I am currently self-studying statistics and I'm confused about the null model in binary logistic regression. I understand that the null model is used to be compared with the model you designed, but what exactly is the null model? Just ln(x)=y?
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5$\begingroup$ It is an intercept only model, where the only parameter is related to the proportion of '1' in the population. $\endgroup$– Michael MCommented Jan 21, 2014 at 18:47
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$\begingroup$ Probably not just $\ln(x)=y$, unless you've defined those variables rather unusually. (And please do define variables in questions.) $\endgroup$– Scortchi ♦Commented Jan 21, 2014 at 19:19
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1 Answer
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The full model is $$\ln \frac {\pi}{1-\pi}=\beta_0 +\beta_1 x_1 +\beta_2 x_2+\ldots$$ where $x_i$ is the $i$th predictor, $\beta_i$ its coefficient, & $$\pi=\Pr(Y=1)$$ where $Y$ is the response (coded 1 for "success" & 0 for "failure")
The null model, as @Michael says, contains just the intercept: $$\ln \frac {\pi}{1-\pi}=\beta_0$$ So the intercept is the log-odds of "success", estimated without reference to any predictors.
answered Jan 21, 2014 at 19:18