Notes:
1. The data were randomly generated similar to the
schedules
of the real data set analyzed in Ding
and Wu (2000, Biostatistics).
2. A total of 32 patients are contained in this study,
indicated by
the ID variable. The Day variable reflects
the time for viral
load measurement time after initiation
of antiviral treatment.
The RNA variable contains the viral load
measurement in unit of
copies per ml blood. The patients are
separated into two
treatment groups (Group 1 and Group 2)
with 16 patients each,
and is identified in the Group variable.
3. Detection limit of the viral load (HIV RNA copies)
assay
is 400 copies per ml blood. If it is
below detectable, the data
is reflected as 400. We imputed 200 for
these data points in the
following data analysis. If more than
one measurement below
detectable level for an individual, we
just impute the first
measurement and exclude the rest of them
in our analysis (otherwise
it may result in misleading dynamic patterns).
4. Due to convergence problems, multiple starting values
(or global search algorithms) should
be used to fit the
models. The starting point in the following
Splus code was
chosen after comparing the results using
many different
starting values.
######################### Beginning of Splus Code ########################
# Input Data: assume that the data at the end is stored in an ASCII
file
# named "data".
workd<-read.table(file="data",header=T,row.names=NULL)
# Impute below detectable viral measurement by half of the detection
limit.
workd$RNA[workd$RNA==400]<-200
# For patient number 27, the viral measurements are below detectable
level on
# both 14th and 28th day. So the latter point is excluded from the
analysis.
workd<-workd[(workd$ID!=27 | workd$Day!=28),]
# Define a function representing the 2 phrase decay model (1.2)
f.twoe<-function(p1, d1, p2, d2, X)
{
P1 <- exp(p1)
P2 <- exp(p2)
P1 * exp( - d1 * X) + P2
* exp( - d2 * X)
}
# Fit the PMEM
# We used a fixed starting values of p1=12, d1=0.65, p2=8 and d2=0.03
# on this data set.
# For other data sets, several starting values should be tried and
# then the best fit chosen.
tmp.fit<-nlme(Y ~ log10(f.twoe(p1, d1, p2, d2, X)),
fixed = list(p1~., d1~., p2~., d2~.),
random = list(p1~., d1~., p2~., d2~.),
cluster = ~ Z,
data = data.frame(Y=log10(workd$RNA),
X=workd$Day, Z=workd$ID),
start = list(fixed= c(12, 0.65, 8, 0.03)))
# Get the EBEs for the decay rates.
tmp.rate<-t(t(tmp.fit$coe$random)+tmp.fit$coe$fixed)[,c("d1","d2")]
# Separate the EBEs into the groups to prepare for hypothesis testing
patid<-sort(unique(workd$ID))
np<-length(patid)
group<-(1:np)
for (i in 1:np) {group[i]<-workd$Group[workd$ID==patid[i]][1]}
np1<-sum(group==1)
np2<-sum(group==2)
d1.grp1<-tmp.rate[group==1,"d1"]
d1.grp2<-tmp.rate[group==2,"d1"]
d2.grp1<-tmp.rate[group==1,"d2"]
d2.grp2<-tmp.rate[group==2,"d2"]
