The versatility of multi-state models for the analysis of longitudinal data with unobservable features

The Whitehall longitudinal cohort included 10,308 men and women aged 35–55 years who were recruited between 1985 and 1988 (Marmot et al. 1991). Data used here were from four follow-up study phases, the last ending in 1999. One outcome of interest is the occurrence of serious coronary heart disease (CHD) events. These events can be either fatal (\(F\)) or non-fatal (NF). A link to the National Health Service Central Registry provided accurate information on the date and cause of death for all CS who died. Thus complete information is available on \(F\) events but data on NF events are subject to interval censoring when a CS is lost to follow-up.

A key question of interest was the relationship between the grade of employment ‘level’ of a civil servant and CHD risk. It was recognized however that loss to follow-up may be linked to health status and thus that censoring could be informative. Therefore it was important to understand how robust any findings related to the risks associated with civil service grade were to informative censoring, and a sensitivity analysis to address this was desired.

If we regard the two expressions of CHD (NF and \(F\)) as two separate, but perhaps dependent, events, then a model for two processes, NF and \(F\), operating simultaneously is needed. During the observation period, each CS can experience none, one or even both of the events, the latter being possible only if an NF event happens first. NF events can also occur after the CS has been lost to follow-up (LTF) for the NF-process. In this case the NF event will be unobserved. No loss to follow-up is possible for the mortality process.

To model the two processes, the multi-state model presented in Fig. 1 can be used. This model, with five possible states, allows all possible combinations of events to be represented. Although we have complete information for the \(F\)-process, we have no information from a CS after they are LTF to the NF-process until the end of follow-up for the \(F\)-process (either death or end of study). In particular, we have no information on NF events that happen during that period of time. However, we allow for the possibility of such an event through inclusion of an unobservable state. The five states in the model are the healthy state, ‘H’, the fatal state, ‘F’, which is absorbing, a state, ‘NF’, to represent an observed non-fatal event, a state, ‘LTF’, to represent loss- to-follow-up for the NF-process, and the unobservable state, ‘NF(LTF)’, that represents an NF event experienced after the CS is lost-to-follow-up for the NF-process.

Note that for simplicity, we have described this multi-state model without regard to the non-CHD related deaths. In the subsequent analysis of the data, these deaths are treated as arising from a competing risk for the CS in the Whitehall II Study. Thus the estimated CHD-related hazard functions in our model are actually cause-specific hazards in a slightly more complex model. As in a typical competing risks analysis, we estimate these hazard functions by treating death from non-CHD related deaths as independently right censored.

Likelihood estimation for the model in Fig. 1 is outlined in Siannis et al. (2007), where the transition rates are assumed to have a proportional hazards structure with a Weibull baseline hazard function, with the time from entry into the study, denoted by \(t,\) taken as the time scale. However, for this model to be identifiable, at least two assumptions, that relate to the baseline transition intensity functions \(\lambda _5(t),\,\lambda _6(t)\) and \(\lambda _7(t),\) are required. The model fitting will be dependent on these assumptions, and hence they need to be plausible and subject to some level of sensitivity analysis.

The first assumption is that \(\lambda _3(t) = \lambda _7(t).\) This implies that whether or not a subject is censored, the risks of death after having an NF event are the same. The second is thatwhich implies that, although the risk, or hazard, associated with having either an \(F\) or NF event could be different after a subject is LTF than before, the ratios of these hazards before and after LTF are assumed proportional, and, when \(k=1,\) equal.

The proposed assumptions are motivated by the particular multi-state model and are not data-driven. They provide a means to identify the transition rates introduced because an unobservable state has been added to the model. The parameter \(k\) cannot be estimated and can serve only as a sensitivity parameter. However, it may be valuable to compare predictions from these alternative multi-state models with those from dynamic predictions made through alternative methods such as landmarking (van Houwelingen and Putter 2008).

