Living with P Values: Resurrecting a Bayesian Perspective on Frequentist Statistics : Epidemiology

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Living with P Values

Resurrecting a Bayesian Perspective on Frequentist Statistics

Greenland, Sandera,b; Poole, Charlesc

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Epidemiology 24(1):p 62-68, January 2013. | DOI: 10.1097/EDE.0b013e3182785741
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P values for null hypotheses of no association or no effect (null P values, which we denote by P0) still thrive in most of the literature. Despite being data frequencies under a hypothetical sampling model, they are routinely misinterpreted as probabilities of a “chance finding” or of the null, which are Bayesian probabilities. To quote one biostatistics textbook: “The P value is the probability of being wrong when asserting that a true difference exists.”1 Similar misinterpretations remain in guides for researchers. For example, Cordova2 states the null P value “represents the likelihood that groups differ after a treatment due to chance.”

It is thus unsurprising that many call for improved education regarding P values.3–11 Others discourage specific uses of P values12–15 or, occasionally, all uses (as in the EPIDEMIOLOGY instructions for authors: “We strongly discourage the use of P values and language referring to statistical significance…”), regarding them as a confounded mix of estimate size and precision.16,17

Many advocates of educational reform specifically decry use of P values in Neyman-Pearson hypothesis testing, in which results are placed in bins labeled “association” and “no association” based on whether the P value is below or above a prespecified alpha level (the maximum tolerable type-I or false-positive error probability), usually 0.05. Neyman-Pearson testing has often been criticized as groundless and arbitrary9,17–20 and even mindless3,5—aptly in our view. Unfortunately, Neyman-Pearson testing has also been called “significance testing,” and the alpha level has been called the “significance level” of the test. This secondary terminology has fed the oft-lamented confusion of Neyman-Pearson testing with the significance-testing approach of Fisher.21,22 In the latter approach, significance level refers to the P value itself, which is treated as a continuous measure of evidence against the tested hypothesis, with lower P values corresponding to greater evidence.6,9,23–26

This article is in the reform tradition. We first consider some arguments against P values. We then review ways in which P values and other ordinary frequentist statistics can be given correct Bayesian interpretations. We focus on how a one-sided P value can provide a lower bound on the posterior probability that the point estimate is on the wrong side of the prior median (the prior odds that the true parameter is above vs. below this median is 50:50).27 In particular, when one sees a two-sided null P value P0 for a difference or log ratio measure, one can immediately say that if one starts with a normal or uniform prior centered on the null, the posterior probability that the point estimate is in the wrong direction (ie, on the wrong side of the null) is no less than P0 / 2.


It is sometimes said that confidence intervals render P values redundant because the two give “equivalent information,” with no indication of what “equivalent” means. True, for any parameter value, its P value is under 0.05 if and only if the parameter value is outside the 95% confidence interval computed by the same method. But this equivalence argument seems to restore 0.05-level testing to the primacy it retains in most medical journals.28

One can compute a P value from a confidence interval; therefore, the abstract (mathematical) information in a P value is indeed redundant given the confidence interval. But, unless the P value is 1, we can compute a confidence interval from the point estimate and P value, rendering the confidence interval redundant in the same technical sense. Furthermore, just as the estimate and P value do not convey a confidence interval well, a confidence interval does not convey desired P values well.10 For instance, we have found colleagues who believe that if a set of risk ratios (RRs) all have lower confidence limits above 1, the RR with the highest lower limit must have the smallest null P value P0, especially if that RR has the largest point estimate. Others seem to think that if the lower limits are the same, the null P values are the same, or the one with the higher point estimate has the lower null P value. These beliefs are often the opposite of reality; for example, if the estimate

is 1.6 with lower limit 1.2, P0 is 0.001, whereas for

= 4.8 with lower limit 1.3, P0 is 0.020.

Equivalence and redundancy are thus not good arguments for suppressing P values as supplements to point and interval estimates. There are, however, good psychological arguments for that suppression. They are based on the observation that null P values are almost always interpreted badly, even by statisticians. For example, a small P0 is routinely taken as refuting the null and a large P0 is routinely taken as supporting the null; however, both interpretations are usually far from correct.8,11–13,25,26,28–37 We will explain how these problems could be avoided if P values were interpreted as distance measures and posterior probabilities rather than as tests of hypotheses.

