Prospective trials of inpatient care for forensic mental health patients are rare. As such, it is seldom possible to carry out an economic evaluation of these services. When it is not possible to analyse trial data, methods of decision modelling represent a viable alternative. Decision-analytic modelling is a technique that employs both mathematical and statistical methods to simulate real life events (Briggs et al., 2006). Decision models enable decision makers to form policies under conditions of uncertainty in a real world context. Models present us with probabilities that a given event will occur under a set of pre-defined conditions. Events are associated with costs and benefits, enabling us to calculate expected costs and benefits of different interventions, the intervention with the greatest expected net benefit being the optimal choice. An important benefit of decision-analytic modelling is that, by making all parameters probabilistic, it allows for uncertainty around all inputs to the model and all decisions. This provides us with information regarding the level of uncertainty in our data, without needing to make arbitrary distinctions. Furthermore, modelling facilitates complete transparency in the assumptions made. There is good scope for economic modelling to be used in the evaluation of personality disorder treatments. Modelling techniques, such as Markov modelling, are used in many areas of health service evaluation, including chronic conditions with complex long-term outcomes such as rheumatoid arthritis (Chen et al., 2006) and diabetes (National Institute for Health and Clinical Excellence, 2008). Such modelling methods have, however, rarely been used in the evaluation of interventions for personality disorders. A recent study by Barrett and Byford (2012) used Markov modelling to evaluate the dangerous severe personality disorder programme, and there has been one other economic model developed to evaluate the cost-effectiveness of psychotherapy for personality disorder (Soeteman et al., 2010, 2011).
A model-based analysis of treatment non-completion has not previously been tried, nor has one related to the economic consequences of personality disorder more generally. This means that there are no existing models upon which to build our analysis. In our analysis, we use the technique of Markov modelling (Sonnenberg and Beck, 1993). Markov modelling has become popular in the evaluation of physical conditions. The operation of a Markov model is dependent on the condition that subjects can be defined as existing in one of a finite number of mutually exclusive and collectively exhaustive ‘states’. Subjects can then make transitions between these states based upon probabilities, with transitions occurring at the beginning of each Markov cycle. Each cycle spent in any one state is then associated with costs and outcomes, which can either be fixed or sampled from a probability distribution. Costs and outcomes are then accumulated over time. Markov models have a number of advantages over simpler methods in that they allow for a wider set of consequences and more realistic temporal dynamics.
Major costs of treatment of offenders with personality disorder follow from the location of the individual. Individuals may reside in the community, prison or hospital. Hospitals may be non-secure or high, medium or low secure. Including the state of being dead, this gives seven possible independent Markov states, as shown in Figure 1. The focus of our study is on costs, so specific outcome measures are not included. The aim of personality disorder services, however, is to rehabilitate people in order to return them safely to a community setting, so location, as defined by Markov states, may be considered an outcome in this respect.
Individuals in our model may move in any direction between any of the six living states, as well as remain in them for successive periods. ‘Dead’ is an absorbing state, transitions to which can be made from any other state, but from which no transitions can be made. Our model depends on the inclusion of two types of parameter: transition probabilities – the probability of moving from one state to another – and state costs. Other types of parameters, such as the quality of life associated with different states or the costs associated with state changes, might have been included but were not for the purpose of this analysis. Our transition probability and cost parameters were obtained from a number of sources, as described later.
Following discharge from a secure unit, individuals with personality disorder may move between states very frequently, so a Markov cycle of more than a month would be insensitive to changes in costs and outcomes and would be misrepresentative of reality. We implemented a model with a weekly Markov cycle. As we are interested in the longer-term consequences of non-completion, we used a 10-year time horizon for our model. Our model is divided into two arms: one for those who complete treatment and one for those who do not. Individuals are assumed not to die during the pre-discharge stage, as this would render them neither a completer nor a non-completer. After discharge, each arm of our model enters separate Markov simulations of post-discharge activity, with identical states and costs but different transition probabilities. Simulations begin from the point at which an individual is first admitted to hospital. We used time-to-event analysis to estimate the probability of discharge in a given Markov cycle. Statistical and regression analyses were carried out using Stata 12 (Stata Statistical Software, College Station, Texas, USA), and the Markov model was constructed using Microsoft Excel 2010 (Redmond, Washington, USA). The complete model Excel file is available in the supplementary online material.
Our analysis used data from a specialist National Health Service (NHS) service at Arnold Lodge, a medium-security hospital facility in Leicester, England. This service provides treatment for offenders with a diagnosis of personality disorder and offers structured cognitive behavioural interventions alongside a therapeutic community approach modified for the secure setting (McMurran et al., 2001; McCarthy and Duggan, 2010). The sample was of 95 patients, each of whom met diagnostic criteria for personality disorder according to the International Personality Disorder Examination (Loranger, 1994). Data were collected between 1999 and 2009, with a follow-up period of up to 10 years after discharge.
