#
# Analysis of longitudinal dropout data with OSWALD (in Splus).
#
#
# The window version 3.2 or 3.4 can be downloaded at
# http://www.maths.lancs.ac.uk/Software/Oswald/. Unzip the file to the
"library" folder
# of Splus 2000, for example, C:\Program Files\sp2000\library\oswald.
#############################################################
############################
# Invoke the library “Oswald” in Splus:
>library(Oswald, lib.loc="C:\\Program
Files\\sp2000\\library\\oswald")
The milk data
- comprise 19 weekly measurements of the protein content of
milk samples from each of 79 cows.
- three treatment (diet) groups: barley (cows 1-25), mix of barley
and lupins (cows 26-52) and lupin (cows 53-79).
- missing data: drop out at weeks 15, 16, 17 and 19. No dropout
at week 18.
>milk<-read.balanced("C:\\Program
Files\\sp2000\\library\\oswald\\DATA\\milk.dat",19,nacode
=9.99)
>summary(milk)
Balanced LDA matrix:
Number of time series: 79
Number of observations per series: 19
Total number of observations: 1501 (164 missing)
>groups(milk)<-rep(c("Barley","Mixed","Lupins"),c(25,27,27))
>milk[1:4]
GROUP [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[,10] [,11]
TIME
1
2
3
4
5
6
7
8
9
10
11
TS1 Barley 3.63
3.82 3.83
TS2 Barley 3.24
3.32 3.31
TS3 Barley 3.98
3.22 2.93
TS4 Barley 3.66
3.18 3.14
[,12] [,13]
TIME
12
13
TS1 3.71 4.10
TS2 3.27 3.41
TS3 2.92 2.82
TS4 3.18 3.24
3.57 3.47 3.65 3.89 3.73 3.77 3.90 3.78
3.25 3.29 3.09 3.38 3.33 3.00 3.16 3.34
3.60 3.43 3.30 3.29 3.25 2.93 3.20 3.27
3.50 3.05 2.90 2.72 3.11 3.05 2.80 3.20
[,14] [,15] [,16] [,17] [,18] [,19]
14
15
16
17
18
19
4.02 4.13 4.08 4.22 4.44 4.30
3.45 3.12 3.42 3.40 3.17 3.00
2.64
NA
NA
NA
NA
NA
3.37 3.30 3.40 3.35 3.28
NA
> milk[c(1, 26, 53)]
GROUP [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[,10] [,11]
TIME
1
2
3
4
5
6
7
10
11
TS1 Barley 3.63 3.57 3.47 3.65 3.89 3.73 3.77
3.82 3.83
TS26 Mixed 3.38 3.38 3.10 3.09 3.15 2.77 3.40
3.57 3.80
TS53 Lupins 3.69 3.38 3.00 3.50 3.09 3.30 3.07
3.60 3.42
[,12] [,13] [,14] [,15] [,16] [,17] [,18]
TIME
12
13
14
15
16
17
18
TS1 3.71 4.10 4.02 4.13 4.08 4.22 4.44
TS26 3.70 3.44 3.51 3.78 3.52 3.26 3.52
TS53 3.59 3.77 3.74 3.70 3.78 3.78 3.77
[,8] [,9]
8
9
3.90 3.78
3.25 3.35
3.22 2.97
[,19]
19
4.30
3.25
3.53
