not consistent with external evidence or the author's beliefs. If a

*P*value of 0.12 is found for an a priori unsuspected difference, an author often says that the groups are “equivalent” or that there was “no difference.” But the same

*P*value found for an expected difference results in the use of words such as “trend” or “suggestion,” a claim that the study was “not significant because of small sample size,” or an intensive search for alternative explanations. On the other hand, an unexpected result with a

*P*value of 0.01 may be declared a statistical fluke arising from data dredging or perhaps uncontrolled confounding. Perhaps worst is the practice that is most common: accepting at face value the significance verdict as a binary indicator of whether or not a relation is real. What drives all of these practices is a perceived need to make it appear that conclusions are being drawn directly from the data, without any external influence, because direct inference from data to hypothesis is thought to result in mistaken conclusions only rarely and is therefore regarded as “scientific.” This idea is reinforced by a methodology that puts numbers—a stamp of legitimacy—on that misguided approach.

*P*values offers no way to accomplish this critical task.

## Proposed Solutions

*P*values and hypothesis tests by promoting a consideration of the size of the observed effect. They are cited more often in medical research reports today than in the past, but their impact on the interpretation of research is less clear. Often, they are used simply as surrogates for the hypothesis test (75); researchers simply see whether they include the null effect rather than consider the clinical implications of the full range of likely effect size. The few efforts to eliminate

*P*values from journals in favor of confidence intervals have not generally been successful, indicating that researchers' need for a measure of evidence remains strong and that they often feel lost without one (76, 77). But confidence intervals are far from a panacea; they embody, albeit in subtler form, many of the same problems that afflict current methods (78), the most important being that they offer no mechanism to unite external evidence with that provided by an experiment. Thus, although confidence intervals are a step in the right direction, they are not a solution to the most serious problem created by frequentist methods. Other recommended solutions have included likelihood or Bayesian methods (6, 19, 20, 79-84). The second article will explore the use of Bayes factor—the Bayesian measure of evidence—and show how this approach can change not only the numbers we report but, more important, how we think about them.

## A Final Note

*P*values, very differently from how most nonstatisticians do (67, 85, 86). But in a world where medical researchers have access to increasingly sophisticated statistical software, the statistical complexity of published research is increasing (87-89), and more clinical care is being driven by the empirical evidence base, a deeper understanding of statistics has become too important to leave only to statisticians.

## Appendix: Calculation of *P* Value in a Trial Involving Six Patients

*Null hypothesis:*Probability that treatment A is better = 1/2

*The*n =

*6 design:*The probability of the observed result (one treatment B success and five treatment A successes) is 6 × (1/2) × (1/2)

^{5}. The factor “6” appears because the success of treatment B could have occurred in any of the six patients. The more extreme result would be the one in which treatment A was superior in all six patients, with a probability (under the null hypothesis) of (1/2)

^{6}. The one-sided

*P*value is the sum of those two probabilities:

*“Stop at first treatment B preference” design:*The possible results of such an experiment would be either a single instance of preference for treatment B or successively more preferences for treatment A, followed by a case of preference for treatment B, up to a total of six instances. With the same data as before, the probability of the observed result of 5 treatment A preferences − 1 treatment B preference would be (1/2)

^{5}× (1/2) (without the factor of “6” because the preference for treatment B must always fall at the end) and the more extreme result would be six preferences for treatment As, as in the other design. The one-sided

*P*value is:

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## Information & Authors

### Information

#### Published In

#### History

**Published in issue**: 15 June 1999

**Published online**: 15 August 2000

#### Keywords

- Behavior
- Careers in research
- Cognition
- Cognitive psychology
- Encephalomyelitis
- Health care
- Infectious diseases
- Medical ethics, humanities, and education
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Toward Evidence-Based Medical Statistics. 1: The

*P*Value Fallacy. Ann Intern Med.1999;130:995-1004. doi:10.7326/0003-4819-130-12-199906150-00008

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