summary(lm(SpermTotal ~ CW + factor(Color)) Coefficients: Estimate Std. dat file at the text website shows their weights before and after an experimental period. 5 shows the format of the data. The girls were randomly assigned to receive one of three therapies during this period. A control group received the standard therapy, which was compared to family therapy and cognitive behavioral therapy.

This is not surprising, since ???? affects the response variable only through ???? = X????. To illustrate, for the one-way layout, consider the over-parameterization ????ij = ????0 + ????i . Then, ????0 + ????i = ????i as well as contrasts such as ????h − ????i = ????h − ????i are estimable. Any sole element in ???? is not estimable. When X has full rank, ???? is identifiable, and then all linear combinations ???? T ???? are estimable. ) The estimates do not depend on which constraints we employ, if necessary, to obtain identifiability.

Show the form of a full-rank model matrix X and corresponding parameter vector ???? for the model, constraining ????1 = ????1 = 0 to make ???? identifiable. Explain how to interpret the elements of ????. b. Show the form of a full-rank model matrix and corresponding param∑ ∑ eter vector ???? when you constrain i ????i = 0 and j ????j = 0 to make ???? identifiable. Explain how to interpret the elements of ????. c. In the full-rank case, what is the rank of X? 14 For the model in the previous exercise with constraints ????1 = ????1 = 0, generalize the model by adding an interaction term ????ij .