Introducción

The data in Table 3.15 are based on a randomized, double-blind, crossover clinical trial of diclofenac + misoprostol(Arthrotec) compared to acetaminophen in patients with osteoarthritis of the hip or knee (Pincus et al., 2001). There are 227 patients randomly assigned to the two treatment sequence groups. The patients in the Arthrotec:Acetaminophen (T:C) group received test drug (diclofenac + misoprostol) for the first period of treatment for osteoarthritis during the course of the study and control medication (Acetaminophen) for the second period; the Acetaminophen:Arthrotec (C:T )group received the opposite regimen. Also, the two evaluation periods were separated by 6 weeks to minimize the potential carryover effects. All patients in this protocol have 5 study visits, including screening visit (visit 1), baseline visits for each period (visit 2 and visit 4), and end of treatment visits for each period (visit 3 and visit 5). One of the primary outcome measures was the continuous visual analog pain scale of the Multidimensional Health Assessment Questionnaire, and it was missing for 9 patients at visit 3, 40 patients at visit 4 and 45 patients at visit 5. For analysis purposes here, every missing value is replaced by using a single / multiple sequental imputation method. In the subsequent discussion, the response variable of each patient at visit 3 and visit 5 is simplified to the trichotomy of pain scores as (0,1,2) according to the outcome as (_>60, (35,60),

6 35) as an ordering from worst to best, and the baseline variables at visit 2 and visit 4 are maintained as continuous. The primary objective of the illustrative analysis of this clinical trial is to demonstrate whether patients with osteoarthritis of the hip or knee have significantly better pain scores with diclofenac + misoprostol than with acetaminophen, and if the period is associated with the outcome.

Table 3.15: Data from a crossover clinical trial for patients with osteoarthritis with single imputation for missing data.

Treatment Frequencies of response at (visit 3 , visit 5) Number Sequence (0,0) (0,1) (0,2) (1,0) (1,1) (1,2) (2,0) (2,1) (2,2) of patients

T:C 7 6 3 12 4 12 21 18 29 112

C:T 13 8 17 3 9 19 3 7 36 115

* T:C group received test drug (diclofenac + misoprostol) followed by control (Acetaminophen); C:T group received control (Acetaminophen) followed by test drug (diclofenac + misoprostol).

Table 3.16: Means, standard errors, and p-values for (visit 2 - visit 4) differences of continuous baseline measurements with (visit 3 - visit 5)6= 0 for trichotomous response, with single imputation for missing data.

Sequence N Mean standard errors P-value

T:C 72 4.674 25.410 0.123

C:T 57 0.338 19.844 0.898

Pooled 129 2.758 23.132 0.178

(T:C - C:T) NA 2.168 2.050 0.292

Table 3.17: Goodness-of-Fit statistics (Deviance and corresponding p value) from Logistic Regression for equal adjacent odds models for comparing treatment periods in sequences, estimates and standard errors (SE), with single imputation for missing data.

Treatment Goodness-of-Fit

Sequence Deviance P value Estimate SE Parameter

T:C 0.713 0.490 0.756 0.202 β+γ

C:T 0.034 0.967 0.921 0.243 β−γ

* T:C group received test drug (diclofenac + misoprostol) followed by control (Acetaminophen); C:T group received control (Acetaminophen) followed by test drug (diclofenac + misoprostol).

Table 3.18: Goodness-of-Fit (Deviance) for models for both sequences and p-values, estimates and corresponding standard errors (SE) for treatments ( Arthrotec vs. Acetaminophen ) and periods ( 1 vs. 2 ) effects, with/without adjusting for the differences between baseline measurements, with single imputation for missing data.

Model Treatment Effect Period Effect Deviance

Estimate SE P-value Estimate SE P-value P-value With Baseline

Adjusting 0.826 0.159 <0.001 -0.100 0.160 0.532 NA

Without Baseline

Adjusting 0.839 0.158 <0.001 -0.083 0.158 0.601 0.828

Table 3.16 shows that baseline variables for the two sequence groups do not differ very much. Goodness-of-Fit statistics for each treatment sequence for the equal adjacent odds ratio model, estimates and corresponding standard errors (SE) for comparing treatment periods within sequences (i.e.; Arthrotec1vs.Acetaminophen2 and Arthrotec2 vs.Acetaminophen1) are displayed in Table 3.17. Table 3.18 shows the adjusted results of the stratified analyses for the ordinal response. After adjusting for the baseline measurements, the estimates of period effect (as sequence) produced p-value of 0.532, indicating no statistical significance at the 0.05 level, and similar results were produced without baseline covariates adjustment. The assessments of treatment effect were very similar from either controlling the baseline covariates or not adjusting for them, and so were their standard errors. The treatment of diclofenac + misoprostol for improvement of pain scores had 2.284 (= e0.826) times higher odds for better outcome than acetaminophen, indicating that the treatment (diclofenac + misoprostol) was better than the control (acetaminophen). Table 3.19 and Table 3.20 show the estimates of treatment and period effects for the dataset where the missing values have been replaced by using multiple imputation method, with / without adjusting for the baseline measurements, and they are very similar as the results shown in Table 3.17 and Table 3.18.

