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The p-Value makes it possible to ensure the robustness of the test and the associated results.

## Introduction

The p-Value is a statistical datum introduced by Gibbons and Pratt in 1975. They propose a measure that calculates the smallest value of α to reject of the null hypothesis : « The smallest level at which the observations are significant in a particular direction ».

Regardless of the statistical tool used and the type of study, the role of the statistician is to ensure that the result obtained is significantly viable. In other words,

Is the result obtained real or only due to chance ?

So, from the test statistic, the p-Value calculates the level of risk α real test. We interpret a p-Value of 0.03 as « We have a 3% chance to reject the null hypothesis wrongly ».

We compare p-Value with the α risk we have chosen (usually 1 or 5%), and interpret it as follows :

• p-Value < α : it can be concluded that at the p-Value% risk level, to reject the null hypothesis H in favor of the alternative hypothesis H1.
• p-Value > α : we accept the null hypothesis H0.

Nevertheless, if p-Value > α it does not necessarily mean that there is no difference between groups. This may also be due to sampling too small to prove a difference.

## p-Value VS Critical value

In the end, these two values are closely linked. Both give the decision level of the test. Where the critical value gives a value only for comparison, the p-Value will be more accurate and will give the exact value of the significance of the test.

Their interpretation differs a little :

• The comparison with the critical value only gives the fact that the test is valid or not.
• The analysis of the p-Value gives the power of the test and therefore the level of « security » that we can give to the result.

Example

We perform a comparison of averages via a law of Student, and we propose 2 different cases of results.

 Cas 1 Cas 2 Results for a bilateral test Practical value : 68,67 Critical value : 2,037 p-Value : 0,0000000… Practical value : 2,0384 Critical value : 2,0369 p-Value : 0,0498 Interpretation of the critical value Critical value < Practical value we reject H0 Critical value < Practical value  We reject H0. Interpretation of the p-Value p-Value < α (5%) We reject H0 with a risk of being wrong to see an event when there is no 0,00000000… % (Error of first kind). p-Value = α (5%) P-Value is similar to the 5% risk we chose. The test is therefore inconclusive because the level of risk is too high.

## Calculation of the p-Value

The p-Value depends on 2 parameters: the practical value of our test statistic and the law that follows the test statistic. From these data, the calculation consists in not looking in the table for the critical value according to the level of risk, but in seeking the value of the risk according to our practical value. The reverse course of the calculation of the critical value is carried out.

For example, a Student test is used to compare averages. We obtain a practical value of 12.09 for a number of degrees of freedom of 23. With regard to the Student’s table, we see that for 23 degrees of freedom and a practical value of 3.485, we have a level of risk lower than 0.1%. Via the Excel spreadsheet, we get the exact value that is 1,9E-11.

## Source

J. D. Gibbons, J. W. Pratt (1975) – P-values : interpretation and methodology

A. Méot (2003) – Introduction aux statistiques inférentielles : de la logique à la pratique