Implementing Worst Rank Imputation Using SAS

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1 Paper SP12 Implementing Worst Rank Imputation Using SAS Qian Wang, Merck Sharp & Dohme (Europe), Inc., Brussels, Belgium Eric Qi, Merck & Company, Inc., Upper Gwynedd, PA ABSTRACT Classic designs of randomized clinical trials quite often require repeated measurements over the treatment period. However, some patients may experience terminal events which prevent their continuation in the treatment. Although no measurements can be collected after those patients leave the study, the drop-out reasons (e.g. drug-related adverse experience or lack of efficacy) and time (e.g. early or late in the study) might be an indication of the effect of the treatment. One approach to incorporate this piece of information into the analysis is to impute, for the missing observation, a worstrank score which is worse than any values actually collected at that time point. The transformation from observed data into rankings for analysis also has the advantage of being less sensitive to data distribution assumption and less affected by the outliers. Nevertheless, how to implement and program the worst rank imputation remains a challenge. This paper provides one possible definition of the worst rank imputation and describes in details how to implement it using SAS. INTRODUCTION In a clinical trial which requires repeated response measurements over a period of time, it s possible that patients discontinue the treatment due to treatment-related reasons (e.g. lack of efficacy or adverse experience) which prevent their physical evaluation before the end of the study. The observations after discontinuation, however, might not be missing at random and might, on the contrary, provide insights to the treatment effect. For example, the discontinuation due to lack of efficacy is a strong negative message and the missing observation is an indication of treatment failure. In this case, the missing observation is informative. One approach to incorporate this informative missingness into the analysis is to impute a worst-rank score for the missing data. This score is worse than any values actually observed. The transformation from observed data into rankings for analysis also has the advantage of being less sensitive to data distribution assumption and less affected by the outliers. However, the exact definition of worst-rank imputation and its implementation remain a challenge. This paper provides one possible definition of the worst rank imputation approach and describes in detail its SAS program architecture as well as step-by-step implementation. Please note that the paper focuses on an application and implementation of the worst-rank imputation instead of its statistical soundness. WORST RANK IMPUTATION Example Study Design A simulated trial (please refer to table 1) is used in this paper to illustrate the worst rank imputation method. The study consists of 14 patients with unique subject ID (USUBJID) 101 to 114. Ideally, 5 assessments of the endpoint of interest should be collected for each patient at the baseline visit (Visit 0) and 4 post-baseline visits (Visit 1 to 4). Patient 101, 102, 103 complete the treatment with measurement collected at every time point. Patient 104 and 107 are also completers, but fail to provide response information for all time points. These 5 patients are labeled as COMPLETER. The rest of the patients drop out of the study towards the end due to various reasons (e.g. WITHDRAWAL OF CONSENT, LOST TO FOLLOW- UP, ADVERSE EVENT, LACK OF EFFICACY, etc). The reasons and time of discontinuation are recorded. As an example, patient 106 stops treatment after Visit 2 due to lack of efficacy. Therefore, this patient only has assessments before drop-out at Visit 0, 1 and 2. No information is available for his Visit 3 and 4. It also happens that some patients skip assessments before they leave the study. One example given here is patient 110 who is missing Visit 2 measurement although he only stops treatment after Visit 3.

2 Table 1 Original Data Collected Endpoint Value at Visit USUBJID Visit Reason NA COMPLETER NA COMPLETER NA COMPLETER NA COMPLETER WITHDRAWAL OF CONSENT LACK OF EFFICACY NA COMPLETER LOST TO FOLLOW-UP ADVERSE EVENT WITHDRAWAL OF CONSENT LACK OF EFFICACY ADVERSE EVENT ADVERSE EVENT LACK OF EFFICACY Without losing generality, change from baseline analysis is adopted for this study. Let s assume that the ranking and the collected assessments have the following relationships: Rank 1 (Best) Maximum (Worst) Assessment Value Smallest Largest And the ranking and the change from baseline values have the following relationships: Rank 1 (Best) Maximum (Worst) Change From Baseline Value (Visit 1 to 4) Smallest (Highly Negative) Largest (Less Negative or Positive) The ranking of change from baseline during the follow-up visits (computed as Visit i value - Visit 0 value) is constructed in a way that the rank value corresponds to the treatment effect (the smallest rank value 1 indicating the best effect and the largest value the worst). In this example, the smallest value of change from baseline (the most negative value which indicates the largest decrease from baseline) corresponds to the best treatment effect, hence rank number 1. Baseline Ranking Although the worst-rank imputation approach only applies to the follow-up time points, the baseline ranking is also calculated for this example since it s sometimes needed to account for baseline levels by means of a covariate in the repeated measurement analysis. The ranking is calculated solely based on the baseline endpoint values. Table 2 gives the ranking of all patients at Visit 0. The best rank (for patient 111) corresponds to the smallest endpoint value (2.0), and the worst rank (for patient 108) corresponds to the largest assessment (222.0). Table 2 Visit 0 (Baseline) Ranking Visit 0 USUBJID Rank Value Visit Reason LACK OF EFFICACY ADVERSE EVENT WITHDRAWAL OF CONSENT WITHDRAWAL OF CONSENT NA COMPLETER ADVERSE EVENT NA COMPLETER LACK OF EFFICACY NA COMPLETER LACK OF EFFICACY NA COMPLETER ADVERSE EVENT NA COMPLETER LOST TO FOLLOW-UP

