Society of Actuaries (SOA) PA Practice Exam

What role does the mean and variance play in a generalized linear model (GLM)?

1. They provide equal influence in prediction

2. They depend on different parameters during modeling

3. They are related and depend on the same predicting parameter

4. They are both independent variables

The correct answer is: They are related and depend on the same predicting parameter

In the context of a generalized linear model (GLM), the mean and variance play a crucial role in defining the relationship between the predictor variables and the response variable. The correct answer indicates that the mean and variance are related and depend on the same predicting parameter. This relationship is foundational in GLMs, where the mean of the response variable is modeled as a function of the predictors through a link function, and the variance is often expressed as a function of the mean. Typically, in GLMs, we assume that the response variable follows a certain distribution from the exponential family (such as normal, binomial, or Poisson). The mean is linked to the linear combination of predictors, while the variance is often a function of the mean, which can be seen in models like the Poisson regression or the binomial regression. For example, in Poisson regression, the variance of the response variable is equal to its mean, which illustrates the dependency of the variance on the predicted mean. Therefore, the connection between mean and variance through predictors in GLMs is central to their formulation and application. This captures the essence of how we model different types of data appropriately, allowing for flexible and powerful relationships in statistical modeling.