Table 3.19: Estimates and standard errors (SE) from Logistic Regression for equal adjacent odds models for comparing treatment periods in sequences, with/without adjusting for the baseline measurements, with multiple imputation for missing data.

Treatment With Adjusting Without Adjusting

Sequence Estimate SE Estimate SE Parameter

T:C 0.749 0.228 0.777 0.224 β+γ

C:T 1.090 0.299 1.017 0.266 β−γ

* T:C group received test drug (diclofenac + misoprostol) followed by control (Acetaminophen); C:T group received control (Acetaminophen) followed by test drug (diclofenac + misoprostol).

*β denotes parameter for treatment comparison andγ denotes parameter for period comparison.

Table 3.20: Estimates, corresponding standard errors (SE) and p-values for treatments ( Arthrotec vs. Acetaminophen ) and periods ( 1 vs. 2 ) effects, for models, with/without adjusting for the baseline measurements, and with multiple imputation for missing data.

Model Treatment Effect Period Effect

Estimate SE P-value Estimate SE P-value With Baseline

Adjusting 0.883 0.182 <0.001 -0.136 0.173 0.434

Without Baseline

Adjusting 0.897 0.178 <0.001 -0.120 0.169 0.479

The results from the randomization based analysis of covariance (RANCOVA) for the equal adjacent odds ratio model based on Appendix 3.5.2 are shown in Table 3.21, with single imputation for missing data. With respect to Appendix 3.5.2, for T:C group, the maximum likelihood estimate

ˆ

β and the mean of baseline covariates for paried observations with different values ¯U_g as vector ˆ F is Fˆ = ( ˆβ,U¯0_g)0 =    0.756 4.674  

   0.038 0.049 0.049 8.968  

. Correspondingly for C:T group,

ˆ F=    0.922 −0.338   ,VFˆ =    0.063 0.073 0.073 6.909   .

In this regard, the standard errors for the results with baseline adjustment are somewhat smaller than their counterparts in Table 3.18. Additionally, with adjustment for the baseline covariates, the adjusted treatment difference estimate from single imputation 0.828 is similar to the counterpart 0.826 that pertains to Table 3.18. The results with multiple imputation for missing data are shown in Table 3.22. With using the combination of maximum likelihood estimation and weighted least squares, the standard errors for the results with baseline adjustment are a bit smaller than their counterparts in Table 3.20. Additionally, with adjustment for the baseline covariates, the adjusted treatment difference estimate from multiple imputation 0.870 are close to their counterpart 0.883 that pertains to Table 3.20. The treatment of diclofenac + misoprostol for improvement of pain scores had 2.387 (=e0.870) times higher odds for better outcome than acetaminophen, indicating that the treatment (diclofenac + misoprostol) was better than the control (acetaminophen).

Table 3.21: Estimates, corresponding standard errors (SE) and p-values for treatments ( Arthrotec vs. Acetaminophen ) and periods ( 1 vs. 2 ) effects, using RANCOVA method based on Appendix 3.5.2, adjusting for the baseline measurements, and with single imputation for missing data.

Sequence T:C Sequence C:T Treatment Period

(β+γ ) (β−γ ) β γ

Estimate 0.731 0.925 0.828 -0.097

Standard Error 0.194 0.250 0.158 0.158

P-value <0.001 <0.001 <0.001 0.270

* T:C group received test drug (diclofenac + misoprostol) followed by control (Acetaminophen); C:T group received control (Acetaminophen) followed by test drug (diclofenac + misoprostol).

Table 3.22: Estimates, corresponding standard errors (SE) and p-values for treatments ( Arthrotec vs. Acetaminophen ) and periods ( 1 vs. 2 ) effects, using RANCOVA method based on Appendix 3.5.2, adjusting for the baseline measurements, and with multiple imputation for missing data.

Treatment Effect Period Effect Imputation Estimate SE P-value Estimate SE P-value

1 0.828 0.158 <0.001 -0.0977 0.158 0.270 2 0.880 0.170 <0.001 -0.163 0.170 0.169 3 0.937 0.178 <0.001 -0.093 0.178 0.301 4 0.893 0.176 <0.001 -0.134 0.176 0.223 5 0.813 0.155 <0.001 -0.184 0.155 0.118 Combined 0.870 0.177 <0.001 -0.134 0.173 0.439

The results from the randomization based analysis of covariance (RANCOVA) for equal adjacent odds ratio via weighted least squares based on Appendix 3.5.3 are shown in Table 3.23 and Table 3.24. With adjustment for the baseline covariates, the adjusted treatment difference estimate from single imputation for missing data 0.815 and the estimate from multiple imputation for missing data 0.825 are very close to their counterparts that pertain to Table 3.21 and Table 3.22. In this regard, the standard errors from multiple imputation for the results with baseline adjustment are a bit smaller than their counterparts in Table 3.22. The treatment of diclofenac + misoprostol for improvement of pain scores had 2.282 (= e0.825) times higher odds for better outcome than acetaminophen, indicating that the treatment (diclofenac + misoprostol) was better than the control (acetaminophen).