As mentioned earlier, the role of employment grade is of primary interest. A relevant test of significance can be based on the multi-state model. A further simplifying assumption can be made that the explanatory variable effects related to all transitions to the \(F\) state are the same. Then, to carry out the significance test, the multi-state models with the grade effect allowed to vary across the different hazards (six df for grade, two for each freely varying transition rate) and the grade effect completely removed (zero df for grade) are fitted. A likelihood ratio (LR) test to compare the two models leads to the test statistic \(LR=142.68,\) which, when compared to the \(\chi ^{2}\) distribution with six df, is highly significant (\(p<0.0001\)). This demonstrates a significant role for grade level.

The possible inadequacies of analyses based on the time-to-first event, either NF or \(F,\) and the time-to-NF event in the competing risks setting where we ignore the \(F\)-process subsequent to occurrence of the NF event, were also a motivating factor for the semi-competing risks multi-state model developed since the follow-up of subjects after an NF event can and should be incorporated. To illustrate the differences in results obtained from performing a time-to-first event Cox regression analysis and the proposed multi-state analysis, we present the cumulative incidence functions from both these analyses by grade level, restricted to males aged 45–49 years. Figure 2a presents the results for all grade levels but with the multi-state results restricted to \(k=1.\) The remaining figures (Fig. 2b–d) present for each grade level in turn, the Cox results and the multi-state results from sensitivity analyses with \(k=0.5, 1\) and \(2.\) It can be seen that the multi-state model estimates more of an increased risk for males associated with grade 3 than does the time-to-first event analysis. The differences for the other grade levels are much less marked.

Standard methods, such as those based on Cox’s relative risk regression model, can provide analyses of the (semi-)competing risks separately and investigate, to an extent, the relationship between the risk of the terminal and non-terminal event through the use of time-dependent covariates. However such methods do not provide a comprehensive joint analysis for semi-competing risks data, as they do with standard competing risks data. An analysis of time-to-first event is particularly problematic using standard methods since some potentially informative data must be ignored.

The multi-state model presented here, with an unobservable state, provides another approach to the analysis of semi-competing risks data. While two model-based assumptions are necessary to ensure model identifiability, the model does provide a structure that clearly reflects the observed data and allows full use to be made of them. Explanatory variables are easily incorporated and estimation of various quantities of interest is possible. Importantly, the association between the risks and potentially informative LTF is introduced in a rather general way.

The next section presents another example of joint modelling through a multi-state structure but where the focus is more specifically on the relationship between processes and where a specific form of measurement error is a particular concern. It, and all our subsequent examples, will derive from the analysis of data from psoriatic arthritis (PsA) patients.

Table 1 presents selected results from fitting two multi-state models to the Toronto PsA data. The first fits the misspecified three-state model for physical functioning (HAQ) alone with updating of both disease activity and damage only at the times of clinic visits. The second fits the expanded multi-state model of Fig. 3b where the fluctuating nature of disease activity is reflected. Both models included age and arthritis duration as explanatory variables but results are only presented for sex and damaged joint count, along with the estimated disease activity effects on fatigue transitions derived from the logarithms of ratios of baseline rates for the expanded model and from the regression coefficients for the misspecified three-state model.

While attenuation of activity effects from this misspecified model is quite marked and would be expected based on earlier work (Raboud et al. 1993; Andersen and Liestøl 2003), this will not necessarily be the case for other explanatory variables. For example, consider the effect of gender on the \(1\rightarrow 2\) transition corresponding to an upward move from none or low disability to moderate disability and the effect of the number of clinically damaged joints on the \(3 \rightarrow 2\) transition corresponding to a downward move from high disability to moderate disability. For the former, even though it is a time-independent variable, we observe an 18 % inflation in the absolute effect size for being male compared to female when we use the misspecified model instead of the expanded model, with the statistical significance of this effect becoming stronger (\(Z\)-statistic = 5.36 vs. 4.40). For the latter, we observe a 31 % reduction in the absolute effect of a one clinically damaged joint increase on the instantaneous transition from high disability to moderate disability, with a somewhat larger \(p\)-value (\(p\) = 0.077 vs. 0.014). The possibility of both attenuation and strengthening of effects has also been demonstrated in simulation studies (Tom and Farewell 2011).

