Stat 431/831
ASSIGNMENT 1 SOLUTIONS
1. Recall that you were given the likelihood:
n
f (ti ; )i [1 F (ti ; )](1i )
L(; t, ) =
i=1
(a) Since the Ti are exponentially distributed we know that the pdf and cdf are given by:
f (t; ) = exp(ti )
1 F (t, ) = P [Ti
STAT 431/831
ASSIGNMENT 1
DUE: OCTOBER 2, 2008
1. [9 points]
For an individual i, i = 1, 2, . . . , n, denote
t
i = I (Ti Ci )
Ti = remission time
S (t) = P (T > t) = 1
f (s; )ds
0
t
Ci = censoring time
exp(s)ds = exp(t)
=1
ti = min(Ti , Ci )
0
From the
Stat 431/831
ASSIGNMENT 1
Due: September 27, 2012
Reminder: Your assignment must be handed in by 1:00pm on the due date in MC 4045.
Note: Be sure to include all R code and relevant output for all questions of this assignment.
1. The following data are the
Stat 431/831
ASSIGNMENT 2 SOLUTIONS
1. Let Yk Bin(mk , k ) iid k = 1, 2 and xk = I [k = 1]. Consider the logistic regression model
log(k /(1 k ) = 0 + 1 xk
(a) The likelihood in terms of (1 , 2 ) is given by:
y
y
L(1 , < 2) 1 1 (1 1 )(m1 y1 ) 2 2 (1 2 )(m
Stat 431
Winter 2011
Assignment 2
Assignment due at 10:30 a.m., Friday February 4.
1. The following data are from a study of the eects of viruses on chicken eggs. Eggs were
injected with various dilutions of a virus and were monitored daily up to day 18 a
STAT 431
ASSIGNMENT 1
DUE: Wed, January 25, 2012, AT 10:30 AM
1. (a) The likelihood is
n
n
L() = (2 2 ) 2 exp
(yi )2 /2 2 .
i=1
The log likelihood is
n
n
() = log (2 2 ) (yi )2 /2 2 .
2
i=1
(b) The score function is
n
(yi )/ 2 = n( )/ 2 .
y
S () =
i=1
Th
Stat 431/831
ASSIGNMENT 3
Due: November 1, 2012
Reminder: Your assignment must be handed in by 1:00pm on the due date in MC 4045.
Note: Be sure to include all R code and relevant output for all questions of this assignment. Be
kind to your TA, dont make t
STAT 431
ASSIGNMENT 2
DUE: Monday, February 13, 2012, AT 10:30 AM
1. In a biomedical study of an immuno-activating ability of two agents TNF (tumor necrosis
factor) and IFN (interferon), to induce cell dierentiation, the number of cells that exhibit
marke
Stat 431/831
ASSIGNMENT 2
Due: October 14, 2010
Note: Be sure to include all R code and relevant output for questions 2-4 of this assignment.
1. Suppose Yk Bin(mk , k ), k = 1, 2 are two independent binomial random variables. Let xk
denote an explanatory
Stat 431/831
ASSIGNMENT 3 SOLUTIONS
1. (a) It seems reasonable to assume that the number of claims Yl observed on a singly policy
has a poisson distribution with mean l and that policies are independent. Since the data
is aggregated over age/volume/distri
Stat 431/831
ASSIGNMENT 2
Due: October 11, 2012
Reminder: Your assignment must be handed in by 1:00pm on the due date in MC 4045.
Note: Be sure to include all R code and relevant output for all questions of this assignment.
1. Wilson and Mandelbrote (1978
Stat 431
Winter 2011
Assignment 1
Assignment due at 10:30 a.m., Friday January 21.
1. Consider a 2 2 contingency table from a prospective study in which people who
were or were not exposed to some pollutant are followed-up and, after several years,
catego
Stat 431/831
MIDTERM SOLUTIONS
1. 431 Version
(a) [2 marks] The likelihood for the rate functions = (1 , . . . , n )T is
n
(i tij )yij ei tij
yij !
i=1
L() =
(b) [6 marks] Recall that Yi P OI (i ti ). The likelihood for Yi is thus
(i ti )yi ei ti
yi !
