## GEE with Missingness

As discussed previously, GEE is a useful method for analyzing discrete and continuous longitudinal data when the mean response is of primary interest. Unlike likelihood-based methodology, however, GEE produces biased parameter estimates when data are missing at random. Rotnitzky and Wypij (1994) developed expression which can be used to quantify this bias in various settings. Robins et al. (1995) proposed an extension of GEE that allows for unbiased parameter estimation for MAR data. They...

## Stratified Data

In clinical studies when survival changes with important prognostic factors, stratification on different level of those factors is often done either at the design stage to ensure treatment balance in each stratum, or at the analysis stage. The ordinary logrank test, ignoring strata effect, is conservative and is biased when there is treatment imbalance in each prognostic subgroup. The stratified logrank test, on the other hand, is unbiased and retains high efficiency as long as the number of...

## Bayesian Stopping Rule For Safety

The trial described in Section 2 was monitored by a data and safety monitoring board, which expressed a concern that engraftment by day 42 was not sufficient to assure the longer term safety of the patients who underwent this procedure. They requested a stopping rule for a longer term endpoint to monitor safety. We chose 100 day transplant-related mortality (TRM) as the safety endpoint. TRM encompasses multiple causes of death and so serves as a suitable safety endpoint. Thus patients who died...

## One Sided Bootstrap Tests

Bloch, Lai, and Tubert-Bitter (2001) consider alternatives of the form HA Aim > A2m for some m i.e., treatment 1 is superior to treatment 2 on at least one of the endpoints and all of the other endpoints for treatment 1 are noninferior to those of treatment 2. They consider the intersection of the rejection region for the likelihood ratio test of H0 (Hotelling's T2) with the rejection region for the non-inferiority region (a set of univariate tests). They show that this results in a level a...

## Shared Random Effects Models and Informative Missingness

In longitudinal clinical trials, repeated measures are often highly variable over time. Rather than modeling the missing data mechanism directly as a function of Yt, as in the selection models discussed in the previous section, Wu and Carroll (1988) exploit the fact that we have repeated responses on each individual to model the missing data mechanism in terms of features of an individual's underlying response process. They propose methodology in which informative dropout is accounted for by...

## Sample Size Calculation

Sample size calculation for cluster randomization trials is discussed in Donner et al. (1981). The main point of note is the need to take into account the variance inflation factor (IF) arising from the within-cluster dependence. In sample size formulas for continuous or binary data based on the usual asymptotic normal approximation, the sample size must be multiplied by the IF. Consider, for example, a two-arm trial with equal allocation of clusters to arms and with (approximately) the same...

## Null Partial and Full Compliance

In Table 6 the cholestyramine data is presented with dichotomized response and the compliance in three categories defined by the cutpoints 20 and 60 for percentage experimental and placebo dose, respectively. As before, we are not interested in the placebo compliance categories, and the structure for the observed data used for modeling is given in Table 7. The three treatment compliance categories in Table 7 are referred to as null, partial, and full compliance. The null compliance in the...

## Example The Catch Trial

As as example of a cluster randomized trial, we describe the Child and Adolescent Trial for Cardiovascular Health (CATCH). Details concerning the design of this trial are reported in Zucker et al. (1995), and the main trial results are reported in Luepker et al. (1996). The CATCH study investigated a school-based educational and environmental intervention aimed at promoting heart-healthy habits in elementary school children. Because the intervention was implemented the school level, a cluster...

## Example 1 Gastrointestinal Tumor Study

These data were previously analyzed by Stablein and Koutrouvelis (1985). The study reported on the results of a trial comparing chemotherapy versus combined chemotherapy and radiation therapy in the treatment of locally unresectable gastric cancer. There are 45 patients in each treatment group. Figure 1 displays the Kaplan-Meier survival curves. The two survival curves cross, suggesting that the DPT using the DiM functional may not be able to detect the difference, because the means, which are...

## References

Optimal permutation tests for the analysis of group randomized trials. Journal of the American Statistical Association 96 14241432. Bozivich, H., Bancroft, T. A., Hartley, H. O. (1956). Power of analysis of variance test procedures for certain incompletely specified models. Annals of Mathematical Statistics 27 1017-1043. Cochran, W. G. (1977). Sampling Techniques. 3rd ed. New York John Wiley. Cohen, J. (1960). A coefficient of agreement for nominal data. Educational...

## Resampling Methods Westfall and Young 1993 Troendle 1995 1996

Consider the univariate comparisons on the K individual endpoints and order the test statistics Let t(1) V t(2) V V t(K) be the ordered values of the T and H0(i), H0(2), , H0(K) be the corresponding ordered null hypotheses. Determine constants q1, q2 , qK and sequentially accept hypotheses H0(1), H0(2), until the first time t(i) > q,, at which point reject H0(1), , H0(K). The constants q, are determined so that the probability of rejecting any true hypothesis under any parameter configuration...

## Analysis of Biological Markers as Failure Time Data

Rather than comparing levels of biological markers over time or at arbitrarily chosen time points, biological markers can also be used to define time-to-event endpoints, for example, time to achieve either a prespecified fixed level, or percentage change from baseline. For example, a successful outcome can be defined as time to reach a threshold such as< 50 copies of HIV RNA per milliliter (or undetectable viral load on the assays), or a treatment failure can be defined as time to a rebound...

## Contributors

Mathematical Statistician, Biometrics Research Branch, Division of Cancer Treatment and Diagnosis, National Cancer Institute, National Institutes of Health, Bethesda, Maryland, U.S.A. James S. Babb, Ph.D. Department of Biostatistics, Fox Chase Cancer Center, Philadelphia, Pennsylvania, U.S.A. Abdel G. Babiker, Ph.D. Head, Division of HIV and Infections, and Professor of Medical Statistics and Epidemiology, Medical Research Council Clinical Trials Unit, London, England...

## Statistical Methodology Analysis

The analysis of this trial has been performed on an intention to treat basis including as many eligible patients as possible for each endpoint. The statistical analyses were conducted using the SPSS and SAS statistical packages. Survival curves were calculated using the Kaplan-Meier method. The Mantel-Cox version of the logrank statistic was calculated to provide the likelihood portion of the Bayesian analysis see below . Estimated hazard ratios are used to compare treatments Parmar and Machin,...

## Implementation An Example

That useful results can be obtained is illustrated by the blood pressure example previously analyzed by Goetghebeur and Lapp 1997 . A new treatment is compared with placebo in a double blind randomized blood pressure reduction trial, assigning patients to take one tablet daily. Drug intake is electronically monitored over the experimental study period which follows a run-in on placebo. The dosing experience is subsequently summarized in a scalar D 1 representing the percentage of prescribed...

## Example

The Tamoxifen Prevention Trial evaluated the effectiveness of Tamoxifen for preventing breast cancer in high-risk women Fisher et al., 1998 . Tamoxifen was found to be effective, but also led to side effects, including increasing the risk of endometrial cancer. Consequently, there is interest in the identification of other antiestrogen drugs that are as effective as Tamoxifen for preventing breast cancer but with fewer side effects. The Tamoxifen trial randomized 13,388 women and found 175...

## Null Hypothesis Example

g - x ln w 1 - w g x,- - 140 ln 9 1 - 9 g The marginal posterior distribution Pk represents a probabilistic summary of all the information about the MTD that is available after the observation of k patients. Figure 2 shows the marginal posterior distribution of the MTD given the data shown in Table 1. As each patient is accrued to the trial, a decision must be made regarding the dose level that the patient is to receive. In a strict Bayesian setting, the decisions are made by minimizing the...