Objetivos de la Negociación

Usually, we have the data with numbers of observational units with favorable outcome for patients in each treatment group for the strata according to the number of affected anatomical sites; or we have distributions of sums of dichotomous specifications for favorable outcome cross-classified by treatments and number of affected sites. For each treatment, the proportion of observational units with good outcome is of interest. An underlying assumption for such estimators is that the patients with each treatment in each stratum represent a corresponding target population in the sense of a simple random sample.

Weighted averages of proportions of observational units with favorable outcome are produced across the strata through the mean number of observational units with outcomes≥j for the ith treatment; also the weights can depend on the sample sizes forc observational units for treatmenti, or they can be comparable to the weights for the Mantel-Haenszel statistic, although both weights are applicable. An unbiased estimator for the proportion of observational units with outcome≥j for the ith treatment is p∗ij =

Pq c=1nciy¯c∗ij Pq c=1cnci = Pq c=1yc+ij Pq c=1cnci

. Thus, a corresponding 0.95 confidence interval isp∗ij±1.96√υp,ij, whereυp,ij =

Pq c=1n

2

ciυcij

(Pq_c=1cnci)2 is an unbiased estimator for its variance and υcij =

Pnci

h=1(ychij+−y¯c∗ij) 2

nci(nci−1) is the unbiased estimator for the variance of ¯yc∗ij in (A.4.9). Sincep∗ij is a weighted average of y¯c∗ij

c with weights{cnci}, how it differs for the two treatments can depend

on how the {nci} differ for them. A way to avoid this issue is to consider ˜p∗ij = Pq c=1w˜cy¯c∗ij Pq c=1w˜cc , for which ˜υp,ij = Pq c=1w˜ 2 cυcij

(Pq_c=1w˜cc)2 is an unbiased estimator for the applicable variance and ˜wc=

nc0nc1 nc0+nc1 is

the Mantel-Haenszel weight; and so the corresponding 0.95 confidence interval is ˜p∗ij ±1.96pυ˜p,ij.

Also, ˜f∗j = (˜p∗1j−p∗˜0j) = Pq

c=1w˜cfcj Pq

c=1w˜cc

, and (˜υp,1j+ ˜υp,0j) is an unbiased estimator of its variance,

where fcj= (¯yc∗1j−y¯c∗0j) is shown below (A.4.9).

These weighted estimators are used to produce stratificaton-adjusted differences in averages between treatment groups. Regardless of whetherH0 applies, if the patients within each treatment group for each stratum are comparable to a simple random sample from a corresponding population, then the ¯yc∗ij are independent across all (c, i) for each j; and the υcij in (A.4.9) are unbiased

estimators for their variances, where υcij = Pnci

h=1(ychij+−y¯c∗ij) 2

nci(nci−1) . For the comparison between two treatments, the fcj can be specified with (¯yc∗1j −y¯c∗0j); and so the unbiased estimator for

the variance is υcj = (υc0j +υc1j). Also, υ∗jw = Pq

c=1w˜ 2

cυcj

(Pq_c=1w˜c)2 is an unbiased estimator for the variance off∗jw=

Pq c=1w˜cfcj Pq

c=1w˜c

with respect to the standardized weightswc= Pqw˜c c=1w˜c

. Here, f∗jw

represents the difference between treatments for the Mantel-Haenszel weighted means across the strata for the numbers of observational units with outcome≥j. A corresponding 0.95 two-sided confidence interval is f∗jw±1.96

√

υ∗jw. With the Mantel-Haenszel weights ˜wc = nc_n0n_cc1, where

nc = (nc0+nc1), it follows that f∗j = Pqc=1

nc0nc1(¯yc∗1j−y¯c∗0j)

nc = f∗jw

Pq

c=1w˜c; and from (A.4.5),

υ∗j0 =var{f∗j|H0} =Pqc=1

nc0nc1υcj0

nc , under the hypothesis of no difference between treatments. Thus,Qj has the extended Mantel-Haenszel structure shown in (A.4.6) in accordance with Stokes

