Tanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 DOI 10.1186/s12874-016-0251-yRESEARCH ARTICLE Open AccessAutoregressive transitional ordinal modelto test for treatment effect in neurologicaltrials with complex endpointsLorenzo G. Tanadini1*, John D. Steeves2, Armin Curt3 and Torsten Hothorn1AbstractBackground: A number of potential therapeutic approaches for neurological disorders have failed to provideconvincing evidence of efficacy, prompting pharmaceutical and health companies to discontinue their involvementin drug development. Limitations in the statistical analysis of complex endpoints have very likely had a negativeimpact on the translational process.Methods: We propose a transitional ordinal model with an autoregressive component to overcome previouslimitations in the analysis of Upper Extremity Motor Scores, a relevant endpoint in the field of Spinal Cord Injury.Statistical power and clinical interpretation of estimated treatment effects of the proposed model were compared toroutinely employed approaches in a large simulation study of two-arm randomized clinical trials. A revisitation of a keyhistorical trial provides further comparison between the different analysis approaches.Results: The proposed model outperformed all other approaches in virtually all simulation settings, achieving onaverage 14 % higher statistical power than the respective second-best performing approach (range: -1 %, +34 %).Only the transitional model allows treatment effect estimates to be interpreted as conditional odds ratios, providingclear interpretation and visualization.Conclusion: The proposed model takes into account the complex ordinal nature of the endpoint underinvestigation and explicitly accounts for relevant prognostic factors such as lesion level and baseline information.Superior statistical power, combined with clear clinical interpretation of estimated treatment effects and widespreadavailability in commercial software, are strong arguments for clinicians and trial scientists to adopt, and further extend,the proposed approach.Keywords: Upper extremity motor scores, Summed overall score, Multivariate ordinal endpoints, Proportional oddsmodel, Statistical power, Spinal cord injury, Sygen®trial, Rasch models, Latent variable modelsBackgroundNeurological research is responsible for the investigationof many devastating disorders such as stroke, Alzheimer’sand Parkinson’s diseases. In terms of health costs, brain-related disorders are a greater socio-economic burdenthan cancer, cardiovascular diseases and diabetes com-bined [1], with yearly costs for the European societyestimated at almost 400 billion e [2].*Correspondence: lorenzo.tanadini@uzh.ch1Department of Biostatistics; Epidemiology, Biostatistics and PreventionInstitute; University of Zurich, Hirschengraben 84, 8001 Zurich, SwitzerlandFull list of author information is available at the end of the articleDespite several therapeutic approaches [3–6] based onrecent discoveries of cellular and molecular processes ofdegeneration, but also spontaneous regeneration follow-ing injury, pharmaceutical and health companies havebeen withdrawing from neuroscience, as a number oftrials intended to show efficacy of treatments for neu-rological disorders failed [7]. In the field of Spinal CordInjury (SCI), four decades after the first pharmacologicaltreatment of acute injuries [8], the promises of preclini-cal discoveries have yet to be translated into a standardtreatment [9].To streamline the translational process, the Interna-tional Campaign for Cures of Spinal Cord Injury Paralysis© The Author(s). 2016 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to theCreative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.Tanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 Page 2 of 13(ICCP) appointed in 2007 an international panel with thetask to reviewing strengths and weaknesses of clinicaltrials in spinal cord injury. Their recommendations forthe planning and conduction of future trials were con-densed in a series of publications [10–13], which stronglyinfluenced the conception of clinical trials thereafter [14].Nonetheless, the ICCP reviews [10–13] did not solicitthe application of the most appropriate and recent statis-tical techniques available for the analysis of complex SCItrial endpoints, and many clinical trials failed to do sotoo [15–19].In fact, virtually all routinely performed clinical assess-ments in spinal cord injury are measured on ordinalscales, which are characterized by an arbitrary numer-ical score establishing a ranking of observations. Thedifference between two following ranks is by no meansbound to be equivalent across the range of the scale,preventing standard operations such as addition, andmaking the use of statistical methods developed forcontinuous endpoints inappropriate. Despite this, clini-cal trials designed and powered for a primary ordinalendpoint often resorted to adding several ordinal end-points to form a single overall summed score, which is insome cases subsequently collapsed to a binary outcome[15–19]. These approaches have been shown to be inap-propriate in a number of aspects [20], and practical con-sequences such as biased parameter estimates, misleadingassociations and loss of power are some of the knownconsequences of assuming metric properties for ordinalendpoints [21–23].In this study, we propose for the first time in SCI atransitional ordinal model with an autoregressive com-ponent for testing for treatment effect on a multivariateordinal endpoint such as the Upper Extremity MotorScores (UEMS), while comparing it to current analy-sis approaches in terms of statistical power and clinicalinterpretation of treatment effect estimates.MethodsThe objective was to propose a new approach to theanalysis of complex ordinal endpoints in neurologicalclinical trials, and provide statistical power comparisonsof procedures for treatment effect testing. Two-armedRandomized Clinical Trials (RCT) with specific levelsof experimental conditions were generated and analysed.Current approaches to the analysis of multivariate ordi-nal endpoints such as the Upper Extremity Motor Scores(UEMS) were compared to the