Most school-based smoking prevention studies employ designs in which schools or classrooms are as... more Most school-based smoking prevention studies employ designs in which schools or classrooms are assigned to different treatment conditions while observations are made on individual students. This design requires that the treatment effect be assessed against the between-school variance. However, the between- school variance is usually larger than the variance that would be obtained if students were individually randomized to different
In analysis of binary data from clustered and longitudinal studies, random effect models have bee... more In analysis of binary data from clustered and longitudinal studies, random effect models have been recently developed to accommodate two-level problems such as subjects nested within clusters or repeated classifications within subjects. Unfortunately, these models cannot be applied to three- level problems that occur frequently in practice. For example, multicenter longitudinal clinical trials involve repeated assesslnents within individuals and individuals
The present study evaluated the predictive validity of individual early emerging nicotine depende... more The present study evaluated the predictive validity of individual early emerging nicotine dependence symptoms in adolescence on smoking behavior in young adulthood. A total of 492 adolescents who, at baseline, had not smoked more than 100 cigarettes in their lifetime and 123 adolescents who smoked more than 100 cigarettes lifetime, and who participated in the 6-year follow-up assessment were included in the present analyses. Predictive validity of 10 nicotine dependence items administered at baseline was evaluated at the 6 year follow-up when the sample had entered young adulthood (mean age=21.6). Among adolescents who had smoked fewer than 100 cigarettes, experiencing higher levels of overall nicotine dependence as well as individual symptoms at baseline longitudinally predicted an increase in risk for daily smoking in young adulthood, after controlling for baseline smoking and other tobacco use. For adolescents who had smoked more than 100 cigarettes at baseline, level of nicotine...
Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco, Jan 5, 2015
The present study sought to identify time-dependent within-participant effects of CYP2A6 genotype... more The present study sought to identify time-dependent within-participant effects of CYP2A6 genotypes on smoking frequency and nicotine dependence in young smokers. Predicted nicotine metabolic rate based on CYP2A6 diplotypes (CYP2A6 diplotype predicted rate [CDPR]) was partitioned into Normal, Intermediate, and Slow categories using a metabolism metric. Growth-curve models characterized baseline and longitudinal CDPR effects with data from eight longitudinal assessments during a 6-year period (from approximately age 16-22) in young smokers of European descent (N = 296, 57% female) who had smoked less than 100 cigarettes lifetime at baseline and more than that amount by Year 6. Phenotypes were number of days smoked during the previous 30 days and a youth version of the Nicotine Dependence Syndrome Scale (NDSS). A zero-inflated Poisson growth-curve model was used to account for the preponderance of zero days smoked. At baseline, Intermediate CDPR was a risk factor relative to both Norma...
Mixed-effects logistic regression models are described for analysis of longitudinal ordinal outco... more Mixed-effects logistic regression models are described for analysis of longitudinal ordinal outcomes, where observations are observed clustered within subjects. Random effects are included in the model to account for the correlation of the clustered observations. Typically, the error variance and the variance of the random effects are considered to be homogeneous. These variance terms characterize the within-subjects (i.e., error variance) and between-subjects (i.e., random-effects variance) variation in the data. In this article, we describe how covariates can influence these variances, and also extend the standard logistic mixed model by adding a subject-level random effect to the within-subject variance specification. This permits subjects to have influence on the mean, or location, and variability, or (square of the) scale, of their responses. Additionally, we allow the random effects to be correlated. We illustrate application of these models for ordinal data using Ecological M...
