Tree-Structured Mixed-Effects Regression Modeling for Longitudinal Data

Subgroup detection in linear growth curve models with generalized linear mixed model (GLMM) trees

Growth curve models are popular tools for studying the development of a response variable within subjects over time. Heterogeneity between subjects is common in such models, and researchers are typically interested in explaining or predicting this heterogeneity. We show how generalized linear mixed-effects model (GLMM) trees can be used to identify subgroups with different trajectories in linear growth curve models. Originally developed for clustered cross-sectional data, GLMM trees are extended here to longitudinal data. The resulting extended GLMM trees are directly applicable to growth curve models as an important special case. In simulated and real-world data, we assess performance of the extensions and compare against other partitioning methods for growth curve models. Extended GLMM trees perform more accurately than the original algorithm and LongCART, and similarly accurate compared to structural equation model (SEM) trees. In addition, GLMM trees allow for modeling both discrete and continuous time series, are less sensitive to (mis-)specification of the random-effects structure and are much faster to compute.

A wide range of missing imputation approaches in longitudinal data: a simulation study and real data analysis

BACKGROUND

Missing data is a pervasive problem in longitudinal data analysis. Several single-imputation (SI) and multiple-imputation (MI) approaches have been proposed to address this issue. In this study, for the first time, the function of the longitudinal regression tree algorithm as a non-parametric method after imputing missing data using SI and MI was investigated using simulated and real data.

METHOD

Using different simulation scenarios derived from a real data set, we compared the performance of cross, trajectory mean, interpolation, copy-mean, and MI methods (27 approaches) to impute missing longitudinal data using parametric and non-parametric longitudinal models and the performance of the methods was assessed in real data. The real data included 3,645 participants older than 18 years within six waves obtained from the longitudinal Tehran cardiometabolic genetic study (TCGS). The data modeling was conducted using systolic and diastolic blood pressure (SBP/DBP) as the outcome variables and included predictor variables such as age, gender, and BMI. The efficiency of imputation approaches was compared using mean squared error (MSE), root-mean-squared error (RMSE), median absolute deviation (MAD), deviance, and Akaike information criteria (AIC).

RESULTS

The longitudinal regression tree algorithm outperformed based on the criteria such as MSE, RMSE, and MAD than the linear mixed-effects model (LMM) for analyzing the TCGS and simulated data using the missing at random (MAR) mechanism. Overall, based on fitting the non-parametric model, the performance of the 27 imputation approaches was nearly similar. However, the SI traj-mean method improved performance compared with other imputation approaches.

CONCLUSION

Both SI and MI approaches performed better using the longitudinal regression tree algorithm compared with the parametric longitudinal models. Based on the results from both the real and simulated data, we recommend that researchers use the traj-mean method for imputing missing values of longitudinal data. Choosing the imputation method with the best performance is widely dependent on the models of interest and the data structure.

Predictions of machine learning with mixed-effects in analyzing longitudinal data under model misspecification

We consider predictions in longitudinal studies, and investigate the well known statistical mixed-effects model, piecewise linear mixed-effects model and six different popular machine learning approaches: decision trees, bagging, random forest, boosting, support-vector machine and neural network. In order to consider the correlated data in machine learning, the random effects is combined into the traditional tree methods and random forest. Our focus is the performance of statistical modelling and machine learning especially in the cases of the misspecification of the fixed effects and the random effects. Extensive simulation studies have been carried out to evaluate the performance using a number of criteria. Two real datasets from longitudinal studies are analysed to demonstrate our findings. The R code and dataset are freely available at https://github.com/shuwen92/MEML.

Mixed-effect models with trees

Tree-based regression models are a class of statistical models for predicting continuous response variables when the shape of the regression function is unknown. They naturally take into account both non-linearities and interactions. However, they struggle with linear and quasi-linear effects and assume iid data. This article proposes two new algorithms for jointly estimating an interpretable predictive mixed-effect model with two components: a linear part, capturing the main effects, and a non-parametric component consisting of three trees for capturing non-linearities and interactions among individual-level predictors, among cluster-level predictors or cross-level. The first proposed algorithm focuses on prediction. The second one is an extension which implements a post-selection inference strategy to provide valid inference. The performance of the two algorithms is validated via Monte Carlo studies. An application on INVALSI data illustrates the potentiality of the proposed approach.

Development and validation of prognostic implications of chromosome abnormalities algorithm for newly diagnosed multiple myeloma

Fluorescence in situ hybridization (FISH) evaluation is essential for initial risk stratification in multiple myeloma (MM). The presence of specific cytogenetic abnormalities (CA) confers a heterogeneity impact on prognosis. However, the cutoff values among different centers are not uniform. Therefore, we conduct this study to better predict the prognosis of newly diagnosed MM patients based on FISH results. The Kaps method was used to calculate the chromosomal abnormal cutoff values. A total of 533 participants were included in the study. The best cutoff value of overall survival were as follows: 17p− 20.1%, 13q− 85%, 1q21+ 39%, t(11;14) 55.5%, t(14;16) 87%, and t(4;14) 53.5%. The survival analysis showed that 17p− and 1q21+ were the independent factors affecting both OS and progress free survival (PFS) among CA. The analysis based on the cutoff value obtained by Kaps suggested that 13q−, t(14;16), 17p−, and 1q21+ were independent factors affecting OS among CA; t(14;16), 17p−, and 1q21+ were independent factors affecting PFS among CA. The prognostic model was constructed by the Kaps method with the Harrell concordance index (c-index) at 0.719 (95% CI, 0.683–0.756; corrected 0.707), which was higher than that calculated by the European Myeloma Network criteria (0.714; 95% CI, 0.678–0.751; corrected 0.696). In conclusion, chromosomal abnormalities in different proportions and combinations can affect the prognosis of MM patients. Therefore, effective criteria should be formulated to evaluate the prognosis of MM patients better.

