Let us consider \(N\) subjects (or clusters). For each subject \(i\), \(1\leq i\leq N\), there are \(m\) binary response values \(y_{i}=(y_{i1},\dots,y_{im})'\) and the covariate matrix \(x_{i}=(x_{i1},\dots,x_{im})'\). For \(i\neq j\), \(y_{i}\) and \(y_{j}\) are independent, but generally the components of each \(y_{i}\) are correlated. The marginal logistic regression model specifies that \(\text{logit}(\pi_{it})=x_{it}'\beta\), where \(\pi_{it}= E(y_{it}\mid x_{it})\) and \(\text{var} (y_{it}\mid x_{it})=\pi_{it} (1-\pi_{it})\). This paper deals with two residual based normal goodness-of-fit test statistics: the Pearson chi-square and an unweighted sum of residual squares. The Pearson chi-square statistic is \(G=\sum_{i=1}^{N}\sum_{j=1}^{m} (y_{ij}- \hat\pi_{ij})^2/\hat\pi_{ij}(1-\hat\pi_{ij})\). The statistic based on an unweighted sum of residual squares is \(U=\sum_{i=1}^{N}\sum_{j=1}^{m} (y_{ij}-\hat \pi_{ij})^2\).
Easy-to-calculate approximations to the mean and variance of either statistic are given. The author emphasizes that the proposed tests are intended to be used with ungrouped binary data, where response is binary and the observed covariates for different subjects are essentially different.