Marginal Models
The analysis presented in Section 8.3.1.1 allows evaluating both the individual- and trial-level surrogacy. However, fitting the joint model (8.1)-(8.2) is not trivial and requires specialized software.
A simpler analysis would result from using marginal models for the longitudinal surrogate and failure-time endpoint. In particular, using the psadat dataset described in Section 8.3.1.1, the LMM (8.9)-(8.10) can be fitted by using a call to PROC MIXED similar to the following one:
PROC MIXED data=psadat method=ml;
TABLE 8.1
Prostate Cancer. Estimates of treatment effects.
Group |
N |
«0,4 |
&l,i |
«2,! |
A |
1 |
51 |
1.07150 |
0.29793 |
-1.82227 |
-0.11320 |
2 |
64 |
-0.06582 |
4.27175 |
-4.02507 |
-0.39519 |
3 |
19 |
-0.11073 |
1.04748 |
1.45070 |
1.43370 |
4 |
12 |
0.97395 |
5.84993 |
-3.62203 |
-0.13982 |
6 |
16 |
-0.02703 |
1.49523 |
-3.39183 |
-1.18884 |
7 |
17 |
-0.04868 |
-2.31853 |
0.48394 |
0.39394 |
8 |
68 |
0.05255 |
3.41043 |
-4.13950 |
-1.10833 |
9 |
57 |
0.01571 |
-0.85608 |
0.87997 |
-0.12267 |
12 |
15 |
-0.46289 |
4.13031 |
-1.80918 |
3.05214 |
13 |
35 |
-1.33082 |
-3.02613 |
1.33767 |
-1.15655 |
14 |
24 |
0.49202 |
5.14484 |
-2.54004 |
1.35772 |
15 |
37 |
0.30646 |
-2.10418 |
1.40429 |
0.27368 |
17 |
16 |
-0.26281 |
-6.36071 |
3.66200 |
1.06424 |
18 |
30 |
-0.08881 |
2.08421 |
-2.24416 |
-0.45748 |
20 |
19 |
-1.23598 |
4.66113 |
-3.34404 |
-1.25542 |
21 |
37 |
-1.13152 |
-2.69907 |
3.09017 |
0.46413 |
24 |
50 |
0.71554 |
0.72275 |
0.90372 |
1.00531 |
CLASS patid countryn timyrcls;
MODEL logpsa = countryn timeyr*countryn timysqrt*countryn treat*countryn treat*timeyr*countryn treat*timysqrt*countryn / noint;
RANDOM int timeyr timysqrt / subject = patid type = un; REPEATED timyrcls / subject = patid;
RUN;
Note that we use the ML estimation (method = ml) for compatibility with the joint-model analysis presented in Section 8.3.1.1, which was based on using the EM algorithm.
Treatment effects on OS can be estimated by the following SAS syntax:
PROC SORT data=psadat;
BY patid timeyr;
RUN;
DATA survdat;
SET psadat;
BY patid timeyr;
FIGURE 8.3
Prostate Cancer. Scatter plot matrix of the estimated treatment effects for the longitudinal log-PSA measurements (aog, ag, a^g) and overall survival time (вг). The circles’ areas are proportional to the trial size.
IF first.patid=1;
RUN;
PROC PHREG data=survdat;
CLASS countryn;
MODEL survyr*survcens(0) = treat(countryn);
STRATA countryn;
RUN;
Note that, in the syntax, we specify trial-by-country-specific hazards (STRATA countryn) rather than assuming that the hazards are proportional, as in (8.2).
The estimated treatment effects for the LMM and the PH model are given in Table 8.2.
The value of RLiaip), obtained from regressing вг on (Soy, Sgj, a2,_{i})^{/}, is equal to 0.33 (delta-method-based 95% CI: [-0.05, 0.70]). When the group- specific sample sizes are used as weights, the resulting value of R2ri_{a}i(f) is equal to 0.40 (delta-method-based 95% CI: [0.03,0.77]). The estimates are consistent with, though somewhat smaller than, the corresponding estimates obtained in the joint-model analysis presented in Section 8.3.1.1.
TABLE 8.2
Prostate Cancer. Estimates of the marginal treatment effects.
Group |
N |
«0,* |
&l,i |
«2,* |
A |
1 |
51 |
1.0667 |
0.2083 |
-1.7529 |
-0.37639 |
2 |
64 |
-0.06095 |
4.5312 |
-4.1454 |
-0.29845 |
3 |
19 |
0.07563 |
0.2193 |
1.5656 |
0.54536 |
4 |
12 |
0.8381 |
4.7874 |
-2.5423 |
0.00404 |
6 |
16 |
-0.2011 |
-0.2152 |
-2.0851 |
-0.48535 |
7 |
17 |
-0.06936 |
-2.2591 |
0.5050 |
0.26265 |
8 |
68 |
0.09522 |
3.5037 |
-4.2436 |
-0.52851 |
9 |
57 |
0.02962 |
-1.0893 |
1.0013 |
-0.08884 |
12 |
15 |
-0.4634 |
4.6271 |
-2.1271 |
2.23486 |
13 |
35 |
-1.3106 |
-2.2500 |
0.9558 |
-0.85353 |
14 |
24 |
0.5362 |
4.9382 |
-2.6709 |
0.85144 |
15 |
37 |
0.3170 |
-2.1101 |
1.3651 |
-0.12038 |
17 |
16 |
-0.3220 |
-6.0554 |
3.6978 |
1.10103 |
18 |
30 |
-0.09799 |
2.2145 |
-2.3323 |
-0.02217 |
20 |
19 |
-1.2290 |
5.8949 |
-3.9451 |
-1.42056 |
21 |
37 |
-1.1031 |
-4.1257 |
3.7887 |
0.35782 |
24 |
50 |
0.7034 |
0.6324 |
0.9446 |
0.49801 |