STAT 5900 F: Longitudinal Data Analysis

STAT 5900 F: Longitudinal Data Analysis

Assignment 1

Due Friday, October 4, 2024

Instructions: Use R for statistical analysis. Show details of your work for full marks. Scan your answer sheet as a single PDF file with the correct page order in a clear manner, and submit it to Brightspace.

1. The data file “stat5900.assign1.datafile.txt” uploaded in Brightspace contains bivariate data on the response y and covariate x obtained from 20 subjects, each with four measurements. Consider fitting the linear mixed effects model

yij = β0 + β1xij + ui + ϵij   (i = 1, . . . , 20; j = 1, . . . , 4),                         (1)

where ui ’s are independent N(0, σ2u), ϵij ’s are independent N(0, σϵ 2 ), and ui and ϵij are independent of each other.

(a) Derive Var(yij ), Cov(yij , yij′) and Cov(yi ′j , yij ).

(b) For simultaneous maximum likelihood (ML) estimation of τ = (β t , θ t ) t with β = (β0, β1) t and θ = (σ2u , σϵ 2) t , derive the Fisher Information matrix (show details of your calculation)

(c) Compute the ML estimate of τ using the Newton-Raphson iterative equation

τ(m+1) = τ(m) + [I(τ)]−1U(τ),

where the score function U(τ ) = ∂ log L/∂τ and the Fisher Information I(τ ) are evaluated at τ(m) (show details of your computation).

(d) Compute the standard errors of the ML estimates from the Fisher Information matrix.

(e) Test H0 : β1 = 0 versus H1 : β1 ≠ 0 using asymptotic properties of ML estimators. What is the p-value of the test?

(f) Compute the efficiency of the ML estimates of (β0, β1) under a “misspecified” model with σ2u = 0.

2. Recall the Sitka spruce data as discussed in Example 4.1 in the textbook (also, see the R code given in class). Refit the model below using data from only Chambers 1 (treated with ozone) and 4 (control) for year 1988:

yij = β1xi1 + . . . + β5xi5 + ηTreati + γTimej + ϵij ,

where xij = 1 for time tj and 0, otherwise (i = 1, . . . , m, j = 1, . . . , 5), Treati is 1 for control and 0 for treatment, and Timej = dayj/100 for control and 0 for treatment.

(a) Find the REML estimate of the variance-covariance matrix V0 of the response vector yi .

(b) Find estimates of the regression coefficients and their standard errors.

(c) Assuming normality of the errors ϵij , test the hypothesis of no treatment effect, that is test H0 : η = γ = 0. What is the p-value of the test? Draw conclusions of the test.