# Plot a sample of 20% of the
trajectories from each treatment group
>plot(milk,p=0.2)
>ldamat.legend(milk,10,4.5)
4.5
3.5
2.5
3.0
milk
4.0
Barley
Lupins
Mixed
5
10
15
Time
# Plot the group means over time.
>milk.means <- olsfit(milk)
> milk.means[c(1, 26, 53)]
GROUP
[,1]
[,2]
[,3]
[,4]
[,5]
[,6]
TIME
1
2
3
4
5
6
TS1 Barley 3.886800 3.642500 3.498000 3.376400 3.484400
3.386400
TS26 Mixed 3.861111 3.540000 3.345556 3.277778 3.337778
3.392593
TS53 Lupins 3.758148 3.427778 3.372963 3.294074 3.238077
3.280370
[,7]
[,8]
[,9]
[,10]
[,11]
[,12]
[,13]
TIME
7
8
9
10
11
12
13
TS1 3.468800 3.503200 3.511739 3.518800 3.455000 3.429200
3.512400
3.8
TS26 3.333333 3.397692 3.435185 3.437037 3.354815 3.374074
3.410769
TS53 3.187200 3.310000 3.347037 3.268846 3.232593 3.213704
3.333704
[,14]
[,15]
[,16]
[,17]
[,18]
[,19]
TIME
14
15
16
17
18
19
TS1 3.506800 3.541579 3.601765 3.682000 3.640667 3.640000
TS26 3.372222 3.446000 3.570588 3.511250 3.450625 3.395714
TS53 3.254074 3.263500 3.265000 3.252000 3.302000 3.205714
> plot(milk.means[c(1, 26, 53)], ylab="Average Milk
Protein")
> ldamat.legend(milk.means, 10, 3.8)
3.6
3.4
3.2
Average Milk Protein
Barley
Lupins
Mixed
5
10
15
Time
A flexible model for fitting longitudinal
data
Yij  ij  U i  Wi (tij )   ij ,
Where ij  X ij  models the mean structure;
N (0, 2 ) is the random effect (random intercept),
Ui
Wi (t )
N (0,  2 ) is the realization from a stationary Gaussian process
and it models the serial correlation:
corr (Wi (t ),Wi ( s ))   (t  s )  exp( | t  s |2 ).
 ij
N (0, 2 ) is the measurement error.
The set of model parameters is  =( , 2 , 2 , 2 , ).
Model the dropout process
log it{ pr ( yk missing|y1 , y2 ,..., yk ; yk 1 is observed)}
q
= 0  1 yk   j yk 1 j ,
j 2
Data are
- completely random dropout (CRD) if
 j  0, j  1, 2,...., q.
- random dropout (RD) if  1
 0.
- informative dropout (ID) if  1
Note that the intercept
 0.
 0 is often modelled as depending on the occasion k , i.e., 0k .
Fitting the model
Given the form of the data, a sensible model for the mean
may be to
fit a straight line for the first three week’s data (for the
settling-in period),
and a polynomial of degree at most 2 for the remainder of
the experiment,
i.e., fit for each group g  1, 2,3 :
t 3
0 g  1t ,
 g (t )  
2