proposed autoregressivetransitional ordinal model. The proposed approach mod-els the transition, e.g. the change in UEMS distribution,from trial baseline to trial end. The autoregressive termof the model describes the anatomical structure of thespinal cord by postulating a direct dependency betweencontiguous segments.Data source and trial endpointThe data utilized in this study was extracted from theEuropean Multicenter Study about Spinal Cord Injury(EMSCI, ClinicalTrials.gov Identifier: NCT01571531,www.emsci.org). EMSCI tracks the functional and neu-rological recovery of patients during the first year afterspinal cord injury in a highly standardized manner. Allpatients gave written informed consent. The ethical com-mittee of the Canton of Zurich, Switzerland, has pre-viously approved the EMSCI project, upon which thisproject is based, and the approval is also valid for anystatistical analysis/re-analysis.To reflect the time frame of a possible future clinicaltrial, we considered baseline (within 2 weeks after injury,t = 1) and one follow-up (6 months after injury, t =2) examination. For this simulation study, we extractedand utilized records of N=405 patients with a MotorLevel (ML) defined between spinal segments C5-T1 (seeAdditional file 1 for details) and with available baselineinformation.The trial endpoint considered is the Upper ExtremityMotor Scores. UEMS represents a subset of the Interna-tional Standards for Neurological Classification of SpinalCord Injury (ISNCSCI) [24] and describes themuscle con-traction force for 10 key muscles on the arms and hands (5on each body side), each one being rated on a 6-point ordi-nal scale (0: total paralysis, through 5: active movementagainst full resistance, see Additional file 1 for details).Accordingly, Yi,m,t is the muscle contraction score forpatient i (i = 1, . . . , n) and key muscle m (m = 1, . . . , 10)measured at time point t (t = 1, 2). Each key muscleYi,m,t is therefore an ordinal variable with k = 6 levels0 < 1 < . . . < 5, and UEMS is a multivariate ordi-nal endpoint. The chosen endpoint is particularly relevantin SCI. A change in total UEMS over trial period hasbeen employed repeatedly in clinical trials [15, 19] andhas been suggested to correlate with changes in activitiesof daily living that rely on recovery of upper extremityfunction [25].RCT simulationAn autoregressive transitional ordinal model of the form:logit[P(yi,m,2 ≤ k)] = αj + βlev xlev,i,m,1 + βbase ybase,i,m,1+ βauto yauto,i,m−1,2(1)was fitted on the EMSCI data. αj are the k − 1 = 5 inter-cept parameters, xlev is a 10-level nominal factor denotingthe combination of Motor Level and the distance fromthe Motor Level to the key muscle m being analysed,expressed as number of key muscles along the spine (ref-erence: motor level: cervical C5, distance: -1 (first muscleTanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 Page 3 of 13below the level)), ybase,i,m,1 is the ordered factor for base-line motor score of key muscle m, and yauto,i,m−1,2 is theordered factor for motor score of the key muscle justabove the one being analysed at t = 2. The autoregres-sive term of the model describes the anatomical structureof the spinal cord, and postulates that the motor score ofa given key muscle depends on the Motor Score of the keymuscle just rostral to it. As a consequence, the observedpattern of lower motor scores with increasing distancefrom the ML is reproduced. In accordance with the abovedescription, Eq. 1 simulated and analysed only key mus-cle score below the Motor Level. Motor scores yi,m,2 forkey muscles at ML were multinomially sampled from cor-responding observed EMSCI frequencies at Motor Level,while motor scores yi,m,2 for key muscles above the MLwere given the maximal score.The parameter estimates recovered from the modelspecified in Eq. 1 describe the spontaneous neurologicalrecovery for patients under standard of care and were sub-sequently used to simulate participants in the control armof the trial. From the EMSCI data we also computed theobserved frequencies of Motor Level combinations for theleft and right body side at baseline. Given that patientshaving both left and right ML at the lowest UEMS keymuscles T1 are very rare (3 % in our EMSCI sample) anddo not contribute to the analysis (no key muscles in theUEMS below the ML), they were not included into thesimulation.Equation 1 models the spontaneous neurological recov-ery for patients under standard of care. We introducedan additional parameter βtrt representing a postulatedtreatment effect, leading to an autoregressive transitionalordinal model of the form:logit[P(yi,m,2 ≤ k)] = αj + βlev xlev,i,m,1 + βbase ybase,i,m,1+ βauto yauto,i,m−1,2 + βtrt xtrt,i,1(2)As previously defined, αj are the k − 1 = 5 inter-cept parameters, xlev is a 10-level nominal factor denotingthe combination of Motor Level and the distance fromthe Motor Level to the key muscle m being analysed,expressed as number of key muscles along the spine (ref-erence: Motor Level: C5, distance: -1), ybase,i,m,1 is theordered factor for baseline motor score of key musclem, yauto,i,m−1,2 is ordered factor for motor score of thekey muscle just above the one being analysed at t = 2,and xtrt is an indicator for treatment arm with placebo asreference.The autoregressive term of the model describes theanatomical structure of the spinal cord, and postulatesthat the motor score of a given key muscle dependson the motor score of the key muscle just rostral to it.As a consequence, the observed pattern of lower motorscores with increasing distance from the ML is repro-duced. Besides the postulated treatment effect βtrt, whichis set to different values depending on the simulation set-tings, all other parameters in Eq. 2 were kept equal tothe estimates recovered by fitting Eq. 1 to the EMSCIdata.We thus simulated randomized clinical trials with twotreatment arms and specific levels of experimental condi-tions. To cover possible SCI early phase as well as phaseIII settings, we generated total trial sample sizes of 50, 75,100, 125, 150, 175, 200 participants. To our knowledge,there is to date no publication on the magnitude of possi-ble treatment effects for UEMS