Longitudinal data are repeated measurements from the same subject observed over time. The within-subject measurements (over time) are typically not independent. Although serial correlation may not be of primary interest but it must be taken into account to make proper inferences. In marginal models, the population-averaged effect of covariates on the longitudinal response is directly specified [

1,

2] and the regression coefficients have interpretation for the population rather than for any individual [

3]. In conditional models, the effect of covariates on responses is specified conditional on random effects or previous history of responses. Hence, the population-averaged effect of covariates is indirectly specified [

4,

5]. In this paper, we consider marginal model approaches.

Properly specified probability models lead to efficient estimation even under missing at random (MAR) [

6] and nested models can be compared using likelihood ratio tests and non-nested models by penalized criteria such as Akaike information criterion (AIC) [

7] or Bayesian information criterion [

8]. Recently, marginalized likelihood-based models have been developed for the analysis of longitudinal categorical data [

9–

13]. Heagerty [

9,

10] proposed marginally specified logistic-normal models and marginalized transition models (MTM) for longitudinal binary data. In both models, a marginal logistic-regression model was used for explaining the average response. The models were specified by introducing random effects in the logistic-normal models and Markov dependence for MTM to explain the within-subject dependence. Miglioretti and Heagerty [

12] developed marginalized multilevel models for longitudinal binary data in the presence of time-varying covariates. Lee and Daniels [

13] extended Heagerty's work to accommodate longitudinal ordinal data using Markov dependence. Marginalized models have advantages over conditional models. First, the interpretation of regression coefficients does not depend on specification of the dependence in the model unlike in conditional models. In addition, estimation of covariate effects has been shown to be more robust to mis-specification of dependence [

10,

11,

13].

Models for correlated ordinal data typically fall into two classes based on how the dependence is modeled: via the global odds ratio [

14–

17] or via random effects [

18–

20]. A general overview of models for ordinal categorical data can be found in Liu and Agresti [

21]. The main contribution of this paper will be to introduce a new marginalized model for longitudinal ordinal data.

A common issue in inference from longitudinal studies is potential biases introduced by missing data. Classes of models to accommodate longitudinal data with dropout are summarized in Hogan

*et al*. [

22]. Standard approaches to handle missing data implicitly ‘impute’ values of response after dropout. For quality of life (QOL) data, if a subject drops out due to death, the QOL will not be defined after the dropout time. One way to address the type of dropout is to model the joint distribution of the longitudinal responses and progression/death times [

23–

25]. Hogan and Laird [

23] used a mixture model for the joint distribution of longitudinal measures and progression/death times. Pauler

*et al*. [

24] proposed a pattern mixture model (PMM) for longitudinal QOL data with non-ignorable missingness due to dropout and censorship by death. Recently, Kurland and Heagerty [

25] explored regression models conditioning on being alive as a valid target of inference. They used regression models that condition on survival status rather than a specific survival time. We will use the ideas in Hogan and Laird [

23] similar to the previous work by Pauler

*et al*. [

24].

We implement a principal stratification approach [

26,

27] here to make inference on the causal effect of the treatment on QOL among (potential) survivors on both treatment arms. Frangakis and Rubin [

26] discussed causal effects in studies where the outcome was recorded and unobserved due to death. Rubin [

28] and Hayden

*et al*. [

29] referred to the estimand in Frangakis and Rubin [

26] as ‘the survivors average causal effect (SACE)’. Egleston

*et al*.

27 proposed assumptions to identify the SACE and implemented a sensitivity analysis for some of those assumptions. Rubin [

30] introduced the causal effect of a treatment on a outcome that is censored by death in QOL studies. In this paper, we describe an approach that can be used to obtain the causal effect of treatment in the presence of death based on principal stratification for longitudinal ordinal outcomes.

This paper is arranged as follows. In Section 2, we describe the motivating example. We briefly review marginalized random effects models (MREMs) for longitudinal binary data [

9] in Section 3. In Section 4, we propose an ordinal MREM (OMREM) for the longitudinal data with ignorable dropout. In Section 5, we conduct a simulation study to examine bias and efficiency in estimation of marginal mean parameters. In the context of QOL data collected in a recent colorectal cancer clinical trial [

31], we propose models for the OMREM under dropout due to progression/death and illustrate them on this data in Section 6.