3



(
t

3)


(
t

3)
, t>3.
1
2
3
 0g
# Creating the three variable for such a polynomial trajectory:
> x1 <- c(1, 2, rep(3, 17))
> x1
[1] 1 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
> x2 <- c(0, 0, 0, 1:16)
> x2
[1] 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
> x3 <- x2^2
(1) Assume Completely Random Dropout
> milk.crd <- pcmid(milk ~ group + tex(x1) + tex(x2) +
tex(x3), vparms =
c(0.1, 0.03, 0.4), correxp = 1)
> milk.crd
Longitudinal Data Analysis Model
assuming completely random dropout
Call:
pcmid(formula = milk ~ group + tex(x1) + tex(x2) + tex(x3),
vparms = c(
0.1, 0.03, 0.4), correxp = 1)
Analysis Method: Maximum Likelihood (ML)
Correlation structure: exp(- phi * |u| ^ 1 )
Maximised log-likelihood:
[1] -2866.67
Mean Parameters:
(Intercept)
group1
group2
tex(x1)
tex(x2)
PARAMETER 4.04367915 -0.10606293 0.001496421 -0.22861430
0.007950934
STD.ERROR 0.04418288 0.02592537 0.014771240 0.01564402
0.008077304
tex(x3)
PARAMETER -0.0005958148
STD.ERROR 0.0005070062
Variance Parameters:
nu.sq
sigma.sq
tau.sq
phi
1.527421e-008 0.07214213 0.02376663 0.1518235
Maximisation converged after 185 iterations.
(2) Assume Random Dropout
>
>
>
>
dt <- rep(1:19, 79)
dt[dt < 15 | dt == 18] <- 0
dt <- as.factor(dt)
dt[1:100]
[1] 0 0 0 0 0 0 0 0
19 0 0 0
[23] 0 0 0 0 0 0 0 0
0 0 0 0
[45] 0 0 0 0 0 0 0 0
0 0 0 0
[67] 0 0 0 0 0 15 16 17
0 0 0 0
0
0
0
0
0
0
15 16 17 0
0
0
0
15 16 17 0
19 0
0
15 16 17 0
19 0
0
0
0
0
0
0
0
0
0
0
19 0
0
0
[89] 0
0
15 16 17 0
19 0
0
0
0
0
> milk.rd <- pcmid(milk ~ group + tex(x1) + tex(x2) +
tex(x3),
vparms = c(0, 0.025, 0.15), correxp = 1, drop.parms = c(0,
-6),
drop.cov.parms = c(0, 19, 18, 17, 18), dropmodel = ~ -1
+ dt)
> milk.rd
Longitudinal Data Analysis Model
assuming random dropout based on 1 previous observations
Call:
pcmid(formula = milk ~ group + tex(x1) + tex(x2) + tex(x3),
vparms = c(
0, 0.025, 0.15), drop.parms = c(0, -6), drop.cov.parms =
c(0,
19, 18, 17, 18), dropmodel = ~ -1 + dt, correxp = 1)
Analysis Method: Maximum Likelihood (ML)
Correlation structure: exp(- phi * |u| ^ 1 )
Maximised log-likelihood:
[1] -2933.502
Mean Parameters:
(Intercept)
group1
group2
tex(x1)
tex(x2)
PARAMETER 4.04376266 -0.1056935 0.00149445 -0.22844484
0.007678527
STD.ERROR 0.05821873 0.0401953 0.02290393 0.01823982
0.011424190
tex(x3)
PARAMETER -0.0005799806
STD.ERROR 0.0007106747
Variance Parameters:
nu.sq
sigma.sq
tau.sq
phi
0 0.07208938 0.02377462 0.1519165
Dropout parameters:
dt0
dt15
dt16
dt17
dt19 y.d
y.d-1
0 19.22091 18.51024 17.77181 18.49529
0 -6.254871
Maximisation converged after 1058 iterations.
(3) Assume Informative Dropout
> milk.id <- pcmid(milk ~ group + tex(x1) + tex(x2) +
tex(x3),
Vparms=c(0, 0.025,0.15), correxp = 1, drop.parms = c(6, 12),
drop.cov.parms = c(0, 19, 18, 17, 18), dropmodel = ~ -1
+ dt)
> milk.id
Longitudinal Data Analysis Model
assuming informative dropout based on 1 previous
observations
Call:
pcmid(formula = milk ~ group + tex(x1) + tex(x2) + tex(x3),
vparms = c(
0, 0.025, 0.15), drop.parms = c(6, -12), drop.cov.parms =
c(0,
19, 18, 17, 18), dropmodel = ~ -1 + dt, correxp = 1)
Analysis Method: Maximum Likelihood (ML)
Correlation structure: exp(- phi * |u| ^ 1 )
Maximised log-likelihood:
[1] -2928.264
Mean Parameters:
(Intercept)
group1
group2
tex(x1)
tex(x2)
PARAMETER
4.041186 -0.1069147 0.002143501 -0.2286773
0.007786218
STD.ERROR
NA
NA
NA
NA
NA
tex(x3)
PARAMETER -0.0004352736
STD.ERROR
NA
Variance Parameters:
nu.sq
sigma.sq
tau.sq
phi
0 0.06914706 0.0246771 0.1536183
Dropout parameters:
dt0
dt15
dt16
dt17
dt19
y.d
y.d-1
0 18.62526 17.53009 16.77428 18.0475 4.680672 -10.90307
Maximisation converged after 1797 iterations.