Longitudinal studies are increasingly common in psychological and social sciences research. In th... more Longitudinal studies are increasingly common in psychological and social sciences research. In these studies, subjects are measured repeatedly across time and in- terest often focuses on characterizing their growth or development across time. Mixed-efiects regression models (MRMs) have become the method of choice for modeling of longitudinal data; variants of MRMs have been developed under a variety of names: Random-efiects
Marijuana use is increasingly widespread among adolescents and young adults; however, few studies... more Marijuana use is increasingly widespread among adolescents and young adults; however, few studies have examined longitudinal trajectories of marijuana use during this important developmental period. As such, we examined adolescent trajectories of marijuana use and the psychosocial factors that may differentiate individuals who escalate their marijuana use over adolescence and young adulthood from those who do not. Participants were 1204 9th and 10th graders at baseline who were over-sampled for cigarette use and were followed over 6-years, as part of an extensive longitudinal study, the Social and Emotional Contexts of Adolescent Smoking Patterns (SECASP) study. Growth Mixture Modeling (GMM) was used to model trajectories of marijuana use and Mixed Effects Regression analyses were used to examine psychosocial correlates of marijuana use escalation over time. Our results revealed three trajectories of non-escalating users (low users, medium users, and high users) and one escalating user trajectory. We found that relative to Non-escalators the Escalators had higher cigarette smoking (p<.0001), novelty-seeking (p=.02), aggressive and anti-social behavior (p<.007), and problem behavior related to peer context (p=.04). Moreover, there were important time and Group by Time interactions in some of these relationships. On the other hand, parental control and depression did not differ between escalators and low and medium non-escalating users. Cigarette smoking, novelty-seeking, aggressive and anti-social behavior, and peer influence are related to 'escalating' marijuana use throughout adolescence and young adulthood.
Several approaches have been proposed to model binary outcomes that arise from longitudinal studi... more Several approaches have been proposed to model binary outcomes that arise from longitudinal studies. Most of the approaches can be grouped into two classes: the population-averaged and subject-specific approaches. The generalized estimating equations (GEE) method is commonly used to estimate population- averaged effects, while random-effects logistic models can be used to estimate subject-specific effects. However, it is not clear to
Mixed-effects logistic regression models are described for analysis of longitudinal ordinal outco... more Mixed-effects logistic regression models are described for analysis of longitudinal ordinal outcomes, where observa- tions are observed clustered within subjects. Random ef- fects are included in the model to account for the correla- tion of the clustered observations. Typically, the error vari- ance and the variance of the random effects are considered to be homogeneous. These variance terms characterize the
Despite the highly replicated relationship between depression and nicotine dependence, little is ... more Despite the highly replicated relationship between depression and nicotine dependence, little is known about this association across both time and levels of lifetime smoking exposure. In the present study, we evaluate if symptoms of depression are associated with emerging nicotine dependence after accounting for smoking exposure and whether this relationship varies from adolescence to young adulthood and across increasing levels of smoking. The sample was drawn from the Social and Emotional Contexts of Adolescent Smoking Patterns Study which measured smoking, nicotine dependence and depression over 6 assessment waves spanning 6years. Analyses were based on repeated assessment of 941 participants reporting any smoking 30days prior to individual assessment waves. Mixed-effects regression models were estimated to examine potential time and smoking exposure varying effects in the association between depression and nicotine dependence. Inter-individual differences in mean levels of depre...
Journal of Consulting and Clinical Psychology, 1995
Random regression models (RRMs) were used to investigate the role of initial severity in the outc... more Random regression models (RRMs) were used to investigate the role of initial severity in the outcome of 4 treatments (cognitive–behavior therapy [CBT], interpersonal psychotherapy [IPT], imipramine plus clinical management [IMI-CM], and placebo plus clinical management [PLA-CM]) for outpatients with major depressive disorder seen in the National Institute of Mental Health Treatment of Depression Collaborative Research Program. Initial severity of depression
Longitudinal studies have a prominent role in psychiatric research; however, statistical methods ... more Longitudinal studies have a prominent role in psychiatric research; however, statistical methods for analyzing these data are rarely commensurate with the effort involved in their acquisition. Frequently the majority of data are discarded and a simple end-point analysis is performed. In other cases, so called repeated-measures analysis of variance procedures are used with little regard to their restrictive and often unrealistic assumptions and the effect of missing data on the statistical properties of their estimates. We explored the unique features of longitudinal psychiatric data from both statistical and conceptual perspectives. We used a family of statistical models termed random regression models that provide a more realistic approach to analysis of longitudinal psychiatric data. Random regression models provide solutions to commonly observed problems of missing data, serial correlation, time-varying covariates, and irregular measurement occasions, and they accommodate systematic person-specific deviations from the average time trend. Properties of these models were compared with traditional approaches at a conceptual level. The approach was then illustrated in a new analysis of the National Institute of Mental Health Treatment of Depression Collaborative Research Program dataset, which investigated two forms of psychotherapy, pharmacotherapy with clinical management, and a placebo with clinical management control. Results indicated that both person-specific effects and serial correlation play major roles in the longitudinal psychiatric response process. Ignoring either of these effects produces misleading estimates of uncertainty that form the basis of statistical tests of hypotheses.
