P-values

In statistical hypothesis testing, the p-value of an observed value tobserved of some random variable T used as a test statistic is the probability that, given that the null hypothesis is true, T will assume a value as or more unfavorable to the null hypothesis as the observed value tobserved. "More unfavorable to the null hypothesis" can in some cases mean greater than, in some cases less than, and in some cases further away from a specified center.

In simpler terms, a p-value is the probability of obtaining a finding at least as "impressive" as that obtained, assuming the null hypothesis is true, so that the finding was the result of chance alone. The fact that p-values are based on this assumption is crucial to their correct interpretation.

Example
For example, say an experiment is performed to determine if a coin flip is fair (50% chance of landing heads or tails), or unfairly biased toward heads (> 50% chance of landing heads). The null hypothesis is that the coin is fair, and that any deviations from the 50% rate can be ascribed to chance alone. Suppose that the experimental results show the coin turning up heads 14 times out of 20 total flips. The p-value of this result would be the chance of a fair coin landing on heads at least 14 times out of 20 flips (as larger values in this case are also less favorable to the null hypothesis of a fair coin). The calculated p-value for this is 0.058.

Therefore, in practical terms - The higher the P-value, the higher the probability that the observation(s) you are studying are just chance.

Interpretation
Generally, one rejects the null hypothesis if the p-value is smaller than the test level, often represented by the Greek letter $$\alpha$$ (alpha). If the level is 0.05, then the probability that the p-value is less than 0.05, given that the null hypothesis is true, is 0.05, provided the test statistic has a continuous distribution.

In the above example, the calculated p-value exceeds 0.05, and thus the null hypothesis - that the observed result of 14 heads out of 20 flips can be ascribed to chance alone - is not rejected. Such a finding is often stated as being "not statistically significant at the 5 % level".

However, had a single extra head been obtained, the resulting p-value would be 0.02. This time the null hypothesis - that the observed result of 15 heads out of 20 flips can be ascribed to chance alone - is rejected. Such a finding would be described as being "statistically significant at the 5 % level".

There is often an alternative hypothesis, but the contruction of the test does not allow for 'supporting' a specific alternative.

Critics of p-values point out that the criterion used to decide "statistical significance" is based on the somewhat arbitrary choice of level (Often set at 0.05).

Frequent misunderstandings
There are several common misunderstandings about p-values. All of the following numbered statements are false:


 * 1) The p-value is the probability that the null hypothesis is true, justifying the "rule" of considering as significant p-values closer to 0 (zero).
 * In fact, frequentist statistics does not, and cannot, attach probabilities to hypotheses. Comparison of Bayesian and classical approaches shows that a p-value can be very close to zero while the posterior probability of the null is very close to unity. This is the Jeffreys-Lindley paradox.
 * 1) The p-value is the probability that a finding is "merely a fluke" (again, justifying the "rule" of considering small p-values as "significant").
 * As the calculation of a p-value is based on the assumption that a finding is the product of chance alone, it patently cannot simultaneously be used to gauge the probability of that assumption being true.
 * 1) The p-value is the probability of falsely rejecting the null hypothesis. This error is a version of the so-called prosecutor's fallacy.
 * 2) The p-value is the probability that a replicating experiment would not yield the same conclusion.
 * 3) 1-(p-value) is the probability of the alternative hypothesis being true (see (1)).

Additional reading

 * Historical background to the origins of p-values and the choice of 0.05 as the cut-off for significance
 * Historical background on the widespread confusion of the p-value

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