which could have guidedus in defining more tailored scenarios. We therefore pos-tulated a rather wide range of six possible treatmenteffects (from no treatment effect (βtrt = 0.0 = log(1))to strong treatment effect (βtrt = 0.4055 = log(1.5))in 0.1 steps). A total of 42 scenarios resulted from sim-ulating all possible combinations of the 7 trial samplesizes and 6 possible treatment effects considered. Beinga proportional odds model, the exponentiated βtrt can beinterpreted as conditional Odds Ratio (OR) between trialarms, meaning that, conditional on all other prognosticfactors being equal, it specifies the ratio of the odds for akey muscle to achieve a motor score of less than or equalto k in the treatment arm divided by the same odds inthe control arm. OR is a statistically sensible and clinicalwidely accepted way of quantifying effects of categoricalvariables.The 42 trial scenarios resulting from all combinations of7 trial sample sizes and 6 possible treatment effects weresimulated in the following way:1. Right and left Motor Levels for the hypothesizednumber of trial participants were drawn from amultinomial distribution with category probabilitiesset to the corresponding observed EMSCIfrequencies.2. Baseline UEMS for each trial participant weresampled with replacement from all EMSCI patientshaving the same left-right ML constellation.3. Each simulated participant was randomly allocatedto either the control or the treatment arm with a 1:1allocation scheme.4. UEMS at six months for the key muscle at ML weredrawn from a multinomial distribution with categoryprobabilities set to the corresponding observedEMSCI frequencies.5. UEMS at six months below the ML were simulatedusing the previously fitted model for spontaneousrecovery (Eq. 1) for participants in the control arm,and the same model with the addition of a postulatedtreatment effect (Eq. 2) for participants in thetreatment arm of the trial.Tanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 Page 4 of 136. Each one of the 42 trial scenarios was replicated 1000times.7. A battery of 6 different tests for treatment effect (seebelow “Endpoint analysis approaches” Section) wereapplied to each simulated trial.8. The statistical power = P(reject H0|H1 is true) wasestimated as the fraction of significant tests fortreatment effect at the nominal level 0.05 among the1000 replications.Endpoint analysis approachesIn neurology in general, and SCI in particular, very com-mon approaches to the analysis of UEMS or similar end-points are as the total sum of all motor scores Y ∗i,2 =∑10m=1 Yi,m,2 or as difference between two time pointsY ∗∗i =∑10m=1 Yi,m,2 − Yi,m,1. Accordingly, treatment effectfor UEMS was tested with:t-test: t-test for Y ∗i,2, comparing mean total UEMS in thetwo treatment groups.t-test delta: t-test for Y ∗∗i , comparing the mean differ-ence in total UEMS from baseline to the end of thetrial between the two treatment groups.ANCOVA: Analysis of covariance for Y ∗i,2, comparingmean total UEMS in the two treatment groups withbaseline total UEMS Y ∗i,1 as controlling continuousvariable.Even though not commonly done in SCI, we considerednecessary that the Motor Level should be incorporatedinto the analysis of motor function. In fact, its impor-tance has been reported before [26, 27]. We thereforeapplied a conditional test of independence between out-come and treatment arm which was stratified accordingto the Motor Level of each trial participant. We pre-dicted that this approach would perform better than theprevious, not stratified ones, and explored the possibil-ity to utilise them as “ad hoc” approach for the analysisof UEMS. Accordingly, treatment effect for UEMS wastested with:i-test: stratified independence test for Y ∗i,2, comparingtotal UEMS in the two treatment groups.i-test delta: stratified independence test for Y ∗∗i , com-paring the difference in total UEMS from baselineto the end of the trial between the two treatmentgroups.Both tests are implemented in the R add-on package coin[28, 29].The last approach for the analysis of UEMS in a RCT isa model that takes into account the ordinal nature of eachkey muscle and explicitly incorporates baseline UEMS aswell as ML into the analysis:transitional: transitional ordinal model for Yi,m,2 of theform specified in Eq. 2, comparing the shift in motorscore probabilities associated with treatment.The proposedmodel is a proportional odds model with anautoregressive component. The latter takes into accountthe spatial orientation of the key muscles along the spinalcord by postulating a direct dependency of adjacent spinalsegments. As a consequence, the observed pattern oflower Motor Scores with increasing distance from the MLis reproduced. This model was fitted using function polrfrom the R add-on packageMASS [30, 31].The parameter βtrt, which quantifies the treatmenteffect on the link scale, is the focus of the proposed model.Its significance testing was based on a permutation test[32, 33], where the distribution of the test statistics underH0 (no treatment effect) was based on refitting the samemodel 1000 times after randomly rearranging the labelsfor arm allocation. This type of statistical significance testdoes not rely on any distributional assumption. In addi-tion, by permuting trial arm allocation at participant level,we accounted for the hierarchical structure of the dataanalysed, where multiple key muscles are measured onthe same participant. All computations were performed inthe R system for statistical computing [34], version 3.1.3.The R code implementing the simulation study is availableonline (doi: http://dx.doi.org/10.5281/zenodo.47600).Revisiting a key SCI trialAs a practical application, we analysed a subset of thedata collected during a past clinical trial. The Sygen®trial recruited N=760 SCI participants in 28 centres inNorth-America in a 5-year period between 1992 and 1997[17, 35, 36]. Sygen ®is a naturally occurring compound incell membranes which has been associated with neuro-protective and regenerative effects in a number of exper-imental models and early-phase human trials. The trial isan example where a promising therapeutic approach wasfinally abandoned, as no