Health services & outcomes research methodology, 2014
A bivariate mixed-effects location-scale model is proposed for estimation of means, variances, an... more A bivariate mixed-effects location-scale model is proposed for estimation of means, variances, and covariances of two continuous outcomes measured concurrently in time and repeatedly over subjects. Modeling the two outcomes jointly allows examination of BS and WS association between the outcomes and whether the associations are related to covariates. The variance-covariance matrices of the BS and WS effects are modeled in terms of covariates, explaining BS and WS heterogeneity. The proposed model relaxes assumptions on the homogeneity of the within-subject (WS) and between-subject (BS) variances. Furthermore, the WS variance models are extended by including random scale effects. Data from a natural history study on adolescent smoking are used for illustration. 461 students, from 9(th) and 10(th) grades, reported on their mood at random prompts during seven consecutive days. This resulted in 14,105 prompts with an average of 30 responses per student. The two outcomes considered were ...
Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco, Jan 10, 2014
Physical activity (PA) and smoking are inversely related. However, evidence suggests that some ty... more Physical activity (PA) and smoking are inversely related. However, evidence suggests that some types of PA, namely work-related PA, may show an opposite effect. Despite growing knowledge, there remains a paucity of studies examining the context of these behaviors in naturalistic settings or in young adults, a high-risk group for escalation. Participants were 188 young adults (mean age = 21.32; 53.2% female; 91% current smokers) who participated in an electronic diary week to assess daily smoking and urges and a PA recall to examine daily PA. PA was coded into non-work-related and work-related activity to examine differential effects. We considered both participants' weekly average PA and their daily deviations from their average. Mixed-effects regression models revealed that higher weekly average non-work PA was associated with lower smoking level and urges. Daily deviations in non-work PA did not predict urges; however, increased daily non-work PA relative to participants' ...
Suboptimal diet and inactive lifestyle are among the most prevalent preventable causes of prematu... more Suboptimal diet and inactive lifestyle are among the most prevalent preventable causes of premature death. Interventions that target multiple behaviors are potentially efficient; however the optimal way to initiate and maintain multiple health behavior changes is unknown. The Make Better Choices 2 (MBC2) trial aims to examine whether sustained healthful diet and activity change are best achieved by targeting diet and activity behaviors simultaneously or sequentially. Study Design Approximately 250 inactive adults with poor quality diet will be randomized to 3 conditions examining the best way to prescribe healthy diet and activity change. The 3 intervention conditions prescribe: 1) an increase in fruit and vegetable consumption (F/V+), decrease in sedentary leisure screen time (Sed-), and increase in physical activity (PA+) simultaneously (Simultaneous); 2) F/V+ and Sed- first, and then sequentially add PA+ (Sequential); or 3) Stress Management Control that addresses stress, relaxat...
SUMMARY Three-level data occur frequently in behaviour and medical sciences. For example, in a mu... more SUMMARY Three-level data occur frequently in behaviour and medical sciences. For example, in a multi-centre trial, subjects within a given site are randomly assigned to treatments and then studied over time. In this example, the repeated observations (level-1) are nested within subjects (level-2) who are nested within sites (level-3). Similarly, in twin studies, repeated measurements (level-1) are taken on each twin (level-2) within each twin pair (level-3). A three-level mixed-eects regression model is described here. Random eects at the second and third level are included in the model. Additionally, both proportional odds and non-proportional odds models are developed. The latter allows the eects of explanatory variables to vary across the cumulative logits of the model. A maximum marginal likelihood (MML) solution is described and Gauss-Hermite numerical quadrature is used to integrate over the distribution of random eects. The random eects are normally distributed in this instan...
A mixed-effects multinomial logistic regression model is described for analysis of clustered or l... more A mixed-effects multinomial logistic regression model is described for analysis of clustered or longitudinal nominal or ordinal response data. The model is parameterized to allow flexibility in the choice of contrasts used to represent comparisons across the response categories. Estimation is achieved using a maximum marginal likelihood (MML) solution that uses quadrature to numerically integrate over the distribution of random effects. An analysis of a psychiatric data set, in which homeless adults with serious mental illness are repeatedly classified in terms of their living arrangement, is used to illustrate features of the model.
