MIXED MODEL ANALYSIS OF ORDINAL RESPONSE DATA WITH TWO NESTED LEVELS OF CLUSTERING: AN APPLICATION INVOLVING A CLINICAL TRIAL

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1 Pak. J. Statist Vol. 27(1), MIXED MODEL ANALYSIS OF ORDINAL RESPONSE DATA WITH TWO NESTED LEVELS OF CLUSTERING: AN APPLICATION INVOLVING A CLINICAL TRIAL Sadia Mahmud 1, Ursula Chohan 2, Gauhar Afshan 2 and M. Qamar-ul-Hoda 2 1 Department of Community Health Sciences, The Aga Khan University, Karachi, Pakistan. sadia.mahmud@aku.edu; mahmud_sadia@yahoo.com 2 Department of Anaesthesia, The Aga Khan University, Karachi, Pakistan ABSTRACT In this paper we present analysis of ordinal response data, from a clinical trial, with two nested levels of clustering using the generalized linear mixed model approach. We present a generalization of the continuation ratio model with two nested random effects. The model was fitted using maximum marginal likelihood estimation, assuming the nested random effects are independent and normally distributed. The clinical trial recruited high risk patients coming for unilateral lower limb surgeries. All patients were given unilateral spinal anaesthesia that intends to anesthetize only the leg that has to be operated; motor level was the ordinal response in the trial, measured repeatedly over time for both legs. The objective of the analysis was to evaluate the difference in the severity of motor level paralysis between the operated and the non-operated leg, and also to assess the effect of time on progression of motor level to higher degrees of paralysis. There was a significant interaction between the covariates time and leg; at baseline progression rate to higher degrees of paralysis was more rapid for the operated relative to the nonoperated leg. The progression rate levelled off by the end of the study, at 30 minutes, more prominently for the non-operated leg. The continuation ratio method, that makes a series of comparison of all lower categories on a scale to the next succeeding one, intricately models the progression of motor level paralysis with time to higher degrees, and gives insight into the mechanism of progression. KEY WORDS Continuation ratio model; three-level data; nested random effects; maximum marginal likelihood estimation. 1. INTRODUCTION Ordinal responses are commonly encountered in medical and social science research. An ordinal response variable consists of ordered categories such as level of pain graded as no pain, mild, moderate and severe pain. Different ways of generalizing the logit model to handle ordered categories have been suggested (Allison 1999; Agresti 1990; Lindsey et al. 1997). Moreover, in community based studies, and in clinical trials one 2011 Pakistan Journal of Statistics 31

2 32 Mixed Model Analysis of Ordinal Response Data encounters measurements recorded repeatedly on the same subject over time or some other dimension. In some situations there can be two nested levels of clustering, for example in an ophthalmology trial both eyes of a group of patients can be followed longitudinally. Such a data structure has been referred to as three-level data, the first level corresponds to the longitudinal measurements, the second level to an eye and the third level to the patient. Regression analysis of multilevel data structure for a continuous response is very developed and can easily be conducted using standard software (Littell et al. 1996). Multilevel analysis of nominal and ordinal responses has received increasing attention in recent years (Gibbons and Hedeker 1997; Ten Have et al. 1999; Hartzel et al. 2001; Raman and Hedeker 2005; Liu and Hedeker 2006). For logistic regression models involving three-level data it is important to adjust for both levels of clustering, and ignoring either can lead to bias in estimating regression parameters under certain conditions (Ten Have et al. 1999). Ten Have et al. (1999) adjusted for two nested levels of clustering for binary response data using a mixed effects logistic regression model. Gibbons and Hedeker (Gibbons and Hedeker 1997) suggested mixed effects probit and logistic regression models for three-level binary response data that was extended to an ordinal response by Raman and Hedeker (Raman and Hedeker 2005) and Liu and Hedeker (2006) using cumulative logit link. Hartzel et al. (2001) presented a general approach for logit random effects modeling of clustered ordinal and nominal responses, they used adjacent-categories logit to model the ordinal response and the random effect structure in their model was not nested. In this paper we present regression analysis of ordinal response data with two nested levels of clustering using the generalized linear mixed model approach. We generalized the continuation ratio method that models the ordinal character of the response variable, to adjust for the clustering structure in the data by including two nested random effects in the model. As an example of repeated ordinal responses with two nested levels of clustering, we consider a clinical trial conducted at the Department of Anesthesia, The Aga Khan University. The conventional method to induce spinal anesthesia for lower limb surgeries is bilateral spinal anesthesia that anesthetizes both legs. Cardiovascular system may be profoundly affected by the spinal anesthesia. Hypotension is the most frequent side effect (Carpenter et al. 1992). The unilateral spinal anesthesia, that anesthetizes only the leg that has to be operated, has been suggested as an alternative technique to restrict the undesired effects (Casati et al. 1998). A clinical trial was conducted with the primary objective to evaluate whether unilateral spinal anesthesia can avoid the undesired side effects such as low blood pressure and increased heart-rate accompanying the conventional bilateral spinal anesthesia for high risk patients (Chohan et al. 2002). The subjects comprised of ASA 1 III and IV patients, aged years, undergoing unilateral lower limb surgeries. This group of patients is considered as a high risk group. All patients received unilateral spinal anesthesia. Response variables such as systolic and diastolic blood pressure and heart-rate, measured at baseline and at every 5 minutes for 30 minutes after initiation of spinal anesthesia, are referred to as the haemodynamic variables and were measured on a continuous scale. For a Gaussian response with one level of clustering it is straight 1 American Society of Anesthesiologists

