Stat404 Fall 2009 Lab 3 1. The following are three matrices: 1 2 B = 3 4 1 5 A = 1 2 3 4 5 6 C = 1 3 2 9 a. Give the dimensions of A, B, and C. T T T T b. Show that A + B = B + A . c. Show that A + B A + B . tr ABC = tr BCA = tr CAB . d. Show that T T T e. Show that CA = A C . C I 2 = I 2 C = C , where with dimensions, 2 2 . f. Show that g. Calculate det(C) and find C h. Show that C –1 C = CC 2. Consider the linear model 1 X = 1 1 1 1 3 5 7 –1 –1 T T ˜ T b. Compute X X –1 . . = I2 . y = Xb + e , ˜ ˜ ˜ 34 y = 47 55 ˜ 64 and a. Compute X X , X y , and T I 2 is an identity matrix y y. ˜ ˜ where 1 –1 T T c. Compute b = X X X y . ˜ d. Find ˜ 2 2 1 T n–k–1 ˜ ˜ and ̂ = --------------------- e e . e = y – Xb ˜ ˜ ˜ T e. Compute var b = ̂ X X –1 . 2 Note: Here and in part d ̂ is ˜ a scalar (i.e., a single number). f. Show that T –1 T I 4 – X X X X is idempotent (i.e., that multiplying it by itself yields the original matrix). g. Construct an ANOVA table with sums of squares (namely, the regression-, error-, and total-sums of squares), associated degrees of freedom, and mean squares. Does X explain a significant proportion of the variation in y at the .05 level? Use an F-test to ˜ support your conclusion. Hints: Be sure to show your work on this and all problems (an on exams too). The error and total sums of squares can be found using matrix algebra as follows: T SS ERROR = e e ˜ ˜ SS TOTAL 1 T 1 T = y – Y1 y – Y1 , where Y = --- y 1 , and where 1 = 1 . n 1 ˜ ˜ ˜ ˜ ˜ ˜ ˜ 1 3. Use the following SPSS commands to verify your calculations in problem 2: data list records=1 /1 y 1-2 x 4. variable labels y 'dependent var' x 'independent var'. begin data. 34 1 47 3 55 5 64 7 end data. regression variables=y,x/dependent=y/enter. In doing this problem, circle and assign unique letters to the numbers on the output that correspond to your computations (in the previous 2 2 problem) of b , ̂ , as well as of values in the ANOVA table ˜ and for the F-test. Write these letters (circled to make them obvious, please) next to the corresponding numbers in your computations. Below please find R and SAS code for problem 3: # R # Code: # Create a data set named, “lab3,” with column vectors for “x” and “y” y <- c(34,47,55,64) x <- c(1,3,5,7) lab3 <- data.frame(cbind(y,x)) attach(lab3) # Regression reg1 <- lm(y~x) summary(reg1) # ANOVA anova1 <- aov(y~x) summary(anova1) * SAS * Code: * Create a data set named, “lab3q3,” consisting of the variables, “y” and “x”; DATA lab3q3; INPUT y x; DATALINES; 34 1 47 3 55 5 64 7 ; RUN; * Regression; PROC REG data=lab3q3; MODEL y=x; RUN; 3