We consider the logistic regression of a binary response variable
\(Y\) on a set of predictor variables \(x = (x_1,\ldots,x_K)^T\)
with \(x_1\) being the covariate of interest:
\(\log \frac{P(Y_i=1)}{1 - P(Y_i = 1)} = \psi_0 + x_i^T \psi,\)
where \(\psi = (\psi_1,\ldots,\psi_K)^T\).
Similar to Self et al (1992), we assume that all covariates are
either inherently discrete or discretized from continuous
distributions (e.g. using the quantiles). Let \(m\) denote the total
number of configurations of the covariate values. Let
$$\pi_i = P(x = x_i), i = 1,\ldots, m$$
denote the probabilities for the configurations of the covariates
under independence. The likelihood ratio test statistic for testing
\(H_0: \psi_1 = 0\) can be approximated by a noncentral chi-square
distribution with one degree of freedom and noncentrality parameter
$$\Delta = 2 \sum_{i=1}^m \pi_i [b'(\theta_i)(\theta_i - \theta_i^*)
- \{b(\theta_i) - b(\theta_i^*)\}],$$ where
$$\theta_i = \psi_0 + \sum_{j=1}^{k} \psi_j x_{ij},$$
$$\theta_i^* = \psi_0^* + \sum_{j=2}^{k} \psi_j^* x_{ij},$$
for \(\psi_0^* = \psi_0 + \psi_1 \mu_1\), and \(\psi_j^* = \psi_j\)
for \(j=2,\ldots,K\). Here \(\mu_1\) is the mean of \(x_1\),
e.g., \(\mu_1 = \sum_i \pi_i x_{i1}.\) In addition, by
formulating the logistic regression in the framework of generalized
linear models, $$b(\theta) = \log(1 + \exp(\theta)),$$ and
$$b'(\theta) = \frac{\exp(\theta)}{1 + \exp(\theta)}.$$
The regression coefficients \(\psi\) can be obtained by taking the
log of the odds ratios for the covariates. The intercept \(\psi_0\)
can be derived as $$\psi_0 = \log(\bar{\mu}/(1- \bar{\mu})) -
\sum_{j=1}^{K} \psi_j \mu_j,$$ where \(\bar{\mu}\) denotes the
response probability when all predictor variables are equal to their
means.
Finally, let \(\rho\) denote the multiple correlation between
the predictor and other covariates. The noncentrality parameter
of the chi-square test is adjusted downward by multiplying by
\(1-\rho^2\).