# Likelihood-ratio test or z-test?

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:

1. 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.
2. 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?

Under the null hypothesis $b_3=0$, the Wald z-test assumes Normality of the coefficient estimate $$\DeclareMathOperator{\se}{se} \frac{\hat b_3}{\se\left(\hat b_3\right)}\sim\mathcal{N}(0,1)$$ while Wilk's likelihood ratio test assumes merely that there exists some transformation to Normality$g(\cdot)$ $$\frac{g\left(\hat b_3\right)}{\se\left(g\left(\hat b_3\right)\right)}\sim\mathcal{N}(0,1)$$ Pawitan (2001), In all Likelihood $\S$ 2.9
If you plot the log-likelihood of $b_3$ (in the case you described with the Wald test significant & the LRT not) you'll probably find it's not much like a parabola, & therefore Wald's test would be likely to over-estimate significance compared to Wilk's.
The two tests are asymptotically equivalent, i.e., in large samples, should produce similar answers. See the classic expository note by Buse (1982). The Wald test ($z$-test) relies on large samples just as much as the likelihood ratio does, so one can't say that one is better justified than the other. However, my understanding is that in small samples, likelihood ratio may perform a little better than the Wald test (can't really back this up with references, this is just the word of mouth that I tend to hear). On the other hand, likelihood ratio testing crucially hinges on i.i.d. assumptions, while with Wald test, all you need is a consistent standard error, so Wald tests are applicable in a broader set of situations, including say correlated data and GEE.