STAT/BIOSTAT 571: Coursework 7
To be handed in on Wednesday 5th March, 2008. 1. On the class web page you will nd data on 30 patients with leprosy. Specically, the dataset consists of count data from a placebo-controlled clinical trial of 30 patients with

STAT/BIOSTAT 571: Coursework 1
To be handed in on Friday 16th January, 2009. 1. A Gauss-Markov Theorem for Dependent Data Suppose E[Y ] = x and var(Y ) = V , where Y = (Y 1 , ., Y m ) with Y i = (Yi1 , ., Yini )T , x = (x1 , ., xm )T is N p with xi = (xi1

STAT/BIOSTAT 571: Coursework 3
To be handed in on Friday 30th January, 2009. 1. Prove that if the prior distribution for T = (1 , ., m ) can be written as:
m
p() =
i=1
p(i |)p()d,
then the covariances cov(i , j ) are all non-negative (assume that E[i | ]

STAT/BIOSTAT 571: Coursework 2
To be handed in on Friday 23rd January, 2009. 1. Consider the class of linear predictors b (y) = a+By, where a and B are constants of dimensions (q + 1) 1 and (q + 1) n. Let W = b By, and show that E[(b b)T A(b b)] = cfw_a E

Biostat/Stat 571: Coursework 2
Answer Key January 23, 2009
Problem 1
Consider the class of linear predictors b? (y) = a+By, where a and B are constants of dimensions (q +1) 1 and (q + 1) n. Let W = b By, and show that E[(b? b)T A(b? b)] = fa Bx E[W]gT Afa

Biostat/Stat 571: Coursework 1
Answer Key January 16, 2009
Problem 1
A Gauss-Markov Theorem for Dependent Data. Suppose E[Y] = x and var(Y) = V, where Y = T T (Y1 ; : : : ; Ym )T with Yi P (Yi1 ; : : : ; Yini )T , x = (x1 ; : : : ; xm )T is N p with xi =

Biostat/Stat 571: Coursework 2
Answer Key January 30, 2009
Problem 1
Prove that if the prior distribution for
T
= ( 1; : : : ; Z Y m
m)
can be written as:
p( ) = then the covariances cov( i ; The above implies that E[
j)
i=1
p( i j )p( )d
are all non-nega

STAT/BIOSTAT 571: Coursework 4
To be handed in on Friday 6th February, 2009. 1. For the data of question 2 of coursework 1 we will carry out a Bayesian analysis using WinBUGS. 2 -2 Assume independent priors with an improper flat prior on , an improper pri

Biostat/Stat 571: Coursework 4
Answer Key February 6, 2009
1
Problem 1
For the data of question 2 of coursework 1 we will carry out a Bayesian analysis using WinBUGS. Assume inde2 pendent priors with an improper at prior on , an improper prior ( 2 ) / on

Biostat/Stat 571: Coursework 5
Answer Key February 13, 2009
Problem 1
Install the inla software on your computer and run the seizure data example. Hint: Type ?inla and see the examples section, specically the commands following data(Epil). The xed eect es

STAT/BIOSTAT 571: Coursework 5
To be handed in on Friday 13th Febuary, 2009. 1. Install the inla software on your computer and run the seizure data example. Hint: Type ?inla and see the examples section, specifically the commands following data(Epil). 2.

Biostat/Stat 571: Coursework 6
Answer Key February 21, 2009
Problem 1
(a)
GEE is a marginal model, so all our parameter estimates are population averages. A natural interpretation is as follows: e e e e
0
The population average counts of leprosy bacilli a

2009 Jon Wakefield, Stat/Biostat 571
Logistic Mixed Eects Models A GLMM for binary data takes the binomial exponential family, with canonical link being logistic. We have Stage 1: Yij ind Binomial(nij , pij ) with pij log = xij + zij bi 1 pij Stage 2: bi