3 Sadia Mahmud et al. 33 forward to perform repeated measures Analysis of Variance available on SAS and other statistical packages. The high risk patients, in the present trial, showed minimal haemodynamic changes following unilateral spinal anesthesia (Chohan et al. 2002).. The motor level measured on the Bromage scale 2 was the ordinal response in the above mentioned clinical trial. The Bromage scale measures progression to higher degrees of motor level paralysis starting from No paralysis. Data for 30 patients were available for the analysis. For each patient motor level was measured both for the operated and the non-operated leg at baseline and at every 5 minutes, for 30 minutes after initiation of spinal anesthesia. A secondary objective of the above mentioned clinical trial was to evaluate the difference in the severity of motor level paralysis between the operated and the non-operated leg, and also to assess the pattern of progression to higher degrees of paralysis with time for each leg of a patient receiving unilateral spinal anesthesia. Ideally unilateral spinal anesthesia should paralyze only the leg to be operated, however there could be some spillover effects on the non-operative leg. In this paper we report regression analysis of the ordinal motor level responses, taking the clustering structure of the data into account. In this three-level data structure, the longitudinal measurements (at 7 time points) are at the first level, legs (operated/nonoperated) are at the second level and patients at the third level. Table 1 gives a visual presentation of the data in a contingency table showing how the proportions of measurements in various categories are changing over time for the two legs. At the baseline ( t 0) all the 30 patients were recorded as having No paralysis (category 1) for both legs. With the passage of time the proportions in categories corresponding to higher degrees of paralysis increase much more rapidly for the operated as compared to the non-operated leg. At t 5 minutes about 23% of the patients were recorded as having No paralysis (category 1) for the non-operated leg whereas only 3% had No paralysis for the operated leg. At t 10 minutes 80% of the patients had complete paralysis (category 5) on the operated whereas only about 7% had complete paralysis on the non-operated leg. At the end of the study ( t 30 minutes) about 97% of the patients had complete paralysis on the operated whereas only about 37% had complete paralysis on the non-operated leg. In section 2.1 we report the analysis of the ordinal motor level responses using the generalized linear model approach that is extended to generalized linear mixed model in section 2.2. Section 3 gives a discussion of the analysis approach, noting its limitations and some discussion of alternative methods for regression analysis of clustered categorical data. 2.1 Generalized Linear Model The models that deal with regression analysis involving an ordinal response variable are generalization of logit model for a binary response (Allison 1999; Agresti 1990). The 2 Bromage scale: 1. No Paralysis 2. Inability to raise extended leg. 3. Inability to raise flex knee 4. Inability to do dorsiflexion of foot but can wiggle toes. 5. Inability to move at all.

