### Joint Modeling of Longitudinal Data and Time-to-Event (Survival) Data

Researchers are interested in two primary responses in the analysis of longitudinal data: longitudinal process response and time-to-event response. It is challenging to model these two responses jointly due to the censoring of the time-to-event response. Methodology Center researchers have developed new statistical procedures for joint modeling and applied them to empirical analyses in Das et al. (2013) and Shiyko et al. (2014).

Das, K., Li, R., Sengupta, S., & Wu, R. (2013). A Bayesian semiparametric model for bivariate sparse longitudinal data. *Statistics in Medicine, 32, *3899-3910. doi: 10.1002/sim.5790 PMCID: PMC3740051

Shiyko, M., Burkhalter, J. E., Li, R., & Park, B. J.. (2014). Modeling nonlinear time-dependent treatment effects: An application of the generalized time-varying effect model (TVEM). *Journal of Consulting and Clinical Psychology, 82* (760-772). PMCID: PMC4067470

### Time-Varying Effect Models (TVEM)

The time-varying effect model developed by Methodology Center researchers allows researchers to estimate the *time-varying *effects of coefficients. A natural extension of linear regression models, TVEM allows coefficients to vary over time. This flexible approach allows the mean trajectory and effects of covariates to vary with time without assuming parametric (e.g., linear or quadratic) functions. Tan, Shiyko, Li, Li, & Dierker (2012) provides a detailed introduction to time-varying effect models for audiences in the psychological sciences. An empirical demonstration of the %TVEM macro appears in Shiyko, Lanza, Tan, Li, & Shiffman (2012). Both articles make use of the %TVEM SAS macro, developed by Center scientists to enable applied researchers to employ TVEM.

Tan, X., Shiyko, M. P., Li, R., Li, Y., & Dierker, L. (2012). A time-varying effect model for intensive longitudinal data. *Psychological Methods.* Advance online publication.* *doi: 10.1037/a0025814 PMCID: PMC3288551

Shiyko, M. P., Lanza, S. T., Tan, X., Li, R., & Shiffman, S. (2012). Using the time-varying effect model (TVEM) to examine dynamic associations between negative affect and self-confidence on smoking urges: Differences between successful quitters and relapsers. *Prevention Science.* Advance online publication. doi: 10.1007/s11121-011-0264-z PMCID: PMC3171604

Qu, A. P., & Li, R. (2006). Quadratic inference functions for varying-coefficient models with longitudinal data.* Biometrics, 62,* 379-391. PMCID: PMC2680010

### Functional Hierarchical Linear Model (FHLM)

Functional hierarchical linear models (FHLMs) extend mixed effects models, hierarchical linear models, and multilevel models by allowing both fixed effects and random effects (i.e. individual effects) in these models to vary over time. Li, Root and Shiffman (2006) proposed FHLMs for analysis of ecological momentary assessment (EMA) data collected in smoking cession study. This article makes use of the %FHLMLLR SAS macro (an acronym for functional hierarchical linear model using local linear regression), developed by center scientists in 2009. This macro allows prevention scientists to graphically represent these complex relationships.

Li, R., Root, T. L., & Shiffman, S. (2006). A local linear estimation procedure for functional multilevel modeling. In T. A. Walls & J. L. Schafer (Eds.), *Models for intensive longitudinal data* (pp. 63-83). New York, NY: Oxford University Press.

### Data Spacing

An important issue in collection of ILD is how frequently the data should be collected. If data were collected too intensively or frequently, demands on subjects may be too intense, and excessive space will be required to store the data. On the other hand, some important features may be missed if data were not collected frequently enough. In Tan Dierker, Li, Rose, and TERN (2011), we study the impact of spacing of data on estimation of substance use trajectories.

Tan, X., Dierker, L., Rose, J., Li, R., & Tobacco Etiology Research Network. (2011). How spacing of data collection may impact estimates of substance use trajectories. *Substance Use & Misuse, 46*(6), 758-68. doi: 10.1007/s11121-011-0217-6

### Semi-Time-Varying Effect Models

Semi-time-varying effect models (semi-TVEMs) enable researchers to analyze longitudinal data sampled at irregular time points by using a regression model with both time-varying effects and time-invariant effects of covariates. Center scientists developed several innovative statistical procedures for the semi-TVEMs described in Kai, Li, and Zou (2011), Fan and Li (2004), and Fan, Huang and Li (2007).

Kai, B., Li, R., & Zou, H. (2011). New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. *Annals of Statistics, 39,* 305-332. PMCID: PMC3109949

Fan, J. Huang, T., & Li, R. (2007). Analysis of longitudinal data with semiparametric estimation of covariance function.* Journal of American Statistical Association. 102*, 632-641. PMCID: PMC2730591

Fan, J., & Li, R. (2004). New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. *Journal of the American Statistical Association, 99,* 710-723.

### Software

We distribute free software to researchers so they can use our methods accurately and easily.

The Methodology Center first released %TVEM SAS macro for SAS in 2011, and continues to improve and expand the software. There are currently four versions of the %TVEM macro, all of which are available in a single download.

- %TVEM_normal for data with a normal distribution
- %TVEM_poisson for data with a poisson distribution
- %TVEM_zip for data with a zero-inflated poisson distribution
- %TVEM_logistic for binary data with a logistic distribution

We developed the SAS macro for Estimation of Functional Hierarchical Linear Models (FHLM) Using Local Linear Regression Estimation Procedure in 2009.