Multivariate Statistical Analysis

Multivariate Statistical Analysis

Statistics 4223/5223 — Spring 2018

Assignment 5

Reading:

By Thursday, March 29, read Chapters 11–12 and 16 of Applied Multivariate Statistical Analysis,

fourth edition; by Wolfgang Härdle and Léopold Simar.

By Tuesday, April 3, read Chapter 14 of Härdle and Simar.

Homework 5:

The following problems are nominally due on Thursday, April 5, but can also be submitted in

class on Tue or Thu, April 10 or 12, or to the course mailbox in Room 904 SSW, any time before

5:00pm on Friday, April 13.

1. A sample of n = 140 seventh-grade children received four tests on x1 = reading speed,

x2 = reading power, y1 = arithmetic speed, and y2 = arithmetic power. The correlations

for performance are

R =

( RXX RXY

RY X RY Y

) =

 

1.0000 0.6328 0.2412 0.0586

0.6328 1.0000 −0.0553 0.0655 0.2412 −0.0553 1.0000 0.4248 0.0586 0.0655 0.4248 1.0000

 

(a) Find the sample canonical correlation coefficients and sample canonical variables.

(b) Find the p-value for a test of H0 : ΣXY = 0.

(c) If you reject H0 : ρ1 = ρ2 = 0 at the α = .05 level of significance, find the p-value for

a test of H0 : ρ2 = 0.

(d) Does reading ability (as measured by the two tests) correlate with arithmetic ability

(as measured by the two tests)? Discuss.

1

2. A random sample of n = 70 families will be surveyed to determine the association between

certain “demographic” variables and certain “consumption” variables.

Define the Criterion set of variables by

y1 = annual frequency of dining at a restaurant

y2 = annual frequency of attending movies

and the Predictor set by

x1 = age of head of household

x2 = annual family income

x3 = educational level of head of household

Suppose 70 observations on the preceding variables give the sample correlation matrix

R =

( RXX RXY

RY X RY Y

) =

 

1.00 0.37 0.21 0.26 0.33

0.37 1.00 0.35 0.67 0.59

0.21 0.35 1.00 0.34 0.34

0.26 0.67 0.34 1.00 0.80

0.33 0.59 0.34 0.80 1.00

 

(a) Determine the sample canonical correlation coefficients, and find the p-value for a

test of the null hypothesis H0 : ΣXY = 0.

(b) If H0 : ΣXY = 0 is rejected at the α = .05 level, test the null hypothesis that the

second and higher canonical correlations are all zero.

(c) Using standardized variables, construct the canonical variables corresponding to the

significant (α = .05) canonical correlation(s).

(d) Interpret the canonical variates corresponding to significant (α = .05) canonical cor-

relation(s).

(e) Do the demographic variables have something to say about the consumption vari-

ables? Do the consumption variables provide information about the demographic

variables?

2