## Principia

Moving finally to the content of the ASA statement, after a short introduction, in which it is recognized that ``the p-value ... is commonly misused and misinterpreted,'' and a reminder of what a p-value ``informally''' is (``the probability under a specified statistical model that a statistical summary of the data ...would be equal to or more extreme than its observed value'') a list of six items, indicated as ``principles'', follows (the highlighting is original).
1. P-values can indicate how incompatible the data are with a specified statistical model.
A -value provides one approach to summarizing the incompatibility between a particular set of data and a proposed model for the data. The most common context is a model, constructed under a set of assumptions, together with a so-called ``null hypothesis.'' Often the null hypothesis postulates the absence of an effect, such as no difference between two groups, or the absence of a relationship between a factor and an outcome. The smaller the -value, the greater the statistical incompatibility of the data with the null hypothesis, if the underlying assumptions used to calculate the -value hold. This incompatibility can be interpreted as casting doubt on or providing evidence against the null hypothesis or the underlying assumptions.
2. P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.
Researchers often wish to turn a -value into a statement about the truth of a null hypothesis, or about the probability that random chance produced the observed data. The -value is neither. It is a statement about data in relation to a specified hypothetical explanation, and is not a statement about the explanation itself.
3. Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.
Practices that reduce data analysis or scientific inference to mechanical ``bright-line'' rules (such as ``'') for justifying scientific claims or conclusions can lead to erroneous beliefs and poor decision making. A conclusion does not immediately become ``true'' on one side of the divide and ``false'' on the other. Researchers should bring many contextual factors into play to derive scientific inferences, including the design of a study, the quality of the measurements, the external evidence for the phenomenon under study, and the validity of assumptions that underlie the data analysis. Pragmatic considerations often require binary, ``yes-no'' decisions, but this does not mean that -values alone can ensure that a decision is correct or incorrect. The widespread use of ``statistical significance'' (generally interpreted as '') as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process.
4. Proper inference requires full reporting and transparency
-values and related analyses should not be reported selectively. Conducting multiple analyses of the data and reporting only those with certain -values (typically those passing a significance threshold) renders the reported -values essentially uninterpretable. Cherry-picking promising findings, also known by such terms as data dredging, significance chasing, significance questing, selective inference, and ``-hacking,'' leads to a spurious excess of statistically significant results in the published literature and should be vigorously avoided. One need not formally carry out multiple statistical tests for this problem to arise: Whenever a researcher chooses what to present based on statistical results, valid interpretation of those results is severely compromised if the reader is not informed of the choice and its basis. Researchers should disclose the number of hypotheses explored during the study, all data collection decisions, all statistical analyses conducted, and all -values computed. Valid scientific conclusions based on -values and related statistics cannot be drawn without at least knowing how many and which analyses were conducted, and how those analyses (including -values) were selected for reporting.
5. A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.
Statistical significance is not equivalent to scientific, human, or economic significance. Smaller -values do not necessarily imply the presence of larger or more important effects, and larger -values do not imply a lack of importance or even lack of effect. Any effect, no matter how tiny, can produce a small -value if the sample size or measurement precision is high enough, and large effects may produce unimpressive -values if the sample size is small or measurements are imprecise. Similarly, identical estimated effects will have different -values if the precision of the estimates differs.
6. By itself, a p-value does not provide a good measure of evidence regarding a model or hypothesis.
Researchers should recognize that a -value without context or other evidence provides limited information. For example, a -value near 0.05 taken by itself offers only weak evidence against the null hypothesis. Likewise, a relatively large -value does not imply evidence in favor of the null hypothesis; many other hypotheses may be equally or more consistent with the observed data. For these reasons, data analysis should not end with the calculation of a -value when other approaches are appropriate and feasible.
These words sound as an admission of failure of much of the statistics teaching and practice in the past many decades. But yet I find their courageous statement still somehow unsatisfactory, and, in particular, the first principle is in my opinion still affected by the kind of `original sin' at the basis of p-value misinterpretations and misuse. Many practitioners consider in fact a value occurring several (but often just a few) standard deviations from the `expected value' (in the probabilistic sense) to be a 'deviance' from the model, which is clearly absurd: no value a model can yield can be considered an exception from the model itself (see also footnote 11 - the reason why ``p-values often work'' will be discussed in section 6). Then, moving to principle 2, it is not that ``researchers often wish to turn a -value into a statement about the truth of a null hypothesis'' (italic mine), as if this would be an extravagant fantasy: reasoning in terms of degree of belief of whatever is uncertain is connatural to the `human understanding'[46]: all methods that do not tackle straight the fundamental issue of the probability of hypotheses, in the problems in which this is the crucial question, are destinated to fail, and to perpetuate misunderstanding and misuse.

Giulio D'Agostini 2016-09-06