Statistics 641, Fall 2015
Homework #8
Due Dec 10, 2015
1. Suppose that we conduct a randomized trial two armed trial with a continuous outcome, Y .
The null hypothesis is that the distribution of Y is the same in the two treatment groups,
z = 0, 1. The on
Stat 641 October 6, 2015
Lecture 14Supplement, Page 1
Comment regarding pvalues.
Suppose we have a continuous response (e.g., normal), and our model assumptions are correct.
For example, for normal data, if H0 : = 0 is true,
t=
se()
has a tdistribution
Stat 641 October 13, 2015
6.15.5
Lecture 12Supplement, Page 1
Incomplete Adherence (supplemental)
Additional comments made in class.
In the gure below:
Randomization + ITT gets us to the rst set of scientic hypotheses: the eect of
assignment to treatment
Stat 641 October 8, 2015
6.15
6.15.1
Lecture 11, Page 1
Randomization Model for Causal Inference (continued)
From Miguel Hernns web page
a
In an ideal world, all decisions would be based on randomized experiments. . . . Unfortunately,
randomized experimen
Stat 641 September 29, 2015
5.7.3
Lecture 8, Page 1
More Proportional Hazards Models
Recall proportional hazards assumption:
log (t; z) = log 0 (t) + z
where z = 0, 1 is treatment group, and 0 (t) is the baseline (cumulative) hazard function and is
the lo
Stat 641 October 1, 2015
6
Lecture 9, Page 1
Whats the Question?
6.1
Overview
Simple Version:
Does T
O?
where
T
= Treatment/Intervention
O = Outcome of interest
= Causation
(Closer to) Reality:
Quality
of
Life
Symptoms
Mortality
Hospitalization
Concomita
Stat 641 October 6, 2015
6.11
Lecture 10, Page 1
Formal Basis for Assessing Causality
6.11.1
Hill Criteria
Sir Bradford Hill proposed the following criteria for assessing causality in observational (non
experimental) studies:
Strength of association
Consi
Stat 641 September 24, 2015
5.4
5.4.1
Lecture 7, Page 1
Comparing two groups
Example: 6MP trial
The 6Mercaptopurine in Acute Leukemia trial
Conducted in 19591960
Patients had undergone corticosteroid therapy for acute leukemia
6Mercaptopurine versu
Stat 641 September 17, 2015
5
Lecture 5, Page 1
Survival Analysis
5.1
Example: TNT
See: LaRosa et al. (2005) N Engl J Med
10001 subjects randomized to
10mg atorvastatin dose: 5006
80mg atorvastatin dose: 4995
over a 16 month period
slightly less than
Stat 641 December 3, 2015
10.10.1
Lecture 26, Page 1
Some Observations Regarding Subdensities
N
Sn
0
Sn
0
Sn
N
0.00
0.25
0.50
0.75
1.00
0
Recall the gures from the R demo:
N
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
In each gure, the early stoppi
Stat 641 September 22, 2015
5.3.5
Lecture 6, Page 1
Condence Intervals, Numerical Example
Calculation of 95% Condence Interval
At t = 7, S(7) = 0.807, 2 = Var (7) = 0.01161, Var S(7) = 0.00756.
On the S(t) scale, we have S(7)(1 1.96 ) = 0.807 (1 1.96 0.01
Stat 641 September 15, 2015
4.10
Lecture 4, Page 1
TwoSample Tests
There are three typical approaches for generalizing to twosample problems.
1. Conditional likelihood
reduce to a onesample problem by conditioning on an ancillary statistic as in the
p
Stat 641 September 8, 2015
2.5
Lecture 2, Page 1
Hypothesis Testing
(Traditional frequentist framework) Suppose that Y has distribution f (y; )
The Null Hypothesis, H0 , is a value (or set of values) of , say 0 , that we usually want to rule
out. That is,
Stat 641 September 10, 2015
4
Lecture 3, Page 1
Likelihood
Example 4.1:
In R
> summary(coxph(Surv(week,status)~group,data=mp6,method="exact")
coef exp(coef) se(coef)
z Pr(>z)
group 1.6282
0.1963
0.4331 3.759 0.00017 *
.