4 34 Mixed Model Analysis of Ordinal Response Data proportional odds, adjacent category and continuation ratio models differ with respect to how the ordinal categories are dichotomized into a binary response. The context of the research problem and research questions, and the nature of the ordinal measurements usually motivate the choice of the particular model to be employed for the analysis of ordinal responses. The continuation ratio model is appropriate for situations in which the ordered categories represent a progression through stages, so that individuals pass through each lower stage before they go on to higher stages. This seems to be the appropriate modeling approach with regards to the clinical relevance of the motor level responses measured on the ordinal Bromage scale in our example. Moreover, unlike the proportional odds model, the continuation ratio model possesses the property of conditional independence between the binary partitions that makes it easier to generalize to repeated ordinal responses (Dos Santos et al. 2000). We note that the odds ratio in the continuation ratio model change if continuation odds are set up starting from the top of the scale or the bottom. In our modeling of motor level responses using the continuation ratio method we start from the bottom of the scale, that is, the No Paralysis category. This is clinically relevant as all 30 patients had No Paralysis for both legs at baseline and they progressed to higher degrees of paralysis with passage of time. Consider an ordinal response with c categories denoted C1, C2, Cc and coded as 1,2.c. Suppose the multinomial distribution of the ordinal response has probabilities l 1,2,... c 1. The continuation ratio logits are defined as (Agresti 1990):, l L Let ck 1 or lower; that is k l ck 1 log, k 1,2,... c 1 (2.1) 1... ck denote the conditional probability of response c k 1, given c k 1 = ck 1 ck ck 1, k 1,2,... c 1 (2.2) The continuation ratio logits can then be expressed as ordinary logits of these conditional probabilities: L k ck 1 log, k 1,2,... c 1 (2.3) 1 ck 1 It can be shown that the original multinomial distribution can be expressed as a product of ( c 1) binomial distributions in terms of the conditional probabilities defined above (Agreti, 1990; Linsey et al. 1997). Hence one can apply the conventional logistic regression with the conditionally independent binary partitions of the ordinal response as the dependent, and the potential predictor variables and the partition variable as the independent variables. The partition variable ( k 1, 2,... c 1 ) is the identification variable for each conditionally independent binary partition and can be included in the model by creating dummy variables.

5 Sadia Mahmud et al. 35 In order to write a formal mathematical expression for the likelihood function for the ordinal response in the above mentioned trial, modeled through the continuation ratio method, let us develop the following notation. Consider an ordinal response with c categories denoted C1, C2, Cc and coded as 1,2.c. Assume that the indices i, j, t correspond to the subject, leg and the time point respectively ( i 1,.., 30, j 1, 2, t 0, 5,..30). Denote the ordinal response for the ith subject s jth leg at tth time point as y ijt. Each ordinal response is considered as a series of conditionally independent binary continuation ratio partitions of the ordinal scale. Denote the binary response corresponding to the kth partition Cck 1 : Cck, Cck 1,, C1, k 1,2, c 1, by Y ijt, k defined as and ijt, k Y if y c k 1 ijt, k 1 ijt = 0 if y c k 1 ijt Y is undefined for y c k 1. ijt The contribution to the likelihood by the ordinal measurement yijt is given by: L ijt min( c yijt1, c1) k ijt k 1 Yijt, k exp ' x 1 exp ' x (2.4) k ijt where k is the intercept associated with kth binary partition of the ordinal response; xijt is a vector of regression coefficients. is a vector of explanatory variables and Assuming the measurements on the same subject are independent (we address the nested clustering structure of the data in next section) the full likelihood is L L. The regression coefficients are estimated by maximizing the likelihood. ijt ijt The continuation ratio model can be fitted using any standard statistical software that fits conventional logistic regression model, by creating the extended vectors reducing the ordinal response to binary (Lindsey et al. 1997). Specialized software such as Sabre (Stott et al. 1999) and SAS (Release 9.1, SAS Institute Inc., Cary NC USA) can also be used through straightforward programming. The continuation ratio model for the unilateral spinal anesthesia trial data was fitted on SAS. We had data for 30 patients, with 14 measurements on each patient, thus giving us a total of 420 measurements on the ordinal scale. There were 950 observations in the extended dataset with the ordinal response reduced to binary. Table 2 (Column 1) reports the fit of the Continuation Ratio Logit (CRL) model. Examination of the scale of time through quartile analysis (Hosmer, D.W. and Lemeshow, S. 1989) indicated a logarithmic scale. There was a highly significant interaction between leg and time variable (Wald test Z = 1.94/0.34 = 5.71, p-value < 0.01, likelihood ratio test, , p-value < 0.005). The positive estimated regression

