Question

A MLR model have LIFE (y) as the response variable, and MALE (x1), BIRTH (x2), DIVO (x3), BEDS (x4), EDUC (x5), and INCO (x6), as predictors.

I know you can use first fit the model using lm(y~x) then use anova(model) to check the SSreg,my question is, what is the difference between  SSreg(β2|β0,β3) and SSreg(β1|β0,β3,β2)? What should you put as the argument of lm() function with respect to (β2|β0,β3) and (β1|β0,β3,β2)

Answer 1

SSreg means that sum squares of the regression model. Elaborately, it means how much variability has been explained by the proposed model. The more the SSreg of a model is, the more variability has been explained by the model. SSreg(β2|β0,β3), SSreg(β1|β0,β3,β2) - this notation is called the extra sum of square. For an MLR model the greater the number of covariates, the greater variability is being explained and hence the bigger the value of the SSreg is for the bigger model as compared to the smaller one. Now, the extra sum of square is the additional sum of squares of regression which is obtained when adding a covariate in the model where already some of the covariates are already there.

Here "SSreg(β2|β0,β3)" defines that extra sum of squares of regression obtained by adding x2 as a covariate in the model where x0 and x3 were already present. Mathematically, SSreg(β2|β0,β3) = sum of squares of regression when x0, x3, x2 are the covariates - sum of squares of regression when x0, x3 are the covariates.

Similarly, "SSreg(β1|β0,β3,β2)" defines that extra sum of squares of regression obtained by adding x1 as a covariate in the model where x0, x3, and x2 were already present. Mathematically, SSreg(β1|β0,β3,β2) = sum of squares of regression when x0, x3, x2, x1 are the covariates - sum of squares of regression when x0, x3, x2 are the covariates.

Hence the difference is that, while SSreg(β2|β0,β3) means that the extra variability explained due to the addition of the covariate x2 into the model where x0 and x3 were already present, SSreg(β1|β0,β3,β2) defines the extra variability explained due to the addition of the covariate x1 into the model where x0, x3, and x2 were already present.

• We create some models and then apply the anova() routine. model1 = lm(y~x0+x3) model2 = lm(y~x0+x3+x2) anova(model1, model2) # This will give the extra sum of squares for adding x2 into the model with x0 & x3 model3 = lm(y~x0+x3+x2+x1) anova(model2, model3) #This will give the extra sum of squares for adding x1 into the model with x0, x3, x2