At the outset, we have a response variable denoted as Y and a fundamental mean function in linear regression:
Y = β0 + β1 + ϵ
Now, let’s introduce a second variable, X2, with the aim of understanding how Y depends on both X1 and X2 simultaneously. By incorporating X2 into our analysis, we formulate a mean function that considers the values of both X1 and X2:
Y = β0 + β1 x1 + β2 x2 + ϵ
The primary objective of introducing X2 is to account for the part of Y that hasn’t yet been elucidated by X1.
In the context of diabetes prediction, we establish a relationship between the percentages of inactivity and obesity, serving as predictors or factors, and the percentage of diabetes using the Generalized Linear Model. This model extends the fundamental concepts of linear regression by introducing a link function that connects the linear model to the response variable and by permitting the measurement variance to be influenced by the predicted value of each measurement.