Assumptions and Notation

Throughout, we assume that an analysis goal is inference about the true value θt of a parameter or some simple function of θt (eg, its antilog) in an analysis using standard models and approximations. Usually, θt is a coefficient in a linear, logistic, Poisson, or proportional hazards regression and analysis and thus represents a difference or log ratio; however, when is a product-term coefficient, it measures variation (heterogeneity) in a difference or log ratio.

will denote one of the usual efficient estimators of θt (eg, least-squares, maximum likelihood), with

its estimated standard error. θ will denote the parameter name or a particular value for the parameter, possibly incorrect, but of special contextual interest. In almost all software and reports of P values, the null value 0 for θ is automatically taken to be of key interest, but other values may be as or more important to analyze. For example, in certain legal situations, a risk or rate ratio RR of 2 translates to a 50% lower bound for a causation probability,38 which leads to examining θ = ln(RR = ln(2).


We illustrate concepts using familiar Wald methods derived from the Z score

and its absolute value

which is the distance in standard errors from the observation

to the possibility θ. The two-sided P value Pθ for θ is, then, the relative frequency (over study repetitions) that the absolute Z statistic

would exceed its observed value if indeed θt=θ and all other assumptions used to compute this frequency are correct. In particular, P0 is the usual two-sided null P value for the hypothesis θt = 0, derived from

, and Pln(2) is the two-sided P value for θt = ln(2), derived from

. The usual 95% confidence interval with limits

can be derived by solving

for θ and thus can be defined as all θ for which Pθ. Conversely, given a 95% confidence interval confidence interval

, we can deduce

and thus compute a Z score and P value for any θ of interest.

P values such as Pθ are often described as measures of goodness of fit, distance, consistency, or compatibility between the observed data and the data-generating model, or between a parameter estimate

and a hypothesized parameter constraint such as θt=θ or θtθ.23,39Pθ is the probability transform of the distance from θ to

; in particular, P0 is the transformed distance of

to the null. A small Pθ is taken to indicate a problem with the assumptions used for its computation (eg, perhaps, θtθ, or there is some uncontrolled validity problem, or both).21,22 This interpretation is popular among those seeking to avoid both hypothesis testing and Bayesian interpretations of P values.26

To illustrate, consider a disease indicator Y = 1,0 and a logistic regression model for disease frequency given two binary factors X and Z,

Pr(Y = 1 | X = x, Z = z) = expit(α + βx + γz)

This model asserts that only three parameters (α,β,γ) are needed to perfectly specify (or encode) the disease frequencies at every combination of X = 1,0 and Z = 1,0. There is rarely any justification for this assumption; however, it is routine and usually unmentioned, or else unquestioned if the P value for the test of model fit is “big enough” (usually meaning at least 0.05 or 0.10).

Consider the null P value P0 for θt = 0 when θ is the product-term coefficient in

Pr(Y = 1 | X = x, Z = z) = expit(α + βx + γz + θxz)

θt = 0 yields model 1 and translates into constancy of the odds ratio relating X to Y as Z varies. P0 is a measure of

  • (a). The distance from 0 to
  • ,
  • (b). The goodness of fit of model 1 when taking the more general model 2 as the referent, and
  • (c). The distance from the fitted model 1 to the fitted model 2 (where the fitted models are the above equations with the parameters replaced by their estimates).

For comparison, the non-null value θt = ln(2) translates into a doubling of the odds ratio relating X to Y when we move from Z = 0 to Z = 1 and corresponds to the model:

Pr(Y = 1 | X = x, Z = z) = expit(α + βz + γz + ln(2)xz)

Pln(2) is a measure of

  • (a).The distance from ln(2) to
  • ,
  • (b).The goodness of fit of model 3 when taking model 2 as the referent, and
  • (c).The distance from the fitted model 3 to the fitted model 2.

Finally, the 95% confidence interval for θt is a measure of the spread of the four-dimensional likelihood function derived from the data and model 2 along the θ dimension. Consequently, confidence interval width is sometimes promoted as a measure of study precision in estimating θt.12 This interpretation, however, assumes model 2; using other models, we would expect all the confidence intervals and P values to change, demonstrating that statistical concepts of precision, fit, and distance are relative to a model rather than absolute properties of a study.


