The accompanying table presents data on one dependent variable and five independent variables.
(a) Give the linear model in matrix form for regressing Y on the five independent variables. Completely define each matrix and give its order and rank.
(b) The following quadratic forms were computed.
Use a matrix algebra computer program to reproduce each of these sums of squares. Use these results to give the complete analysis of variance summary.
(c) The partial sums of squares for X1, X2, X3, X4, and X5
are .895, .238, .270, .337, and .922, respectively. Give the R-notation that describes the partial sum of squares for X2. Use a matrix algebra program to verify the partial sum of squares for X2.
(d) Assume that none of the partial sums of squares for X2, X3, and X4
is significant and that the partial sums of squares for X1
and X5
are significant (at α = .05). Indicate whether each of the following statements is valid based on these results. If it is not a valid statement, explain why.
(i) X1
and X5
are important causal variables whereas X2, X3, and X4
are not.
(ii) X2, X3, and X4
can be dropped from the model with no meaningful loss in predictability of Y .
(iii) There is no need for all five independent variables to be retained in the model.