GLM Gaussian vs GLM Binomial vs log-link GLM Gaussian

Question

I am trying to do a study of deaths due to malaria in order to find the best way to predict how dangerous this disease is.

I don't have a strong background in statistics, I am an auto-learner building my knowledge using online courses.

First, I collected the data this way :

Statistics(gender, age, ..)   |   Number_Death

As far as I know, my options are

• GLM with a Binomial distribution: for predicting if this dangerous or not. In this case, I labelled the predictor to be 0 (for no death), 1 for one or more cases.

• GLM with a Poisson distribution: for predicting the number of events based on the predictor.

Now, I am confused. For what purposes would we use a GLM with Gaussian distribution, a GLM with a Gaussian and a log link function, or a GLM with a Gamma distribution?

Answer 1

There are three components to a glm. A probability distribution, a linear predictor, and a link function that relates the linear predictor to the expected value of the probability distribution for the response which I will denote as $Y$. First of all, notice that for both of the gaussian models the outcome is a continuous random variable.

For GLM gaussian, I assume this has the default identity link, so $E(Y)=X\beta$, then this is no different than a regular linear model with $Y \sim N(X\beta, \sigma^2)$. Notice this case assumes constant variance as the mean of $Y$ changes linearly with $X$.

For log-linked GLM gaussian, $log(E(Y))=X\beta$, so $E(Y) = e^{X\beta}$ and $Y \sim N( e^{X\beta}, \sigma^2)$. This example is perhaps the cleanest of the three you asked about that will help elucidate the three components. The link is log, the linear predictor is $X\beta$, and the probability distribution is normal. Using this model would be one way to account for a very particular function form of a non-linear relationship between your predictors $X$ and the response, though it still assumes constant variance around the mean $e^{X\beta}$.

For GLM gamma, see When to use gamma GLMs?