For semi-continuous data which are a mixture of true zeros and continuously distributed positive values, the use of two-part mixed models provides a convenient modelling framework. However, deriving population-averaged (marginal) effects from such models is not always straightforward. Su et al. presented a model that provided convenient estimation of marginal effects for the logistic component of the two-part model but the specification of marginal effects for the continuous part of the model presented in that paper was based on an incorrect formulation. We present a corrected formulation and additionally explore the use of the two-part model for inferences on the overall marginal mean, which may be of more practical relevance in our application and more generally.This work was supported by the Medical Research Council (Unit Programme number U105261167)

Farewell, VT

Su, L

Tom, BD

PubMed

 For semi-continuous data which are a mixture of true zeros and continuously distributed positive values, the use of two-part mixed models provides a convenient modelling framework. However, deriving population-averaged (marginal) effects from such models is not always straightforward. Su et al. presented a model that provided convenient estimation of marginal effects for the logistic component of the two-part model but the specification of marginal effects for the continuous part of the model presented in that paper was based on an incorrect formulation. We present a corrected formulation and additionally explore the use of the two-part model for inferences on the overall marginal mean, which may be of more practical relevance in our application and more generally. </jats:p

Brian DM Tom

Li Su

Vernon T Farewell

Crossref

Statistical Methods in Medical Research

A corrected formulation for marginal inference derived from two-part mixed models for longitudinal semi-continuous data

For semi-continuous data which are a mixture of true zeros and continuously distributed positive values, the use of two-part mixed models provides a convenient modelling framework. However, deriving population-averaged (marginal) effects from such models is not always straightforward. Su et al. presented a model that provided convenient estimation of marginal effects for the logistic component of the two-part model but the specification of marginal effects for the continuous part of the model presented in that paper was based on an incorrect formulation. We present a corrected formulation and additionally explore the use of the two-part model for inferences on the overall marginal mean, which may be of more practical relevance in our application and more generally

