Question

suppose y is the response variable in a study, and there is a covariate factor that has 3 levels labeled by 1, 2, 3. define dummy variables (x1, x2) and a regression model to explain the dependence between the response and the covariate factor. take level 3 to be the reference level. what parameters or functions of the parameters describe the expected responses of the levels 1, 2, and 3 respectively?

Answer 1

In a regression model with a covariate having three levels, two dummy variables (x1, x2) are used, with level 3 as the reference. The model is y = β0 + β1x1 + β2x2 + ε. Parameters β0, β1, and β2 describe the expected responses for levels 3 (reference), 1, and 2.

Step-by-step explanation:

To define dummy variables for a covariate with three levels in a regression model, assuming level 3 as the reference category, we introduce two dummy variables called x1 and x2. The dummy variable x1 will be 1 if the level is 1 and 0 otherwise. Similarly, x2 will be 1 if the level is 2 and 0 otherwise.

The regression model to describe the dependence between the response variable y and the covariate factor will then be:

y = β0 + β1x1 + β2x2 + ε,

where β0 represents the expected value of y when the covariate is at level 3 (the reference level), β1 shows the difference in the expected value of y between level 1 and the reference level, and β2 shows the difference between level 2 and the reference level.

The expected responses for levels 1, 2, and 3 are β0 + β1, β0 + β2, and β0 respectively.