In this post, we will discuss the P-value definition, interpretation, introduction, and some related examples.
P-value Definition
The P-value also known as the observed level of significance or exact level of significance or the exact probability of committing a type-I error (probability of rejecting $H_0$, when it is true), helps to determine the significance of results from the hypothesis. The P-value is the probability of obtaining the observed sample results or a more extreme result when the null hypothesis (a statement about population) is true.
In technical words, one can define the P-value as the lowest level of significance at which a null hypothesis can be rejected. If the P-value is very small or less than the threshold value (chosen level of significance), then the observed data is considered inconsistent with the assumption that the null hypothesis is true, and thus null hypothesis must be rejected while the alternative hypothesis should be accepted. A P-value is a number between 0 and 1 in literature.
Usual P-value Interpretation
- A small P-value (<0.05) indicates strong evidence against the null hypothesis
- A large P-value (>0.05) indicates weak evidence against the null hypothesis.
- p-value very close to the cutoff (say 0.05) is considered to be marginal.
Let the P-value of a certain test statistic is 0.002 then it means that the probability of committing a type-I error (making a wrong decision) is about 0.2 percent, which is only about 2 in 1,000. For a given sample size, as | t | (or any test statistic) increases the P-value decreases, so one can reject the null hypothesis with increasing confidence.
Fixing the significance level ($\alpha$, i.e. type-I error) equal to the p-value of a test statistic then there is no conflict between the two values, in other words, it is better to give up fixing up (significance level) arbitrary at some level of significance such as (5%, 10%, etc.) and simply choose the P-value of the test statistic. For example, if the p-value of the test statistic is about 0.145 then one can reject the null hypothesis at this exact significance level as nothing wrong with taking a chance of being wrong 14.5% of the time someone rejects the null hypothesis.
P-value addresses only one question: how likely are your data, assuming a true null hypothesis? It does not measure support for the alternative hypothesis.
Most authors refer to a P-value<0.05 as statistically significant and a P-value<0.001 as highly statistically significant (less than one in a thousand chance of being wrong).
The P-value interpretation is usually incorrect as it is usually interpreted as the probability of making a mistake by rejecting a true null hypothesis (a Type-I error). The P-value cannot be the error rate because:
The P-value is calculated based on the assumption that the null hypothesis is true and that the difference in the sample is by random chance. Consequently, a p-value cannot tell about the probability that the null hypothesis is true or false because it is 100% true from the perspective of the calculations.
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