Basic Statistics and Data Analysis

Lecture notes, MCQS of Statistics

p-value interpretation, definition, introduction and examples

p-value interpretation, definition, introduction and examples

The p-value also known as observed level of significance or exact level of significance or the exact probability of committing a type-I error (probability of rejecting H0, when it is true), helps to determine the significance of results from 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 actually true.

In technical words, one can define p-value as the lowest level of significance at which a null hypothesis can be rejected. If p-value is very small or less than the threshold value (chosen level of significance), then the observed data is considered as inconsistent with the assumption that the null hypothesis is true and thus null hypothesis must be rejected while the alternative hypothesis should be accepted. The p-value is a number between 0 and 1 and in literature it is usually interpreted in the following way:

  • 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) are 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, that 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 level of significance ($\alpha$) (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 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 of someone reject 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 refers to p-value<0.05 as statistically significant and p-value<0.001 as highly statistically significant (less than one in a thousand chance of being wrong).

p-value is usually incorrectly interpreted as it is usually interpreted as the probability of making a mistake by rejecting a true null hypothesis (a Type-I error). p-value cannot be error rate because:

p-value is calculated based on the assumption that the null hypothesis is true and that the difference in the sample by random chances. Consequently, 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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p-value Interpretation and misinterpretation of p-value

p-value Interpretation

The P-value is a probability, with a value ranging from zero to one. It is measure of how much evidence we have against the null hypothesis. P-value is a way to express the likelihood that $H_0$ is not true. The smaller the p-value, the more evidence we have against $H_0$.

p-value can be defined as

The largest significance level at which we would accept the null hypothesis. It enables us to test hypothesis without first specifying a value for $\alpha$. OR

The probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true.

If the P-value is smaller then the chosen significance level then $H_0$ (null hypothesis) is rejected even when it is true. If it is larger than the significance level $H_0$ is not rejected.

If the P-value is less than

  • 0.10, we have some evidence that $H_0$ is not true
  • 0.05, strong evidence that $H_0$ is not true
  • 0.01, Very strong evidence that $H_0$ is not true
  • 0.001, extremely strong evidence that $H_0$ is not true

Misinterpretation of a P-value

Many people misunderstand P-values. For example, if the P-value is 0.03 then it means that there is a 3% chance of observing a difference as large as you observed even if the two population means are same (i.e. the null hypothesis is true). It is tempting to conclude, therefore, that there is a 97% chance that the difference you observed reflects a real difference between populations and a 3% chance that the difference is due to chance. However, this would be an incorrect conclusion. What you can say is that random sampling from identical populations would lead to a difference smaller than you observed in 97% of experiments and larger than you observed in 3% of experiments.

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Difference between a Probability value and the Significance level

Difference between a probability value and the significance level?

Basically in hypothesis testing the goal is to see if the probability value is less than or equal to the significance level (i.e., is p ≤ alpha). It is also called the size of the test or size of the critical region. It is generally specified before any samples are drawn so that the results obtained will not influence our choice.

  • The probability value (also called the p-value) is the probability of the observed result found in your research study of occurring (or an even more extreme result occurring), under the assumption that the null hypothesis is true (i.e., if the null were true).
  • In hypothesis testing, the researcher assumes that the null hypothesis is true and then sees how often the observed finding would occur if this assumption were true (i.e., the researcher determines the p-value).
  • The significance level (also called the alpha level) is the cutoff value the researcher selects and then uses to decide when to reject the null hypothesis.
  • Most researchers select the significance or alpha level of .05 to use in their research; hence, they reject the null hypothesis when the p-value is less than or equal to .05.
  • The key idea of hypothesis testing it that you reject the null hypothesis when the p-value is less than or equal to the significance level of.05.
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