Intermittent observation of time-varying explanatory variables is common and, to a large extent, pragmatic assumptions are made to deal with this. However, many regression analyses of longitudinal data can be configured in terms of a multi-state model. If panel data is available for the estimation of the model and, in addition, there is one key time-varying explanatory variable which is reasonably represented by an ordinal variable, then the use of an expanded multi-state model may be useful. Minimally, it could act as the basis for a sensitivity analysis with respect to the potential impact of the more commonly made simplifying assumptions with respect to the pattern of the explanatory variable over time. Moreover an advantage, as opposed to other joint modelling approaches to coping with measurement error, is that it allows for the uncertainty in this key explanatory variable without explicitly having to model the rapid changes between visits.

The next section returns to the introduction into a multi-state model of unobservable or partially observable states and illustrates how the incorporation of a misclassification structure can allow the fitting of such a model.

A three-state model that reflects this structure is given in Fig. 4. Suppose that \(S(t)\) represents the state for a patient at time \(t,\) where \(t\) denotes the years since diagnosis of PsA. In this model, state \(1\,(S(t)=S_1\)) corresponds to active disease, state \(2\,(S(t)=S_2\)) corresponds to the early stage of remission and state \(3\,(S(t)=S_3\)) corresponds to an established stage of remission. It should be noted that a patient in either \(S_2\) or \(S_3\) is regarded as in remission. These are conceptual, not observed, states that are introduced in order to give all remissions some duration. This is achieved by the further assumption that patients in \(S_2\) can only move to \(S_3.\) Transitions back to active disease, \(S_1,\) can only occur from \(S_3.\)

This model can be further specified by three transition rates. Associated with the \(S_1 \rightarrow S_2\) transition, the first is the instantaneous rate of progression to \(S_2,\) conditionally on occupying \(S_1\) at \(t\):where \(\lambda _{012}(t)\) is a baseline transition rate, \(\mathbf{{x}}(t)\) represents a vector of (possibly time-dependent) explanatory variables and \(\mathbf{{\beta }}_{12}\) is the corresponding unknown vector of regression coefficients. Typically, this transition rate, (3), into \(S_{2}\) would be the transition of greatest interest as it would provide a model for the distribution of time to remission from an active disease state.

A transition rate for the \(S_3 \rightarrow S_1\) transition, \(\lambda _{31}(t ; \mathbf{{x}}(t)),\) can be defined analogously and corresponds to a model for transitions from remission back to active disease. The final transition, \(S_2 \rightarrow S_3,\) will have a rate of the same form but this conceptual transition is unlikely to be of clinical interest.

In fitting the model, the uncertainty regarding a patient’s state at some time point can be made explicit through the introduction of misclassification probabilities. Suppose that \(S(t)=r\) (\(r=S_1, S_2, S_3\)) represents the true underlying state of a patient at time \(t,\) and \(O(t)=s\) (\(s=O_1, O_2, O_3\)) represents the corresponding observed state at time \(t\) based on active joint counts. Specifically, let \(O_1\) denote the observation of a non-zero active joint count at a visit, \(O_2\) denote the observation of a zero active joint count at a visit not preceded by at least two other such visits, and \(O_3\) denote the observation of the third or subsequent zero active joint counts in a sequence of such visits. Then, based on the assumptions about active joint counts outlined earlier and the assumption that the misclassification probabilities are independent of time \(t,\) we can specify \(\Pr \{O(t)=s \mid S(t)=r\}\) as follows:Essentially, only \(S_1\) is allowed to be misclassified. Thus, at some time point \(t,\) a patient with a clinic visit and an associated zero active joint count not preceded by at least two other such visits can either be in active disease (\(S_1\)) or be in early stage of remission (\(S_2\)). A logistic model is used to investigate the influence of explanatory variables \(\mathbf{{z}}\) on \(\Pr (O_2 \mid S_1).\) Specifically, for illustration, a binary explanatory variable, \(Z,\) will be included. It is coded one for the first visit with a zero joint count not preceded by any other such visits and zero for visits with a zero active joint count only preceded by one such visit. The variable is defined only for visits corresponding to observed state \(O_{2}.\)

