Consider the following two logistic regression models: $$ \begin{aligned} &\text{Model A: }&P(Y=1)&=\frac{\text{exp}\left(b_1+b_2X_2\right)}{1+\text{exp}\left(b_1+b_2X_2\right)} \\ &\text{Model B: }&P(Y=1)&=\frac{\text{exp}\left(b_1+b_2X_2+b_3X_3\right)}{1+\text{exp}\left(b_1+b_2X_2+b_3X_3\right)} \end{aligned} $$ In model A, suppose that both the parameters $b_1$ and $b_2$ appear significant, i.e. $-1,96 > z_k$ or $1,96 < z_k$. When adding a dummy variable $X_3$ to create model B, there seem to be two options if I want to check whether the variable adds more explanatory power to the model:
- Perform a Likelihood-ratio test between model A and model B by calculating the test statistic $G^2=-2\log \left(\frac{L_{\text{A}}}{L_{\text{B}}}\right)$ and compare it to the critical value of the chi-squared distribution with one degree of freedom.
- Perform a z-test of $X_3$ in model B, calculating $z=\frac{b_3}{\text{se}\left(b_3\right)}$ and compare it to the critical values of the standard normal distribution.
The question now is: Is it sufficient in this case to calculate the z-value, if I want to determine whether model B has significant more explanatory power than model A? Is it possible that $X_3$ in a z-test is significant, and at the same time the LR-test implies there is no difference between the two models?
In summary: when adding only one variable to a logistic model, what is the best test to perform, if I want to investigate if the variable added significant explanatory power to the model?