significant treatment effect couldbe assessed on the primary endpoint despite a consid-erable final sample size (N=760). The primary endpointassessed the overall neurological status of a patient andwas defined as a dichotomization derived from an ordinalscale (see [36] for the exact definition). The primary end-point was analysed bymeans of logistic regression. Severalancillary analyses were performed and mostly preferredthe treatment arm, even though the differences were notalways statistically significant. To our knowledge, no anal-ysis performed at the level of motor scores of the upperextremity key muscles UEMS as reported here have beenpublished.We revisited the trial by testing for treatment effect onthe UEMS with all six approaches outlined before (see“Endpoint analysis approaches” Section). The proposedTanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 Page 5 of 13autoregressive transitional ordinal model (Eq. 2) can beeasily fitted as proportional odds model to the segment-wise UEMS data in the long format. The autoregressivecomponent yi,m−1,2 can be incorporated by shifting thesix-month, muscle-wise UEMS entries so as to be alignedto the key muscle yi,m,2 just caudal to them.To reflect our simulation study, we selected participantswith a ML between C5 - C8 (T1 were discarded, becausethere is no key muscle caudal to the ML on the UEMS),and considered only patients treated with a low dosage(the original trial had two treatment doses, the higher ofwhich was abandoned during the study). After patientsselection, we analysed a finale sample of N=284 par-ticipants, 127 (45 %) of which in the control arm. Thisanalysis is intended to give an example of the applica-tion of the proposed transitional ordinal model, but is notintended and should not be taken as a definitive conclu-sion about the value or outcome of the trial. Given thestrongly selected patients sample utilised, the differentendpoint analysed and the different scope of our analysis,generalizations of this type cannot be drawn.ResultsRCT simulationFor the purpose of this study, we simulated 1000 timeseach one of the 42 different combinations of trials sizeand postulated treatment effect. Statistical power, whichis defined as the probability of rejecting the H0 of notreatment when there is in fact a treatment effect, wasestimated as the fraction of this 1000 iterations wherethe test for treatment effect resulted significant at the0.05 level. Table 1 reports the statistical power of alltreatment testing approaches for all simulated settings.Figure 1 shows the statistical power of all six approachesfor the intermediate treatment effect simulated. Figure 2displays graphically the statistical power of all treatmenttesting approaches for all simulated settings. The nomi-nal level 0.05 was maintained by all approaches when notreatment effect was introduced in the simulation, mak-ing further comparisons between different approachesstraightforward.For the smallest treatment effect βtrt = 0.0953 =log(1.1), all six tests for treatment effect showed alow power, never exceeding P(reject H0|H1 is true)≤ 0.135. The transitional ordinal model was nonethe-less superior to all other approaches in virtually everytrial size setting, its power point estimates averaging2.3 % higher than the respective second best-performingapproach.Already at the next higher treatment effect simulatedβtrt = 0.1823 = log(1.2), the transitional ordinal modelshowed roughly twice as much power as the second-bestperforming approach, though it did not exceed P(rejectH0|H1 is true) ≤ 0.36. This held for all simulation settingsexcept the smallest sample size. Statistical power pointestimates for the transitional ordinal model were onaverage 10.3 % higher than the respective second best-performing approach.In the settings with median simulated treatment effectβtrt = 0.2624 = log(1.3) shown in Fig. 1, the transi-tional ordinal model was superior for all trial sizes. Powerpoint estimates for the proposed model were on average19.4 % higher than the respective second best-performingapproach, with this difference in performance increasingwith increasing trial size.With the simulated treatment effect of βtrt = 0.3365 =log(1.4), the transitional ordinal model had superior sta-tistical power of 26.3 % on average, compared to therespective second best-performing approach, with thisdifference increasing with increasing trial size.For the largest simulated treatment effect of βtrt =0.4055 = log(1.5), the transitional ordinal model had anaverage superior statistical power of 27.9 %, compared tothe respective second best-performing approach. The dif-ference in performance increased strongly up to trial sizeN=100, but then declined with larger sizes.Overall, despite a comparably poor performance of allapproaches for small simulated treatment effects, a stablepattern in the ranking of performance emerged: the pro-posed transitional ordinal approach provided best powerresults in virtually all settings. ANCOVA was usuallythe second-best approach, closely followed by the inde-pendence test on the difference of UEMS from baselineY ∗∗i , the similarly performing t-test on the differenceof UEMS from baseline Y ∗∗i and the independence teston the UEMS after six months Y ∗i,2. The t-test on theUEMS after six months Y ∗i,2 performed worst in almost allsettings.Revisiting a key SCI trialWe analysed a subset of the data collected during theSygen ®trial [17, 35, 36]. To our knowledge, no analysison this data has been performed at the level of motorscores of the upper extremity key muscles UEMS asreported here. The results of the six analysis approaches(see Endpoint analysis approaches section) are reportedhere:t-test: No significant difference in the estimated meansμ̂ctrl = 30.370 and μ̂trt = 30.170 of UEMS at 6months between trial arms: t(275)=0.130, p−value =0.896.t-test delta: No significant difference in the estimatedmean change μ̂ctrl = 11.978 and μ̂trt = 10.540 ofUEMS between trial arms: t(259)=1.239, p−value =0.216.ANCOVA: No significant difference in the estimatedmeans of UEMS at 6 months between trial arms,Tanadinietal.BMCMedicalResearchMethodology (2016) 16:149 Page6of13Table 1 Statistical