6 36 Mixed Model Analysis of Ordinal Response Data coefficient of the interaction term indicates that the effect of time on progression to higher degrees of paralysis is more rapid for the operated as compared to the nonoperated leg. 2.2 Generalized Linear Mixed Model The CRL analysis discussed in section 2.1 takes no account of the clustering structure of the data, and hence is inadequate. The purpose of reporting these results is to provide a basis for comparison with the model that adjusts for the two nested levels of clustering. As discussed in the introduction section Ten Have et al. (1999) and Gibbons and Hedeker (1997) proposed random effects (or mixed effects) logistic regression models for binary response data with two nested levels of clustering. Raman and Hedeker (2001) and Liu and Hedeker (2006) extended Gibbons and Hedeker s (1997) model for a binary response to an ordinal response, however they used the cumulative logit link. These authors modeled the nested clustering structure by addition of nested random effects in the linear predictor of the conventional logistic function (that is the logit) and implemented maximum marginal likelihood estimation (MMLE) using numerical integration to approximate the intractable integrals over the random effects. For an adequate analysis of data from the unilateral spinal anesthesia trial we propose a random effects (RE) model with two random effects i and j(i), to account for the nested levels of clustering due to the ith subject and the jth leg nested in subject i respectively. We refer to the notation introduced in section 2.1. For the sequence of ordinal outcomes for the ith subject, the marginal (integrated) likelihood is where *; ; ; L g y x f f d d (2.5) i ijt ijt i j( i) i j( i) i j( i) jt g y ijt. min( c yijt 1, c1) k 1 exp Yijt, k k *' xijt subject i leg( subject) j( i) 1 exp *' x k ijt subject i leg( subject) j( i) k is the intercept associated with the kth binary partition of the ordinal response. i f and f ji effects are the probability distribution functions of the two nested random i and j(i) respectively. We assume that i normal variables, with independence also assumed between leg( subject) leg level respectively. and j(i) are independent standard i and j(i). subject denote the standard deviation of the random effects at the subject and at the xijt is a vector of explanatory variables and * cluster-specific log-odds ratio parameters. The full likelihood is L L. i i and is a vector of

7 Sadia Mahmud et al. 37 Ten Have et al. (1999) implemented the estimation procedure for a logit model with a binary response with two nested levels of clustering, using the maximum marginal likelihood approach in a FORTRAN program. They accounted for the nested clustering by two independent standard normal variables and approximated the intractable integrals over the random effects using numerical integration. They approximated the integration of the likelihood over random effects with weighted sums where weights are binomial mass functions. We used the continuation ratio model to reduce the ordinal response in our example to binary on SAS (as discussed in section 2.1) and implemented maximum marginal likelihood estimation through the FORTRAN program developed by Ten Have et al. (1999). For this implementation we specified the initial values for the fixed effects regression coefficients obtained from the CRL fit, and set the initial values for the standard deviation of the random effects at subject subject and at leg level leg( subject) respectively equal to 0.5. Table 2 (2 nd column) reports the results of the analysis when the two nested level of clustering in the data were taken into account through the random effects (RE) model 2 stated above. The likelihood ratio test, 2 [log likelihood of CRL - log likelihood of CRL + RE ] = , gives an assessment of the relative fit of the data provided by the two models (Hedeker and Gibbons 1994) and clearly indicates the significance of including the random effects in the model df Results from the ordinal CRL+ RE analysis give the same inference as the ordinal CRL model, with a highly significant interaction between the leg and time variables (Wald test statistic Z = 3.92/0.64= 6.125, p-value < 0.01). However, the parameter estimates of fixed effects (that are significantly different from zero) are much larger for the ordinal CRL + RE model. Moreover, the standard errors of the parameter estimates for the CRL + RE model are greater than the corresponding standard errors from the CRL fit. The variability attributable to subjects ( ˆ subject = 2.01) is significant (Wald test statistic Z = 2.01/0.80 = 2.51, p-value= 0.012) and that attributable to legs nested within subjects ( ˆ leg ( subject ) = 2.83) is highly significant (Wald test statistic Z = 2.83/0.53 = 5.34, p-value < 0.01). The continuation ratio model for an ordinal response assumes that the effect of an explanatory variable (on the response) does not depend on the level of the partition variable. This implies absence of interaction between the partition and the explanatory variable. There was a significant interaction between the time and the partition variable at 2 the level of CRL + RE model, the likelihood ratio test, (3df) = 2( ) = [ (3df) = 7.815]. The CRL + RE model including the time and partition variable interaction is reported in Table 2 (3 rd column). This is a departure from the usual assumption that the regression coefficient corresponding to an explanatory variable does not depend on the level of the partition variable. However, Agresti (1990) and other authors (Dos Santos and Berridge 2000; Bender and Bender 2000) do allow for different

