"scale" in logistic regression

Question

I am working on translating some R code into Python's statsmodels package, chiefly some logistic regression work that I've done, when I came across the following in the statsmodels documentation,

WARNING: Loglikelihood and deviance are not valid in models where scale is equal to 1 (i.e., Binomial, NegativeBinomial, and Poisson). If variance weights are specified, then results such as loglike and deviance are based on a quasi-likelihood interpretation. The loglikelihood is not correctly specified in this case, and statistics based on it, such AIC or likelihood ratio tests, are not appropriate.

What is this "scale", and what is the statistical reason why scale=1 invalidates the likelihood ratio test that I want to use and have used in R? (Was it even valid when I did it in R?)

Answer 1

Logistic regression has one canonical parameter, the log odds. So if you use GLM as a maximum likelihood procedure, the linear model for the response is:

$$ \text{logit} \left( Pr(Y = 1) = \mu\right) = \beta_0 + \beta_1 X_1 + \ldots + \beta_p X_p$$

Where $g(\mu) = \text{logit}(\mu) = \log(\mu/(1-\mu))$ is called the "link function".

Consequently the mean-variance relationship is given by $V = \frac{\partial}{\partial \mu} g^{-1}(\mu)$ which in this case is $\mu(1-\mu)$, the readily recognizable variance of Bernoulli random variable.

GLMs are estimated by Fisher Scoring.

A scale family of distributions is any family of probability densities where given $X$ being a member of that family, $Y = \phi X$ is still a member of that same family (someone correct me with a formal definition here). The most famous example is the normal distribution.

The Bernoulli density is not a scale family. That means if you want to estimate a generalization of the logistic model where the linear model for the response is still given by:

$$ \text{logit} \left( \mu \right) = \beta_0 + \beta_1 X_1 + \ldots + \beta_p X_p$$

but the variance is given by:

$$ V = \phi^2 \mu (1-\mu)$$

you need to use non-standard GLM estimation, or you need to use quasilikelihood and calculate the dispersion parameter $\phi$ as a nuissance parameter, using the deviance residuals. This estimating equation is no longer a maximum likelihood procedure, but has many MLE-like properties. Wedderbern consequently coined the process one of quasilikelihood in 1973.