power for all simulation settings. Point estimates, as well as Wilson confidence intervals are reported for all analysis approachesSize Treatment OR T-test CI lower CI upper T-test delta CI lower CI upper I-test CI lower CI upper I-test delta CI lower CI upper ANCOVA CI lower CI upper Transitional CI lower CI upper50 0.0000 1.0 0.053 0.041 0.069 0.052 0.040 0.068 0.051 0.039 0.066 0.042 0.031 0.056 0.046 0.035 0.061 0.050 0.038 0.06575 0.0000 1.0 0.048 0.036 0.063 0.050 0.038 0.065 0.052 0.040 0.068 0.051 0.039 0.066 0.053 0.041 0.069 0.052 0.040 0.068100 0.0000 1.0 0.047 0.036 0.062 0.046 0.035 0.061 0.054 0.042 0.070 0.046 0.035 0.061 0.048 0.036 0.063 0.045 0.034 0.060125 0.0000 1.0 0.049 0.037 0.064 0.052 0.040 0.068 0.040 0.030 0.054 0.056 0.043 0.072 0.056 0.043 0.072 0.057 0.044 0.073150 0.0000 1.0 0.056 0.043 0.072 0.044 0.033 0.059 0.041 0.030 0.055 0.040 0.030 0.054 0.050 0.038 0.065 0.040 0.030 0.054175 0.0000 1.0 0.050 0.038 0.065 0.050 0.038 0.065 0.043 0.032 0.057 0.053 0.041 0.069 0.042 0.031 0.056 0.047 0.036 0.062200 0.0000 1.0 0.051 0.039 0.066 0.052 0.040 0.068 0.046 0.035 0.061 0.053 0.041 0.069 0.056 0.043 0.072 0.048 0.036 0.06350 0.0953 1.1 0.057 0.044 0.073 0.060 0.047 0.076 0.063 0.050 0.080 0.052 0.040 0.068 0.062 0.049 0.079 0.049 0.037 0.06475 0.0953 1.1 0.055 0.042 0.071 0.056 0.043 0.072 0.051 0.039 0.066 0.069 0.055 0.086 0.049 0.037 0.064 0.086 0.070 0.105100 0.0953 1.1 0.057 0.044 0.073 0.071 0.057 0.089 0.061 0.048 0.078 0.071 0.057 0.089 0.071 0.057 0.089 0.106 0.088 0.127125 0.0953 1.1 0.074 0.059 0.092 0.068 0.054 0.085 0.082 0.067 0.101 0.075 0.060 0.093 0.081 0.066 0.100 0.094 0.077 0.114150 0.0953 1.1 0.063 0.050 0.080 0.070 0.056 0.088 0.062 0.049 0.079 0.075 0.060 0.093 0.078 0.063 0.096 0.116 0.098 0.137175 0.0953 1.1 0.066 0.052 0.083 0.071 0.057 0.089 0.069 0.055 0.086 0.079 0.064 0.097 0.073 0.058 0.091 0.117 0.099 0.138200 0.0953 1.1 0.072 0.058 0.090 0.101 0.084 0.121 0.080 0.065 0.098 0.092 0.076 0.112 0.099 0.082 0.119 0.135 0.115 0.15850 0.1823 1.2 0.068 0.054 0.085 0.090 0.074 0.109 0.065 0.051 0.082 0.091 0.075 0.110 0.093 0.077 0.113 0.111 0.093 0.13275 0.1823 1.2 0.096 0.079 0.116 0.095 0.078 0.115 0.106 0.088 0.127 0.100 0.083 0.120 0.107 0.089 0.128 0.164 0.142 0.188100 0.1823 1.2 0.106 0.088 0.127 0.098 0.081 0.118 0.112 0.094 0.133 0.099 0.082 0.119 0.114 0.096 0.135 0.226 0.201 0.253125 0.1823 1.2 0.115 0.097 0.136 0.127 0.108 0.149 0.135 0.115 0.158 0.132 0.112 0.154 0.145 0.125 0.168 0.261 0.235 0.289150 0.1823 1.2 0.134 0.114 0.157 0.155 0.134 0.179 0.138 0.118 0.161 0.167 0.145 0.191 0.171 0.149 0.196 0.298 0.270 0.327175 0.1823 1.2 0.134 0.114 0.157 0.161 0.140 0.185 0.166 0.144 0.190 0.177 0.155 0.202 0.182 0.159 0.207 0.331 0.303 0.361200 0.1823 1.2 0.145 0.125 0.168 0.189 0.166 0.214 0.175 0.153 0.200 0.191 0.168 0.217 0.215 0.191 0.242 0.360 0.331 0.39050 0.2624 1.3 0.106 0.088 0.127 0.128 0.109 0.150 0.101 0.084 0.121 0.127 0.108 0.149 0.142 0.122 0.165 0.226 0.201 0.25375 0.2624 1.3 0.120 0.101 0.142 0.152 0.131 0.176 0.140 0.120 0.163 0.153 0.132 0.177 0.173 0.151 0.198 0.277 0.250 0.306100 0.2624 1.3 0.145 0.125 0.168 0.208 0.184 0.234 0.178 0.156 0.203 0.200 0.176 0.226 0.234 0.209 0.261 0.383 0.353 0.414125 0.2624 1.3 0.185 0.162 0.210 0.214 0.190 0.240 0.204 0.180 0.230 0.237 0.212 0.264 0.261 0.235 0.289 0.474 0.443 0.505150 0.2624 1.3 0.192 0.169 0.218 0.248 0.222 0.276 0.236 0.211 0.263 0.269 0.242 0.297 0.265 0.239 0.293 0.528 0.497 0.559175 0.2624 1.3 0.229 0.204 0.256 0.275 0.248 0.303 0.257 0.231 0.285 0.299 0.271 0.328 0.325 0.297 0.355 0.595 0.564 0.625200 0.2624 1.3 0.280 0.253 0.309 0.329 0.301 0.359 0.321 0.293 0.351 0.367 0.338 0.397 0.392 0.362 0.423 0.673 0.643 0.70150 0.3365 1.4 0.119 0.100 0.141 0.154 0.133 0.178 0.141 0.121 0.164 0.153 0.132 0.177 0.161 0.140 0.185 0.303 0.275 0.33275 0.3365 1.4 0.184 0.161 0.209 0.195 0.172 0.221 0.212 0.188 0.238 0.209 0.185 0.235 0.240 0.215 0.267 0.410 0.380 0.441100 0.3365 1.4 0.221 0.196 0.248 0.253 0.227 0.281 0.260 0.234 0.288 0.288 0.261 0.317 0.302 0.274 0.331 0.580 0.549 0.610125 0.3365 1.4 0.290 0.263 0.319 0.314 0.286 0.343 0.308 0.280 0.337 0.339 0.310 0.369 0.396 0.366 0.427 0.692 0.663 0.720Tanadinietal.BMCMedicalResearchMethodology (2016) 16:149 Page7of13Table 1 Statistical power for all simulation settings. Point estimates, as well as Wilson confidence intervals are reported for all analysis approaches (Continued)Size Treatment OR T-test CI lower CI upper T-test delta CI lower CI upper I-test CI lower CI upper I-test delta CI lower CI upper ANCOVA CI lower CI upper Transitional CI lower CI upper150 0.3365 1.4 0.309 0.281 0.338 0.376 0.347 0.406 0.374 0.345 0.404 0.404 0.374 0.435 0.442 0.411 0.473 0.736 0.708 0.762175 0.3365 1.4 0.329 0.301 0.359 0.399 0.369 0.430 0.396 0.366 0.427 0.434 0.404 0.465 0.463 0.432 0.494 0.800 0.774 0.824200 0.3365 1.4 0.407 0.377 0.438 0.464 0.433 0.495 0.445 0.414 0.476 0.495 0.464 0.526 0.536 0.505 0.567 0.857 0.834 0.87750 0.4055 1.5 0.162 0.140 0.186 0.178 0.156 0.203 0.196 0.173 0.222 0.190 0.167 0.215 0.210 0.186 0.236 0.392 0.362 0.42375 0.4055 1.5 0.238 0.213 0.265 0.263 0.237 0.291 0.281 0.254 0.310 0.291 0.264 0.320 0.318 0.290 0.348 0.592 0.561 0.622100 0.4055 1.5 0.302 0.274 0.331 0.354 0.325 0.384 0.366 0.337 0.396 0.390 0.360 0.421 0.392 0.362 0.423 0.737 0.709 0.763125 0.4055 1.5 0.368 0.339 0.398 0.443 0.412 0.474 0.420 0.390 0.451 0.467 0.436 0.498 0.515 0.484 0.546 0.825 0.800 0.847150 0.4055 1.5 0.397 0.367 0.428 0.509 0.478 0.540 0.467 0.436 0.498 0.546 0.515 0.577 0.583 0.552 0.613 0.891 0.870 0.909175 0.4055 1.5 0.495 0.464 0.526 0.559 0.528 0.589 0.567 0.536 0.597 0.597 0.566 0.627 0.648 0.618 0.677 0.919 0.900 0.934200 0.4055 1.5 0.530 0.499 0.561 0.616 0.585 0.646 0.598 0.567 0.628 0.669 0.639 0.697 0.706 0.677 0.733 0.967 0.954 0.976Tanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 Page 8 of 13Fig. 1 Comparison of statistical power for the median treatmenteffect. The statistical power of all six approaches for treatment effecttesting are plotted against total trial size (1:1 randomization) for themedian simulated treatment effect βtrt = 0.2624 = log(1.3)controlling for baseline UEMS: β̂trt = −1.165, p −value = 0.307.i-test: No significant dependency between UEMS at 6months and treatment arm: Z=0.553, p−value=0.58.i-test delta: No significant dependency between changein UEMS and treatment arm: Z=1.525, p − value =0.127.transitional: No significant shift inmotor score probabil-ities associated with treatment arm: β̂trt = −0.197,p − value = 0.207.Summarizing, all six approached did not show signifi-cant results at the nominal level 0.05, but they all showeda tendency to less positive outcomes for patients in thetreatment arm. This analysis is intended to give an exam-ple of the application for the proposed transitional ordinalmodel, but is not intended and should not be taken as adefinitive conclusion about the value or outcome of thetrial.DiscussionThe aim of this simulation study was to compare severalapproaches of testing for treatment effect in two-armedRCT in a neurological setting. We therefore simulatedclinical trials with cervical SCI participants with specificlevels of experimental conditions and tested for treatmenteffect with six different approaches. Routinely employedanalysis approaches not only rely on strong assumptionsabout the properties of the endpoints being analysed, butwere also outperformed in virtually all settings by the theproposed autoregressive transitional ordinal model for theanalysis of UEMS.Adding ordinal endpoints to form a single overall score isgenerally not validCommon approaches to the analysis of UEMS and similarneurological endpoints are as the total sum of all motorscores Y ∗i,2 =∑10m=1 Yi,m,2 or as difference between twotime points Y ∗∗i =∑10m=1 Yi,m,2 − Yi,m,1.Whether it is appropriate to combine a set of ordinalvariables to generate a total score is usually not checkedin neurology [37]. It should nonetheless be a require-ment, as there are at least two strong assumptions relatedto the analysis of summed motor scores as a metricendpoint: unidimensionality and equal differences. Uni-dimensionality refers to the property of several scores tomeasure a single, common patient’s characteristic. Whilethere is some preliminary evidence that unidimension-ality holds for UEMS [38], the opposite was reportedfor both the Functional Independence Measure FIM [39],the Spinal Cord Independence Measure SCIM [40], asituation which is very likely to be found in functionalendpoints and Patients Reported Outcomes PRO. Equaldifferences imply that a unit change in motor scores rep-resent exactly the same clinical change, independently ofwhere the change took place on the scale (e.g. a changefrom 0 to 1 is assumed to be of the same magnitude asa change from 3 to 4 in motor scores), or of which keymuscle are considered (the previous example is assumedto hold even when the changes took place on differentkey muscles, say e.g. one proximal and one distal from thelesion level).The widely used method of adding up several ordi-nal endpoints to form a single overall score is thereforegenerally not valid with regard to the two assumptionsexemplified above, and has been repeatedly reportedin neurological and related physical functioning settings[39–44]. From a practical point of view, biased parameterestimates, as well as misleading associations and loss ofpower are some of the known consequences of assumingmetric property for ordinal endpoints [21–23]. There istherefore a compelling need to embrace statistical modelsspecifically designed for the analysis of complex ordinalendpoints.RCT simulationThe proposed autoregressive transitional ordinal modelis the first attempt in SCI to model and analyse a com-plex endpoint with a regression model which reflectsits ordinal nature and takes into account importantprognostic factors. The proposed model for the anal-ysis of UEMS in cervical SCI patients outperformedall other approaches in virtually all settings. The sensi-bly lower statistical power achieved by commonly usedapproaches, in addition to their implicit assumptions,indicate that their use as default analysis methods in notjustified.Tanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 Page 9 of 13Fig. 2 Contour plots of statistical power for all simulation settings. The statistical power of all testing approaches is represented using loess smoothapproximation. Contour curves visualize combinations of trial size and treatment effect with equivalent statistical power, which is reported asnumerical value. The colour key differentiates regions of low statistical power (violet) from regions of high statistical power (blue)Contrary to our expectations, a stratification ofthe t-test based on the Motor Level did not pro-vide a discernible improvement in statistical power(Table 1). In fact, even though blocked independencetests showed a slightly higher power than their corre-sponding t-tests (Fig. 2), the gain in power was notsuch that their application as “ad hoc” solution resultedsubstantiated.In terms of clinical interpretation of treatment effectestimates, we note that by applying the proposed model,the exponentiated treatment effect estimate β̂trt can beinterpreted as the conditional odds ratio between thetreatment and control trial arms, which is a common andaccepted way of quantifying treatment effect in the clini-cal setting. Even when the proportional odds assumptionis not fully met, it still provides an interpretable parame-ter that summarizes the treatment effect over all levels ofthe outcome [23]. In addition, the transitional model pro-vides motor score probabilities for each combination ofprognostic variables, making the direct comparison andvisual representation of treated and untreated participantsstraightforward (see Fig. 3).On the contrary, clear interpretation of the results pro-duced by common approaches is precluded by summedscores of suppositional metric endpoints, providing lit-tle insight for trial scientists and clinicians. Importantly,small and possibly localized treatment effects, which area hallmark of many neurological disorders, can be disen-tangled using ordinal approaches for motor scores, butbecome lost in the analysis of summed total scores.Finally, our simulation showed (Table 1) that a statisticalpower of 80 %, which is a common goal