Repeated Measures
Models
Stat 557
Heike Hofmann
Big 12 - 2012
• with 17 games still
missing ...
• can’t estimate home
advantage
(Kansas State hasn’t lost
yet, Kansas hasn’t won)
Outline
• Marginal Homogeneity
• additional covariates
• general log-linear models
Repeated Measures
Models
• Extension of matched pairs data
• Multiple (T ≥ 3) measurements observed
for same individual, e.g. individuals’ weekly
progress
• Measurements for cluster of individuals (T
≥ 3), e.g. one litter, teeth at dentist’s visit, ...
If β1 = β2 = ... = βT = 0 we observe marginal homogeneity, i.e. t
P (YT = 1).
Example: Drug
Comparisons
Example: Crossover-Study of Drugs Drugs A, B, and C are t
disease in a cross-over study, i.e. each individual is treated for some
binary: (success/failure) for each drug, giving a data set of
• Cross-over effect of drugs
A, B, C
• Interested in marginal
distributions
P(A=S), P(B=S), P(C=S)
A B
S S
S S
S F
S F
F S
F S
F F
F F
Total
C
S
F
S
F
S
F
S
F
count
6
16
2
4
2
4
6
6
46
One question of interest for these data is, whether all of the drugs are
difference. This question translates to whether marginal homogeneity
From the raw data we get estimates for the effectiveness of each drug
8
Repeated
Response
Data
d ResponseMultiple
8 Data
Repeated
Response D
Binary
A lot of repeatedly,
studies observe
individuals repeatedly,
e.g.
rve individuals
e.g.
longitudinal
studies.
A data
lot of
studies
observe
individuals
rep
For
these
we
will
be
mainly
interested
in
the
ill be mainly interested in the marginal distributions.
For
these
data
we will
be mainly
intere
Let
(Y
,
Y
,
....,
Y
)
be
a
tuple
of
binary
response
va
1
2
T
be a tuple of binary response variables observed at (time) po
Let
(Y1 , Yin
, ....,
Y
) points
be
of
We Yare
interested
the
probability
of
for e
response
for
time
t=1,
...,
T binary
t binary
2for
Tt,
the probability
of success
each
i.e. awetuple
aresuccess
interested
A as
logit model
is interested
then definedinasthe probability of
We
are
n defined
logit model
logit P (Yt
A logit
is 1)
then
logitmodel
P (Yt =
= αdefined
+ βt , as
estimability:
with
constraint βT = 0 (or α = 0).
= 0 (or α = 0).
= β2 =marginal
... = βThomogeneity,
= 0 we observe
marginal
h
1observe
βT = 0 IfweβMarginal
i.e.
then
P
(Y
withhomogeneity
constraint βT = 0 (or α = 0). 1
P (YT = 1).
Response
•
•
•
If β1 = β2 = ... = βT = 0 we observe
P
(Y
=
1).
T
Example: Crossover-Study of Drugs Drugs
ver-Study of Drugs Drugs A, B, and C are tested on 4
disease
in a individual
cross-over isstudy,
i.e.foreach
individual
er study,
i.e. each
treated
some
time wit
Drugs Crossover
> head(drugs.m)
count id variable value
1
6 1
A
Y
2
16 2
A
Y
3
2 3
A
Y
4
4 4
A
Y
5
2 5
A
N
6
4 6
A
N
glm(formula = value ~ variable - 1, family = binomial(link = logit),
data = drugsm, weights = count)
Deviance Residuals:
Min
1Q Median
-3.698 -2.740 -0.220
Drugs Crossover
3Q
2.152
Max
3.986
Coefficients:
Estimate Std. Error z value Pr(>|z|)
variableA
0.4418
0.3021
1.462
0.1436
variableB
0.4418
0.3021
1.462
0.1436
variableC -0.6286
0.3096 -2.031
0.0423 *
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 191.31
Residual deviance: 182.60
AIC: 188.60
on 24
on 21
degrees of freedom
degrees of freedom
Number of Fisher Scoring iterations: 4
Marginal Homogeneity
> anova(drugs.null, drugs.mh, test="Chisq")
Analysis of Deviance Table
Model 1: value ~ 1
Model 2: value ~ variable - 1
Resid. Df Resid. Dev Df Deviance P(>|Chi|)
1
23
191.05
2
21
182.60 2
8.451
0.01462 *
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’
1
Longitudinal Mental
Depression
• study followed subjects over 4 week period
• measurements taken at baseline, and after
1,2, and 4 weeks: Normal/Abnormal mental
state
• Additionally recorded:
treatment (randomly assigned at baseline)
severity (mild or severe)
Longitudinal Mental
Depression
1
2
3
4
diagnosis treatment N_N_N N_N_A N_A_N N_A_A A_N_N A_N_A A_A_N A_A_A
mild standard
16
13
9
3
14
4
15
6
mild new drug
31
0
6
0
22
2
9
0