Multi-state models with misclassification have been discussed in Jackson et al. (2003) and can be fit using the msm R package under the assumption that transition rates are constant (i.e., time-independent) or piece-wise constant over time. Because the \((O_1, O_2, O_3)\) classification is only observed at clinic visits, the maximum likelihood estimation is based on the panel data arising from this intermittent observation pattern and the interval censoring option of the msm package must be used.

For comparison, two other approaches can be used to define remission in the PsA data. The first treats a patient as in remission at the end of a time period with three consecutive clinic visits with no active joints, while in the second approach, a patient is considered in remission at the beginning of such a period. Following Farewell and Su (2011), the three-state model can be denoted as Model A and the two other possibilities outlined here as Models B and C respectively.

A detailed analysis of data from 790 patients entering the Toronto PsA Clinic in the years 1973–2006 can be found in Farewell and Su (2011). The time scale \(t\) is taken to be time since diagnosis of PsA. Typically patients will come to clinic at some time after diagnosis so likelihood estimation must incorporate left truncation of observation and this is also implemented in the R package msm. After exploratory analyses, baseline rates were taken to be piecewise constant in the two intervals \([0, 15]\) and \([15, 40].\)

Models B and C would both have only two alternating states, corresponding to \(S_{1}\) and \(S_{3}\) in the three-state model, and no misclassification. In Model B, it is assumed that patients were in remission only at the third of three or more consecutive clinic visits with zero active joint counts (either from enrollment or following a visit with a non-zero active joint count). That is, a transition to remission had occurred between the second and third clinic visits of a time period with three consecutive visits with zero active joint counts. In Model C, it is assumed that remission had occurred by the first visit of such a period and after the previous visit at which a non-zero count was observed.

Two explanatory variables were included in the transition rate models for \(\lambda _{12}(t)\) and \(\lambda _{31}(t).\) Regression coefficients and standard errors for these variables for Models A, B and C are given in Table 2.

The models can also be compared when used to estimate the total length of stay in various states over a fixed time period. If a period of 40 years after PsA diagnosis is considered, then the estimated time with active disease is 30.6, 32.5 and 29.5 years in Models A, B and C, respectively. Correspondingly, the time in remission (time spent in either \(S_2\) or \(S_3\) in Model A) for the same patients follows the reverse ordering. The results for female patients who were 35 years old at PsA diagnosis are different with the estimated time with active disease being 34.6, 37.0 and 35.5 years in Models A, B and C, respectively.

With the most conservative approach of defining remission, Model B always gives the least time spent in remission. Model A allows the possibility that, at the visits with zero active joint counts but not preceded by at least two other such visits, the patients were actually in early stage of remission. Therefore the estimated time spent in remission in Model A is expected to be longer than in Model B. In Model C, we allow the patients to be in remission exactly two clinic visits earlier than in Model B, which also makes the estimated time spent in remission longer. The ranking of the corresponding estimates from Models A and C will depend on specific scenarios. Model A allows a patient to possibly be in remission at visits with zero active joint counts but not preceded by at least two other such visits, regardless of the observed states at the following visits. In Model C, patients could be in remission as early as at the first visit of a sequence of three visits with zero counts. However, no possibility is given to remission at those one or two visits with zero active joint counts not preceded by at least two other such visits and immediately being followed by a visit with non-zero active joint count or reaching the study cut-off date. Thus a trade-off between these factors determines the estimated lengths of stay from Models A and C.