3 Follow-up Visit Ranking If there were no missing values, patients at each follow-up visit (Visit 1 to Visit 4) could be ranked against their response change from baseline. However when it s impossible to collect information at all time points, the worst-rank imputation method provides a possible solution. The guiding principle of worst-rank imputation is to rank information in a way which corresponds to the treatment effect. Intuitively, patients who still stay in the study at the specific time point show better response compared to patients who have left the study due to drug-related reasons. Continuing patients are therefore assigned better ranks than patients discontinued for treatment-related causes. For a patient who is known to remain in study but has missing assessment at the time point, his last observation during the treatment is carried forward. So in the case of patient 110 who only leaves study after Visit 3, the missing Visit 2 assessment can be imputed using the Visit 1 observation. Non-dropouts (with/without missing information) are ranked together based on the observed change from baseline (calculated from observed or imputed values). Table 3 lists all the patient assessments after Last Observation Carried-Forward (LOCF). Table 4 shows the computed change from baseline from visit 1 to visit 4 with LOCF. All carried-forward values are highlighted in bold. Table 3 Data After LOCF Imputation Endpoint Value at Visit USUBJID Visit Reason NA COMPLETER NA COMPLETER NA COMPLETER NA COMPLETER WITHDRAWAL OF CONSENT LACK OF EFFICACY NA COMPLETER LOST TO FOLLOW-UP ADVERSE EVENT WITHDRAWAL OF CONSENT LACK OF EFFICACY ADVERSE EVENT ADVERSE EVENT LACK OF EFFICACY Table 4 Calculated Change From Baseline After LOCF Imputation Endpoint Change From Baseline Value at Visit USUBJID Visit Reason NA COMPLETER NA COMPLETER NA COMPLETER NA COMPLETER WITHDRAWAL OF CONSENT LACK OF EFFICACY NA COMPLETER LOST TO FOLLOW-UP ADVERSE EVENT WITHDRAWAL OF CONSENT LACK OF EFFICACY ADVERSE EVENT ADVERSE EVENT LACK OF EFFICACY Discontinued patients are ranked after all patients still under treatments. Those drop-outs are grouped based on the reason of discontinuation. The reasons can be ordered so that drop-outs due to less severe drug-related reasons are ranked before those due to more serious causes. In the example study, one possible ordering might be to consider patients with reasons WITH- DRAWAL OF CONSENT or LOST TO FOLLOW-UP (group 1) as showing better treatment effect, compared to those with