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ArticleA corrected formulation formarginal inference derivedfrom two-part mixed modelsfor longitudinal semi-continuousdataBrian DM Tom, Li Su and Vernon T FarewellAbstractFor semi-continuous data which are a mixture of true zeros and continuously distributed positive values,the use of two-part mixed models provides a convenient modelling framework. However, derivingpopulation-averaged (marginal) effects from such models is not always straightforward. Su et al.presented a model that provided convenient estimation of marginal effects for the logistic componentof the two-part model but the specification of marginal effects for the continuous part of the modelpresented in that paper was based on an incorrect formulation. We present a corrected formulation andadditionally explore the use of the two-part model for inferences on the overall marginal mean, which maybe of more practical relevance in our application and more generally.Keywordsbridge distribution, excess zeros, longitudinal data, random effects1 IntroductionIn Su et al.,1 we described a two-part marginal model for longitudinal semi-continuous data that area mixture of true zeros and continuously distributed positive values. Our likelihood-based modelhad an underlying two-part mixed model, where, in the random effects logistic regression for the firstpart (i.e. the binary part), the random intercept was assumed to follow the bridge distribution ofWang and Louis.2 A zero-mean normal random intercept was included into the linear mixedmodelling structure of the second part (i.e. the continuous part).Our primary focus was to ensure that the regression parameters in the binary part of the two-partmarginal model were interpretable after integration over the random effects distribution. Marginalcovariate effects on the expected value of the response for the population of observed non-zeroMedical Research Council Biostatistics Unit, Institute of Public Health, Cambridge, UKCorresponding author:Brian DM Tom, MRC Biostatistics Unit, Institute of Public Health, University Forvie Site, Robinson Way, Cambridge CB2 0SR, UK.Email: brian.tom@mrc-bsu.cam.ac.ukStatistical Methods in Medical Research2016, Vol. 25(5) 2014–2020! The Author(s) 2013Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0962280213509798smm.sagepub.comresponses may, however, also be of interest. These arise directly from the approaches in Moultonet al.,3 Lu et al.,4 Hall and Zhang5 and Yang and Simpson6 and involve well-defined integrations(see refs7,8).However, when discussing the continuous part of our model, we assumed, as did Tooze et al.,9that integrating out the random effects was straightforward, and that the form of the relationshipbetween covariates and the marginal mean of the response, given that it is positive, was thesame as the conditional mean given a positive response and random effects. Unfortunately,this is not the case. This paper rectifies this error and explores the use of the proposed modelwhen the target of inference is the overall marginal mean, which may be of most practicalrelevance.2 Marginal inference from two-part models2.1 ModelOur two-part marginal model1 is based on the original two-part mixed modelling frameworkintroduced in Olsen and Schafer10 and Tooze et al.9 and the random effects specifications in Linet al.11Briefly, let Yij be a semi-continuous variable for the ith (i ¼ 1, . . . ,N) subject at time tij(j ¼ 1, . . . , ni). Let Xij and Xij be the covariate vectors (possibly overlapping) associated with theith subject at time tij in the two parts of the two-part mixed model. Let Bi and Vi be correlatedsubject-level random intercepts, which are independent of the covariates. Define alsoij ¼ fBi,Vi,Xij,Xijg and ij ¼ fXij,Xijg.Yij can be represented by two variables, the occurrence variable Zij ¼ IðYij 4 0Þ and the intensityvariable gðYijÞ given that Yij 4 0, where gðÞ is a (monotonic) transformation making YijjYij 4 0normally distributed with a subject–time-specific mean.The distribution of Yij is formulated by assuming, firstly, that Zij is specified by a random effectslogistic regression with logitfPrðZij ¼ 1jijÞg ¼ Xije þ Bi, where Xij is a 1 q covariate vector, e is aq 1 regression coefficient vector and Bi is the subject-level random intercept in this first part (i.e.the binary part) assumed to follow the (symmetric) mean zero bridge distribution of Wang andLouis2 with unknown parameter ’ (05 ’5 1). Next, the intensity variable gðYijÞ given Yij 4 0 isassumed to have the linear mixed modelling structure described by gðYijÞjij,Yij 4 0 ¼Xijþ Vi þ ij, where Xij is a 1 p covariate vector,  is a p 1 regression coefficient vector andVi is the subject-level random intercept for the second part (i.e. the continuous part) assumedNð0, 2v Þ. The error term ij is assumed to be Nð0, 2e Þ and independent of the random