The fit of Model A is further characterized by the estimated misclassification probabilities. While \(\Pr \left\{ O(t)=O_2 \mid S(t)= S_1, Z\right\} \) is directly modelled here, the quantity \(\Pr \left\{ S(t)= S_1\mid O(t)=O_2, \mathbf{{x}}, Z \right\} ,\) corresponding to the probability that patients were actually in active disease when observed at clinic visits with zero active joint counts not preceded by at least two other such visits, given the explanatory variables, is more relevant. The latter probability can be calculated via Bayes’ rule if \( \Pr \left\{ S(t)=S_1 \mid \mathbf{{x}} \right\} \) and \(\Pr \left\{ S(t)=S_2 \mid \mathbf{{x}} \right\} \) are approximated by the proportions of estimated total length of stay in \(S_1\) and \(S_2\) for the time periods \(t \in [0, 15)\) and \(t \in [15, 40],\) given the explanatory variables.

This calculation, for male patients diagnosed at age 35, gives misclassification probabilities, \(\Pr \left\{ S(t)= S_1\mid O(t)=O_2 \right\} ,\) of 0.938 and 0.322 for the first (\(Z=1\)) and second consecutive visits (\(Z=0\)) with zero counts during the first 15 years of disease and values of 0.810 and 0.118 subsequently. For females, the comparable numbers are 0.965 and 0.468 in the first time period and 0.887 and 0.199 in the second. As expected, the misclassification probabilities at the second visit with a zero count are much smaller than at the first.

The benefit of using this three-state model for remission is that it avoids ad hoc definitions of time to remission. However, to do this a “hypothetical” transition must be introduced to allow a plausible time in remission. Nevertheless, because misclassification is included in the model formulation, the assignment of observed states known to be problematic is avoided. Minimally, the use of the three-state model could be seen as a check on the robustness of findings from more simplistic approaches which might be adequate for some purposes and is another example of a multi-state model providing an approach to sensitivity analyses.

In the next section, we consider the joint modelling of many processes. This will involve the fitting of many correlated multi-state models, and in the particular application, each of these models will represent two separate processes as well. In addition, we will consider the extent to which causal arguments can be linked to results from estimation of these models.

There are 28 hand joints in total (excluding the wrists) and we consider a model for the 14 pairs of comparable joints on the left and right hands.

A four-state model for damage in each of the 14 pairs of hand joints is depicted in Fig. 5. The four states of this multi-state model are defined asNote that this is a multi-state model at a specific joint location in both left and right hands.

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 sub-processes, whereby no two sub-processes or components can change state ‘simultaneously’), then (in)dependencies between sub-processes can be explicitly expressed through the transition rates of the original Markov process. The model in Fig. 4 is composable, comprising separate damage sub-processes for the left and right hands at the specific joint location, and local dependencies between the sub-processes will be reflected in relationships between the transition rates of this multi-state model. In addition, this model does not allow transitions directly from \(S_{1}\) to \(S_{4}.\) This is a necessary constraint to ensure composability but is not very restrictive for models with transitions in continuous time. Transitions between \(S_{2}\) and \(S_{3}\) are not allowed since damage is irreversible. Furthermore, because the 14 multi-state processes within a patient should be more similar than across patients, we introduce subject-specific random effects into the model. These act multiplicatively on the baseline transition rates of the 517 patients. The random effects, \(U_k,\ k=1,\ldots ,517,\) are assumed to be distributed as independent Gamma random variables with unit mean and variance \(\theta \) which we denote by \(U_{k} \sim \text{ Gamma}(1/\theta , 1/\theta ).\) This multi-state model is a generalization of Schweder’s model to incorporate random effects and is similar to the one proposed by Cook et al. (2004) for clustered progressive multi-state processes and is in the same spirit as earlier work done by Self and Prentice (1986) for multivariate failure time data. Nevertheless, this model still allows us to examine local dependencies through the transition rates.

