This example addresses a multicenter, double-blind, parallel-group, randomized clinical trial to compare a test treatment to placebo for patients with seborrheic dermatitis, and listings of the data appear in Ramaswamy et al. (Ramaswamy et al., 1997). A total of 167 patients were randomized to one of two treatment groups at 8 study centers, where 83 received test drug, and 84 received placebo. All patients in this protocol were required to have facial disease, and other anatomical areas such as scalp or chest were evaluated if treated. Patients presenting at baseline with clinically diagnosed stable or exacerbating inflammatory seborrheic dermatitis (as mild, moderate, severe) were enrolled in the study. The primary objective of this study was to demonstrate whether there is a better response for the test drug than the placebo, in a way that accounted for the number of treated anatomical regions among face, scalp or chest, and associations with the baseline assessments. At the final visit, the response of each patient was classified as ordered outcomes for global improvement of the skin disease condition ranging from 0 (best) to 5 (worst). Here, the response variable is simplified to the dichotomy of not favorable or favorable as (0 vs. 1) for ( (2,3,4,5) vs. (0,1) ) in Table 4.6. We also consider the sum of two specifications for favorable outcome as (0 vs. 1) for (>2 vs. 61 ) and ( >3 vs. 62 ) in Table 4.7, and it equals (0,1,2) for each anatomical region
Table 4.6: Number of anatomical regions with not favorable or favorable outcome (for >2 vs. 61) dichotomy by treatment and number of anatomical regions for patients with seborrheic dermatitis.
One Site
Affected Favorable Outcome
Total No Yes Placebo 16 22 38 Treatment 16 23 39 Total 32 45 77 Two Sites
Affected Number of Regions with Favorable Outcome
Total
None One Two
Placebo 17 8 12 37
Treatment 12 4 22 38
Total 29 12 34 75
Three Sites
Affected Number of Regions with Favorable Outcome
Total
None One Two Three
Placebo 3 1 3 2 9
Treatment 2 0 1 3 6
Table 4.7: Sum of anatomical regions for sum of dichotomous specifications for not favorable or favorable outcome as (>2 vs. 61) and (>3 vs. 62) by treatment and number of anatomical regions for patients with seborrheic dermatitis.
One Site
Affected Sum of Specifications for Favorable Outcome Total 0 1 2 Placebo 11 5 22 38 Treatment 10 6 23 39 Total 21 11 45 77 Two Sites
Affected Sum of Regions and Specifications for Favorable Outcome Total 0 1 2 3 4 Placebo 13 2 4 6 12 37 Treatment 6 5 3 2 22 38 Total 19 7 7 8 34 75 Three Sites
Affected Sum of Regions and Specifications for Favorable Outcome Total 0 1 2 3 4 5 6 Placebo 2 0 1 1 1 2 2 9 Treatment 2 0 0 0 0 1 3 6 Total 4 0 1 1 1 3 5 15
The proportionsp∗ij for favorable response are shown in Table 4.8 for the data in Table 4.6, and
they are 0.62 for the test treatment and 0.48 for the placebo group after adjusting for the strata for number of affected anatomical sites. The corresponding difference between treatments in these
proportions for favorable outcome is 0.14, with 0.08 as its standard error. Table 4.8 also provides the ¯y∗∗i = p∗i1+2p∗i2 and their standard errors for the data in Table 4.7, which are 0.68 and 0.56 with
standard errors of about 0.05; and the corresponding difference between treatments is 0.12 with 0.07 as its standard error.
Table 4.8: Stratified estimators for extent of good outcome, standard error (SE) and 0.95 confidence interval (CI), regardless of whether the null hypothesis applies (re proportions with good outcome asp∗ij and ¯y∗∗i=P2j=1p∗ij in Appendix 4.5.4).
Outcome Treatment Placebo Difference
Based On Estimation 0.617 0.482 0.135 Table 4.6 SE 0.056 0.052 0.076 CI (0.507, 0.727) (0.380, 0.584) (-0.014, 0.284) P-Value 0.076 Estimation 0.677 0.561 0.116 Table 4.7 SE 0.051 0.050 0.071 CI (0.577, 0.776) (0.464, 0.659) (-0.023, 0.254) P-Value 0.102
Table 4.9: Stratified estimators for extent of good outcome, standard error (SE) and 0.95 confidence interval (CI), regardless of whether the null hypothesis applies (re CMH weighted averages as ˜p∗ij
and ˜y∗∗i=P2j=1p∗˜ij in Appendix 4.5.4).