Typical misinterpretations of P values treat them as Bayesian posterior probabilities. For example, one common but extreme mistake interprets Pθ as the probability that θt = θ; however, these two probabilities are usually far apart.29,30 Such misinterpretations may be recognized and avoided by examining situations in which Pθ is indeed a probability (bet) about the true value θt. For a review of basic Bayesian ideas such as prior and posterior probabilities, see for example, Greenland.40–42

Extreme-Prior Interpretation

Under maximum-likelihood analyses, the most direct Bayesian interpretations of P values and confidence intervals arise when only two extreme kinds of prior distributions are allowed: point priors, which express 100% certainty that a parameter is a given value; and equal-odds (uniform) priors on the usual normalizing (“natural parameter”) scale for setting confidence intervals (eg, typically the log scale for ratio parameters, and the coefficient scale in multiplicative models). If τ denotes the prior standard deviation for θ, point priors correspond to τ = 0 and equal-odds priors correspond to τ = ∞.

A parameter θ with a point prior asserting it must be zero is usually inapparent (implicit) in the model because it is replaced in the model by zero and so disappears from sight. Model 1 is an example in which θ is invisible, yet the model is a special case of model 2 with the added constraint θt = 0, which in Bayesian terms corresponds to imposing the point prior Pr(θt = 0) = 1 for θ and placing equal odds on all possible combinations of α, β, and γ. Similarly, model 3 corresponds to imposing the point prior Pr{θt = 1n(2)} = 1 and placing equal odds on all possible combinations of α, β, and γ, and, but θ now remains visible in the model because it is nonzero.

If instead we assume only the more general model 2 and place equal prior odds on all combinations of α, β, γ, and θ, we obtain the following interpretations of frequentist statistics as approximate Bayesian posterior statistics:

  • (a). Pθ is the posterior probability that
  • is closer to θ than to the truth θt (ie, is Pθ the probability that
  • .
  • (b). Pθ / 2 is the posterior probability
  • that is on the wrong side of θ relative to the truth θt; in particular, P0 / 2 is the probability that the observed association is in the wrong direction. Hence, if,
  • > θ, Pθ / 2 is the posterior probability that θt < θ; if,
  • < θ, Pθ / 2 is the probability that θt > θ; in particular, if
  • is positive, P0 / 2 is the probability that θt is negative; if
  • is negative, P0 / 2 is the probability that θt is positive.
  • (c). The 95% confidence interval
  • for θ becomes a 95% posterior probability interval for θ; hence, under model 2 and the prior, the probability that
  • is 0.95 (parallel interpretations extend to other confidence levels).
  • (d).
  • is the posterior median (the odds of θt being above vs. below
  • are equal).

Thus, the equal-odds prior renders correct the usual misinterpretation of confidence intervals and provides Bayesian interpretations of P values.

As an example, suppose

= 1.40 and

= 0.60. Then, the following Bayesian posterior probability statements follow from model 2, the data, and an equal-odds prior:

  • (i).
  • , giving P0 = 0.02 as the probability that 1.40 is closer to 0 than to θt and P0 / 2 = 0.01 as the probability that θt is negative or that eθt < 1.
  • (ii).
  • , giving Pln(2) = 0.24 as the probability that 1.40 is closer to ln(2) than to θt, and Pln(2)/2 = 0.12 as the probability that θt < ln(2) or that eθt < 2.
  • (iii).
  • , 2.58 has 95% probability of containing θt, so that the probability of e0.22 = 1.25 < eθt < e2.58 = 13.2 is 95%.
  • (iv). The odds of θt being above versus below 1.40 is 1, meaning equal probabilities that eθt < e1.40 = 4.06 and eθt > 4.06.

Taking P0 / 2 as the posterior probability that the observed association is in the wrong direction is found among pre-Fisherian writers, including Gossett, the inventor of the t test,4,34,43 and follows from Bayesian 2 × 2 table results published in 1877.44,45 More generally, (a)–(c) follow from taking the posterior distribution of the model parameters as proportional to the likelihood function, approximating the marginal posterior for θt with a normal (


) distribution (hence likelihood-ratio statistics will render a better approximation than Z scores).

