Generalized Linear Models

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Last modified: 2024-12-13: 0:43:41 (AM)

This section is primarily adapted starting from the textbook “An Introduction to Generalized Linear Models” (4th edition, 2018) by Annette J. Dobson and Adrian G. Barnett:

https://doi.org/10.1201/9781315182780

The type of predictive model one uses depends on several issues; one is the type of response.

Measured values such as quantity of a protein, age, weight usually can be handled in an ordinary linear regression model, possibly after a log transformation.
Patient survival, which may be censored, calls for a different method (survival analysis, Cox regression).
If the response is binary, then can we use logistic regression models
If the response is a count, we can use Poisson regression
If the count has a higher variance than is consistent with the Poisson, we can use a negative binomial or over-dispersed Poisson
Other forms of response can generate other types of generalized linear models

We need a linear predictor of the same form as in linear regression $\beta x$. In theory, such a linear predictor can generate any type of number as a prediction, positive, negative, or zero

We choose a suitable distribution for the type of data we are predicting (normal for any number, gamma for positive numbers, binomial for binary responses, Poisson for counts)

We create a link function which maps the mean of the distribution onto the set of all possible linear prediction results, which is the whole real line ($-\infty, \infty$). The inverse of the link function takes the linear predictor to the actual prediction.

Ordinary linear regression has identity link (no transformation by the link function) and uses the normal distribution
If one is predicting an inherently positive quantity, one may want to use the log link since ex is always positive.
An alternative to using a generalized linear model with a log link, is to transform the data using the log. This is a device that works well with measurement data and may be usable in other cases, but it cannot be used for 0/1 data or for count data that may be 0.

R glm() Families
Family	Links
gaussian	identity, log, inverse
binomial	logit, probit, cauchit, log, cloglog
gamma	inverse, identity, log
inverse.gaussian	1/mu^2, inverse, identity, log
Poisson	log, identity, sqrt
quasi	identity, logit, probit, cloglog, inverse, log, 1/mu^2 and sqrt
quasibinomial	logit, probit, identity, cloglog, inverse, log, 1/mu^2 and sqrt
quasipoisson	log, identity, logit, probit, cloglog, inverse, 1/mu^2 and sqrt

R `glm()` Link Functions; $\eta = X\beta = g(\mu)$
Name	Domain	Range	Link Function	Inverse Link Function
identity	$(-\infty, \infty)$	$(-\infty, \infty)$	$\eta = \mu$.	$\mu = \eta$
log	$(0,\infty)$	$(-\infty, \infty)$	$\eta = \text{log}\left\{\mu\right\}$	$\mu = \text{exp}\left\{\eta\right\}$
inverse	$(0, \infty)$	$(0,\infty)$	$\eta = 1/\mu$	$\mu = 1/\eta$
logit	$(0,1)$	$(-\infty, \infty)$	$\eta = \text{log}\left\{\mu/(1-\mu)\right\}$	$\mu = \text{exp}\left\{\eta\right\}/(1+\text{exp}\left\{\eta\right\})$
probit	$(0,1)$	$(-\infty, \infty)$	$\eta = \Phi^{-1}(\mu)$	$\mu = \Phi(\eta)$
cloglog	$(0,1)$	$(-\infty, \infty)$	$\eta = \text{log}\left\{-\text{log}\left\{1-\mu\right\}\right\}$	$\mu = {1-\text{exp}\left\{-\text{exp}\left\{\eta\right\}\right\}}$
1/mu^2	$(0,\infty)$	$(0, \infty)$	$\eta = 1/\mu^2$	$\mu = 1/\sqrt{\eta}$
sqrt	$(0,\infty)$	$(0,\infty)$	$\eta = \sqrt{\mu}$	$\mu = \eta^2$