To investigate the relationship between damage and the dynamic course of disease activity at the individual joint (location) level we define binary indicators of joint-level activity, \(A_{L}(t), A_{R}(t)\) for the left and right joints respectively in a pair at time \(t.\) Here, we make no distinction between activity in the form of tenderness and effusion. However, where a joint makes a transition into a state of damage, we are interested in assessing a possible dose–response relationship between joint activity and the rate at which damage occurs. Thus we also define dynamic binary indicators for joint-level tenderness, \(T_{L}(t)\) and \(T_{R}(t),\) and joint-level effusion (with or without tenderness but usually tender), \(E_{L}(t)\) and \(E_{R}(t),\) for the left (\(L\)) and right (\(R\)) hands at time \(t.\) Hence, assuming that no previous damage has occurred to either joint (\(S_{1}\)), models for the transition rates out of \(S_{1}\) are given by:We further assume that the baseline transition rates are constrained to be the same across the 14 hand-joint locations. That is, \(\lambda ^{(l)}_{012}=\lambda _{012}\) and \(\lambda ^{(l)}_{013}=\lambda _{013},\,\forall l.\) For simplicity, we do not explicitly denote the dependence of the rates on explanatory variables in the left hand side of the defining equations.

When forming models for the transition rates into \(S_{4},\) we choose not to include information (in the form of explanatory variables) on the activity process in the opposite (damaged) joint at time \(t\) because of the dominant effect of symmetrical damage to be discussed later. Hence, our models for the transition rates into \(S_{4}\) are given by:and, once again, we assume that the baseline transition rates are constrained to be the same across the 14 joint locations.

Here, we focus on the relationship between observed transitions in the damage process and the value of these activity variables (represented by activity or joint tenderness and swelling on both the left and right hands) at the last clinic visit. That is, we assume \(A_{L}(t),\,A_{R}(t),\,T_L(t),\,T_R(t),\,E_L(t)\) and \(E_R(t)\) to be piecewise constant between clinic visits. As is done in Sect. 3, this assumption could be relaxed and is further discussed in O’Keeffe et al. (2011). We additionally assume that these activity variables can only change ‘state’ immediately after the time of a clinic visit when damage information becomes available.

It is ‘biologically’ conceivable that this model, given by (4) and (5), can be further constrained to allow \(\alpha _{L12}=\alpha _{R13},\,\tau _{R12}=\tau _{L13},\, \tau _{L24}=\tau _{R34},\) and similarly, \(\varepsilon _{R12}=\varepsilon _{L13},\,\varepsilon _{L24} =\varepsilon _{R34}.\) These constraints are used in the interest of parsimony.

We also include additional binary explanatory variables (for joints on the left and right hands) that indicate whether a joint was ever observed active (either swollen or tender) at the present visit or any of the previous protocol visits. We constrained the associated regression coefficients to be the same for transitions in the left and right hands and allowed only prior activity in the undamaged joint to have an effect on the corresponding transition into \(S_{4}.\) These binary variables attempt to incorporate more of the history of disease activity (i.e. the persistence) in the hand joints of patients into the four-state damage model.

A symmetrical damage pattern in arthritis implies that the tendency for a joint at a specific location to become damaged is increased if the contralateral joint on the other hand is earlier damaged, and this applies at all joint locations. To investigate the extent of disease symmetry in the hands, it is useful to re-parameterize some of the baseline transition rates in terms of others. That is, we specify \(\lambda _{024}^{(l)} = \lambda _{013}^{(l)}\exp (\gamma _{24})\) and \(\lambda _{034}^{(l)}=\lambda _{012}^{(l)}\exp (\gamma _{34}),\) with no constraints placed on the baseline transition rates. Therefore a symmetric pattern would correspond to \(\lambda _{012}^{(l)} < \lambda _{034}^{(l)}\) and \(\lambda _{013}^{(l)} < \lambda _{024}^{(l)},\) or equivalently that \(\gamma _{24}>0\) and \(\gamma _{34}>0.\)

Maximum likelihood estimation of the parameters of the four-state damage model can be implemented and details are given in O’Keeffe et al. (2011).

