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 Hypothesis tests

## General information about how hypothesis tests work

When performing Cox proportional hazards regression, Prism offers the choice of three different hypothesis tests to assess how well the specified model fits the given data. As with all hypothesis tests, the way these tests work is by first defining a null hypothesis (H0). Each test then uses the available information from the data and the model to generate a test statistic and corresponding P value. This P value represents the probability of obtaining a test statistic as large (or larger) than the one calculated assuming that the null hypothesis is true. As such, it is also required to set a (mostly arbitrary) threshold as to when this probability is considered “small enough”. This threshold is known as the alpha (α) level, and is typically set to be equal to 0.05. If the P value obtained is less than α (i.e. the probability that the test statistic would be as large or larger than the one obtained assuming the null hypothesis is true is less than 5%), then we reject the null hypothesis.

REMEMBER: failing to reject the null hypothesis that the parameter estimate is zero, does not confirm this hypothesis!! All that can be said is that the hypothesis cannot be rejected given the data.

For each of the hypothesis tests that Prism offers for Cox proportional hazards regression, the same global null hypothesis is used. Namely, H0:β=0. In words, this says that the null hypothesis is that the parameter coefficients for all predictor variables are zero (or, equivalently, that the hazard ratios are all 1.0). Another way to think about this is that, under the null hypothesis, the best fit model is one where changes in the predictor variables have no effect on the hazard rate.

The three hypothesis tests that Prism offers are:

Partial likelihood ratio test (also called the log-likelihood ratio test or the G test)

Wald test

Score test

Each of these tests utilizes the maximum log partial likelihood estimate (or MPLE), and the mathematics involved are very complex and beyond the scope of this guide. However, as described above, each of these tests provides a similar assessment of the model and the data. Namely, if the test statistic for each of these tests is sufficiently large, we can reject the null hypothesis that the parameter estimates for the model are zero.