# Generalized Linear Models --- {{< include macros.qmd >}} This section is primarily adapted starting from the textbook "An Introduction to Generalized Linear Models" (4th edition, 2018) by Annette J. Dobson and Adrian G. Barnett: <https://doi.org/10.1201/9781315182780> The type of predictive model one uses depends on several issues; one is the type of response. * Measured values such as quantity of a protein, age, weight usually can be handled in an ordinary linear regression model, possibly after a log transformation. * Patient survival, which may be censored, calls for a different method (survival analysis, Cox regression). * If the response is binary, then can we use logistic regression models * If the response is a count, we can use Poisson regression * If the count has a higher variance than is consistent with the Poisson, we can use a negative binomial or over-dispersed Poisson * Other forms of response can generate other types of generalized linear models We need a linear predictor of the same form as in linear regression $\beta x$. In theory, such a linear predictor can generate any type of number as a prediction, positive, negative, or zero We choose a suitable distribution for the type of data we are predicting (normal for any number, gamma for positive numbers, binomial for binary responses, Poisson for counts) We create a link function which maps the mean of the distribution onto the set of all possible linear prediction results, which is the whole real line ($-\infty, \infty$). The inverse of the link function takes the linear predictor to the actual prediction. * Ordinary linear regression has identity link (no transformation by the link function) and uses the normal distribution * If one is predicting an inherently positive quantity, one may want to use the log link since ex is always positive. * An alternative to using a generalized linear model with a log link, is to transform the data using the log. This is a device that works well with measurement data and may be usable in other cases, but it cannot be used for 0/1 data or for count data that may be 0. Family | Links ------ | ------ gaussian | **identity**, log, inverse binomial | **logit**, probit, cauchit, log, cloglog gamma | **inverse**, identity, log inverse.gaussian | **1/mu^2**, inverse, identity, log Poisson | **log**, identity, sqrt quasi | **identity**, logit, probit, cloglog, inverse, log, 1/mu^2 and sqrt quasibinomial | **logit**, probit, identity, cloglog, inverse, log, 1/mu^2 and sqrt quasipoisson | **log**, identity, logit, probit, cloglog, inverse, 1/mu^2 and sqrt : R glm() Families Name | Domain | Range | Link Function | Inverse Link Function ---------| ------------------- | ------------------- | --------------------------- | -------------------------------- identity | $(-\infty, \infty)$ | $(-\infty, \infty)$ | $\eta = \mu$. | $\mu = \eta$ log | $(0,\infty)$ | $(-\infty, \infty)$ | $\eta = \log{\mu}$ | $\mu = \exp{\eta}$ inverse | $(0, \infty)$ | $(0,\infty)$ | $\eta = 1/\mu$ | $\mu = 1/\eta$ logit | $(0,1)$ | $(-\infty, \infty)$ | $\eta = \log{\mu/(1-\mu)}$ | $\mu = \exp{\eta}/(1+\exp{\eta})$ probit | $(0,1)$ | $(-\infty, \infty)$ | $\eta = \Phi^{-1}(\mu)$ | $\mu = \Phi(\eta)$ cloglog | $(0,1)$ | $(-\infty, \infty)$ | $\eta = \log{-\log{1-\mu}}$ | $\mu = {1-\exp{-\exp{\eta}}}$ 1/mu^2 | $(0,\infty)$ | $(0, \infty)$ | $\eta = 1/\mu^2$ | $\mu = 1/\sqrt{\eta}$ sqrt | $(0,\infty)$ | $(0,\infty)$ | $\eta = \sqrt{\mu}$ | $\mu = \eta^2$ : R `glm()` Link Functions; $\eta = X\beta = g(\mu)$

Name	Domain	Range	Link Function	Inverse Link Function
identity	\((-\infty, \infty)\)	\((-\infty, \infty)\)	\(\eta = \mu\).	\(\mu = \eta\)
log	\((0,\infty)\)	\((-\infty, \infty)\)	\(\eta = \text{log}\left\{\mu\right\}\)	\(\mu = \text{exp}\left\{\eta\right\}\)
inverse	\((0, \infty)\)	\((0,\infty)\)	\(\eta = 1/\mu\)	\(\mu = 1/\eta\)
logit	\((0,1)\)	\((-\infty, \infty)\)	\(\eta = \text{log}\left\{\mu/(1-\mu)\right\}\)	\(\mu = \text{exp}\left\{\eta\right\}/(1+\text{exp}\left\{\eta\right\})\)
probit	\((0,1)\)	\((-\infty, \infty)\)	\(\eta = \Phi^{-1}(\mu)\)	\(\mu = \Phi(\eta)\)
cloglog	\((0,1)\)	\((-\infty, \infty)\)	\(\eta = \text{log}\left\{-\text{log}\left\{1-\mu\right\}\right\}\)	\(\mu = {1-\text{exp}\left\{-\text{exp}\left\{\eta\right\}\right\}}\)
1/mu^2	\((0,\infty)\)	\((0, \infty)\)	\(\eta = 1/\mu^2\)	\(\mu = 1/\sqrt{\eta}\)
sqrt	\((0,\infty)\)	\((0,\infty)\)	\(\eta = \sqrt{\mu}\)	\(\mu = \eta^2\)