4 ADVERSE EVENT (group 2). due to LACK OF EFFICACY (group 3) is regarded as the indication of worst response among all reasons. For the patients who drop out due to the same reason, their discontinuation time (recorded as discontinuation visit number) are compared. Patients who drop out later in the study are ranked better (smaller rank values) than those who leave the study earlier. Table 5, 6, 7 and 8 give the worst-rank imputation results at each Visit 1 to 4 respectively. At Visit 1, all patients except 105 and 112 are still in the study, and they are ranked based on the calculated change from baseline value. 105, 112 discontinued from study after baseline visit and are ranked after all other patients. Patient 105 drops out for WITHDRAWAL OF CONSENT (discontinuation group 1) and is considered as showing better treatment response than patient 112 who terminates the study due to ADVERSE EVENT (discontinuation group 2). Table 5 Visit 1 Ranking Visit 1 USUBJID Rank Change From Baseline Visit Reason Group/Description / LOST TO FOLLOW-UP NA COMPLETER / WITHDRAWAL OF CONSENT NA COMPLETER / LACK OF EFFICACY / ADVERSE EVENT / LACK OF EFFICACY NA COMPLETER NA COMPLETER / ADVERSE EVENT NA COMPLETER / LACK OF EFFICACY / WITHDRAWAL OF CONSENT / ADVERSE EVENT At Visit 2, patients 107 and 110 are ranked using their calculated change from baseline with LOCF imputation. Patient 105, 112 and 114 are no longer in the study and are considered as having worse treatment effect than all other patients. They are ordered based on the implication of reason of discontinuation. Table 6 Visit 2 Ranking Visit 2 USUBJID Rank Change From Baseline Visit Reason Group/Description (LOCF) / LOST TO FOLLOW-UP / ADVERSE EVENT NA COMPLETER / LACK OF EFFICACY / WITHDRAWAL OF CONSENT NA COMPLETER NA COMPLETER NA COMPLETER NA COMPLETER / ADVERSE EVENT / LACK OF EFFICACY / WITHDRAWAL OF CONSENT / ADVERSE EVENT / LACK OF EFFICACY At Visit 3, patient 108, 105, 109, 112, 106 and 114 have all dropped out from study. They are grouped based on discontinuation reasons. Within each group, patients are ranked based on the time when they stop the treatment. For example, both 106 and 114 leave the study due to LACK OF EFFICACY. 106 is assigned a better rank because he discontinues later (after Visit 2) than 114 (after Visit 1).

5 Table 7 Visit 3 Ranking Visit 3 USUBJID Rank Change From Baseline Visit Reason Group/Description (LOCF) NA COMPLETER NA COMPLETER NA COMPLETER / WITHDRAWAL OF CONSENT / ADVERSE EVENT NA COMPLETER NA COMPLETER / LACK OF EFFICACY / LOST TO FOLLOW-UP / WITHDRAWAL OF CONSENT / ADVERSE EVENT / ADVERSE EVENT / LACK OF EFFICACY / LACK OF EFFICACY At Visit 4, only patients 103, 104, 107, 101 and 102 still remain in the study and all the rest stop the treatment due to various reasons. Table 8 Visit 4 Ranking Visit 4 USUBJID Rank Change From Baseline Visit Reason Group/Description (LOCF) NA COMPLETER NA COMPLETER NA COMPLETER NA COMPLETER NA COMPLETER / WITHDRAWAL OF CONSENT / LOST TO FOLLOW-UP / WITHDRAWAL OF CONSENT / ADVERSE EVENT / ADVERSE EVENT / ADVERSE EVENT / LACK OF EFFICACY / LACK OF EFFICACY / LACK OF EFFICACY SAS IMPLEMENTATION Figure 1 shows the process flow of the SAS implementation of the worst-rank imputation method described in the previous section. Firstly, all collected patient data are classified based on whether the patient is still continuing in the study at each specific time point. If the patient is a drop-out, the data are grouped based on the reason of discontinuation. Next, if the patient is continuing at a specific visit, and his observation is missing, his last observation prior to the current visit is carried forward. In the 3 rd step, all continuing patients are ranked together according to the assessment values (collected or imputed). The dropout patients due to the same reason are ranked together based on their time of discontinuation, and the whole dropout group is appended after the continuing patients from step 3. The drop-out groups are ordered, and the reason group which reflects worse treatment effect is always positioned after those with better response.

6 Figure 1 Process Flow All observations (missing or not) Step 1: Classify observations based on patient status a. Continueing patients with non-missing observations b. Continueing patients with missing observations c. Dropouts reason 1 d. Dropouts reason 2 e. Dropouts reason 3 Step 2: LOCF Step 4: Rank based on time and append Step 3: Rank based on Observation Ranked data for a, b & c Step 4: Rank based on time and append Ranked data for a,& b Ranked data for a, b, c & d Step 4: Rank based on time and append Ranked data for all patients STEP 1: CLASSIFY OBSERVATIONS BASED ON PATIENT STATUS At each visit time point, whether the patient is still continuing in the study (group a & b in the flow chart) can be determined by 2 criteria: 1. Patient is labeled as a COMPLETER; Or 2. Patient is not a COMPLETER, but his discontinuation visit occurs after the current visit. The rest of the patients are categorized according to the drop-out reasons (e.g. group c for WITHDRAWAL OF CONSENT or LOST TO FOLLOW-UP; group d for ADVERSE EVENT; and group e for LACK OF EFFICACY). Table 9 below lists the SAS datasets after categorizing the observations in the given example. Each dataset contains variables USUBJID, VISITNUM, RESULTN, DSDY and DSDECOD corresponding respectively to the subject identifier, visit number, measurement value during the visit, discontinuation visit number and patient discontinuation reason (or patient status in the case of a COMPLETER).

