The generalized linear models (GLM) can be used when the distribution of the response variable is non-normal or when the response variable is transformed into linearity. The GLMs are flexible extensions of linear models that are used to fit the regression models to non-Gaussian data.

One can classify a regression model as linear or non-linear regression models.

The basic form of a Generalized linear model is

\begin{align*}

g(\mu_i) &= X_i’ \beta \\

&= \beta_0 + \sum\limits_{j=1}^p x_{ij} \beta_j

\end{align*}

where $\mu_i=E(U_i)$ is the expected value of the response variable $Y_i$ given the predictors, $g(\cdot)$ is a smooth and monotonic link function that connects $\mu_i$ to the predictors, $X_i’=(x_{i0}, x_{i1}, \cdots, x_{ip})$ is the known vector having $i$th observations with $x_{i0}=1$, and $\beta=(\beta_0, \beta_1, \cdots, \beta_p)’$ is the unknown vector of regression coefficients.

### Fitting Generalized Linear Models

The glm() is a function that can be used to fit a generalized linear model, using the generic form of the model below. The formula argument is similar to that used in the lm() function for the linear regression model.

mod <- glm(formula, family = gaussian, data = data.frame)

The `family`

argument is a description of the error distribution and link function to be used in the model.

The class of generalized linear models is specified by giving a symbolic description of the linear predictor and a description of the error distribution. The link functions for different families of the probability distribution of the response variables are given below. The family name can be used as an argument in the glm( ) function.

Family Name | Link Functions |
---|---|

`binomial` | `logit ` , `probit` , `cloglog` |

`gaussian` | `identity` , `log` , `inverse` |

`Gamma` | `identity` , `inverse` , `log` |

`inverse gaussian` | $1/ \mu^2$, `identity` , `inverse` ,`log` |

`poisson` | `logit` , `probit` , `cloglog` , `identity` , `inverse` |

`quasi` | `log` , $1/ \mu^2$, `sqrt` |

### Generalized Linear Models Example in R

Consider the “cars” dataset available in R. Let us fit a generalized linear regression model on the data set by assuming the “dist” variable as the response variable, and the “speed” variable as the predictor. Both the linear and generalized linear models are performed in the example below.

data(cars) head(cars) attach(cars) scatter.smooth(x=speed, y=dist, main = "Dist ~ Speed") # Linear Model lm(dist ~ speed, data = cars) summary(lm(dist ~ speed, data = cars) # Generalized Linear Model glm(dist ~ speed, data=cars, family = "gaussian") plot(glm(dist ~ speed, data = cars)) summary(glm(dist ~ speed, data = cars))

Diagnostic Plots of Generalized Linear Models