Likelihood ratio test= 16.25
Wa
Stat 641 November 24, 2015
10
Lecture 24, Page 1
Interim Monitoring
10.1
Data Monitoring Committees/Trial Organization
A Data Monitoring Committee (DMC), otherwise known as
Independent Data Monitoring Committee (IDMC),
Data and Safety Monitoring Board (
Stat 641 October 15, 2015
6.17.4
Lecture 13, Page 1
Continuous Outcomes
Suppose that we observe Yij where
j = A, B (control, experimental treatment)
i = 1, . . . , nj ,
increasing Y implies better outcome
Whats the (statistical) question?
E[YiB ] > E[
Stat 641 October 13, 2015
6.15.8
Lecture 12, Page 1
Incomplete Adherence (Continued)
Common alternatives to ITT. I like to call these, and analyses like them, Hopebased Analyses.
We make certain untestable (and almost certainly false!) assumptions, and h
Statistics 641, Fall 2015
Homework #7
Due Dec 3, 2015
1. Suppose that we observe the following continuous responses for groups A and B.
A: 3.0, 1.7, 4.6, 3.6, 1.2, 3.1
B: 3.9, 6.7, 7.5, 3.4, 8.8, 9.6
Let be then dierence in means, = X B X A . Compute a tw
Statistics 641, Fall 2015
Homework #6
Due Nov 19, 2015
1. The dataset data6.csv contains data collected from a crossover study with 40 subjects per
sequence. The variables in the dataset are:
seq
y
id
period
z
Assigned treatment sequence
Response
Subject
Statistics 641, Fall 2015
Homework #5
Due Nov 5, 2015
1. Suppose all subjects in a randomized trial are followed for 1 year, and at the end of that time
they either survive disease free (DFS), survive but experience a recurrence of disease, or die.
Note t
Statistics 641, Fall 2015
Homework #3
Due Oct 8, 2015
1. For the following use the data from the le data3.csv
The variables in the dataset are:
trt
days
status
sex
age
Treatment group (0/1)
Followup time in days
censoring/failure indicator (1=failure, 0=
Statistics 641, Fall 2015
Homework #2
Due October 1, 2015
1. The data le data2.csv (in csv format, comma delimited) contains columns
z: treatment variable (0,1).
w: categorical baseline covariate with four levels, 14.
x: value of response variable
Assu
Statistics 641, Fall 2015
Homework #1
Due Sept 17, 2015
1. Suppose in a randomized trial, we observe a baseline (prerandomization) variable, w, and
two response variables, x and y. The data le data1.csv (in csv format, comma delimited)
contains columns
Stat 641 October 20, 2015
6.17.5
Lecture 14, Page 1
Ordinal Outcomes
Ordered categorical/ordinal. (Usually small) number of ordered categories:
NYHA Class (I, II, III, IV)
better/same/worse
ECOG performance status
0
1
2
3
4
5

asymptomatic
symptomatic
Stat 641 November 17, 2015
9.3.5
Lecture 22, Page 1
Randomization Tests (Continued)
Example:
Two treatments,
2N subjects,
Continuous response,
Random allocation (N subjects randomly assigned each treatment)
T = rank sum for subjects in group A.
Rand
Stat 641 November 10, 2015
3
Lecture 20, Page 1
Sample Size
3.1
Example
Example:
Suppose we have observations Y1 , Y2 , . . . with Yj N (, 2 ), and assume 2 is known.
Let Y n
1
=
n
n
Yi
i=1
We reject H0 : 0 if Y n > Cn for some Cn
Fix type I error rat
Stat 641 November 19, 2015
4.6
Lecture 23, Page 1
Assignment of best treatment to the most subjects
Responseadaptive allocation:
Goal: Balance between
Need for generalizable knowledge regarding best treatments for future patients
Desire for optimal tr
Stat 641 November 12, 2015
8.4
8.4.1
Lecture 21, Page 1
Survival Data (continued)
No Censoring
EI(0) = ET 0 1 =
n
0 1 = n0 1
Z1 + Z1
2
1 0 1
n=
2
.
Because we have assumed no censoring, n is the required number of events.
8.4.2
Common Censoring Time
Suppo