Most school-based smoking prevention studies employ designs in which schools or classrooms are as... more Most school-based smoking prevention studies employ designs in which schools or classrooms are assigned to different treatment conditions while observations are made on individual students. This design requires that the treatment effect be assessed against the between-school variance. However, the between- school variance is usually larger than the variance that would be obtained if students were individually randomized to different
In analysis of binary data from clustered and longitudinal studies, random effect models have bee... more In analysis of binary data from clustered and longitudinal studies, random effect models have been recently developed to accommodate two-level problems such as subjects nested within clusters or repeated classifications within subjects. Unfortunately, these models cannot be applied to three- level problems that occur frequently in practice. For example, multicenter longitudinal clinical trials involve repeated assesslnents within individuals and individuals
The present study evaluated the predictive validity of individual early emerging nicotine depende... more The present study evaluated the predictive validity of individual early emerging nicotine dependence symptoms in adolescence on smoking behavior in young adulthood. A total of 492 adolescents who, at baseline, had not smoked more than 100 cigarettes in their lifetime and 123 adolescents who smoked more than 100 cigarettes lifetime, and who participated in the 6-year follow-up assessment were included in the present analyses. Predictive validity of 10 nicotine dependence items administered at baseline was evaluated at the 6 year follow-up when the sample had entered young adulthood (mean age=21.6). Among adolescents who had smoked fewer than 100 cigarettes, experiencing higher levels of overall nicotine dependence as well as individual symptoms at baseline longitudinally predicted an increase in risk for daily smoking in young adulthood, after controlling for baseline smoking and other tobacco use. For adolescents who had smoked more than 100 cigarettes at baseline, level of nicotine...
Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco, Jan 5, 2015
The present study sought to identify time-dependent within-participant effects of CYP2A6 genotype... more The present study sought to identify time-dependent within-participant effects of CYP2A6 genotypes on smoking frequency and nicotine dependence in young smokers. Predicted nicotine metabolic rate based on CYP2A6 diplotypes (CYP2A6 diplotype predicted rate [CDPR]) was partitioned into Normal, Intermediate, and Slow categories using a metabolism metric. Growth-curve models characterized baseline and longitudinal CDPR effects with data from eight longitudinal assessments during a 6-year period (from approximately age 16-22) in young smokers of European descent (N = 296, 57% female) who had smoked less than 100 cigarettes lifetime at baseline and more than that amount by Year 6. Phenotypes were number of days smoked during the previous 30 days and a youth version of the Nicotine Dependence Syndrome Scale (NDSS). A zero-inflated Poisson growth-curve model was used to account for the preponderance of zero days smoked. At baseline, Intermediate CDPR was a risk factor relative to both Norma...
Mixed-effects logistic regression models are described for analysis of longitudinal ordinal outco... more Mixed-effects logistic regression models are described for analysis of longitudinal ordinal outcomes, where observations are observed clustered within subjects. Random effects are included in the model to account for the correlation of the clustered observations. Typically, the error variance and the variance of the random effects are considered to be homogeneous. These variance terms characterize the within-subjects (i.e., error variance) and between-subjects (i.e., random-effects variance) variation in the data. In this article, we describe how covariates can influence these variances, and also extend the standard logistic mixed model by adding a subject-level random effect to the within-subject variance specification. This permits subjects to have influence on the mean, or location, and variability, or (square of the) scale, of their responses. Additionally, we allow the random effects to be correlated. We illustrate application of these models for ordinal data using Ecological M...