8 38 Mixed Model Analysis of Ordinal Response Data regression parameters for the effect of a covariate for different binary continuation ratio partitions of the ordinal response. Fitting the model including the interaction between the partition variable and leg (operated/non-operated) at CRL level leads to an unbelievable large estimated standard error for an estimated regression coefficient involving the interaction term. This is an indication of the presence of a zero cell count due to spreading the data over too many cells. The unstable result prevents us from ascertaining if in fact this interaction is significant (Hosmer and Lemeshow 1989). Due to the unstable result at the CRL level this interaction was not further evaluated at CRL + RE level. The estimated effects of the explanatory variables, leg and time (for different binary partitions) calculated from the final CRL + RE model (Table 2, 3 rd column) are shown in Fig. 1. Fig. 1 demonstrates the interaction between time and leg; at baseline, when the spinal anesthesia was initiated, progression rate to higher degrees of motor-level paralysis was more rapid for the operated as compared to the non-operated leg (for all binary partitions). The progression rate seems to level off by the end of the study, at 30 minutes, more so for the non-operated leg. The significant interaction between time and partition variable indicates that the effect of time is different for different binary partitions of the ordinal response. The effect of time on the ordinal response for the four binary partitions is shown in Fig. 2; for each leg the four intersecting curves corresponding to each binary partition illustrate this interaction. We note that for each leg at t 0 (baseline) the progression rate to the higher degree of paralysis was most rapid for partition 4 (category 2 versus 1) as compared to partition 3 (category 3 versus 2 &1), 2 (category 4 versus 3, 2 & 1) and 1 (category 5 versus 4,3,2 & 1), with the progression rate leveling off at later times for all four partitions. As all the patients started out with no paralysis (category 1) on both legs at baseline and progressed to higher degrees of motor level paralysis with passage of time, the most rapid progression rate for partition 4 (category 2 versus 1) at baseline seems plausible. 3. DISCUSSION In this paper we have presented an analysis of repeated ordinal response data with two nested levels of clustering, from a clinical trial involving unilateral spinal anaesthesia, using the generalized linear mixed model (GLMM). The ordinal character of the response is modeled using the continuation ratio method that reduces the ordinal response to conditionally independent binary continuation ratio partitions. The continuation ratio method was used to model the ordinal responses as it seemed clinically relevant with regards to the context of the problem, and the nature of the ordinal responses measured on the Bromage scale, assessing the degree of motor level paralysis. All patients initially had no paralysis on both legs and progressed to higher degrees of motor level paralysis with time, the progression rate being more rapid for the operated leg. For both legs the progression rate leveled off by the end of follow up at 30 minutes. After the initiation of spinal anaesthesia the severity of paralysis on the operated leg was greater than that on the non-operative leg, with the difference between the severity of paralysis increasing with time.

