# Wald Test for Logistic Regression vs T-Test for Linear Regression

Why does Logistic Regression use the Wald test, whereas Linear Regression uses the t-test?

What causes the difference?

Below is a short explanation. If you have questions, I am happy to elaborate or go into further detail.

Linear Regression:

The use of t-tests is linear regression comes from the distribution of normally distributed error terms:

$$y_i=X_i'\beta + \epsilon_i$$

where $$\epsilon_i \sim N(0,1)$$ iid. It follows that

$$\frac{\hat{\beta_j}-\beta_{j0}}{se(\hat{\beta_j})} \sim t(N-K),$$

where $$N$$ is the sample size and $$K$$ is the length of the vector $$\beta$$.

Note that the default in most regression software packages test the hypothesis that $$\hat{\beta_j}=0$$, i.e. setting $$\beta_{j0}$$ equal to zero.

Logistic Regression:

Logistic regression assumes errors follow the logistic distribution. Consequently, the term $$\frac{\hat{\beta_j}-\beta_{j0}}{se(\hat{\beta_j})}$$ does not follow a t-distribution. Instead, we can use the Wald test, which relies on asymptotic normality as is implied by the Central Limit Theorem.

• Logistic regression assumes errors follow the logistic distribution. Do you know of the derivation/explanation for this? – Trajan Apr 21 '20 at 11:10
• Logistic regression is motivated by having a continuous, unobserved (latent) variable that determines the binary outcome: $y^*=X\beta+\epsilon.$ When the latent variable is positive (negative), the binary dependent variable takes a value of 1 (0). Logistic regression assumes $\epsilon_i$ follows standard logistic distribution, which is used to derive the likelihood function and estimate the model parameters $\beta.$ – dlnB Apr 21 '20 at 14:23
• I can elaborate further if you wish. Please accept the answer if you are satisfied. – dlnB Apr 21 '20 at 14:24