Stat 216 Final Solutions

Transcription

1 Stat 16 Final Solutions Name: 5/3/05 Problem 1. (5 pts) In a study of size and shape relationships for painted turtles, Jolicoeur and Mosimann measured carapace length, width, and height. Their data suggest an analysis in terms of logarithms. The covariance of the natural logarithms of the dimensions of 4 male turtles is S = The following maximum likelihood estimates of the factor loadings for an m = 1 model were obtained: Variable Estimated Factor Loadings 1. ln(length).10. ln(width) ln(height).0765 Using the estimated factor loadings, obtain the maximum likelihood estimates of each of the following. a. (5 pts) Specific variances ˆψ 1 = s 11 ˆl 11 = ˆψ = s ˆl 1 = ˆψ 3 = s 33 ˆl 31 = b. (5 pts) Communalities ĥ 1 = ˆl 11 =.0104 ĥ = ˆl 1 =.0057 ĥ 3 = ˆl 31 =

2 c. (5 pts) Proportion of variance explained by the factor. ˆl 11 + ˆl 1 + ˆl 31 s 11 + s + s 33 =0.905 That is, 90.5% of the total variance is explained by the estimated factor. d. (5 pts) The residual matrix S n ˆLˆL ˆΨ ˆLˆL = ĥ 1 ˆl 11ˆl1 ˆl11ˆl1 ĥ ˆl11ˆl31 ˆl 1ˆl31 ˆl11ˆl31 ˆl1ˆl31 ĥ 3 Hence, from a. and b. we obtain 0 S n ˆLˆL ˆΨ = e. (5 pts) Indicate why a test of H 0 : Σ = LL + Ψ (with m = 1) versus H 1 : Σ is unrestricted, cannot be carried out for this problem. The distribution of the test statistic under the null is χ with (p m) p m degrees of freedom. In this case, p =3,m= 1, so the distribution has 0 d.f.

3 Problem. (0 pts) Suppose X comes from one of two populations: π 1 : normal with mean µ 1 and covariance matrix Σ 1 π : normal with mean µ and covariance matrix Σ 1. (10 pts) If the respective density functions are denoted f 1 (x) and f (x), find the expression for the quadratic discriminator Q =ln [ f 1 (x)] f (x) See solutions to Homework 5.. (10 pts) If Σ 1 = Σ = Σ, verify that Q becomes (µ 1 µ ) Σ 1 x 1 (µ 1 µ ) Σ 1 (µ 1 + µ ) See solutions to Homework 5. 3

4 Problem 3. (5 pts) The attached computer output shows the results of a discriminant analysis designed to discriminate Egyptian individuals into two periods B.C. and A.D. The data consist of 150 cases with four skull measurements: MB (Maximum B readth), BH (Basibregmatic Height), BL (Basialveslour Length) and NH (Nasal Height). The approximate year of skull formation is also included. Negative years represent the B.C. period and positive years the A.D. period. Year was used to assign the observations to two groups: 1= B.C. and = A.D.. Consider a skull with the following measurements: Variable Value MB 137 BH 130 BL 90 NH 50 a. (5 pts) Evaluate the linear discriminant functions for both groups of skulls. ˆd 1 (x) = MB +4.80BH +3.19BL +.1NH ˆd (x) = MB +4.70BH +3.08BL +.17NH b. (5 pts) Assuming equal priors, into which group would you classify this observation? For this specific skull, ˆd 1 (x) = 910. and ˆd (x) = Obviously, ˆd (x) > ˆd 1 (x) but they are very close. The result is too close to the boundary, but in principle we should classify the observation to the A.D. group. c. (5 pts) Assuming equal priors, compute the posterior probability that the observation belongs to group 1. exp 910. P (π 1 X) = exp exp = In other words, the measurements are not very discriminatory for the two populations. d. (5 pts) Estimate the probability that a B.C. skull will be misclassified. From the output, the misclassification probability is e. (5 pts) What assumptions underlie this discriminant analysis? Is there evidence in the output to indicate they are met? Two main assumptions: a. two populations are normal, b. the two covariance matrices are equal. Both are satisfied: a. The multivariate normality assumption is satisfied 4

5 (see p-values for Mardia s tests). b. Also Bartlett s test for covariance equality has a p-value of Problem 4. (0 pts) A random sample of 70 families is surveyed to determine the association between certain demographic variables and certain consumption variables. The attached computer output shows the results of a canonical correlation analysis of these two groups of variables: (1) Consumption: annual frequency of dining at a restaurant, annual frequency of attending movies, () Demographic: age of head of household, annual family income, educational level of head of household. 1. (10 pts) How many canonical variate pairs are significantly correlated? Provide justification. The likelihood ratio test yieds one significant canonical variate, since the p-value for testing ρ 1 = ρ = 0 is.001 and the p-value for testing ρ = 0 is.533. Note that there can be at most two.. (5 pts) What is the estimated correlation of the first canonical pair? from page 5 in the output. 3. (5 pts) What can you conclude from the first canonical variate pair? From the coefficients and the correlations between the first canonical variate pair (V 1,W 1 ) and the respective variables in the two groups, we conclude that income is the most important variable to explain dining frequency and at a lesser degree going to the movies. Age and education seem to be almost irrelevant. 5