Similarly, for the comparison between the test treatment and the control treatment with respect to the means ¯yc∗i = Pnci h=1ychi nci = Pr

j=1y¯c∗ij for sums acrossj, theυci∗ =

Pnci

h=1(ychi−¯yc∗i) 2

nci(nci−1) are unbiased estimators for their variances as shown in (A.4.11) ; and υc= (υc0∗+υc1∗) are unbiased estimators for the variances of the fc∗ = (¯yc∗1−y¯c∗0). Then the applicable extended Mantel-Haenszel statistic is Q = ( Pq c=1w˜c(¯yc∗1−y¯c∗0)) 2 Pq c=1w˜cυc∗0

as shown in (A.4.8); and it applies to fw = Pq

c=1w˜cfc∗ Pq

c=1w˜c

with the υc∗0 shown below (A.4.8) pertaining to its variance underH0.

4.2.2 Randomization-Based Analysis Of Covariance (RANCOVA)

The nonparametric, randomization-based analysis of covariance methodology uses weighted least squares on the treatment differences of response and covariates in ways like that discussed in Koch et al.(Koch et al., 1998). Covariance adjustment in the treatment effect estimates is based on a minimal statistical assumption of no difference in covariate means, which is a consequence of randomization to treatment.

Computations for randomization-based covariance adjustment can be made with the %N P arCov4 SAS macro discussed in Zink et al.(Zink et al., 2017). Usually, hypothesis is specified as either NULL or ALT and impacts whether the covariance matrices of responses and covariates in each stratum are computed across treatments (under the null) or within treatments (under the alternative). In general, the variance under the null would be the principal structure for producing p-values for hypothesis testing under the null corresponding to no difference between treatments. The variance under the alternative is used for computing confidence intervals.

For this macro, theychij+ are the response variables, and the¯zchiare the covariates. By weighted

least squares, b_w,0 is obtained as b_w,0 =

Pq

c=1w˜cbc,0 Pq

c=1w˜c

, for theb_c,0 in (A.4.14) ; and its corresponding covariance matrix is V_bw,0 =

Pq c=1w˜

2

cVbc,0

(Pq_c=1w˜c)2 , whereVbc,0 is shown below (A.4.14). Thus, Qbw,0 = b0_w,0V−_b1

w,0bw,0 is a multivariate Mantel-Haenszel test statistic for H0 with randomization-based covariance adjustment within the respective strata. UnderH0,Qbw,0 approximately has the chi- squared distribution with d.f.=r. Also, withL=10_r,QLbw,0 =b

0

w,0L0(LVbw,0L 0₎−1_Lb

w,0, the test statistic in (A.4.15) pertains to theychi, andd.f.= 1; and withL=δj, whereδj is a vector with the

jth element equal to 1 and the other elements equal to 0, the test statistic pertains to theychij+ for outcome≥j. The methods for confidence intervals can be applied using b∗w =

Pq c=1w˜cbc Pq

c=1w˜c

as shown below (A.4.16), and its covariance matrix is Vb∗w =

Pq c=1w˜

2

cVbc

(A.4.16). Thus, linear functionsLb∗w with consistent covariance matrix estimatorVLb∗w =Lb∗wL 0

can be used to produce covariance adjusted confidence intervals as counterparts to those based on thef∗jw orfw.

4.3

Examples

Two examples illustrate the methods in this article. In Section 4.3.1, the first example has an ordinal response variable for the comparison of two randomized treatments for a disorder of eyes, and it has two strata according to the number of enrolled eyes. There is also an ordered categorical response variable for the second example in Section 4.3.2, and it pertains to the comparison of two randomized treatments for a dermatology disorder; and there are three strata according to the number of affected anatomical sites, two covariates for baseline severity, and no missing data (because the original data are no longer available, and so any missing values for a patient have been replaced by some logically comparable values based on their measurements from other affected anatomical sites).