for clinical trialsplanners, is reached by the ordinal model only for largetrial size and large postulated treatment effects. As a totaltrial size of N=200 seems to currently represent the prac-tical upper limit for conducting SCI trials, the statisticaldetection of an existing treatment effect seems to rely ona rather strong effect. Further improvements of the ordi-nal model will likely result in lowered requirements fortreatment detection.Tanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 Page 10 of 13Fig. 3 Visualization of median treatment effect βtrt = 0.2624 = log(1.3). In contrast to all other analysis approaches, the transitional ordinal modelallows to graphically represent shifts in motor score distributions for any constellation of relevant prognostic factors, permitting a much moredetailed investigation of treatment effect. As illustrative example, represented is the distribution of motor score probabilities for participants in thecontrol (left panel) and treatment arm (right panel). Lower scores became less, while higher score became more probable in the treatment arm. Thetreatment effect βtrt = 0.2624 = log(1.3) corresponds to an Odds Ratio of OR=1.3.The specific constellation of prognostics factor represented refersto a C8 key muscle, with a Motor Level C5 (xlev=C5.-3), a baseline motor score of ybase,i,m,1 = 1, and an autoregressive component yauto,i,m−1,2 = 3 forthe motor score of the key muscle just above the one being reportedRevisiting a key SCI trialTo provide a concrete application of our approach, weanalysed a subset of participants of the Sygen® trial[17, 35, 36]. Many ancillary analyses in the original publi-cation were based on t-test and ANCOVA approaches andfavoured the treatment group over placebo [17]. In par-ticular, treated participants showed a faster initial recov-ery than control subjects, who nonetheless caught up atslightly later time points.On the subsample of patients we considered, no oneof the six approaches was significant at the conventionalnominal level 0.05. Nonetheless, all approaches showeda tendency towards negative effect of treatment on theUEMS, meaning that treated patient showed on average aslightly worse recovery than patients in the control arm.Especially for the ordinal approach, the results imply thatthe odds of participants in the treatment group of achiev-ing up to a given motor score were only eβ̂trt = 0.82 timesthe odds of a participant with similar characteristics inthe control arm, indicating a worse recovery for treatedpatients.The negative estimate of treatment effect in cervical par-ticipants is rather unexpected. The observed unbalancetoward more severe lesions in the treatment arm mayexplain at least in part these results, which nonethelessmight be examined more closely to rule out potentiallyunintended detrimental effects. Nevertheless, we retainthat generalizations of our results to the overall validity ofthe trial and its compound cannot be drawn.Are summed overall scores not “good enough” ?In our application, all six approaches presented deliveredcomparable results, namely statistically non-significantnegative trends for participants in the treatment arm. Onemay therefore wonder what the added value of an ordinalapproach like the proposed transitional ordinal model is.Briefly, routinely employed approaches based on summedoverall scores imply:• Unmet assumptions: adding ordinal endpoints toform a single overall score requires equal differencesacross all ordinal scales as well as unidimensionality.Tanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 Page 11 of 13Both assumptions are usually not further investigated[37], but the first can be rejected on medical reasonsonly, while the latter does not hold for several SCIendpoints (e.g. FIM [39], SCIM [40]).• Flawed inference and estimation: known practicalconsequences of assuming metric property forordinal endpoints are biased parameter estimates andmisleading associations [21–23].• Reduced statistical power: small and possibly localisedeffects are expected to be the hallmark of spinal cordinjury rehabilitation strategies. The simulationreported provide evidence for a much lower capacityof approaches based on summed scores to detectexisting treatment effects. Lower power alsotranslates in higher requirement for trial participants.• Unclear interpretation of treatment effect: a clearinterpretation of treatment effect estimates asconditional OR, which can be visualised for each keymuscle separately (see Fig. 3), is not possible forsummed scores.• Limited future extensions: future refinement ofroutinely employed approaches are strongly limitedby the underlying, inappropriate analysis approach.Instead, ordinal approaches, which are based on aregression framework, easily accommodates forextensions (e.g. further prognostic factors,interactions, localised effects).Concluding, from a theoretical point of view, routinelyemployed approaches have little scientific validity andhave been replaced by more rigorous approaches. Evenmore importantly, they are also potentially misleading onpractical terms. Our flexible model represents thereforean improved and pragmatic solution to the analysis of thistype of complex ordinal endpoints.Brain Injury: similar issues, similar solutionsWe observe that most of the discussion points we raisedlink to the report by the International Mission on Prog-nosis and Clinical Trial Design in Traumatic Brain InjuryTBI [45]. TBI is a related clinical field which faced verysimilar challenges, mainly related to the heterogeneity ofthe patient population, and had a similar history of clinicaltesting as SCI.In fact, TBI also experienced a disappointing progres-sion of clinical testing of treatment interventions in spiteof extremely promising pre-clinical data and early phasetrials. Maas et al. [45] reported that a key difficulty hasbeen the inherent heterogeneity TBI subjects, and that theobserved development was due, at least to some extent, tolimitations in the trial designs and analyses. Both aspectshave also been reported as hallmarks of SCI research.Summarizing, The TBI Mission solicited the TBI com-munity to [45]:• provide details of the major baseline