# GLM Gaussian vs GLM Binomial vs log-link GLM Gaussian

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?

• Wouldn't the number of deaths depend on the population at-risk in each case? Are these figures in a specific location (say a country) over time (in which case, population changes over time), or are they different regions (population changes by region), ... or something else? This information will be important in several ways. Commented Dec 28, 2014 at 0:01
• I suggest you do some reading on time-to-event or survival models. Poisson models fit in there as one option, but there are others. It is worth your time to look into the theory and practice. Commented Feb 1, 2015 at 14:33

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 $e^{X\beta}$.