9 Sadia Mahmud et al. 39 To account for the clustering structure of the data we use a generalization of the continuation ratio model with two nested random effects. We fitted the model using maximum marginal likelihood estimation (MMLE) as developed by Ten Have et al. (1999) for a binary response with two nested levels of clustering. We note a limitation of the dataset used in this paper to exemplify the GLMM with regards to its size; 30 subjects at 3 rd level, each with 2 legs (2 nd level) and longitudinal measurements at 7 time points on each leg. Validity of statistical inferences constructed from logistic-normal models requires large number of clusters. However, keeping in mind that one cannot always have perfect dataset to exemplify methodologies, noting the limitation of moderate number of 3 rd level clusters in our dataset, we have demonstrated the generalization of the continuation ratio model with two nested random effects, fitting the model using MMLE. A referee noted that limited sample size limits the number of covariates that can be allowed into the model. However, in our example there were only two covariates of interest; operated versus non-operated leg and time. In addition a partition variable had to be introduced in the regression model to classify the four binary continuation ratio partitions. Due to only three covariates in the model, the above mentioned limitation of small sample size was not serious. Our final model including the three covariates as main effects and two interactions, as discussed in the results section, was numerically stable. However, we note that the interaction between the variable leg and the partition variable could not be analyzed due to the zero cell count problem that occurs when the data is spread over too many cells in case of limited sample size. In the GLMM analysis we have presented in this paper we assumed that the two nested random effects, at the subject and at the leg level respectively, are distributed as independent standard normal variables. The distribution of random effects is routinely assumed to be normal mainly for mathematical convenience (Song P, 2007). Recent research has focused on the impact of misspecification of random effects distribution on the maximum likelihood estimates for generalized linear mixed models (GLMM). For the case of linear mixed models (that corresponds to the identity link function for GLMM) the parameter estimates are rather robust with respect to deviation from normality of random effects. However, for random-intercept logistic models the estimates of the mean structure parameters can have substantial bias upon misspecification of random effects distribution in case of large random effects variance (Alonso A. et al. 2008). In our present analysis the maximum likelihood estimates of the variances of the two random intercepts are moderate (of the order of magnitude of 2 at the subject level and 3 at the leg level respectively) thereby suggesting that the potential bias in the estimates of fixed effects in case of misspecification of random effects distribution could be small. However, there can be considerable bias in the estimate of variance components in case of misspecification of the random effects distribution, thereby making it difficult to distinguish between the small or large variance scenarios (Alonso A. et al. 2008). This suggests the need for further research to investigate the impact of misspecification of random effects distribution on the estimates of fixed and random effects in a GLMM with two nested levels of clustering. Investigating the impact of the distribution of random effects on estimation of parameters in a GLMM with two nested levels of clustering will be taken up as future research.

10 40 Mixed Model Analysis of Ordinal Response Data An alternative approach to the solution of generalized linear mixed models (GLMM) is to turn to approximate quasi-likelihood solutions (Rodriguez and Goldman 1995, Breslow and Clayton 1993, Wolfinger and O Connell 1993). A SAS macro called GLIMMIX (Littell et al. 1996) uses a quasi-likelihood approach - restricted pseudolikelihood (REPL) suggested by Wolfinger and O Connell (1993) to find parameter estimates for the generalized linear mixed models and allows for multiple nested random effects. Other commercially available software such as MLwin (Goldstein et al. 1998) also use quasi-likelihood solutions of GLMM allowing for multiple random effects. However, the approximate quasi-likelihood solution can lead to downward bias in the estimates of fixed and random effects (variance of the distribution of random effects) especially in case of small cluster size (Rodriguez and Goldman 1995). Sabre (Stott et al. 1999) fits GLMM for 2- level binary, ordinal and count data using MMLE but allows only for a single case-specific random effect and does not allow for multiple nested random effects. PROC NLMIXED that is a recent update on SAS performs MMLE for random effects or mixed models, however it does not allow for nested random effects. Few authors Ten Have et al. (1999), Raman and Hedeker (2005) and Liu and Hedeker (2006) have presented analysis of binary or ordinal responses with two nested levels of clustering using MMLE, however as mentioned earlier the latter two references used a cumulative logit model for the ordinal responses. In this paper we have presented an analysis of ordinal response data with two nested levels of clustering using the continuation ratio method, and fitted the model using MMLE. Moreover, we have also explored interaction between an explanatory variable of interest and the partition variable in the continuation ratio model. The results of our GLMM analysis of the unilateral spinal anesthesia trial data indicate a significant interaction between factors leg and time. The operated leg progressed to higher degrees of paralysis more rapidly than the non-operated leg immediately after initiation of spinal anesthesia, with both legs leveling off at later times. The GLMM approach for the analysis reported in this paper intricately models the progression with time to higher degrees of motor level paralysis for both legs through the continuation ratio method, starting with no paralysis at baseline, and gives insight into the mechanism of progression through the interaction of time with different binary partitions of the ordinal response. As an alternative to GLMM for accounting the clustering structure of the data, one may fit a population-averaged model, such as Generalized Estimating Equations (GEE) (Mancl and DeRouen 2001, Qaqish and Liang 1992).The focus of a population averaged model is the marginal expectation of the response and the parameters are interpreted regardless of cluster membership. The mixed model is a subject-specific model that accounts for the heterogeneity across subjects (and across legs in our example) by addition of random effects in the logit. The parameter estimates from a subject-specific model are interpreted in terms of effect of covariates on an average (or a typical) subject (Omar and Thompson 2000, Vonesh and Chinchilli 1997, Zeger et al. 1988). A subjectspecific interpretation seems relevant with regards to the objective of our analysis to assess the effect of operated versus non-operated leg and of time on the motor level of an average (or a typical) patient. A subject specific model stratifies subjects according to