severe standard
2
2
8
9
9
15
27
38
severe new drug
7
2
5
2
31
5
32
6
Treatment Effects: Percentage of “normal”
1
2
3
4
diagnosis treatment
week1
week2
week4
mild standard 51.25000 58.75000 67.50000
mild new drug 52.85714 78.57143 97.14286
severe standard 19.09091 25.45455 41.81818
severe new drug 17.77778 50.00000 83.33333
Marginal Model
logit P (Yt = 1) = α + β1 s + β2 d + β3 t
• s = 1 for severe cases
• d = 1 for new drug
• t = 0,1,2 week (log of actual treatment
2
time)
Data
treatment diagnosis lweek normal abnormal
1
standard
mild
0
41
39
2
standard
mild
1
47
33
3
standard
mild
2
54
26
4
standard
severe
0
21
89
5
standard
severe
1
28
82
6
standard
severe
2
46
64
7
new drug
mild
0
37
33
8
new drug
mild
1
55
15
9
new drug
mild
2
68
2
10 new drug
severe
0
16
74
11 new drug
severe
1
45
45
12 new drug
severe
2
75
15
12 Binomials
Model
depress.main.sm <- glm(cbind(normal,abnormal)~diagnosis+treatment+lweek,
data=depress.sm, family=binomial(link=logit))
Data
1
2
3
4
diagnosis treatment N_N_N N_N_A N_A_N N_A_A A_N_N A_N_A A_A_N A_A_A
mild standard
16
13
9
3
14
4
15
6
mild new drug
31
0
6
0
22
2
9
0
severe standard
2
2
8
9
9
15
27
38
severe new drug
7
2
5
2
31
5
32
6
4 Multinomials
Model
depress.main.sm <- glm(cbind(normal,abnormal)~diagnosis+treatment+lweek,
data=depress.sm, family=binomial(link=logit))
glm(formula = cbind(normal, abnormal) ~ diagnosis + treatment +
lweek, family = binomial(link = logit), data = depress.sm)
Deviance Residuals:
Min
1Q
Median
-2.4263 -1.2754 -0.1555
3Q
1.8274
Max
2.8076
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)
-0.50671
0.14605 -3.470 0.000521 ***
diagnosissevere
-1.36798
0.14457 -9.462 < 2e-16 ***
treatmentnew drug 0.97070
0.14156
6.857 7.03e-12 ***
lweek
0.88920
0.08939
9.947 < 2e-16 ***
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 275.760
Residual deviance: 36.314
AIC: 98.727
on 11
on 8
degrees of freedom
degrees of freedom
Number of Fisher Scoring iterations: 4
Model Interaction
logit P (Yt = 1) = α + β1 s + β2 d + β3 t + β4 dt
depress.two.sm <- glm(cbind(normal,abnormal)~diagnosis+treatment*lweek,
data=depress.sm, family=binomial(link=logit))
Analysis of Deviance Table
Model 1: cbind(normal, abnormal) ~
Model 2: cbind(normal, abnormal) ~
Resid. Df Resid. Dev Df Deviance
1
8
36.314
2
7
3.433 1
32.881
--Signif. codes: 0 ‘***’ 0.001 ‘**’
diagnosis + treatment + lweek
diagnosis + treatment * lweek
P(>|Chi|)
9.797e-09 ***
0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
glm(formula = cbind(normal, abnormal) ~ diagnosis + treatment *
lweek, family = binomial(link = logit), data = depress.sm)
Deviance Residuals:
Min
1Q
Median
-0.6792 -0.3597 -0.1609
3Q
0.4793
Max
0.8948
Coefficients:
Estimate Std. Error z value
(Intercept)
-0.042175
0.163022 -0.259
diagnosissevere
-1.398773
0.145777 -9.595
treatmentnew drug
-0.006181
0.221929 -0.028
lweek
0.468408
0.112983
4.146
treatmentnew drug:lweek 1.046819
0.188310
5.559
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’
Pr(>|z|)
0.796
< 2e-16 ***
0.978
3.39e-05 ***
2.71e-08 ***
0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 275.7599
Residual deviance:
3.4329
AIC: 67.846
on 11
on 7
degrees of freedom
degrees of freedom
Number of Fisher Scoring iterations: 3
Difference between Agresti and our fit:
> subset(depress, (treatment=="standard") &(diagnosis=="mild"))
week4 week2 week1 treatment diagnosis count
1
N
N
N standard
mild
16
2
A
N
N standard
mild
13
3
N
A
N standard
mild
9
4
A
A
N standard
mild
3
5
N
N
A standard
mild
14
6
A
N
A standard
mild
4
7
N
A
A standard
mild
15
8
A
A
A standard
mild
6
overall n = 80
> subset(depress.sm, (treatment=="standard") &(diagnosis=="mild"))
treatment diagnosis lweek normal abnormal n
1 standard
mild
0
41
39 80
2 standard
mild
1
47
33 80
3 standard
mild
2
54
26 80
overall n = 240
Generalized Log-Linear
Models
• Extend concept of log-linear model:
log µ = Xβ
C log Aµ = Xβ
• regular loglinear models are special case,
when A and C are identity matrices