### Testing the Global Null Hypothesis

The following statistics can be used to test the global null hypothesis . Under mild assumptions, each statistic has an asymptotic chi-square distribution with degrees of freedom given the null hypothesis. The value is the dimension of . For clustered data, the likelihood ratio test, the score test, and the Wald test assume independence of observations within a cluster, while the robust Wald test and the robust score test do not need such an assumption.

#### Robust Score Test

where is the score residual of the ith subject at ; that is, , where the score process is defined in the section Residuals.

#### Robust Wald’s Test

where is the sandwich variance estimate (see the section Robust Sandwich Variance Estimate for details).