In clinical trials with repeated measurements, the time-averaged difference (TAD) may provide a more powerful evaluation of treatment efficacy than the rate of changes over time when the treatment effect has rapid onset and repeated measurements continue across an extended period after a maximum effect is achieved (Overall and Doyle,
In clinical trials with repeated measurements, comparing treatments based on the time-averaged difference (TAD), defined as the difference in the average of longitudinally measured responses between treatment groups, is often considered a meaningful metric for the treatment effect. Overall and Doyle (1994) suggested that TAD can provide a more powerful evaluation of treatment efficacy than the rate of changes over time when the treatment effect has rapid onset and repeated measurements are obtained across an extended period after the maximum effect has been achieved. Many sample size formulas have been developed for the inference of TAD between two treatment groups (Overall and Doyle, 1994; Diggle
In this paper, we investigate sample size calculation for the comparison of time-averaged responses among
Suppose in a clinical trial a total of
We have
The GEE estimator of
which is derived based on an independent working correlation structure. Liang and Zeger (1986) showed that as
and
Here
To compare the time-averaged responses among
where
As
Missing data are frequently encountered in clinical trials with repeated measurements. We now show that a closed-form extension of (
and
We assume that the missing probabilities only depend on time. In addition, we define
and
The general variance
Plugging
To assess the performance of the proposed sample size method, we conduct simulation under different parameter settings. Suppose subjects are randomly assigned to one of four treatment groups (
In simulation studies, we set
These settings correspond to four scenarios. We assume no missing data at
The simulation study also explores two correlation structures: compound symmetry (CS,
Calculate the required sample size (
Generate
Create incomplete data sets. Generate missing indicators based on the specified missing pattern and the marginal observation probabilities.
Calculate
Repeat Steps 2–4 for
Tables 1 and 2 summarize the required sample sizes and their corresponding empirical Type I errors and empirical powers under different combinations of simulation parameters. Table 1 is obtained under alternative hypotheses
PASS sample size software manual (Pass14, 2015) illustrated the sample size estimation to test proportions in a repeated measurement design between two treatment groups. Here, we show sample size estimation to test proportions among three treatment groups. An investigator wants to design a study that compares the efficacy of a prophylactic treatment for the common cold with two active drugs and a placebo. The null hypothesis is that there is no difference in the proportion of patients who get sick among three treatment groups. Patients will be randomly assigned to one of three treatment groups with an equal probability, and followed monthly from September to April (beginning in October, hence
In this study, we derived a sample size formula to compare the time-averaged responses of repeated binary outcomes among
Required sample size (empirical power, empirical Type I error) under
CS | AR(1) | ||||
---|---|---|---|---|---|
284 (0.8074, 0.0508) | 397 (0.7996, 0.0546) | 188 (0.8083, 0.0550) | 266 (0.8032, 0.0544) | ||
IM | 300 (0.8060, 0.0546) | 413 (0.8042, 0.0507) | 205 (0.8047, 0.0547) | 283 (0.8035, 0.0519) | |
295 (0.8043, 0.0522) | 408 (0.8065, 0.0511) | 201 (0.8098, 0.0573) | 280 (0.7983, 0.0524) | ||
305 (0.8096, 0.0526) | 418 (0.8032, 0.0481) | 209 (0.8056, 0.0555) | 286 (0.8055, 0.0532) | ||
MM | 312 (0.8079, 0.0555) | 433 (0.8068, 0.0499) | 212 (0.8081, 0.0524) | 297 (0.8037, 0.0526) | |
301 (0.8082, 0.0521) | 417 (0.8041, 0.0517) | 205 (0.8038, 0.0518) | 287 (0.8060, 0.0520) | ||
323 (0.8049, 0.0488) | 449 (0.8076, 0.0542) | 219 (0.8095, 0.0562) | 307 (0.8021, 0.0486) | ||
MIX | 306 (0.8099, 0.0547) | 423 (0.8035, 0.0551) | 208 (0.8053, 0.0515) | 290 (0.8043, 0.0503) | |
298 (0.8070, 0.0481) | 413 (0.7985, 0.0505) | 203 (0.8037, 0.0561) | 283 (0.8060, 0.0535) | ||
314 (0.8079, 0.0563) | 433 (0.8014, 0.0534) | 214 (0.8076, 0.0563) | 297 (0.8094, 0.0506) |
CS = compound symmetry; AR = auto-regressive; IM = independent missing; MM = monotone missing; MIX = mixed missing.
Required sample size (empirical power, empirical Type I error) under
CS | AR(1) | ||||
---|---|---|---|---|---|
285 (0.8005, 0.0516) | 399 (0.8054, 0.0501) | 189 (0.8045, 0.0536) | 267 (0.8050, 0.0507) | ||
IM | 301 (0.8000, 0.0529) | 414 (0.8067, 0.0504) | 205 (0.8026, 0.0567) | 284 (0.8087, 0.0504) | |
296 (0.8032, 0.0500) | 410 (0.8088, 0.0522) | 201 (0.8024, 0.0536) | 281 (0.8090, 0.0549) | ||
306 (0.8020, 0.0537) | 419 (0.8010, 0.0497) | 209 (0.8078, 0.0561) | 287 (0.8046, 0.0518) | ||
MM | 312 (0.8027, 0.0523) | 434 (0.8055, 0.0530) | 212 (0.8059, 0.0575) | 297 (0.8002, 0.0504) | |
301 (0.8094, 0.0489) | 419 (0.8091, 0.0551) | 205 (0.8087, 0.0526) | 288 (0.8020, 0.0518) | ||
324 (0.8077, 0.0527) | 450 (0.8008, 0.0500) | 220 (0.8075, 0.0531) | 308 (0.8033, 0.0545) | ||
MIX | 307 (0.8035, 0.0512) | 424 (0.8001, 0.0505) | 209 (0.8039, 0.0529) | 291 (0.8084, 0.0505) | |
299 (0.8079, 0.0495) | 414 (0.8089, 0.0503) | 203 (0.8058, 0.0551) | 284 (0.8010, 0.0559) | ||
315 (0.8083, 0.0522) | 435 (0.8074, 0.0536) | 215 (0.8085, 0.0525) | 297 (0.8077, 0.0506) |
CS = compound symmetry; AR = auto-regressive; IM = independent missing; MM = monotone missing; MIX = mixed missing.