Objections to Equal-Odds Priors and Responses

One objection to the equal-odds prior underlying (a)–(c) is that it is improper (it has infinite instead of 100% total probability), which leads to undefined prior probabilities even though prior odds are defined. Although some Bayesians defend direct use of improper priors,46 one can instead replace the equal-odds prior by proper priors that are weak enough to leave properties (a)–(c) approximately correct in practice, as is done in “objective” and “reference” Bayesian methods.47 We will call all these priors “weak priors,” whether proper or improper.

A more serious objection is that weak priors are usually contextually absurd.40,41 Suppose θt is a log death-rate ratio for an Food and Drug Administration–approved glioma chemotherapy versus radiation therapy. Then, an equal-odds prior says ln(10−100), 0, and ln(10100) are all equally credible values for θt. But (10−100) implies complete prevention and is disproved by any death with chemotherapy, whereas ln(10100) implies complete causation and is disproved by any survivor of chemotherapy.

One response to these criticisms is to take interpretations (a)–(c) as Bayesian measures of the information content of the data relative to the model, before prior information about the model parameters is added. This approach provides a contrast against Bayesian results derived from informative priors and is supported by noting that (a)–(c) are precisely the Bayesian results one obtains when no prior data are added to the actual data.40,41 Reference-Bayes analyses produce similar interpretations in that they correspond to adding almost no data.

Bounding Results from More Informative Priors

Weak priors are most absurd when all the values of θ being debated (and thus all serious candidates for a prior median) are so close to the null that it is difficult to distinguish among them or distinguish them from the null.40,41,48,49 Many, if not most, modern controversies (eg, long-term nutrient effects and drug side effects) involve debates within ranges like ¼ < RR < 4. In these settings, confidence intervals (and thus posterior intervals under weak priors) typically cover a substantial part of the range under debate, and no one would take seriously claims of (say) RR > 100.

Even without controversy, the effort and discomfort of specifying detailed priors is daunting. Not only must we rationalize all choices to ourselves and the reader, but we may have to allow for the fact that readers may have different priors from ours. Anticipating the full spectrum of these differences with sensitivity analyses of various priors is even more demanding. Thus, in a world of sharply constrained time resources, demands for informative priors leads instead to avoidance of Bayesian analysis.

An alternative, however, is to offer bounds on posterior probabilities when the prior is restricted to a given class. These bounds show the range that a sensitivity analysis over that class would produce and thus may address concerns of those wary of detailed priors. In particular, suppose one would regard it reasonable to consider as a possible prior for θt a distribution that is symmetric around a single mode (maximum) at its median θm and weak for the remaining model parameters. This class includes the most common coefficient priors, including normal, t, logistic, and certain reference distributions with median θm, and implies that exp(θt) has prior median exp(θm). Of special interest, normal (Gaussian) priors arise when the prior information about θt is derived from a number of sources or studies, none of which contributes a dominant fraction of the information; this prior produces a lognormal prior for exp(θt).

Let Pm denote the two-sided P value for the prior median θm. Then, Pm / 2 approximates the smallest possible posterior probability that the observed estimate

is on the wrong side of θt27,50; in addition,

is the furthest the posterior median could be from the prior median θm. These interpretations also apply using uniform priors on θt with median θm. Thus, Pm / 2 provides a statistic with a correct Bayesian interpretation, without demanding detailed specification of a prior or computation beyond Pm / 2. Again, suppose

= 1.40 and

= 0.60, with weak priors for all parameters but θ. Then, P0 / 2 is the smallest possible posterior probability that θt is negative when the θt prior is unimodal and symmetric (eg, normal) around 0; Pln(2)/2 = 0.12 is the smallest possible posterior probability that θt < ln(2) when the θt prior is unimodal and symmetric around ln(2); and 1.40 is the furthest the posterior median could be from any prior median.

Some Limitations

Within the above class of priors, the discrepancy between the lower bound Pm / 2 and the posterior probability increases as the spread of the prior decreases, albeit the discrepancy is not large until the prior becomes fairly informative relative to the data and model. To illustrate, suppose we have a normal prior with mean 0 and standard deviation τ for a log rate ratio θt = ln( RR t). This is a symmetric, unimodal prior with median 0 for θt and median 1 for RRt = exp(θt). The Table shows the resulting posterior probabilities of for RRt < 1 various estimates

= exp(

), standard errors


, and prior standard deviations τ for θt expressed in multiples of

. The final column uses an infinite τ, which gives back P0 / 2.


