## 1. Introduction

From a clinical perspective, it is generally held that better understanding of a disease process will lead to more appropriate treatment and disease management of patients. Often multistate models (Hougaard, 1999; Commenges, 1999; Andersen and Keiding, 2002; Meira-Machado *et al.*, 2009) are particularly useful for this. In this paper we illustrate their use for the study of disease progression in psoriatic arthritis.

In his 1970 seminal paper, Schweder (1970) introduced the concept of local (in)dependence between components of a composable finite Markov process. He felt that many studied phenomena can be realistically described by time continuous finite Markov processes. If, in addition, the Markov process representing the phenomenon under study could be defined to be composable (i.e. represented as a vector of distinct subprocesses, whereby no two subprocesses or components can change state ‘simultaneously’), then (in)dependences between these subprocesses can be explicitly expressed through the transition intensities of the original Markov process.

More explicitly, let **Y** be a composable finite Markov process with components *Y*_{1},…,*Y*_{p}, which we denote as **Y**∼(*Y*_{1},…,*Y*_{p}), and let *V*={1,…,*p*}. Here the dependence on time in the notation is left out for convenience but is implicit. A component *Y*_{k} is said to be locally independent of *Y*_{j}, *j*≠*k*, given the remaining components **Y**_{V∖{j,k}}, if and only if the transition intensities of **Y** corresponding to transitions only between states in *Y*_{k} are constant functions for any state in *Y*_{j}, within a specified infinitesimal time interval. If *Y*_{k} is not locally independent of *Y*_{j}, it is said to be locally dependent on *Y*_{j}.

The above relationship is a ‘local’ property as it holds in an infinitesimal time interval. Furthermore, recall that, because **Y**∼(*Y*_{1},…,*Y*_{p}) is composable, no two components of **Y** can change state at the same time over this infinitesimal interval and therefore the transition intensities for such simultaneous transitions in more than one component are zero. Moreover, local independence is an asymmetric relationship, i.e. it has a direction, and so, *Y*_{k} being locally independent of *Y*_{j} does not necessarily imply that *Y*_{j} is locally independent of *Y*_{k} over the same infinitesimal time interval.

This important concept of local independence was extended in Aalen (1987) to apply to more general stochastic processes that admit a Doob–Meyer decomposition, with unrelated innovations. Aalen stressed the usefulness of ‘dynamic’ models where dynamic refers to how the future relates to the past, and it is this dynamic nature of Schweder's work, rather than the Markov assumption *per se*, which he suggested is important for statistical analysis. Earlier works by Cox (1972) and Aalen *et al.* (1980) also reflect this dynamic viewpoint. More recently, Didelez (2007, 2008) introduced dynamic graphical models to describe these local dependences, which further allowed local independences to be read off. In our discussion of the progression of psoriatic arthritis disease that follows, we use Schweder's local (in)dependence concept as a means of characterizing the findings from dynamic analyses based on multistate models.

Because we focus on arthritic disease progression at the individual joint level, our analysis is based on the use of correlated multistate models. Cook *et al.* (2004) considered such an analysis for progressive processes but with a discrete multivariate random-effects distribution used to account for correlation. For a four-state progressive model, we extend this to allow the use of gamma-distributed random effects. In addition, for a three-state model with some reversible transitions, we outline an approach that is based on generalized estimating equations to account for correlation between processes.

Our reported investigations are based on observational clinical data and it is well recognized that causal relationships can never be proved with such data. However, as Weiss (1986) argued,

‘it is necessary to attempt to draw inferences of cause and effect, even from inevitably incomplete data, for the alternative is to make no inference at all, which would preclude taking preventive or therapeutic action’.

Therefore, after presenting results of our analyses, we consider the extent to which they might allow inference concerning causal relationships. In doing so, we take up the implied challenge of Aalen (Aalen *et al.* (2008), page 348) who wrote:

‘One major danger of avoiding the subject of causality in statistical education and statistical literature, is that one never gets any insight into this fascinating concept, which has such an old history in philosophy and science. The fact is that statistics plays a major role in looking for causal connections in many fields and statisticians who know next to nothing about causality as a larger concept will be far less useful than they could have been.’

After 30 years of data collection and 20 years of previous analyses, it seems particularly appropriate to take up this challenge with the psoriatic arthritis cohort data that are discussed in the next section.