A clinical judgement on completion status was made at the time of discharge for purposes separate from this study. We used this information to separate our sample into two subgroups: ‘completers’ (n = 24) and ‘non-completers’ (n = 71). The former group comprised those who completed treatment and the subsequent planned discharge. Members of the non-completer group had been discharged early because of poor behaviour (for example, verbal aggression, violence or drug use) or premature disengagement or were unable to engage in the treatment because of other previously undetected mental health problems.
The transitional probabilities of our model are used to determine which Markov states individuals are likely to move from and to in any Markov cycle. These probabilities were calculated by eliciting the rate of each possible transition between the community, prison and hospital states in every week of the observed 10-year follow-up. These rates were then averaged and converted to weekly probabilities. Transition probabilities were calculated separately for treatment completers and non-completers. Using this method, we were able to estimate the standard errors for transition rates, allowing us to incorporate a probabilistic sensitivity analysis as discussed in succeeding text. As stated earlier, all transitions are possible in practice; however, because of the size of our sample, some transitions were not observed. Our model assumes that, in all states, individuals face an equivalent risk of death. A recent study of post-discharge outcomes from Arnold Lodge found that individuals with personality disorder had a fourfold greater risk of dying than the general population (Davies et al., 2007), but the Davies study was based on a small and potentially high-risk group. In their study of mortality rates for those with mental disorder, Harris and Barraclough (1998) found that individuals with personality disorder have, on average, a standardised mortality ratio of 1.84. As such, we chose to use age-specific and gender-specific rates from English standard life tables, inflated by 84%. A sensitivity analysis was carried out to test the effect on our results of using standard mortality rates. Discharge locations and length of stays were known for each individual. Using these data, we estimated a time-to-event model, where the event of interest was discharge from Arnold Lodge, dependent on an individual's completion status. We implemented a parametric Weibull regression model, which enabled us to obtain time-dependent discharge probabilities that increased or decreased over time, as appropriate, depending on completion status.
Our main aim was to demonstrate how, for a given population of forensic inpatients with personality disorder, non-completion of treatment may lead to differing costs after discharge. The sole focus of this model is therefore upon costs, from a combined perspective of the NHS and criminal justice system. Costs were assigned using a number of different sources and attached to resource use. NHS Reference Costs (Department of Health, 2011) were used for the cost-per-bed of hospital stays. Our figure for the average cost per prisoner is taken from a recent report by the Prison Reform Trust (2009). An accurate measurement of the cost of individuals in the community was less readily available, and we have chosen to use that found by a 2002 study of the economic impact of personality disorders in the UK (Rendu et al., 2002) (inflated to 2010 prices). We assumed there to be no cost associated with an individual being dead. The mean cost of one Markov cycle in each state, and its standard error, is shown in Table 1. To take into account the diminished present value of future costs, and in line with UK National Institute for Health and Clinical Excellence guidelines (NICE, 2009), costs were discounted at an annual rate of 3.5% with model progression. Pre-discharge costs were estimated using the figure for a medium secure stay shown in Table 1, and these were applied to an individual's length of stay at Arnold Lodge before entering the Markov model.
Table 1. Weekly Markov state costs
| ||Community||Prison||Low secure||Medium secure||High secure||Non-secure||Dead|
Key outputs of the model include the cost incurred by completers and non-completers, on average, after 10 years from admission. In addition to this, we present results in terms of cohort state distributions; the proportion of individuals residing in different states. We also present the results from a number of sensitivity analyses. It was important, in the case of our study, to take into account the considerable uncertainty around our parameters. We therefore, as far as possible, implemented a probabilistic sensitivity analysis (Doubilet et al., 1985). In the case of transition probabilities between community, prison and hospital states, we used standard errors to sample from a distribution of values. For probabilities, we use a beta distribution, which is bound by 0 and 1, as is customary (Briggs et al., 2006). Mortality rates were not directly made probabilistic but age was, with a normal distribution assumed. For costs relating to hospital stays, we used Excel's Solver (Frontline Systems, Inc., Incline Village, Nevada, USA) function to fit an estimated standard error of the unit cost from the lower-quartile and upper-quartile values provided in the reference costs. As is recommended (Briggs et al., 2006), we assumed a gamma distribution and sampled probabilistically using our estimated standard errors. The only cost for which we were not able to sample from a distribution was the cost of prison. Our Weibull time-to-event estimations for length of stay were made probabilistic using the Cholesky decomposition method.
Our model implemented 10,000 simulations, as we wanted to be sure to take account of the high level of uncertainty in our data. Each simulation samples a random value from the parameter's assumed distribution. By making our model probabilistic, our results are less likely to be skewed by outliers and extreme observations. Further sensitivity analyses were carried out on specific parameters. We investigated the effect of allowing transitions to occur that were not observed in the data, but that we would expect to occur occasionally in reality. Under ideal circumstances it would have been possible for us to carry out a cost-effectiveness analysis, though without data on health states or other specific outcomes this was not possible.
In addition to our primary results, we present results of a post-discharge model. We believe this to be a relevant perspective for practitioners and commissioners, as there is likely to be on-going provision of specialist hospital-based treatment for high-risk personality disordered offenders. This perspective may inform existing services focused on enhancing completion.