Originally, the idea of testing independence between single nonparametric covariate and random error in a partially linear regression model was furnished by Das and Maiti (2022). The most powerful test in that context is executed by the nonparametric measure of association \(\tau ^*\) , originally proposed by Bergsma and Dassios (2014) which involves paired observations on second order difference of response and sole nonparametric covariate. In this article, the authors are motivated to perform a similar testing procedure in a generalized partially linear logistic regression model with regression function \(\displaystyle {L(\mathbf{X},\mathbf{W})=\left( 1+e^{-(\mathbf{X}^T \underset{\sim }{\beta }+m(\mathbf{W}))}\right) ^{-1}}\) where \(\mathbf{X}\) is a p-tuple random vector and \(\mathbf{W}\) is a q-tuple random vector; by testing independence between \(\displaystyle {(\mathbf{X},\mathbf{W})}\) and error. Additionally, we assume \(m(\cdot )\) to satisfy Lipschitz continuity on \(\mathbb {R}^q\) . Eventually, we develop some U-statistics based on \(\tau ^*\) to carry out the prescribed testing of independence scheme. As a consequence of the null hypothesis suggesting independence among the covariates and error, we can establish independence between a general order difference of estimated response and the actual response. Then, \(\tau ^*\) is defined on the bivariate observations on the general order difference of estimated response and the response itself. \(\tau ^*=0\) straightaway implies independence between the general order difference of estimated response and the response. Furthermore, for different order difference, various modified test statistics are proposed. Their asymptotic powers are evaluated as well as compared for different order difference by taking a sequence of contiguous alternatives, instead of considering the sole alternative hypothesis citing dependence between the variables under consideration.