prognosticcharacteristics• broaden inclusion criteria as much as is it compatiblewith the current understanding of the mechanisms ofaction of the intervention• incorporate pre-specified covariate adjustment intothe statistical analysis• use an ordinal approach for the statistical analysisA part from the first recommendation, which is mainlyconcerned with the way clinical studies are reported, thefollowing three points regard the planning and especiallythe analysis of clinical trials in TBI, and are implementedin this publication. Selection of patients is based only onthe initial Motor Level, which relates to the understand-ing of motor function. The proposed model (see Eq. 2)both include the most relevant covariates adjustment,namely baseline motor scores as well as motor lesion, anduses and ordinal approach for ordinal data based on theproportional odds model.Latent variable models: an improved, readily availableframeworkMore generally speaking, the statistical foundations ofregression models for ordinal endpoints were developedmore than 4 decades ago [46–48], and have ever sinceundergone a steady development. There is a huge body ofliterature pertaining to the analysis of ordinal variables,including Item Response Theory IRT and mixed-effectsmodels for ordinal variables [49]. Despite this develop-ment, most clinical trials in neurology still rely on sur-passed approaches [44], corroborating the negative trendof methodological errors related to the analysis of ordinalscales in medical research [50].The proposed transitional ordinal model (Eq. 2) is anextension of the well known proportional odds model(e.g. [51]). The latter can be seen as an important spe-cial case within the IRT framework, and is closely relatedto the Rasch model [46]. All these statistical models aregenerally referred to as latent variable models, becausethey find application in situations where a set of ordi-nal variables are seen as indicators of a latent variable.This latent variable is the main interest of the analysis,and, although it cannot be measured directly, it can beinferred from the available ordinal variables. The latentvariable approach seems both appropriate and appealingfor applications in the clinical setting, and the transitionalordinal model proposed draws a concrete link from SCI tolatent variable models. Further extensions of our approachcan be tailored to the analysis of other endpoints suchas functional assessments and PROs. In fact, the analy-sis of PRO, and the related trial powering based on Raschmodels has recently received much attention [52, 53].We believe that the transition from currently employedTanadini et al. BMCMedical ResearchMethodology  (2016) 16:149 Page 12 of 13analysis approaches to more sophisticated models withinthe readily available framework of latent variable mod-els would represent a great scientific progression for theplanning and analysis of complex neurological endpoints.ConclusionWe propose an autoregressive transitional ordinal modelfor the analysis of a specific SCI endpoint which takesinto account the complex ordinal nature of the endpointunder investigation and explicitly accounts for relevantprognostic factors. Superior statistical power in virtuallyall settings, combined with a clear clinical interpretationof treatment effect and widespread availability on com-mercial softwares, are strong arguments for clinicians andtrial scientists to adopt, and further refine, the proposedapproach.Additional fileAdditional file 1: International Standards for Neurological Classification ofSpinal Cord Injury. (PDF 935 kb)AbbreviationsEMSCI: European multicenter study about spinal cord injury; FIM: Functionalindependence measure; ICCP: International campaign for cures of spinal cordinjury paralysis; IRT: Item response theory ISNCSCI: Int. standards forneurological classification of spinal cord injury; ML: Motor level; OR: Odds ratio;PRO: Patient reported outcomes; RCT: Randomized clinical trial; SCI: Spinalcord injury; SCIM: Spinal cord independece measure; TBI: Traumatic braininjury; UEMS: Upper Extremity Motor ScoresAcknowledgementsWe appreciate the continuous assistance of René Koller with the EMSCIdatabase.FundingLGT was partially financially supported by the International Foundation forResearch in Paraplegia. The Foundation had no influence on any aspect of thispublication.Availability of data andmaterialsThe datasets supporting the conclusions of this article are not publiclyavailable. Interested researcher may apply for data access to the responsibleorganization, which is usually granted for research-only purposes. The R codeimplementing the simulation study is freely available (doi:http://dx.doi.org/10.5281/zenodo.47600).Authors’ contributionsLGT conceived the study, implemented the simulation and performed theanalysis, and drafted the manuscript. JDS participated in the interpretation ofthe analyses and revision of the manuscript. AC participated in theinterpretation of the analyses and revision of the manuscript. TH conceivedthe study, and participated in the simulation, interpretation, and drafting ofthe manuscript. All authors read and approved the final manuscript.Competing interestsThe authors declare that they have no competing interests.Consent for publicationNot applicable.Ethics approval and consent to participateThe data utilized in this study was extracted from the European MulticenterStudy about Spinal Cord Injury (EMSCI, ClinicalTrials.gov Identifier:NCT01571531, www.emsci.org). All patients gave written informed consent.The ethical committee of the Canton of Zurich, Switzerland, has previouslyapproved the EMSCI project, upon which this project is based, and theapproval is also valid for any statistical analysis/re-analysis.Author details1Department of Biostatistics; Epidemiology, Biostatistics and PreventionInstitute; University of Zurich, Hirschengraben 84, 8001 Zurich, Switzerland.2ICORD, University of British Columbia and Vancouver Coastal Health,Vancouver, Canada. 3Spinal Cord Injury Center, Balgrist University Hospital,Zurich, Switzerland.Received: 31 May 2016 Accepted: 19 October 2016References1. 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