To illustrate the use of the four-state damage model, we use data extracted from 517 of the 790 patients in the Toronto PsA clinic who entered before January, 2007. These 517 patients correspond to those who at clinic entry had no clinical damage in any of the hand joints on either the left or right hands and therefore information on all these patients is comparable.

Based on these data, and with no explanatory variables introduced into the transition rate models, the parameters, \(\gamma _{24}\) and \(\gamma _{34}\) defined above to characterize symmetry are estimated to be 1.82 and 1.39 respectively with corresponding 95 % confidence intervals of (1.49, 2.14) and (1.01, 1.78). Both of these confidence intervals indicate substantial departures from zero to the right, and therefore present strong evidence for symmetry. This symmetry, in turn, implies that there are local dependencies in both direction between the two damage sub-processes (i.e. the left and the right) of the composable multi-state joint location process.

Table 3 highlights estimation results related to the explanatory variables in the multi-state model defined in Subsect. 5.1 to investigate the relationship between disease activity and damage progression. Seven patients were excluded due to missing information on disease activity. In this table ‘transitive’ joint refers to the joint undergoing the transition to a state of damage (i.e. to \(S_{2},\,S_3\) or \(S_4\)) and ‘opposite’ joint refers to the same joint in the opposite hand. From this table, where there is no previous damage in either joint (i.e. the model is currently in \(S_{1}\)), we observe increases in the rates of transitions into a state of damage when there is current activity (in the form of both tenderness and effusion) and when there has been some past activity in the transitive joint, compared to when no activity has been seen. We note that effusion shows a larger positive effect on transition to damage than tenderness in this situation. Conversely, activity in the opposite joint appears not to significantly affect the transition rate to damage.

In the situation where the opposite joint is already damaged, we again see an increase in the transition to damage both where there is current tenderness and where there is current effusion as well as where past activity has occurred in the transitive joint. We note that the effects of tenderness and effusion are similar; we no longer see an apparent dose–response relationship for the covariates representing activity in the transitive joint. Presumably this is related to the additional effect of symmetrical joint damage.

The log-rate ratios of the six transitive association effects described previously, are all positive and large, with probable differential effects observed between the comparable transitions for tender only and the more severe effused (usually also tender) joints, ipsilaterally, where no damage has occurred to the opposite joint. The effects corresponding to effusion tend to be larger than the comparable ones for tender only, ipsilaterally, where no damage has occurred to the opposite joint. These six association effects are all the statistically significant ones found and the four corresponding to current activity suggest that the link between activity and damage is local/specific to the joint on the particular hand being considered. These four associations are what we would consider as a local dependence/influence of activity on damage at a joint. The two significant associations corresponding to past activity in the transitive joint may indicate that the persistence of activity is also important in predicting future damage progression. We do not observe any statistically significant association, or evidence of a possibly substantive effect, of having or not having activity in a joint on one hand with having or not having damage on the contralateral joint of the other hand.

Schweder (1970) regarded his concept of local (in)dependence as a potential aid in addressing causal questions and it is closely linked to the concept of Granger causality (Granger 1969). We note that Granger causality was initially defined in the context of discrete time series, and local (in)dependence may be viewed somewhat as an extension of the Granger causality/non-causality concept to processes in continuous time. The asymmetry of the local independence concept makes it particularly attractive, as having one sub-process locally influencing a change in another, but the other not having any influence on the first, is precisely how we would like to characterize a causal effect of one sub-process on another. However, a one-sided local dependence relationship between two sub-processes of a composable Markov process is not sufficient to imply causation. Aalen (1987), when extending the concept, stressed that local (in)dependence was a dynamic statistical approach which, by incorporating time explicitly, offers a natural way to model potential causal relationships.