Outcome Treatment Placebo Unadjusted Adjusted
Based On Difference Difference
Estimation 0.616 0.482 0.134 0.111 Table 4.6 SE 0.057 0.052 0.077 0.073 CI (0.504, 0.728) (0.380, 0.584) (-0.017, 0.285) (-0.031, 0.257) P-Value 0.082 0.131 Estimation 0.676 0.561 0.115 0.105 Table 4.7 SE 0.053 0.049 0.072 0.067 CI (0.572, 0.779) (0.464, 0.657) (-0.026, 0.256) (-0.025, 0.237) P-Value 0.110 0.115
In Table 4.9, counterparts to the results in Table 4.8 are provided with respect to the ˜p∗ij. Also,
shown there are estimated treatment differences, standard errors, and 0.95 confidence intervals from the application of randomization-based analysis of covariance to the ˜p∗ij through the %N P arCov4
macro. For this method, treatment is coded 1 for active treatment, 0 for placebo, while stratum is coded as 1,2 or 3 corresponding to the number of treated anatomical sites. An indicator for "moderate" or "severe" baseline severity and an indicator of "severe" baseline severity measurements are the covariates. The outcome is numeric, and responses are coded with scores for (outcome / number of affected sites). The hypothesis is also specified as either NULL or ALT in %N P arCov4 macro coding, for producing p-values for hypothesis testing under the null or for computing confidence intervals under the alternative. Also, the estimate and standard error pertaining to Table 4.7 are based on additional division by r = 2 to account for the sum of two dichotomous specifications being it basis.
Table 4.10: Estimated treatment effects and 0.95 confidence interval (CI) by %NParCov4 method.
Outcome Difference Standard
Based On Method Estimate Error P-Value Confidence Interval
Null:Unadjusted 0.218 0.124 0.079 Alternative:Unadjusted 0.218 0.125 0.081 (-0.027, 0.463) Table 4.6 Null:Adjusted 0.232 0.123 0.059 Alternative:Adjusted 0.236 0.124 0.057 (-0.007, 0.478) Null:Unadjusted 0.374 0.231 0.106 Alternative:Unadjusted 0.374 0.234 0.111 (-0.086, 0.833) Table 4.7 Null:Adjusted 0.413 0.228 0.070 Alternative:Adjusted 0.415 0.232 0.073 (-0.039, 0.869)
The results from the randomization based method with %N P arCov4 for the f∗2w andfw are
shown in Table 4.10, both without and with adjustment for the covariates. In this regard, the standard errors for the results with adjustment are somewhat smaller than their counterparts without adjustement. Also, the p-values based on thef∗2w in Table 4.10 are somewhat more suggestive of a
difference between treatment and placebo than their counterparts based on the ˜p∗i2 in Table 4.9. Additionally, for fw
2 for the average of the two specifications that pertain to the results in Table 4.10 for the data in Table 4.7, the unadjusted difference estimate 0.187 and the adjusted estimate difference 0.207 are somewhat smaller than their counterparts that pertain to Table 4.6, although their standard errors near 0.116 are somewhat smaller.
4.4
Discussions
The studies with multiple anatomical regions are cluster randomization trials where subjects are randomized into two or more treatment groups, and assessments are made for multiple observational
units from each patient. For example, in the dermatology clinical trial for the skin disorder, each patient contributes data for one or more anatomical sites among the face, scalp, or chest. In this case, a patient corresponds to a cluster, and the responses of observational sites within a patient are usually correlated. The extensions and applications of the Mantel-Haenszel (CMH) methodology have been applied successfully here for studies with multiple anatomical regions. The study involves subjects who can be randomized into two or more treatment groups (which form the rows in a contingency table), and the dichotomous or ordinal response outcomes form the columns so as to account for the scope of anatomical regions for which the outcomes (e.g., improvement of the disease in the study) may not be independent of each other. The proposed extended CMH approach has all the same features of CMH tests. Namely, row sample sizes are adequate for asymptotic properties to apply (e.g.,≥5). CMH and extended CMH tests have many favorable properties, particularly by having minimal assumptions for randomized studies.
The comparison between test and control treatments can have better power through adjustment for baseline covariates with randomization-based methods. Randomization-Based Analysis Of Covariance (RANCOVA) is based on multivariate extensions of the Mantel-Haenszel test statistic with randomization-based covariance adjustment within the respective strata. The weighted least squares estimators can be combined across the strata to provide stratified counterparts that address comparisons between the two treatment groups using Mantel-Haenszel weights for the strata. The resulting estimators are stratified estimators with randomization-based covariance adjustment. With sufficiently large sample sizes, such estimators have an approximately multivariate normal distribution with the covariance matrix being essentially known through its corresponding consistent estimator. Accordingly, confidence intervals can be constructed for linear functions of such fully adjusted estimators (with respect to both stratification and covariates), and the scope of such linear functions can include the separate response variables, and sums (or averages) across the response variables. Also, multi-visit studies with a repeated measures data structure for both visits and multiple anatomical sites can be addressed by extensions of the methods in this paper (Li et al., 2017).