effects. Therandom effects, Bi and Vi are assumed to be correlated with their joint distribution specified througha Gaussian copula transformation model, where the correlation of the underlying Gaussian randomvariables is  (see Supplementary Material). The covariate vectors Xij, Xij may coincide, but this isnot required.2.2 Marginal covariate effectsThe main benefit of the bridge density, stressed in Su et al.,1 is that after integration over the randomintercepts, ðBi,ViÞ, the marginal probability PrðZij ¼ 1jijÞ relates to the linear predictors throughthe same logit link function as for the corresponding conditional probability, PrðZij ¼ 1jijÞ.Furthermore, if we specify the marginal regression structure of the binary part aslogitfPrðZij ¼ 1jijÞg ¼ Xij, then the marginal covariate effects  are proportional to the subject-specific conditional covariate effects e, with  ¼ ’e.Tom et al. 2015However, although in Su et al.1 we claimed that the marginal mean of gðYijÞjij,Yij 4 0, foundafter integrating gðYijÞjij,Yij 4 0 over ðBi,ViÞ, is Xij, this is not the case generally. The correctform of the marginal mean of gðYijÞjij,Yij 4 0 isEfgðYijÞjij,Yij 4 0g ¼ Xijþ EðVijij,Yij 4 0Þ ð1Þwhich will be dependent on the impact of covariates, Xij, on the marginal and conditionalprobabilities of occurrence (see Supplementary Material).As the integral given by EðVijij,Yij 4 0Þ has no closed-form solution, an exact analyticalexpression for (1) is not available. However, bounds on (1) are available. Specifically, we canshow, after some algebraic manipulations (see Supplementary Material), that for   0Xij  Eð gðYijÞjij,Yij 4 0Þ  Xijþvffiffiffiffiffi2p ð1þ eXijÞand for   0Xij  Eð gðYijÞjij,Yij 4 0Þ  Xijþvffiffiffiffiffi2p ð1þ eXijÞAlthough an exact analytical expression is not available, numerically solving (1) at the maximum-likelihood estimates is straightforward as only a single integral is involved. This integral can beevaluated using adaptive Gaussian quadrature techniques. The estimation of the parameters, , 2b , 2v , 2e and  is based on maximizing the likelihood presented in Su et al.12.3 Interpretation of the marginal effects in the continuous partAs noted earlier and in Su et al.,1 the interpretation of  is straightforward as these parameters aresimply (population-averaged) log-odds ratios. In contrast, assessment of the impact of a covariateon the marginal mean (given being positive), EðYijjij,Yij 4 0Þ, depends on whether or not thatcovariate is also involved in the binary part of the two-part model. If the covariate is not included inthe binary part or Bi and Vi are uncorrelated (i.e.  ¼ 0), then the interpretation of its effect onEðYijjij,Yij 4 0Þ can be quantified through just the appropriate element of . However when Bi andVi are correlated and, in addition, the covariate of interest is in both regression components of themodel, then a simple interpretation is not readily obtainable because of the non-linearity of (1) inthis covariate.In such a case, the impact of a covariate could be assessed through plotting the relationshipbetween this covariate and EðYijjij,Yij 4 0Þ, with other covariates held fixed, or alternatively bydescribing the local changes (i.e. through the derivative or the difference) in EðYijjij,Yij 4 0Þ withrespect to the covariate.7 However, the clinical relevance of EðYijjij,Yij 4 0Þ has been questioned,as discussed by Albert12 in light of work by Lu et al.4 and Williamson et al.13 On the other hand, theoverall marginal mean of Yij as the target of inference is more easily justified clinically. Thecalculation of this overall mean is addressed in Section 2.4 and Section 2.5 illustrates its use.2.4 Overall marginal meanWhen gðÞ is the identity function, the overall marginal mean, EðYijÞ  EðYijjijÞ, is given byEðYijjYij ¼ 0ÞPrðYij ¼ 0Þ þ EðYijjYij 4 0ÞPrðYij 4 0Þ ¼ EðYijjYij 4 0ÞPrðYij 4 0Þ2016 Statistical Methods in Medical Research 25(5)where we have suppressed the dependence on the covariate vectors, ij, for convenience. Although aclosed form for the overall marginal mean is not available, the analyst can easily numericallyevaluate it (as is done in the subsequent Health Assessment Questionnaire (HAQ) analysis).From Section 2.2, bounds on the overall marginal mean can be obtained asPrðYij 4 0ÞXij  EðYijÞ  PrðYij 4 0ÞXijþvffiffiffiffiffi2pwhen   0, andPrðYij 4 0ÞXij  EðYijÞ  PrðYij 4 0ÞXijþvffiffiffiffiffi2pwhen   0, where PrðYij 4 0Þ ¼ ð1þ eXijÞ1. Similar bounds can be derived for other commonmonotonic transformation functions for gðÞ. For example, bounds on EðYijjijÞ when gðÞ islogarithmic are shown in the Supplementary Material.3 The HAQ data revisitedIn this section, we revisit the HAQ data described in Su et al.1 The objective is to examine theassociation between alleles that code for human leukocyte antigen (HLA) proteins and disabilitylevel in a psoriatic arthritis (PsA) patient cohort. R code for