glms.qmd

# 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; $\eta = X\beta = g(\mu)$