ST 790, Homework 3
Spring 2017
1. Standard errors for covariance parameters. Ordinarily, for population-averaged models, the
primary focus is on inference on the parameter β characterizing population mean response,
and the goal of appropriate modeling and fitting of the overall covariance structure is mainly
to achieve the most efficient inference on β. Here, then, inference on the covariance parameter ξ is generally not of direct interest.
From a subject-specific point of view , where a hierarchical mixed effects model has been
specified, questions of scientific interest are often not only about the fixed effects parameter
β in (6.43) on page 185 of the notes characterizing the mean value of the individual-specific
parameters β i . Investigators also may be interested in inference on the covariance matrix
D of the random effects b i given x i , which we take in this problem to be the same for all
individuals as in (6.43).
For example, in a simple linear individual model as in (6.25) for the dental study, the diagonal
elements of D (2 × 2) are the variances of individual-specific intercepts and slopes and thus
characterize the extent to which these vary in the populations of boys and girls, and the
off-diagonal element represents the extent to which “high” or “low” intercepts are associated
with “steeper” or “shallower” individual trajectories. Investigators are often interested in these
phenomena, as they are features of the population of interest.
Accordingly, assuming that the matrix D is positive definite, it is sometimes of interest to
obtain standard errors for the estimators for its distinct components, which are of course
elements of the covariance parameter ξ. In this problem, you will derive the form of the large
sample covariance matrix of the maximum likelihood estimator b
ξ under the assumption that
the models for E(Y i |x i ) = X i β and the overall covariance structure var(Y i |x i = V i (ξ, x i ) are
correctly specified, with true values β 0 and ξ 0 . In a subject-specific model, V i (ξ, x i ) is a
correctly specified model for overall covariance structure if the models for the among- and
within-individual covariance matrices are correctly specified.
Let the dimension of ξ be Q = r + s.
(a) Via the usual M-estimation argument, i.e., a Taylor series of (5.35) on page 133 of the
b and b
notes evaluated at β
ξ jointly solving (5.32) and (5.35),
(1/2)
m X
b T V −1 (b
b
(Y i − X i β)
ξ, x i ){∂/∂ξk V i (b
ξ, x i )}V −1
i
i (ξ, x i )(Y i − X i β)
i=1
h
i
b
b
− tr V −1
(
ξ,
x
){∂/∂ξ
V
(
ξ,
x
)}
= 0, k = 1, ... , r + s,
i
k i
i
i
(1)
find the form of the matrix Λ such that
L
m1/2 (b
ξ − ξ 0 ) −→ N (0, 2Λ−1 )
under the assumption that the distribution of Y i |x i is normal.
Hint: You may find the following results useful: If Z ∼ N (µ, V ) and A, A1 , and A2 are
symmetric matrices, then
var(Z T AZ ) = 2tr(AV AV ) − 4µT AV Aµ,
1
and
cov(Z T A1 Z , Z T A2 Z ) = 2tr(A1 V A2 V ) − 4µT A1 V A2 µ.
Note: Both SAS proc mixed and R lme() allow the user to request the estimator of 2m−1 Λ−1
and associated standard errors (square roots of the diagonal elements), thus providing approximate assessment of uncertainty of estimation of the components of ξ under normality.
(b) You will now investigate what happens if the distribution of Y i |x i is not normal. To make
this simple, assume that ni ≡ 1 for all i = 1, ... , m, so that Y i = Yi , X i β, and V i = Vi are
scalars, and define the excess kurtosis of the distribution of Yi |x i to be the value κ such that
(Yi − X i β 0 )2
var
= 2 + κ,
Vi (ξ 0 , x i )
where κ = 0 for a normal distribution. Under these conditions, show that
L
m1/2 (b
ξ − ξ 0 ) −→ N (0, (2 + κ)Λ−1 ),
where now Λ is as above when ni ≡ 1.
Result: This demonstrates that the standard errors for elements of b
ξ output by standard software, which are based on the assumption of normality, will underestimate the true sampling
variation if κ > 0, which is the case for distributions with heavier tails than the normal.
2. Balanced data. Consider the linear mixed effects model given in equations (6.1) and (6.16).
On page 188 of the notes, we noted that, if the data are balanced, so that each of m individuals is observed at the same n time points, under the hierarchical model formulation, the
matrix C i is identical for all individuals, so that the matrix Z i = Z ∗ is identical for all individuals. We also noted that, under certain conditions, the usual estimator for β and the OLS
estimator are equivalent. In this problem, you will show this under the following conditions:
• Z ∗ (n × q) is of full column rank
• There are two groups, where there are m1 individuals in group 1 and m − m1 individuals
in group 2, so that
Xi
= ( Z ∗ 0 ),
= ( 0 Z
∗
i = 1, ... , m1
),
i = m1 + 1, ... , m,
where 0 is a (n × q) matrix of all zeros, and the model is parameterized so that the first
q entries of β pertain to the first group and the second q entries pertain to the second
group, as for the model for the dental study data on pages 177-178 and in equation
(6.30).
• The within-individual covariance matrix R i (γ) = σ 2 I n for all individuals
• The covariance matrix of the random effects is taken to be different in each group, so
that
var(b i |ai ) = D 1 ,
i = 1, ... , m1
= D2,
i = m1 + 1, ... , m,
so that
Vi
= V ∗1 = Z ∗ D 1 Z ∗T + σ 2 I n ,
=
V ∗2
∗
= Z D2Z
2
∗T
2
+ σ I n,
i = 1, ... , m1
i = m1 + 1, ... , m.
Suppose D 1 , D 2 , and σ 2 are either known or estimated. Show that the usual estimator
b=
β
m
X
!−1
X Ti V −1
i Xi
i=1
m
X
X Ti V −1
i Yi
i=1
is equal to the the ordinary least squares estimator
b
β
OLS =
m
X
!−1
X Ti X i
i=1
m
X
X Ti Y i .
i=1
3. Age-related macular degeneration clinical trial, continued. Recall the data from the clinical
trial in patients with age-related macular degeneration (AMD), which are in the file armd.dat
on the class webpage. There were 240 patients, each of whom was randomized to receive
either a placebo or active treatment (interferon-α). We again consider the visual acuity outcome discussed in the notes, which was intended to be ascertained for each subject at
weeks 0 (baseline), 4, 12, 24, and 52. The data set has the following columns:
1
2
3
4
5
patient ID
baseline lesion grade (1, 2, 3, 4 representing increasing severity)
treatment group (0 = placebo, 1 = interferon-α)
time (weeks)
visual acuity
See Homework 2 for the full description. As discussed in the notes, some patients are
missing data from some of the intended visits. Thus, be careful to account for this and to
state any assumptions necessary to justify your analysis.
The investigators are interested in the following questions: (i) Do either of the treatments
improve visual acuity or at least arrest its decline? (ii) Is the typical pattern of change in
visual acuity over the study period different between placebo and interferon-α? How? (iii)
What is the mean rate of change in visual acuity over the study period for each treatment?
(iv) Is the typical value of baseline visual acuity in the population different depending on
baseline lesion severity? (v) Is the typical pattern of change in visual acuity over the study
period associated with baseline lesion severity?
These questions are subject-specific in nature. Using methods in Chapter 6 of the notes,
carry out analyses to address these questions and write a brief report summarizing what you
did and the results. Be sure to describe how you interpreted and formalized the questions of
interest from the SS perspective. Comment on how confident you feel about the reliability of
the inferences and conclusions.
Please turn in your code and output along with your report (you can edit the output to include
only the portions that pertain directly to your report and embed it in your report if you like).
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