15.5 - Generalized Linear Models

All of the regression models we have considered (including multiple linear, logistic, and Poisson) actually belong to a family of models called generalized linear models. (In fact, a more "generalized" framework for regression models is called  general regression models, which includes any parametric regression model.)  Generalized linear models provides a generalization of ordinary least squares regression that relates the random term (the response Y) to the systematic term (the linear predictor \(\textbf{X}\beta\)) via a link function (denoted by \(g(\cdot)\)). Specifically, we have the relation

\(\begin{equation*}
\mbox{E}(Y)=\mu=g^{-1}(\textbf{X}\beta),
\end{equation*}\)

so \(g(\mu)=\textbf{X}\beta\). Some common link functions are:

Identity Link

The identity link:

\(\begin{equation*}
g(\mu)=\mu=\textbf{X}\beta,
\end{equation*}\)

which is used in traditional linear regression.

Logit Link

The logit link:

\(\begin{align*}
&g(\mu)=\log\biggl(\frac{\mu}{1-\mu}\biggr)=\textbf{X}\beta\\
&\Rightarrow\mu=\frac{e^{\textbf{X}\beta}}{1+e^{\textbf{X}\beta}},
\end{align*}\)

which is used in logistic regression.

Log Link

The log link:

\(\begin{align*}
&g(\mu)=\log(\mu)=\textbf{X}\beta\\
&\Rightarrow\mu=e^{\textbf{X}\beta},
\end{align*}\)

which is used in Poisson regression.

Probit Link

The probit link:

\(\begin{align*}
&g(\mu)=\Phi^{-1}(\mu)=\textbf{X}\beta\\
&\Rightarrow\mu=\Phi(\textbf{X}\beta),
\end{align*}\)

where \(\Phi(\cdot)\) is the cumulative distribution function of the standard normal distribution. This link function is also sometimes called the normit link. This also can be used in logistic regression.

Complementary Log-Log Link

The complementary log-log link:

\(\begin{align*}
&g(\mu)=\log(-\log(1-\mu))=\textbf{X}\beta\\
&\Rightarrow\mu=1-\exp\{-e^{\textbf{X}\beta}\},
\end{align*}\)

which can also be used in logistic regression. This link function is also sometimes called the gompit link.

Power Link

The power link:

\(\begin{align*}
&g(\mu)=\mu^{\lambda}=\textbf{X}\beta\\
&\Rightarrow\mu=(\textbf{X}\beta)^{1/\lambda},
\end{align*}\)

where \(\lambda\neq 0\). This is used in other regressions which we do not explore (such as gamma regression and inverse Gaussian regression).

Also, the variance is typically a function of the mean and is often written as

\(\begin{equation*}
\mbox{Var}(Y)=V(\mu)=V(g^{-1}(\textbf{X}\beta)).
\end{equation*}\)

The random variable Y is assumed to belong to an exponential family distribution where the density can be expressed in the form

\(\begin{equation*}
q(y;\theta,\phi)=\exp\biggl\{\dfrac{y\theta-b(\theta)}{a(\phi)}+c(y,\phi)\biggr\},
\end{equation*}\)

where \(a(\cdot)\), \(b(\cdot)\), and \(c(\cdot)\) are specified functions, \(\theta\) is a parameter related to the mean of the distribution, and \(\phi\) is called the dispersion parameter. Many probability distributions belong to the exponential family. For example, the normal distribution is used for traditional linear regression, the binomial distribution is used for logistic regression, and the Poisson distribution is used for Poisson regression. Other exponential family distributions lead to gamma regression, inverse Gaussian (normal) regression, and negative binomial regression, just to name a few.

The unknown parameters, \(\beta\), are typically estimated with maximum likelihood techniques (in particular, using iteratively reweighted least squares), Bayesian methods, or quasi-likelihood methods. The quasi-likelihood is a function which possesses similar properties to the log-likelihood function and is most often used with count or binary data. Specifically, for a realization y of the random variable Y, it is defined as

\(\begin{equation*}
Q(\mu;y)=\int_{y}^{\mu}\dfrac{y-t}{\sigma^{2}V(t)}dt,
\end{equation*}\)

where \(\sigma^{2}\) is a scale parameter. There are also tests using likelihood ratio statistics for model development to determine if any predictors may be dropped from the model.