Solved – Why ChiSq test and Likelihood Ratio Test are the same when comparing to logistic regression models

anovachi-squared-testlikelihood-ratiologistic

I am trying to understand avnoa on two logistic regression models. I noticed that the chisq (Chisq) test and likelihood ratio test (LRT) are returning the exact same results in this following toy example. Why?

> fit1=glm(am~mpg+hp,mtcars, family = binomial())
> fit2=glm(am~mpg+hp+wt,mtcars, family = binomial())
> anova(fit1,fit2, test="Chisq")
Analysis of Deviance Table

Model 1: am ~ mpg + hp
Model 2: am ~ mpg + hp + wt
  Resid. Df Resid. Dev Df Deviance Pr(>Chi)   
1        29    19.2326                        
2        28     8.7661  1   10.466 0.001216 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> anova(fit1,fit2, test="LRT")
Analysis of Deviance Table

Model 1: am ~ mpg + hp
Model 2: am ~ mpg + hp + wt
  Resid. Df Resid. Dev Df Deviance Pr(>Chi)   
1        29    19.2326                        
2        28     8.7661  1   10.466 0.001216 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Best Answer

The likelihood ratio test is a (type of) Chi-square test.

The documentation for ?anova.glm says:

You can also choose "LRT" and "Rao" for likelihood ratio tests and Rao's efficient score test. The former is synonymous with "Chisq" (although both have an asymptotic chi-square distribution).

Plainly, it is just a poor choice of notation. "Chisq" test is more succinctly said to be a likelihood ratio test.

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