Society of Actuaries (SOA) PA Practice Exam

Which link function is paired with the identity distribution in GLMs?

1. Log

2. Probit

3. Identity

4. Logit

The correct answer is: Identity

In Generalized Linear Models (GLMs), the link function serves to establish a relationship between the linear predictor and the mean of the distribution function of the response variable. The identity link function is specifically designed to maintain a direct correspondence between the linear predictor (the combination of independent variables weighted by their coefficients) and the expected value of the dependent variable. This means that, under the identity link, the model predicts the actual observed values without any transformation. When dealing with the identity link, the expected value of the response variable is modeled directly as a linear combination of the predictors. This is ideal for situations where the response variable is continuous and unbounded, such as in a standard linear regression model. By using the identity link, any predictions made will fall within the same scale as the original response data. Thus, the identity link function is essential when the assumption is that the mean of the response variable can be expressed directly as a linear function of the predictors, which aligns perfectly with the characteristics of the identity distribution. This understanding is crucial for model selection and implementation in GLMs.