Mathematical formalization of the causality concept, in itself, is not enough to allow a causal relationship to be inferred from an observed association. From an epidemiological viewpoint, Hill (1965) discussed aspects of an association that should be considered when attempting to infer causation. These are known as the Bradford Hill criteria and are (i) strength of association; (ii) consistency; (iii) specificity; (iv) temporality; (v) biological gradient; (vi) biological plausibility; (vii) coherence; (viii) experimental evidence (when available); and (ix) analogy. Hill stressed that these are only ‘viewpoints’ to be considered however and are not necessary and/or sufficient conditions to declare causation from an observed association, although temporality is a necessary condition as a cause must precede its effect.

Analysis of the Toronto PsA data provided evidence of a symmetric pattern of joint damage. For a pair of joints at the same location in the two hands, local dependencies were identified in both directions between the damage sub-processes in the left and right hands. While these results represent an important finding, these local dependencies do not immediately warrant a causal explanation, as they do not produce a one-sided (local dependence) asymmetric relationship between the two damage sub-processes. It is quite plausible that the same underlying biological mechanism is driving these two damage sub-processes, although a robust biological explanation for damage symmetry has not yet been established.

However, the analyses also suggest (i) joint-specificity of the relationship between activity and damage; (ii) strong associations for the four statistically significant and biologically plausible local effects obtained; and, where no previous damage has occurred in either joint of a pair, (iii) a dose–response relationship of activity with damage at the joint level (i.e. a biological gradient). These relationships represent local dependence of activity on damage although the activity process is not formally modelled. Here the determination of asymmetry of local dependence relationships is not specifically addressed and is less critical because any influence of permanent damage on subsequent activity in a joint is of less clinical interest or, at least, represents a very different clinical question. However, as Aalen argues, the dynamic perspective of Schweder’s work may be most important and these analyses, because of the longitudinal nature of the data, do help to characterize the temporal relationship between activity and damage.

These results are also consistent with the majority of the Bradford Hill Criteria: specificity, strength of association, biological gradient, temporality and biological plausibility, at the joint level. Others such as consistency and analogy have been shown at the patient (systemic) level in other PsA populations and in RA populations respectively. Moreover our results do not, in any way that we know of, conflict with generally known facts regarding the biology and natural history of disease progression in PsA patients, thus suggesting coherence. Furthermore recent clinical trials on biologic agents have shown the effectiveness of these therapies in slowing the disease progression. These agents calm the inflammation of arthritis by inhibiting immunological pathways that trigger inflammatory response. Therefore these results do provide support for a putative causal relationship between activity and damage. Also, we believe that, from the perspective of both biological plausibility and temporal ordering, this demonstrated relationship at the joint level offers more support for a causal link than the relationships at the patient level seen in previous investigations.

The study of the progression of PsA using multi-state models provided an intuitive way of examining the disease process from a dynamic perspective and afforded a straightforward way to assess local (in)dependencies in dynamic processes. The introduction of random effects allowed a natural way to model correlated multi-state processes so that individual joint data could be appropriately examined. Thus, it is possible to consider the extent to which evidence of local dependence between the activity and damage processes, together with the Bradford Hill Criteria, permits a causal link between activity and damage in this observational setting. However as Prentice and Thomas (1987) writes:

‘\(\ldots \) any reported associations from an observational study will be subject to some uncertainty concerning causality. Associations that are strong, that exhibit “regular” dose–response relationships, and that can be replicated in a range of study populations come, in time, to be regarded as causal.’

Therefore it is over time and with replication and mechanistic understanding that the link between activity and damage will truly be regarded as causal.