= e
Stat 431/831
ASSIGNMENT 1
Due: September 30, 2010
1. Consider a series of Bernoulli random variables where the probability of success for each trial
is and the probability of failure is 1 . The random variable Y , which represents the
number of failures b
Stat 431/831
ASSIGNMENT 2 SOLUTIONS
1. (a) Using the raw data we estimate the odds ratio of having a conviction for someone with
a long education versus someone with a short education to be 0.4167.
6
L = P [conviction|long education] =
24
16
S = P [convic
Stat 431, Winter 2015 - Assignment 4
Instructor Dr. Schonlau, due on Friday April 3, 2015 at noon. Either submit a softcopy online into LEARN
or as a hardcopy (Box 15 located on the 4th floor of MC). Someone will collect the assignments at that
time or af
Stat 431/831
Assignment 3
Due: 12pm November 1st, 2013
Note: Use cover page. Include all R code and relevant output. At the same time, you should
properly organize/summarize your results; we are not supposed to search for your results in R
output.
1. In a
STAT 431
ASSIGNMENT 1
DUE: Wednesday, January 25, 2012, AT 10:30 AM
1. Suppose that Yi , i = 1, . . . , n, are independent and identically distributed normal random
variables with unknown mean and known variance 2 . The probability density function of
eac
1
STAT 431
ASSIGNMENT 2
DUE: Monday, February 13, 2012, AT 10:30 AM
QUESTION 1
(a) We start our investigation by tting the main eects model, that is, the model with only TNF
(denoted by x1 ) and IFN (denoted by x2 ) doses included, and the model that incl
(Please do not post online)
Matthias Schonlau Solution Stat 431 Winter 2015, A2
Q1 Surgical Risk Solution
(a)
i. H 1 : There is no association between treatment and outcome in any category of risk.
log i = o 2 x2 3 x3
1
i
with degrees of freedom 6 3
Please do not post on the web.
Stat 431, Winter 2015, A3 , solutions.
Question Fabric solution
(a) Under a time homogenous Poisson model we would have Yi Poisson(ti ) , i = 1, ,32 , and we
can find the log-likelihood through:
n
L( ) =
i =1
(ti ) i e
yi !
Stat 431, Winter 2015 - Assignment 2
Instructor Dr. Schonlau, due on Wednesday Feb 4, 2015 at noon. Either submit a softcopy online into
LEARN or as a hardcopy (Box 15 located on the 4th floor of MC). Someone will collect the assignments
at that time or a
Stat 431/831
ASSIGNMENT 4
Due: November 22, 2012
Reminder: Your assignment must be handed in by 1:00pm on the due date in MC 4045.
Note: Be sure to include all R code and relevant output for all questions of this assignment. Feel
free to print you output
1
A Brief Review of Likelihood Methods
Problem 1.1.
Suppose that Yi , i = 1, . . . , n, are independent and identically distributed
normal random variables with unknown mean and known variance 2 . The
probability density function of each Y is then given b
Stat 431/831
TERM TEST 2 SOLUTIONS
Question 1
[20/25 marks]
(a) Here we are modelling the death counts (ij ) in various age and smoking groups: [4 marks]
log(ij ) = log(ij ij ) = log ij + log ij = x + log(ij )
It is important to include the oset term beca
STAT 431/831
ASSIGNMENT 2
DUE: OCTOBER 21, 2008
1. [9 points] A sample of 146 ve-year-old children had their teeth examined and those with decayed,
missing, or lled teeth (dmft) were noted. From their address it was also determined whether their
drinking
STAT 431/831
ASSIGNMENT 2
DUE: OCTOBER 16, 2008
1. A sample of 146 ve-year-old children had their teeth examined and those with
decayed, missing, or lled teeth (dmft) were noted. From their address it was
also determined whether their drinking water was u
STAT 431
SKETCH SOLUTIONS OF TERM EXAM 1
FEBRUARY 1, 2012
1. (a) Recall that in the case of a continuous random variable y :
f (y ; , )dy = 1
f (y ; , )dy =
1
f (y ; )dy = 0
assuming we can bring the dierential operator inside the integral. Since
1
log f