this new analysis is located in theSupplementary Material.Table 2 of Su et al.1 presented results from fitting the two-part mixed model to the data,where the third column shows the conditional covariate effects in the continuous part. As notedearlier, the corresponding marginal covariate effects are generally not equal to these conditional effects.In this particular application, it is perhaps more natural to examine the association between theHLA alleles and the overall expected disability level of the patients over the study period, instead ofthe association when some disability is present. This is because disability, as measured by HAQ, forpatients can vary over time and, for example, at one visit a patient can have mild disability, but atthe next visit his/her situation may be improved resulting in a zero value of HAQ. We conjecturethat it will often be felt to be clinically more informative to present the marginal covariate effects onthe overall expected disability level together with the marginal covariate effects on the probability ofhaving any level of disability.For the HAQ example, we sample from the asymptotic distribution of the parameters based onthe estimates in Table 2 of Su et al.1 and calculate the contrasts of overall expected HAQ with andwithout specific HLA alleles, controlling for other covariates. For presentation purposes, we fix theage at PsA diagnosis at 35 years and disease duration at 15 years, which correspond to zero values instandardized versions of the two variables. These contrasts represent the effects of HLA alleles onthe overall expected disability level (controlling for other covariates) in the PsA patient cohort.The top panels of Figure 1 show the HLA-B27 effects given other alleles, sex, age at diagnosis anddisease duration. Because the overall mean of HAQ is not directly parametrized in the fitted model,the corresponding covariate effects are not the same for all values of the other variables. However,the HLA-B27 effects are approximately the same across different combinations of other covariates,and the 95% confidence intervals do not include zero. This demonstrates a significant associationbetween HLA-B27 and overall expected HAQ.In Su et al.,1 we found a significant interaction between the effects of HLA-DQW3 and HLA-DR7 in the binary part of the two-part mixed model (p ¼ 0:035), while the same interaction wasTom et al. 2017non-significant in the continuous part (p ¼ 0:85). The estimated marginal (log-odds ratio) effect ofthis interaction in the binary part was 0.8089 with 95% confidence interval [0.0565, 1.5613].The middle and bottom panels of Figure 1 reflect the possible interaction between HLA-DQW3and HLA-DR7 on the overall marginal mean of HAQ stratified by gender and absence/presence ofthe HLA-B27 allele. Age at PsA diagnosis is fixed at 35 years and disease duration at 15 years.For illustrative purposes, considering the left middle (or bottom) panel of Figure 1 for femaleswith the presence of HLA-B27, we estimate that the difference in the HLA-DQW3 (or, alternatively,HLA-DR7) effects on the overall marginal mean of HAQ between those with the presence of0.00.51.0Female, B27 effectsDifference in average HAQA B C DABCDDQw3=0,DR7=0DQw3=0,DR7=1DQw3=1,DR7=0DQw3=1,DR7=10.00.51.0Male, B27 effectsDifference in average HAQA B C DABCDDQw3=0,DR7=0DQw3=0,DR7=1DQw3=1,DR7=0DQw3=1,DR7=10.00.51.0Female, DQW3 effectsDifference in average HAQA BC DABCDB27=0,DR7=0B27=1,DR7=0B27=0,DR7=1B27=1,DR7=10.00.51.0Male, DQW3 effectsDifference in average HAQA BC DABCDB27=0,DR7=0B27=1,DR7=0B27=0,DR7=1B27=1,DR7=10.00.51.0Female, DR7 effectsDifference in average HAQA BC DABCDB27=0,DQw3=0B27=1,DQw3=0B27=0,DQw3=1B27=1,DQw3=10.00.51.0Male, DR7 effectsDifference in average HAQA BC DABCDB27=0,DQw3=0B27=1,DQw3=0B27=0,DQw3=1B27=1,DQw3=1Figure 1. Contrasts (with 95% confidence intervals) of overall mean of HAQ for different combinations of thecovariates (controlling for being 35 years old at PsA diagnosis and having a disease duration of 15 years).HAQ: Health Assessment Questionnaire.2018 Statistical Methods in Medical Research 25(5)HLA-DR7 (or HLA-DQW3) allele and those with it absent (i.e. contrast D–B in figure) is 0.0564with 95% confidence interval [–0.2062, 0.3232]. For females with HLA-B27 absent, the estimate ofthis difference in the HLA-DQW3 (or, alternatively, HLA-DR7) effects on the overall marginalmean of HAQ between those with and without the HLA-DR7 (or HLA-DQW3) allele (i.e. contrastC–A) is 0.0648 with 95% confidence interval [–0.1971, 0.3158]. These estimates of the HLA-DQW3and HLA-DR7 interaction for females, with and without HLA-B27 present, are similar and bothnon-significant statistically. Conclusions based on these results are similar to those found for thecontinuous part in the two-part marginal model (data not shown).4 DiscussionIn this article, we have corrected the formulation for