10_generalized_linear_models.qmd

# Generalized Linear Models (GLM)

In practice, GLMs are often used when the response variable is not
continuous. Several examples of GLMs include

-   Logistic Regression (Y is binary)

-   Multinomial Logistic Regression (Y is nominal)

-   Poisson Regression (Y is a count)

-   Negative Binomial Regression (Y is a count with over-dispersion)

**[R]{.sans-serif}** uses the function `glm` to fit GLMs. We will use
logistic regression to illustrate `glm`.

## Logistic Regression

Logistic regression is a specific type of GLM used to model a binary
outcome. If we label a success "1" and a failure "0", the *odds* of
success are defined as $P(Y=1)/P(Y=0)$. Logistic regression models the
log odds of success as a linear combination of the predictors.
$${\rm log}\left(\frac{P(Y=1|X)}{P(Y=0|X)}\right) =  \beta_0 + \beta_1X_1 + ... + \beta_pX_p$$

In logistic regression, $\beta_i$ is the effect of $X_i$ on the log odds
of success ($Y=1$).

For an example, we use the "Mroz\" data in the *car* library. Let's
first review the "Mroz" data set.

`> library(car)`

`> help(Mroz)`

The syntax for `glm` is nearly the same as the syntax for `lm`. One
important additional argument is *family*, which specifies what type of
GLM will be fit. Logistic regression models a binary response, so we use
the binomial family.

`> mroz.mod = glm(lfp `$\sim$` I(k5==0) + age + wc + hc + lwg + inc, data=Mroz,`

`+   family=binomial)`

`> summary(mroz.mod)`

The fitted object `mroz.mod` is of a new type of class.

`> class(mroz.mod)`

We can extract the estimated coefficients using `coef`.

`> coef(mroz.mod)`

When reporting the effects of a logistic regression analysis, it is
common to include the effect on the *odds* of success. Since $\beta_i$
is the effect of $X_i$ on the log odds of success, $e^{\beta_i}$ is the
effect of $X_i$ on the odds of success[^1]. We can estimate the effects
of the predictors on the odds by exponentiating the coefficients.

`> exp(coef(mroz.mod))`

We can also obtain confidence intervals for both the log odds and odds
effects.

`> confint(mroz.mod)`

`> exp(confint(mroz.mod))`