Precision medicine: Subgroup identification in longitudinal trajectories

In clinical studies, the treatment effect may be heterogeneous among patients. It is of interest to identify subpopulations which benefit most from the treatment, regardless of the treatment's overall performance. In this study, we are interested in subgroup identification in longitudinal studies when nonlinear trajectory patterns are present. Under such a situation, evaluation of the treatment effect entails comparing longitudinal trajectories while subgroup identification requires a further evaluation of differential treatment effects among subgroups induced by moderators. To this end, we propose a tree-structured subgroup identification method, termed "interaction tree for longitudinal trajectories", which combines mixed effects models with regression splines to model the nonlinear progression patterns among repeated measures. Extensive simulation studies are conducted to evaluate its performance and an application to an alcohol addiction pharmacogenetic trial is presented.

Random forests for high-dimensional longitudinal data

Random forests are one of the state-of-the-art supervised machine learning methods and achieve good performance in high-dimensional settings where p, the number of predictors, is much larger than n, the number of observations. Repeated measurements provide, in general, additional information, hence they are worth accounted especially when analyzing high-dimensional data. Tree-based methods have already been adapted to clustered and longitudinal data by using a semi-parametric mixed effects model, in which the non-parametric part is estimated using regression trees or random forests. We propose a general approach of random forests for high-dimensional longitudinal data. It includes a flexible stochastic model which allows the covariance structure to vary over time. Furthermore, we introduce a new method which takes intra-individual covariance into consideration to build random forests. Through simulation experiments, we then study the behavior of different estimation methods, especially in the context of high-dimensional data. Finally, the proposed method has been applied to an HIV vaccine trial including 17 HIV-infected patients with 10 repeated measurements of 20,000 gene transcripts and blood concentration of human immunodeficiency virus RNA. The approach selected 21 gene transcripts for which the association with HIV viral load was fully relevant and consistent with results observed during primary infection.

A New Multilevel CART Algorithm for Multilevel Data with Binary Outcomes

The multilevel logistic regression model (M-logit) is the standard model for modeling multilevel data with binary outcomes. However, many assumptions and restrictions should be considered when applying this model for unbiased estimation. To overcome these limitations, we proposed a multilevel CART (M-CART) algorithm which combines the M-logit and single level CART (S-CART) within the framework of the expectation-maximization. Simulation results showed that the proposed M-CART provided substantial improvements on classification accuracy, sensitivity, and specific over the M-logit, S-CART, and single level logistic regression model when modeling multilevel data with binary outcomes. This benefit of using M-CART was consistently found across different conditions of sample size, intra-class correlation, and when relationships between predictors and outcomes were nonlinear and nonadditive.

Regression Trees for Longitudinal Data with Baseline Covariates

Longitudinal changes in a population of interest are often heterogeneous and may be influenced by a combination of baseline factors. In such cases, traditional linear mixed effects models (Laird and Ware, 1982) assuming common parametric form for the mean structure may not be applicable. We show that the regression tree methodology for longitudinal data can identify and characterize longitudinally homogeneous subgroups. Most of the currently available regression tree construction methods are either limited to a repeated measures scenario or combine the heterogeneity among subgroups with the random inter-subject variability. We propose a longitudinal classification and regression tree (LongCART) algorithm under conditional inference framework (Hothorn, Hornik and Zeileis, 2006) that overcomes these limitations utilizing a two-step approach. The LongCART algorithm first selects the partitioning variable via a parameter instability test and then finds the optimal split for the selected partitioning variable. Thus, at each node, the decision of further splitting is type-I error controlled and thus it guards against variable selection bias, over-fitting and spurious splitting. We have obtained the asymptotic results for the proposed instability test and examined its finite sample behavior through simulation studies. Comparative performance of LongCART algorithm were evaluated empirically via simulation studies. Finally, we applied LongCART to study the longitudinal changes in choline levels among HIV-positive patients.

Recursive Partitioning with Nonlinear Models of Change

In this article, we introduce nonlinear longitudinal recursive partitioning (nLRP) and the R package longRpart2 to carry out the analysis. This method implements recursive partitioning (also known as decision trees) in order to split data based on individual- (i.e., cluster) level covariates with the goal of predicting differences in nonlinear longitudinal trajectories. At each node, a user-specified linear or nonlinear mixed-effects model is estimated. This method is an extension of Abdolell et al.'s (2002) longitudinal recursive partitioning while permitting a nonlinear mixed-effects model in addition to a linear mixed-effects model in each node. We give an overview of recursive partitioning, nonlinear mixed-effects models for longitudinal data, describe nLRP, and illustrate its use with empirical data from the Early Childhood Longitudinal Study-Kindergarten Cohort.