7 Table 9 SAS datasets after step 1 Group a: Continuing patients with non-missing observations USUBJID VISIT- RESULTN DSDY DSDECOD NUM COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER WITHDRAWAL OF CONSENT LACK OF EFFICACY LACK OF EFFICACY LACK OF EFFICACY LOST TO FOLLOW-UP LOST TO FOLLOW-UP LOST TO FOLLOW-UP ADVERSE EVENT ADVERSE EVENT ADVERSE EVENT LACK OF EFFICACY LACK OF EFFICACY LACK OF EFFICACY LACK OF EFFICACY ADVERSE EVENT ADVERSE EVENT ADVERSE EVENT ADVERSE EVENT ADVERSE EVENT LACK OF EFFICACY LACK OF EFFICACY Group b: Continuing patients with missing observations USUBJID VISIT- RESULTN DSDY DSDECOD NUM COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER COMPLETER WITHDRAWAL OF CONSENT WITHDRAWAL OF CONSENT WITHDRAWAL OF CONSENT WITHDRAWAL OF CONSENT Group c: Missing information due to reason 1 USUBJID VISITNUM RESULTN DSDY DSDECOD WITHDRAWAL OF CONSENT WITHDRAWAL OF CONSENT WITHDRAWAL OF CONSENT WITHDRAWAL OF CONSENT LOST TO FOLLOW-UP LOST TO FOLLOW-UP WITHDRAWAL OF CONSENT Group d: Missing information due to reason 2 USUBJID VISITNUM RESULTN DSDY DSDECOD ADVERSE EVENT ADVERSE EVENT ADVERSE EVENT ADVERSE EVENT ADVERSE EVENT ADVERSE EVENT ADVERSE EVENT Group e: Missing information due to reason 3 USUBJID VISITNUM RESULTN DSDY DSDECOD LACK OF EFFICACY LACK OF EFFICACY LACK OF EFFICACY LACK OF EFFICACY LACK OF EFFICACY LACK OF EFFICACY STEP 2: LOCF IMPUTATION FOR CONTINUING PATIENTS WITH MISSING VALUES For the continuing patients with missing observations (group b dataset from step 1), a Last Observation carried Forward approach (LOCF) is applied to impute missing data. Below is the SAS implementation of LOCF. %macro LOCF(inds= /* Input dataset name */,outds= /* Output dataset name */,SubjidVar= /* Patient identifier */,TimeVar= /* Time point variable */,Var= /* Name of the variable to be carried forward */,CrfwdVar= /* Name of the variable for the carried-forward values */ ); proc sort data=&inds out=&outds; by &SubjidVar &TimeVar;