the continuous part of the two-part marginalmodel presented in Su et al.1 We show that the (marginal) mean of gðYijÞjij,Yij 4 0 is not the fixedeffects predictor, Xij, as originally reported, but is non-linear in the covariates included in thebinary part of the model. Thus, interpretation of the impact of a covariate on the marginal meangiven being positive cannot be made from only considering the relevant component of  when therandom effects are correlated and that covariate is also included in the binary part.In some contexts, the logit may not be the preferred link function in the binary part. For example,in dilution and serological studies the cloglog link may be more appropriate. In psychometrics, theprobit may be more convenient. For either of these alternatives, a two-part marginal formulationcan be derived. For instance, if the logit link is replaced with the probit and Bi is assumed Nð0, 2bÞinstead of from the bridge distribution when formulating the binary part of the two-part mixedmodel, then the link function of the marginal regression structure for the binary part, afterintegrating out Bi, remains probit.2 Furthermore, in the binary part, the marginal covariateeffects are proportional to their subject-specific conditional covariate effects, with constant ofproportionality 1=ffiffiffiffiffiffiffiffi1þ2bp. Under the same linear mixed effects structure considered earlier for thecontinuous part (i.e. Vi normal and gðÞ the identity function), a closed-form solution for themarginal mean of Yij, given Yij 4 0, (and therefore for the overall marginal mean) can be derivedin terms of the standard normal density and cumulative distribution function (see SupplementaryMaterial). Unfortunately, this closed-form solution is again non-linear in the covariates associatedwith the binary part, and therefore interpretation of marginal covariate effects on the continuouspart (and on the overall marginal mean) will not generally be straightforward.In conclusion, care should be taken when using two-part models for semi-continuous data thatare longitudinal or otherwise clustered. Both the specification of random effects structures, asdiscussed in Su et al.,14 and the interpretation and calculation of marginal effects, as discussed inthis paper, require careful attention to assumptions.Declaration of conflicting interestsThe author(s) declared no potential conflicts of interest with respect to the research, authorship, and/orpublication of this article.FundingThe author(s) disclosed receipt of the following financial support for the research, authorship, and/orpublication of this article: This work was supported by the Medical Research Council (Unit Programmenumber U105261167).Tom et al. 2019References1. Su L, Tom B and Farewell V. A likelihood-based two-partmarginal model for longitudinal semicontinuous data. StatMethods Med Res 2013; DOI: 10.1177/0962280211414620.2. Wang Z and Louis T. Matching conditional and marginalshapes in binary mixed-effects models using a bridgedistribution function. Biometrika 2003; 90: 765–775.3. Moulton LH, Curriero FC and Barroso PF. Mixturemodels for quantitative HIV RNA data. Stat Methods MedRes 2002; 11: 317–325.4. Lu SE, Lin Y and Shih WCJ. Analyzing excessive nochanges in clinical trials with clustered data. Biometrics2004; 60: 257–267.5. Hall DB and Zhang Z. Marginal models for zero inflatedclustered data. Stat Model 2004; 4: 161–180.6. Yang Y and Simpson D. Unified computational methodsfor regression analysis of zero-inflated and bound-inflateddata. Comput Stat Data Anal 2010; 54: 1525–1534.7. Liu L, Cowen M, Strawderman RL, et al. A flexible two-part random-effects model for correlated medical costs. JHealth Econom 2010; 29: 110–123.8. Li X, Bandyopadhyay D, Lipsitz S, et al. Likelihoodmethods for binary responses of present components in acluster. Biometrics 2011; 67: 629–635.9. Tooze JA, Grunwald GK and Jones RH. Analysis ofrepeated measures data with clumping at zero. StatMethods Med Res 2002; 11: 341–355.10. Olsen MK and Schafer JL. A two-part random-effectsmodel for semicontinuous longitudinal data. J Am StatAssoc 2001; 96: 730–745.11. Lin L, Bandyopadhyay D, Lipsitz SR, et al. Associationmodels for clustered data with binary and continuousresponses. Biometrics 2010; 66: 287–293.12. Albert PS. Letter to the editor. Biometrics 2005; 47:879–881.13. Williamson JM, Datta S and Satten GA. Marginal analysisof clustered data when cluster size is informative.Biometrics 2003; 59: 36–42.14. Su L, Tom BD and Farewell VT. Bias in 2-part mixedmodels for longitudinal semicontinuous data. Biostatistics2009; 10: 374–389.2020 Statistical Methods in Medical Research 25(5)

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A corrected formulation for marginal inference derived from two-part mixed models for longitudinal semi-continuous data

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