Table 3.23: Estimates, corresponding standard errors (SE) and p-values for treatments ( Arthrotec vs. Acetaminophen ) and periods ( 1 vs. 2 ) effects, using RANCOVA method based on Appendix 3.5.3, adjusting for the baseline measurements, and with single imputation for missing data.

Sequence T:C Sequence C:T Treatment Period

(β+γ ) (β−γ ) β γ

Estimate 0.708 0.922 0.815 -0.107

Standard Error 0.215 0.239 0.161 0.161

P-value <0.001 <0.001 <0.001 0.253

* T:C group received test drug (diclofenac + misoprostol) followed by control (Acetaminophen); C:T group received control (Acetaminophen) followed by test drug (diclofenac + misoprostol).

*β denotes parameter for treatment comparison andγ denotes parameter for period comparison.

Table 3.24: Estimates, corresponding standard errors (SE) and p-values for treatments ( Arthrotec vs. Acetaminophen ) and periods ( 1 vs. 2 ) effects, using RANCOVA method based on Appendix 3.5.3, adjusting for the baseline measurements, and with multiple imputation for missing data.

Treatment Effect Period Effect Imputation Estimate SE P-value Estimate SE P-value

1 0.815 0.161 <0.001 -0.107 0.161 0.253 2 0.813 0.166 <0.001 -0.143 0.166 0.194 3 0.862 0.165 <0.001 -0.038 0.165 0.409 4 0.846 0.167 <0.001 -0.129 0.167 0.220 5 0.788 0.156 <0.001 -0.195 0.156 0.106 Combined 0.825 0.166 <0.001 -0.122 0.175 0.484

The results from RANCOVA for means for randomized conditions for paired observations based on Appendix 3.5.4 are shown in Table 3.25, with single or multiple imputation for missing data separately. Since it is a randomized clinical study, we force the baseline covariates to be same for two paired observations. The treatment of diclofenac + misoprostol for improvement of pain scores had 0.403 higher values for better outcome than acetaminophen, with 0.067 as standard error, indicating that the treatment (diclofenac + misoprostol) was better than the control (acetaminophen).

Table 3.25: Estimates, corresponding standard errors (SE) and p-values for treatments ( Arthrotec vs. Acetaminophen ) and periods ( 1 vs. 2 ) effects, using RANCOVA method based on Appendix 3.5.4, adjusting for the baseline measurements, and with single/multiple imputation for missing data.

Single Imputation Multiple Imputation Treatment Period Treatment Period

Estimate 0.403 0.021 0.403 0.010

Standard Error 0.066 0.066 0.067 0.065

P-value <0.001 0.624 <0.001 0.874

3.4

Discussions

When the response variables are on ordinal scales and there are only two treatment conditions, the data for matched pairs can be summarized in a series of 2×r tables that correspond to the respective matched pairs. The extended Mantel-Haenszel test statistic is robust for tesing for association in an average sense across the matched pairs, even with sample sizes of 1 per treatment in matched pair studies, provided that there are enough matched sets. Nevertheless, if the association is slight but consistent across the tables, this procedure will be effective in detecting that association. A Randomization Based Analysis Of Covariance (RANCOVA) counterpart to the extended Mantel-Haenszel test statistic is specified in Appendix 3.5.4, which is for difference in mean scores between paired observations, forcing the means of baseline covariate distributions to be equal for two paired observations, and it approximately has the chi-squared distribution with d.f.= 1.

Matched pairs mentioned in this paper are studies where individuals are randomized to treatment or control group within a matched set. We can further address extensions of covariance adjustment for ordinal outcomes through using both non-parametric strategies and conditional logistic regression methods. The Randomization Based Analysis Of Covariance (RANCOVA) methodology is useful for providing covariate-adjusted estimates of treatment effects for data from a randomized clinical trial (Hussey et al., 2016). With respect to the model based on Appendix 3.5.2 as RANCOVA for the equal adjacent odds ratio for randomized conditions (or treatments) for paired observations, the

odds ratio estimate is obtained by applying both the maximum likelihood estimate and weighted least squares. Also, the odds ratio estimate based on Appendix 3.5.3 as obtained only through weighted least squares is similar to the result based on Appendix 3.5.2. Both can provide more powerful statistical tests (or narrower confidence intervals) for comparisons between treatments through variance reduction. They also permit the random imbalances between treatment groups in terms of the distributions of the covariables to be adjusted to equivalence. In this way, there can be a clarification of the degree to which detected differences between randomized groups for response variables are due to treatment rather than random imbalances for the distributions of patients’ background or baseline characteristics. The non-parametric methods have essentially no assumptions for a randomized clinical trial, and for a log-linear model can produce results with expected properties.

CHAPTER 4: METHODS FOR TREATMENT COMPARISONS FOR