Longitudinal studies are increasingly common in psychological and social sciences research. In th... more Longitudinal studies are increasingly common in psychological and social sciences research. In these studies, subjects are measured repeatedly across time and in- terest often focuses on characterizing their growth or development across time. Mixed-efiects regression models (MRMs) have become the method of choice for modeling of longitudinal data; variants of MRMs have been developed under a variety of names: Random-efiects
Marijuana use is increasingly widespread among adolescents and young adults; however, few studies... more Marijuana use is increasingly widespread among adolescents and young adults; however, few studies have examined longitudinal trajectories of marijuana use during this important developmental period. As such, we examined adolescent trajectories of marijuana use and the psychosocial factors that may differentiate individuals who escalate their marijuana use over adolescence and young adulthood from those who do not. Participants were 1204 9th and 10th graders at baseline who were over-sampled for cigarette use and were followed over 6-years, as part of an extensive longitudinal study, the Social and Emotional Contexts of Adolescent Smoking Patterns (SECASP) study. Growth Mixture Modeling (GMM) was used to model trajectories of marijuana use and Mixed Effects Regression analyses were used to examine psychosocial correlates of marijuana use escalation over time. Our results revealed three trajectories of non-escalating users (low users, medium users, and high users) and one escalating user trajectory. We found that relative to Non-escalators the Escalators had higher cigarette smoking (p<.0001), novelty-seeking (p=.02), aggressive and anti-social behavior (p<.007), and problem behavior related to peer context (p=.04). Moreover, there were important time and Group by Time interactions in some of these relationships. On the other hand, parental control and depression did not differ between escalators and low and medium non-escalating users. Cigarette smoking, novelty-seeking, aggressive and anti-social behavior, and peer influence are related to 'escalating' marijuana use throughout adolescence and young adulthood.
Several approaches have been proposed to model binary outcomes that arise from longitudinal studi... more Several approaches have been proposed to model binary outcomes that arise from longitudinal studies. Most of the approaches can be grouped into two classes: the population-averaged and subject-specific approaches. The generalized estimating equations (GEE) method is commonly used to estimate population- averaged effects, while random-effects logistic models can be used to estimate subject-specific effects. However, it is not clear to
Mixed-effects logistic regression models are described for analysis of longitudinal ordinal outco... more Mixed-effects logistic regression models are described for analysis of longitudinal ordinal outcomes, where observa- tions are observed clustered within subjects. Random ef- fects are included in the model to account for the correla- tion of the clustered observations. Typically, the error vari- ance and the variance of the random effects are considered to be homogeneous. These variance terms characterize the
Despite the highly replicated relationship between depression and nicotine dependence, little is ... more Despite the highly replicated relationship between depression and nicotine dependence, little is known about this association across both time and levels of lifetime smoking exposure. In the present study, we evaluate if symptoms of depression are associated with emerging nicotine dependence after accounting for smoking exposure and whether this relationship varies from adolescence to young adulthood and across increasing levels of smoking. The sample was drawn from the Social and Emotional Contexts of Adolescent Smoking Patterns Study which measured smoking, nicotine dependence and depression over 6 assessment waves spanning 6years. Analyses were based on repeated assessment of 941 participants reporting any smoking 30days prior to individual assessment waves. Mixed-effects regression models were estimated to examine potential time and smoking exposure varying effects in the association between depression and nicotine dependence. Inter-individual differences in mean levels of depre...
Journal of Consulting and Clinical Psychology, 1995
Random regression models (RRMs) were used to investigate the role of initial severity in the outc... more Random regression models (RRMs) were used to investigate the role of initial severity in the outcome of 4 treatments (cognitive–behavior therapy [CBT], interpersonal psychotherapy [IPT], imipramine plus clinical management [IMI-CM], and placebo plus clinical management [PLA-CM]) for outpatients with major depressive disorder seen in the National Institute of Mental Health Treatment of Depression Collaborative Research Program. Initial severity of depression
Longitudinal studies have a prominent role in psychiatric research; however, statistical methods ... more Longitudinal studies have a prominent role in psychiatric research; however, statistical methods for analyzing these data are rarely commensurate with the effort involved in their acquisition. Frequently the majority of data are discarded and a simple end-point analysis is performed. In other cases, so called repeated-measures analysis of variance procedures are used with little regard to their restrictive and often unrealistic assumptions and the effect of missing data on the statistical properties of their estimates. We explored the unique features of longitudinal psychiatric data from both statistical and conceptual perspectives. We used a family of statistical models termed random regression models that provide a more realistic approach to analysis of longitudinal psychiatric data. Random regression models provide solutions to commonly observed problems of missing data, serial correlation, time-varying covariates, and irregular measurement occasions, and they accommodate systematic person-specific deviations from the average time trend. Properties of these models were compared with traditional approaches at a conceptual level. The approach was then illustrated in a new analysis of the National Institute of Mental Health Treatment of Depression Collaborative Research Program dataset, which investigated two forms of psychotherapy, pharmacotherapy with clinical management, and a placebo with clinical management control. Results indicated that both person-specific effects and serial correlation play major roles in the longitudinal psychiatric response process. Ignoring either of these effects produces misleading estimates of uncertainty that form the basis of statistical tests of hypotheses.