The Table shows a general property of normal priors: in practical terms, when the prior mean is θm, Pm / 2 will notbe far below the posterior probability that

is on the wrong side of θm if τ is over twice (so that the weight 1 / τ2 for the prior mean θ is less than a quarter the weight 1/

2 for the ordinary estimate

). Of course, if one focuses on a particular normal prior, it can be easily used to calculate directly all desired posterior intervals and probabilities40,41,51; Pm / 2 then remains a reference point showing what weaker normal priors with the same mean could yield.

The bounding interpretation also assumes weak priors on the rest of the parameters. Such priors are usually unrealistic. Nonetheless, provided care is taken in specifying the regression model (in particular, by centering the regressors51), posterior probabilities for the target parameter tend to be much less sensitive to reasonably imprecise priors on the remaining parameters than they are to the target (θt) prior. Regardless, stronger priors on all parameters require much more specification effort than most are willing to provide; but again, if provided, one can use them directly in Bayesian procedures,51 and P values can serve as reference values for readers wary of more informative priors.

All the above approximations assume use of efficient methods such as maximum likelihood under standard asymptotics52 and that the data distribution satisfies certain conditions met by ordinary epidemiologic models.27 For less efficient methods (like inverse-probability weighting), the interpretations have to be qualified technically by stating they are approximate posterior probabilities if one is given only the estimating function, rather than the complete likelihood function.


There is a huge literature on Bayesian interpretation of P values, most of which demands nonlinear transforms and concepts unfamiliar in basic statistics. We describe the most common example,29,30,33,34,52–55 although others exist.56

Your Bayes factor for a hypothesis is the ratio of your posterior odds on the hypothesis to your prior odds on the hypothesis; thus, it is how much your odds change (in multiplicative terms) in light of the data.49,57,58 Suppose your prior odds on the null versus all other possibilities combined was 1 and your posterior odds on the null was ¼ (you would bet only 20% on it after seeing the data); then, your Bayes factor for the null would be ¼/1 = ¼. When the odds are comparing simple point hypotheses, such as θt = 0 versus θt = ln(2), the Bayes factor simplifies to the likelihood ratio.

Consider the class of “spike and smear” priors for which there is a point probability q that θt = θm (ie, a spike or point mass of probability at θm of size q), and for the remaining possibilities θtθm, the prior follows a symmetric unimodal distribution around θm. This makes q / (1−q) the prior odds on θt = θ versus θtθ. Sellke et al34 show that, when Pm < 1 / e=0.37, −ePmln(Pm), is an approximate lower bound on the Bayes factor; in particular, for Pm = 0.10, 0.05, and 0.01, lower bounds on the Bayes factor are 0.63, 0.41, and 0.12, respectively. They further show that for normal priors on θtθm, these bounds are even higher: 0.70, 0.47, and 0.15. Thus, if P0= 0.05 a Bayesian analysis with q = ½ = 50% prior probability on the null and mean-zero normal otherwise leaves a posterior probability for the null of at least 0.47 / (1 + 0.47) = 32%; this maximum drop from 50% to 32% probability for θt = 0 is nowhere near as impressive as the usual “borderline significant” description of P0 = 0.05. We can also find the highest value of q for which P0 would deserve its common misinterpretation as the posterior probability of the null. For P0 = 0.05, this value is a prior probability of q = 10% for θt =0.

Transforming P0 into a Bayes factor thus reveals a serious weakness of conventional interpretations of null P values. Bayes factors have several disadvantages, however. They cannot be read directly off tabulated estimates or P values; they require the conceptual steps of thinking in terms of Bayes factors; and they do not provide any posterior probabilities unless we specify q, an arbitrary analysis constant that represents commitment of an appreciable fraction of prior probability to a single specific value of θ. In contrast, the interpretations given earlier use only confidence intervals and P values, with no need for Bayes factors or probability spikes.

There are serious objections to the use of probability spikes in population research because of their contextual meaning.27,59 For example, a null spike represents an assertion that, with prior probability q, we have background data that prove θt=0 with absolute certainty q = ½; thus represents a 50–50 bet that there is decisive information literally proving the null. Without such information (such as a point prediction by a highly plausible physical law), a probability spike at the null is an example of “spinning knowledge out of ignorance.”31 This prejudice favoring the null characterizes common misinterpretations of frequentist tests37 and yet can lead to Bayesian conflicts with those tests.60 In contrast, weak priors have no spike.