2009 Jon Wakefield, Stat/Biostat 571
Bayesian Inference for the LMEM Consider the model yi = xi + zi bi + i
2 with bi iid N(0, D), i ind N(0, I ni ), with bi and i independent.
The form of the posterior follows from exploiting conditional independencies:

2009 Jon Wakefield, Stat/Biostat 571
BINARY DATA MODELS We devote an entire chapter to binary data since such data are challenging, both in terms of modeling the dependence, and parameter interpretation. We again consider mixed effects models approaches (

STAT/BIOSTAT 571: Final Takehome Exam
To be handed in to my Biostatistics mailbox by 2pm on Friday 20th March, 2009. No collaboration! The Six Cities Study of Air Pollution and Health was a longitudinal study designed to characterize lung growth as measur

STAT/BIOSTAT 571: Coursework 8
To be handed in on Wednesday 12th March, 2008. You will find data on the 275 children in the Indonesian study at http:/faculty.washington.edu/heagerty/Books/AnalysisLongitudinal/datasets.html The description of the variables

Biostat/Stat 571: Coursework 8
Answer Key March 13, 2009 1.
We have E[Y jb] = Hence E[Y ] = Eb fPr(z < cx + cbjb)g where z N(0; 1). Hence E[Y ] = Z
1 1
1 ex +b = 1 + ex +b 1+e x
b
= G(x + b)
(c[x + b])
(Z
c(x +b)
(1 )
1
1=2
e
z 2 =2
dz
)
(2
2
)
1=2
e
b2 =

Biostat/Stat 571: Coursework 7
Answer Key March 6, 2009
Table 1 contains pharmacokinetic data on 10 subjects who have been administered a dose of D = 30 mg of the drug Cadralazine. Let Zij denote the drug concentration (in mg/ml) at time tij , j = 1; :; n

STAT/BIOSTAT 571: Final Takehome Exam
1. 10 marks Summarize the data, including the use of informative plots. 2. 6 marks We can write Median(FEV1ij ) = exp(0 + 1 (Ageij - IAgei1 ) + 2 IAgei1 Hence, exp(1 ) describes the longitudinal change in the populati

2009 Jon Wakefield, Stat/Biostat 571
CHAPTER 14: MISSING DATA A serious problem in data analysis is the existence of missing data. We concentrate on missing responses in a dependent data situation. Implications of missing data: 1. Data are unbalanced not

2009 Jon Wakefield, Stat/Biostat 571
Further notes on GEE Intuitively: to restore the unbiasedness of the estimating equation for the complete population we need to weight the contribution of Yij by the inverse of ij . For unbiasedness of the estimating e

2009 Jon Wakefield, Stat/Biostat 571
GENERAL REGRESSION MODELS We consider the class of Generalized Linear Mixed Models (GLMMs) and non-linear mixed effects models (NLMEMs). In this chapter we will again consider both a conditional approach to modeling, v

2009 Jon Wakefield, Stat/Biostat 571
Stat/Biostat 571 Statistical Methodology: Regression Models for Dependent Data
Jon Wakefield Departments of Statistics and Biostatistics, UW Lectures: Monday/Wednesday/Friday 1.302.20, T473. Coursework: (and approximat

2009 Jon Wakefield, Stat/Biostat 571
Covariance Models for Clustered Data Whether we take a GEE or LME approach (with inference from the likelihood or from the posterior) we require flexible yet parsimonious covariance models. With LME we have so far assu

2009 Jon Wakefield, Stat/Biostat 571
CHAPTER: MULTILEVEL MODELS We have so far considered dependencies within data, when given a single set of random eects units were viewed as independent in these situations we have a single layer of clustering. In many

2008 STAT/BIOSTAT 571: Coursework 6
To be handed in by the start of the lecture on Wednesday 27th February, 2008. Table 1 contains pharmacokinetic data on 10 subjects who have been administered a dose of D = 30mg of the drug Cadralazine. Let Zij denote th