Health services & outcomes research methodology, 2014
A bivariate mixed-effects location-scale model is proposed for estimation of means, variances, an... more A bivariate mixed-effects location-scale model is proposed for estimation of means, variances, and covariances of two continuous outcomes measured concurrently in time and repeatedly over subjects. Modeling the two outcomes jointly allows examination of BS and WS association between the outcomes and whether the associations are related to covariates. The variance-covariance matrices of the BS and WS effects are modeled in terms of covariates, explaining BS and WS heterogeneity. The proposed model relaxes assumptions on the homogeneity of the within-subject (WS) and between-subject (BS) variances. Furthermore, the WS variance models are extended by including random scale effects. Data from a natural history study on adolescent smoking are used for illustration. 461 students, from 9(th) and 10(th) grades, reported on their mood at random prompts during seven consecutive days. This resulted in 14,105 prompts with an average of 30 responses per student. The two outcomes considered were ...
Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco, Jan 10, 2014
Physical activity (PA) and smoking are inversely related. However, evidence suggests that some ty... more Physical activity (PA) and smoking are inversely related. However, evidence suggests that some types of PA, namely work-related PA, may show an opposite effect. Despite growing knowledge, there remains a paucity of studies examining the context of these behaviors in naturalistic settings or in young adults, a high-risk group for escalation. Participants were 188 young adults (mean age = 21.32; 53.2% female; 91% current smokers) who participated in an electronic diary week to assess daily smoking and urges and a PA recall to examine daily PA. PA was coded into non-work-related and work-related activity to examine differential effects. We considered both participants' weekly average PA and their daily deviations from their average. Mixed-effects regression models revealed that higher weekly average non-work PA was associated with lower smoking level and urges. Daily deviations in non-work PA did not predict urges; however, increased daily non-work PA relative to participants' ...
Suboptimal diet and inactive lifestyle are among the most prevalent preventable causes of prematu... more Suboptimal diet and inactive lifestyle are among the most prevalent preventable causes of premature death. Interventions that target multiple behaviors are potentially efficient; however the optimal way to initiate and maintain multiple health behavior changes is unknown. The Make Better Choices 2 (MBC2) trial aims to examine whether sustained healthful diet and activity change are best achieved by targeting diet and activity behaviors simultaneously or sequentially. Study Design Approximately 250 inactive adults with poor quality diet will be randomized to 3 conditions examining the best way to prescribe healthy diet and activity change. The 3 intervention conditions prescribe: 1) an increase in fruit and vegetable consumption (F/V+), decrease in sedentary leisure screen time (Sed-), and increase in physical activity (PA+) simultaneously (Simultaneous); 2) F/V+ and Sed- first, and then sequentially add PA+ (Sequential); or 3) Stress Management Control that addresses stress, relaxat...
SUMMARY Three-level data occur frequently in behaviour and medical sciences. For example, in a mu... more SUMMARY Three-level data occur frequently in behaviour and medical sciences. For example, in a multi-centre trial, subjects within a given site are randomly assigned to treatments and then studied over time. In this example, the repeated observations (level-1) are nested within subjects (level-2) who are nested within sites (level-3). Similarly, in twin studies, repeated measurements (level-1) are taken on each twin (level-2) within each twin pair (level-3). A three-level mixed-eects regression model is described here. Random eects at the second and third level are included in the model. Additionally, both proportional odds and non-proportional odds models are developed. The latter allows the eects of explanatory variables to vary across the cumulative logits of the model. A maximum marginal likelihood (MML) solution is described and Gauss-Hermite numerical quadrature is used to integrate over the distribution of random eects. The random eects are normally distributed in this instan...
A mixed-effects multinomial logistic regression model is described for analysis of clustered or l... more A mixed-effects multinomial logistic regression model is described for analysis of clustered or longitudinal nominal or ordinal response data. The model is parameterized to allow flexibility in the choice of contrasts used to represent comparisons across the response categories. Estimation is achieved using a maximum marginal likelihood (MML) solution that uses quadrature to numerically integrate over the distribution of random effects. An analysis of a psychiatric data set, in which homeless adults with serious mental illness are repeatedly classified in terms of their living arrangement, is used to illustrate features of the model.
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Papers by Donald Hedeker