For more than 70 years, null P values have been the most common and yet most controversial inferential statistic, primarily because of their use for hypothesis testing against fixed alpha (significance) levels such as 0.05. A point often lost in statistical training, however, is that P values can be divorced from decision rules, testing, arbitrary cutoffs, and null hypotheses. In ordinary epidemiologic analyses, two-sided P values can instead be used as probability measures (with both frequentist and Bayesian meaning) of the distance from (or fit of) the entire set assumptions (including model and validity assumptions and the explicit assumption of θt = θ) to the data. One-sided P values such as P0 / 2 can be used to measure support for or probability of their one-sided hypotheses, given the remaining assumptions. Large P values are then as telling as small ones; for example, if P0 / 2 is large, then, considered in isolation, the study seems vague about the direction of an association when viewed through the same model lens used to compute the estimate

and the confidence interval. Furthermore, if one uses an informative prior to derive the posterior probability of the point estimate being in the wrong direction, P0 / 2 provides a reference point indicating how much the prior information influenced that posterior probability.

Confidence intervals provide valuable information about the precision implied by the data12,17 given the assumed model; hence, we would certainly not wish to see them replaced by P values. However, confidence intervals are rarely precise about the relation of observations to prespecified parameter values, which in part explains why null P values can be viewed as defensible supplements to confidence intervals when the null has legitimate claim to our attention (as in exploratory and many highly controversial settings).4,10 Furthermore, the frequency behavior and posterior interpretation of P0 are more robust to model misspecification than are confidence intervals and non-null P values, which is worth noting given that we are always uncertain about the model form.5 They are also more robust to validity problems that produce bias toward the null, especially certain simple types of measurement error or misclassification of the exposure or disease,61 where, under a weak prior, P0 / 2 remains the posterior probability that the estimate is in the wrong direction,62 whereas confidence intervals become invalid.

Even when the null is an arbitrary hypothesis unworthy of special attention, null-related questions are encouraged by automatic software emission of P0. Again, incorrect Bayesian interpretations of P0 remain the norm, encouraged by some instructional sources.1,2,63 It thus seems important to displace these misinterpretations with correct ones. With correct interpretation of P0 in mind, extensions of P values to θ of greater contextual interest is simple, and extensions to more realistic priors can also be obtained by translating the priors to data and adding them to the conventional analysis.41,51

After one separates P values from arbitrary testing cutoffs such as 0.05 and goes beyond the null, one may see confidence intervals as problematic for encouraging degraded binary inferences.28 Asking whether a value θ is in or out of the confidence interval turns the confidence interval into a 0.05-level test of that value. If we try to be more precise, we hit failings of intuition. For example, if the 95% confidence limits for RR are 1.8 and 3.3, what is the one-sided P value forRR ≤ 2? Few would recognize quickly that it is Pln(2)/2 = 0.10, giving a 10% posterior probability that under a weak prior for all model parameters, and at least 10% if the prior for RR was more sharply specified as lognormal with median at 2.

Failure to appreciate the complementarity of confidence intervals and P values may stem more from historical accidents in how the two statistics entered common usage.22 The chief disaster is that for generations Bayesian perspectives were banished from introductory statistics. Much like what happened with other attempts at knowledge suppression, users have filled the gap with interpretations that miss crucial subtleties and are often simply false. A common false interpretation is that P0 is the probability that chance alone produced the observed association. Unfortunately, the premise that chance alone operated is logically the same as the null hypothesis (“chance alone” implies there is no effect and no bias), and thus, this misinterpretation is the same as claiming P0 is the probability of the null.11,37 Correct Bayesian interpretations can address this problem.

Given the extensive misinterpretations of frequentist statistics and the enormous (some would say impossible)64 demands made by fully Bayesian analyses, a serious argument can be made for de-emphasizing (if not eliminating) inferential statistics in favor of more data presentation such as tables of raw numbers.65 After all, the only purpose of a single study is to contribute its description and data to a broad pool of relevant information.66 Without an inferential ban, however, an improvement of practice will require re-education, not restriction.


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