# Conduct EBE-based tests:
# t-test and rank-sum test for AH1
# t-test and rank-sum test for AH2
# T^2-test (MANOVA) and O'Brien rank-sum test for AH3
# AH1. Testing one-sided alternative here: d1 is bigger in Group 2.
t.test(d1.grp1,d1.grp2,alternative="less")$p.value
wilcox.test(d1.grp1,d1.grp2,alternative="less")$p.value
# Resulting p-values: t.test 0.049997, wilcox.test 0.09175338
# AH2. Testing one-sided alternative here: d2 is bigger in Group 2.
t.test(d2.grp1,d2.grp2,alternative="less")$p.value
wilcox.test(d2.grp1,d2.grp2,alternative="less")$p.value
# Resulting p-values: t.test 0.4378882, wilcox.test 0.4482666
# AH3. Testing two-sided alternative: d1 and/or d2 are different.
# MANOVA
summary(manova(rbind(cbind(d1.grp1,d2.grp1),cbind(d1.grp2,d2.grp2))
~ group))$Stats[,1,5]
# Resulting p-value 0.2614775
# O'Brien test
tmp<-rank(c(d1.grp1,d1.grp2))
rd1.grp1<-tmp[(1:np1)]
rd1.grp2<-tmp[-(1:np1)]
tmp<-rank(c(d2.grp1,d2.grp2))
rd2.grp1<-tmp[(1:np1)]
rd2.grp2<-tmp[-(1:np1)]
rd.grp1<-rd1.grp1+rd2.grp1
rd.grp2<-rd1.grp2+rd2.grp2
wilcox.test(rd.grp1,rd.grp2)$p.value
# Resulting p-value 0.2344624
###################### End of Splus Code #####################################
ID Day RNA Group
1 0.000 67324 1
1 0.125 171810 1
1 0.333 39576 1
1 1.000 56412 1
1 3.000 5688 1
1 7.000 1956 1
1 14.000 1379 1
1 28.000 400 1
2 0.000 569633 1
2 0.125 248890 1
2 0.333 425369 1
2 1.000 119297 1
2 3.000 94723 1
2 7.000 9045 1
2 14.000 867 1
2 28.000 805 1
3 0.000 398899 1
3 0.125 219505 1
3 0.333 178366 1
3 1.000 73744 1
3 3.000 35458 1
3 7.000 51040 1
3 14.000 1701 1
3 28.000 1078 1
4 0.000 610484 1
4 0.125 323648 1
4 0.333 271977 1
4 1.000 598412 1
4 3.000 72110 1
4 7.000 7902 1
4 14.000 4696 1
4 28.000 761 1
5 0.000 253495 1
5 0.125 432427 1
5 0.333 367006 1
5 1.000 50973 1
5 3.000 41538 1
5 7.000 9444 1
5 14.000 991 1
5 28.000 711 1
6 0.000 1541533 1
6 0.125 860756 1
6 0.333 593412 1
6 1.000 339032 1
6 3.000 181938 1
6 7.000 76764 1
6 14.000 19341 1
6 28.000 5218 1
7 0.000 23678 1
7 0.125 70048 1
7 0.333 138385 1
7 1.000 32492 1
7 3.000 11417 1
7 7.000 1931 1
7 14.000 421 1
7 28.000 400 1
8 0.000 744825 1
8 0.125 521558 1
8 0.333 325766 1
8 1.000 356948 1
8 3.000 52540 1
8 7.000 11829 1
8 14.000 3254 1
8 28.000 12596 1
9 0.000 1252328 1
9 0.125 261470 1
9 0.333 320139 1
9 1.000 142677 1
9 3.000 14074 1
9 7.000 12346 1
9 14.000 8613 1
9 28.000 3363 1
10 0.000 329416 1
10 0.125 168726 1
10 0.333 291034 1
10 1.000 111688 1
10 3.000 70522 1
10 7.000 14427 1
10 14.000 1548 1
10 28.000 1282 1
11 0.000 157635 1
11 0.125 85405 1
11 0.333 97043 1
11 1.000 66889 1
11 3.000 6094 1
11 7.000 2765 1
11 14.000 995 1
11 28.000 400 1
12 0.000 463754 1
12 0.125 1111460 1
12 0.333 461062 1
12 1.000 340430 1
12 3.000 166770 1
12 7.000 57106 1
12 14.000 9993 1