8 data &outds; set &outds; by &SubjidVar &TimeVar; retain &CrfwdVar; if first.&subjidvar then &CrfwdVar=.; if &var >.z then &CrfwdVar=&var; %mend locf; In this example, the macro can be invoked in the following way. %locf( inds=group_b,outds=group_blocf,subjidvar=usubjid,timevar=visitnum,var=resultn,crfwdvar=resultn_locf ); STEP 3: RANK BASED ON OBSERVATIONS FOR CONTINUING PATIENTS Patients from group a and group b (with LOCF) are ranked based on the values of assessment (observed or imputed) using the SAS procedure RANK. The RANK procedure computes ranks for one or more numeric variables across all observations within each by-variable group and outputs the ranks to a new SAS data set. The following program shows how to rank the values of RANKVAR at each visit (as indicated by the by VISITNUM statement) using a simple PROC RANK step. In this example, RANKVAR refers to the baseline assessment during Visit 0, and change from baseline during the follow-up Visit 1 to Visit 4. The ranking results within each visit are stored in the variable WR. proc RANK data=group_a_blocf out=rk_group_a_blocf; by VISITNUM; var RANKVAR; ranks WR; By default, PROC RANK assigns rank number 1 to the smallest variable value. To reverse the order, you may specify the option DESCENDING so that the largest value corresponds to rank number 1. The procedure also allows flexibility in ranking tied values. The valid options are TIES=HIGH LOW MEAN where TIES=HIGH assigns the largest of the corresponding ranks to all the tied records; TIES=LOW assigns the smallest of the corresponding ranks to all the tied records; and TIES=MEA'N is the default and assigns the mean of the corresponding ranks to all the tied records. In an example dataset with only one variable and 4 observations of values 4, 5, 5 and 6, the default PROC RANK assigns rank 1, 2.5, 2.5 and 4 to these 4 observations respectively. If option TIES=HIGH is specified, the procedure calculates ranks as 1, 3, 3 and 4. When TIES=LOW, the observed values are ranked as 1, 2, 2 and 4. STEP 4: RANK BASED ON REASON AND TIME FOR DROP-OUT PATIENTS At each visit, the patients who have stopped treatment (group c, d, e from step 1) are ranked after those still in the study (group a and b). Based on the treatment effect (as implied by the discontinuation reason), patients with more serious dropout reason are ordered towards the bottom. Therefore, in this example, group e patients will be ranked worse than group d, and group d worse than group c. Within each group, patients will be ranked based on their drop-out time, the earlier they drop, the worse they will be ranked. The following APPENDRANK macro is defined to rank patients in a discontinuation group (c, d or e) and append their rank to the previous group. %macro appendrank( processds= /* Dataset to be ranked */

9 ,previousds= /* Dataset to append to */,byvar= /* By variable */,var = /* Name of the variable to be ranked */,rankvar= /* Name of the variable for ranking results */,outds= /* Output dataset */ ); proc sort data=&processds out=_ds1; by &byvar; %**** Rank based on drop-out time **; proc rank data=_ds1 out=_ds2 DESCENDING; by &byvar; var &var; ranks &rankvar; %**** get the largest rank in previous data; proc SQL; create table _rkmax as select &byvar, MAX(&rankvar) as _MAX from &previousds group by &byvar; create table _ds3 as select a.*, _MAX from _ds2 as a, _rkmax as b where a.&byvar=b.&byvar ; quit; data _ds4; set _ds3; &rankvar=&rankvar+_max; drop _MAX; data &outds; set &previousds _ds4; proc sort data=&outds; by &byvar wr; %**** Delete temporary datasets created within macro; proc datasets library=work memtype=data nolist ; delete _ds1 _ds2 _ds3 _ds4 _rkmax; quit ; %mend appendrank; To rank group c patients and consolidate the ranking with the previous group a and b patients (specified by previousds=rk_group_a_blocf), simply call the macro with the following parameters: %appendrank( Processds = group_c,previousds = rk_group_a_blocf,byvar = visitnum,var = dsdy,rankvar = WR,outds = rk_group_a_blocf_c ); The output dataset rk_group_a_blocf_c contains the ranking for all group a, b and c patients using worst rank imputation. It can then be passed into the appendrank macro again as previousds for the ranking of group d together. And similarly, the final ranking is generated by a 3rd call to the same macro. CONCLUSIONS In clinical trials, quite often repeated measurements are required for the analysis, however, patients may drop out in the middle of the trial due to various reasons. This paper described one possible solution, a worst-rank imputation approach, which transforms the collected observations into ranks to smooth the extreme outliers, and accounts for missing information

10 based on the reason and time of dropout. A step-by-step explanation of its SAS coding was also provided. REFERENCES: John M. Lachin Worst-Rank Score Analysis with Informatively Missing Observations in Clinical Trials Controlled Clinical Trials Volume 20, Issue 5, October 1999, Pages SAS Institute Inc., SAS Language Reference Version 6 First Edition Copyright 1990 by SAS institute, Cary, NC, USA ACKNOWLEDGMENTS The authors greatly acknowledge the review and candid feedback from Margaret Coughlin, Frederic Coppin, Kristel Vandormael and Cindy White. TRADEMARKS SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. CONTACT INFORMATION Qian Wang Scientific Programming Biostatistics and Research Decision Sciences Merck Sharp & Dohme (Europe), Inc. Clos du Lynx 5 B-1200 Brussels, Belgium Qian_Wang@merck.com Eric Qi Scientific Programming Biostatistics and Research Decision Sciences Merck & Co. (UG1D-88) Upper Gwynedd, PA Eric_Qi@merck.com

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