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What's the relationship between p value, hypothesis, and alpha?

From Greenland et al:

**Uncertainty, probability, and statistical significance**

A more refined goal of statistical analysis is to provide an evaluation of certainty or uncertainty regarding the size of an effect. It is natural to express such certainty in terms of “probabilities” of hypotheses. **In conventional statistical methods, however, “probability” refers not to hypotheses, but to quantities that are hypothetical frequencies of data patterns under an assumed statistical model.** These methods are thus called frequentist methods, and the hypothetical frequencies they predict are called “frequency probabilities.” Despite considerable training to the contrary, many statistically educated scientists revert to the habit of misinterpreting these frequency probabilities as hypothesis probabilities. (Even more confusingly, the term “likelihood of a parameter value” is reserved by statisticians to refer to the probability of the observed data given the parameter value; it does not refer to a probability of the parameter taking on the given value.)

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**Not only does a P value not tell us whether the hypothesis targeted for testing is true or not; it says nothing specifically related to that hypothesis unless we can be completely assured that every other assumption used for its computation is correct—an assurance that is lacking in far too many studies.**

Nonetheless, the P value can be viewed as a continuous measure of the compatibility between the data and the entire model used to compute it, ranging from 0 for complete incompatibility to 1 for perfect compatibility, and in this sense may be viewed as measuring the fit of the model to the data. Too often, however, the P value is degraded into a dichotomy in which results are declared “statistically significant” if P falls on or below a cut-off (usually 0.05) and declared “nonsignificant” otherwise. **The terms “significance level” and “alpha level” (α) are often used to refer to the cut-off; however, the term “significance level” invites confusion of the cut-off with the P value itself. Their difference is profound: the cut-off value α is supposed to be fixed in advance and is thus part of the study design, unchanged in light of the data. In contrast, the P value is a number computed from the data and thus an analysis result, unknown until it is computed.**

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