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13 Sadia Mahmud et al. 43 Table 1: Contingency table of ordinal Bromage scale categories crossed with Leg for each of 7 time points: cell counts (and percentages) TIME (minutes) 0 LEG 5 LEG 10 LEG 15 LEG 20 LEG 25 LEG 30 LEG MOTOR LEVEL non-operated 30 (100%) 0(0%) 0(0%) 0(0%) 0(0%) operated 30(100%) 0(0%) 0(0%) 0(0%) 0(0%) non-operated 7(23.3%) 10(33.3%) 11(36.7%) 2(6.7%) 0(0%) operated 1(3.3%) 0(0%) 2(6.7%) 10(33.3%) 17(56.7%) non-operated 6(20.0%) 6(20.0%) 11(36.7%) 5(16.7%) 2(6.7%) operated 0(0%) 0(0%) 1(3.3%) 5(16.7%) 24(80%) non-operated 5(16.7%) 4(13.3%) 9(30.0%) 6(20.0%) 6(20.0%) operated 0(0%) 0(0%) 0(0%) 1(3.3%) 29(96.7%) non-operated 5(16.7%) 3(10.0%) 9(30.0%) 5(16.7%) 8(26.7%) operated 0(0%) 0(0%) 0(0%) 1(3.3%) 29(96.7%) non-operated 5(16.7%) 3(10.0%) 8(26.7%) 3(10.0%) 11(36.7%) operated 0(0%) 0(0%) 0(0%) 1(3.3%) 29(96.7%) non-operated 5(16.7%) 3(10.0%) 8(26.7%) 3(10.0%) 11(36.7%) operated 0(0%) 0(0%) 0(0%) 1(3.3%) 29(96.7%)

14 44 Mixed Model Analysis of Ordinal Response Data Table 2: Parameter (and standard Error) Estimates for continuation ratio logit (CRL), CRL + random effects (RE) CRL CRL + RE CRL+RE (with partition*log e (time+1)) Intercept (0.43) (1.46) (2.10) opleg a (0.75) (1.35) 0.61 (1.56) log e (time+1) 0.96 (0.14) ** 3.01 (0.33) ** 3.24 (0.58) ** opleg* log e (time+1) 1.94 (0.34) ** 3.92 (0.64) ** 4.02 (0.73) ** partition2 b (0.26) 1.78 (0.41) ** 5.09 (1.78) ** partition3 b 1.35 (0.26) ** 5.15 (0.60) ** 5.62 (1.86) ** partition4 b 1.26 (0.31) ** 7.15 (0.86) ** 6.17 (1.84) ** partition2 * log e (time+1) (0.62) * partition3 *log e (time+1) (0.65) partition4 *log e (time+1) 0.98 (0.71) subject 2.01 (0.80) * 2.21 (0.69) ** legsubject 2.83 (0.53) ** 3.42 (0.60) ** Log Likelihood *p-value < 0.05 **p-value < 0.01 a. opleg = 1 for operated leg; 0 otherwise, b. dummy variables for the partition variable, partition level 1 (k=1) is taken as reference.

15 Sadia Mahmud et al. 45 Fig. 1: The interaction between time and leg; continuation ratio logit versus time 1 Continuation ratio logit versus time partition leg non-operated operated operated non-operated logit time

16 46 Mixed Model Analysis of Ordinal Response Data Fig. 2: The interaction between time and partition variable; continuation ratio logit versus time. Continuation ratio logit versus time non-operated leg operated partition logit time