Joint model for longitudinal mixture of normal and zero-infated power series correlated responses Abbreviated title:mixture of normal and zero-inflated power series random-effects model Nastaran Sharifan � , Ehsan Bahrami Samani, and Mojtaba Ganjali Department of Statistics, Shahid Beheshti University, Tehran, Iran ABSTRACT In this paper, a joint model is presented for analyzing longitudinal contin- uous and count mixed responses. The frequency distribution of continuous longitudinal response variable for each subject at any time has a skewed and or multi-modal form. Then, a suitable fnite mixture of normals is used as its distribution. It seems that the continuous response comes from several distinct sub-populations. The number of zeros of the count response is infated. Also, a zero-infated power series (ZIPS) distribution is applied as its distribution in order to model the count response. The correlation of longitudinal responses through time and that of mixed continuous and count responses are modeled by utilizing the random-efects vectors in the fnite mixtures of regression (FMR) models. Further, a full likelihood-based approach is used to obtain the maximum likelihood estimates of parameters via the EM algorithm. Then, some simulation studies are performed for assessing the performance of the model. Additionally, an application is illustrated for joint analysis of the number of days during the last month that the individual drank alcohol, as well as the respondents’ weight. Finally, the two frst times of the Americans Changing Lives survey are evaluated. ARTICLE HISTORY Received 25 September 2019 Accepted 24 July 2020 KEYWORDS Mixed correlated responses; finite mixture distributions; joint model; the finite mixture of normals; zero- inflated; longitudinal studies; random effect; the EM algorithm 1. Introduction 1.1. Motivation Correlated continuous and count responses have been recorded in many longitudinal studies in biology, public health, and medical studies. Sometimes, it seems that the continuous response comes from several distinct subpopulations. Also, the count response has inflation on zeros. The outcomes related to the mixed correlated mixture of normals and count data are pervasive, and some research should be conducted for promoting their analysis. Consider the following example to clarify the issue. The national longitudinal study of American’s Changing Lives survey series (ACL survey) is an ongoing nationally representative longitudinal study that began in 1986 with 3617 adults aged 25 and up. The present study aims to evaluate the correlated continuous and count responses as the weight (W), measured in pounds for each respondent, and the number of days in the last month that the individual drank alcohol (DD), respectively. Further, 2795 subjects with no missingness in their responses through time were selected. Figures 1 and 2, respectively, present bar charts for DD, and histograms for W at the first two waves of the studies. 1 Furthermore, Table 1 indicates the descriptive statistics for these variables. As shown, the bar charts are accumulated in the zero value for both waves (about 50% during the first time and more during the second time), and the histograms have positive skewness. Many individuals never CONTACT Ehsan Bahrami Samani ehsan_bahrami_samani@yahoo.com Department of Statistics, Shahid Beheshti University, Tehran, Iran � JOURNAL OF BIOPHARMACEUTICAL STATISTICS https://doi.org/10.1080/10543406.2020.1814798 © 2020 Taylor & Francis Group, LLC