12 28.000 1856 1
13 0.000 225887 1
13 0.125 177991 1
13 0.333 109232 1
13 1.000 249347 1
13 3.000 42956 1
13 7.000 2712 1
13 14.000 1344 1
13 28.000 400 1
14 0.000 5354718 1
14 0.125 10156869 1
14 0.333 8465124 1
14 1.000 6276483 1
14 3.000 1759661 1
14 7.000 372656 1
14 14.000 67606 1
14 28.000 33587 1
15 0.000 162043 1
15 0.125 47688 1
15 0.333 114639 1
15 1.000 67453 1
15 3.000 16189 1
15 7.000 4507 1
15 14.000 1269 1
15 28.000 441 1
16 0.000 466821 1
16 0.125 323978 1
16 0.333 288308 1
16 1.000 134974 1
16 3.000 33357 1
16 7.000 6074 1
16 14.000 2573 1
16 28.000 3130 1
17 0.000 766694 2
17 0.125 595493 2
17 0.333 287100 2
17 1.000 435330 2
17 3.000 28789 2
17 7.000 2615 2
17 14.000 2145 2
17 28.000 1867 2
18 0.000 154188 2
18 0.125 109839 2
18 0.333 200805 2
18 1.000 151538 2
18 3.000 37298 2
18 7.000 2224 2
18 14.000 490 2
18 28.000 616 2
19 0.000 70841 2
19 0.125 252774 2
19 0.333 192880 2
19 1.000 151659 2
19 3.000 43354 2
19 7.000 6732 2
19 14.000 1738 2
19 28.000 1377 2
20 0.000 158011 2
20 0.125 183884 2
20 0.333 114733 2
20 1.000 139015 2
20 3.000 29253 2
20 7.000 4070 2
20 14.000 1868 2
20 28.000 672 2
21 0.000 934817 2
21 0.125 1352673 2
21 0.333 1192666 2
21 1.000 142216 2
21 3.000 55779 2
21 7.000 32402 2
21 14.000 3525 2
21 28.000 1039 2
22 0.000 227665 2
22 0.125 123610 2
22 0.333 98786 2
22 1.000 60500 2
22 3.000 13351 2
22 7.000 1044 2
22 14.000 691 2
22 28.000 400 2
23 0.000 2961627 2
23 0.125 554609 2
23 0.333 902410 2
23 1.000 676630 2
23 3.000 164742 2
23 7.000 45623 2
23 14.000 13675 2
23 28.000 8694 2
24 0.000 421949 2
24 0.125 136907 2
24 0.333 732840 2
24 1.000 133798 2
24 3.000 16982 2
24 7.000 9185 2
24 14.000 3639 2
24 28.000 1401 2
25 0.000 926362 2
25 0.125 1488839 2
25 0.333 364706 2
25 1.000 663691 2
25 3.000 100743 2
25 7.000 22729 2
25 14.000 7972 2
25 28.000 1218 2
26 0.000 302650 2
26 0.125 359009 2
26 0.333 599914 2
26 1.000 323553 2
26 3.000 116214 2
26 7.000 2533 2
26 14.000 6498 2
26 28.000 2381 2
27 0.000 59804 2
27 0.125 22617 2
27 0.333 22926 2
27 1.000 22569 2
27 3.000 3523 2
27 7.000 1507 2
27 14.000 400 2
27 28.000 400 2
28 0.000 284082 2
28 0.125 101536 2
28 0.333 134319 2
28 1.000 57690 2
28 3.000 20888 2
28 7.000 1340 2
28 14.000 634 2
28 28.000 524 2
29 0.000 50026 2
29 0.125 146659 2
29 0.333 64786 2
29 1.000 43144 2
29 3.000 12406 2
29 7.000 2341 2
29 14.000 640 2
29 28.000 486 2
30 0.000 83640 2
30 0.125 111536 2
30 0.333 84267 2
30 1.000 91913 2
30 3.000 24772 2
30 7.000 6968 2
30 14.000 1028 2
30 28.000 911 2
31 0.000 345629 2
31 0.125 233055 2
31 0.333 287857 2
31 1.000 52375 2
31 3.000 38088 2
31 7.000 4804 2
31 14.000 3548 2
31 28.000 2749 2
32 0.000 867389 2
32 0.125 523124 2
32 0.333 244840 2
32 1.000 174122 2
32 3.000 44033 2
32 7.000 7083 2
32 14.000 1445 2
32 28.000 421 2