# Using Parameter Constraints to Choose State Structures in Cost-Effectiveness Modelling

## Abstract

### Background

This article addresses the choice of state structure in a cost-effectiveness multi-state model. Key model outputs, such as treatment recommendations and prioritisation of future research, may be sensitive to state structure choice. For example, it may be uncertain whether to consider similar disease severities or similar clinical events as the same state or as separate states. Standard statistical methods for comparing models require a common reference dataset but merging states in a model aggregates the data, rendering these methods invalid.

### Methods

We propose a method that involves re-expressing a model with merged states as a model on the larger state space in which particular transition probabilities, costs and utilities are constrained to be equal between states. This produces a model that gives identical estimates of cost effectiveness to the model with merged states, while leaving the data unchanged. The comparison of state structures can be achieved by comparing maximised likelihoods or information criteria between constrained and unconstrained models. We can thus test whether the costs and/or health consequences for a patient in two states are the same, and hence if the states can be merged. We note that different structures can be used for rates, costs and utilities, as appropriate.

### Application

We illustrate our method with applications to two recent models evaluating the cost effectiveness of prescribing anti-depressant medications by depression severity and the cost effectiveness of diagnostic tests for coronary artery disease.

### Conclusions

State structures in cost-effectiveness models can be compared using standard methods to compare constrained and unconstrained models.

## Keywords

Akaike Information Criterion Exit State HAMD Score Merge State Decision Recommendation#### Key Points for Decision Makers

State-transition cost-effectiveness models with different state structures can give different recommendations on treatment decisions or research prioritisation. To date, there have been no formal statistical methods described for comparing different state structures. |

Merging two states in a transition model, such as similar types of event, is practically equivalent to constraining the outward transition probabilities, costs and utilities to be equal for the two states. Thus, the state structures can be compared by assessing whether these constraints are reasonable. This can be done using standard methods for comparing statistical models, and suitable data. |

For example, comparing transition probabilities requires data consisting of the numbers of patients observed to transition out of the states of interest to each potential destination. To compare costs and utilities between states, individual-level samples are required. Maximum likelihood and Akaike’s information criterion can then be used to assess the constraints. If such data are not available, they might be derived from published summaries, or the comparison can be made informally. |

## 1 Introduction

Health economic evaluations rely on cost-effectiveness models, such as Markov multi-state models [1], to produce accurate comparative assessments of the costs and health effects of different interventions for the management of disease. Given a cost-effectiveness model, there may be uncertainty about the correct transition probabilities, costs or utilities. This is commonly termed parameter uncertainty and managed by probabilistic sensitivity analysis [2, 3]. Research recommendations can also be guided by the expected value of perfect information (EVPI) and expected value of partial perfect information, comparing the benefits in terms of costs and monetised health effects gained from a decision based on evidence, where parameter uncertainty is removed or reduced, with that based on current evidence [4].

Formal statistical approaches for comparing model structures against the data used to build them include the Akaike information criterion (AIC) [10, 11] and, for Bayesian models, the deviance information criterion [12, 13]. These trade off the fit to the data, represented by the likelihood, with the complexity, related to the effective number of parameters, of the models. Thus, they can find the optimal balance between the risk of bias (from excluding important events or predictors) and the reduced uncertainty in the estimates from a smaller model. These criteria can also be used to construct weighted averages over the possible structures [11, 14]. However, these methods are only valid for models fit to the same datasets and it has been shown that multi-state models with different state structures use different datasets [15]. Another approach is to split the model into a series of sub-functions and add discrepancy parameters to the outputs of these functions to represent state structure uncertainty [16]. However, the discrepancies do not indicate which assumptions are more plausible and can be difficult to interpret for complex models. A further approach is to compare the ability of the models to predict the events represented by both models [15, 17], in the above CAD example, this would be CAD and death. However, calculating the appropriate measure of fit for the restricted information criteria described in this article is technically demanding. State structures might also be compared by informal validation against external data if available.

We propose a method that allows the choice between state structures to be parameterised, and for which standard likelihood-based model selection criteria are valid. This enables us to compare structures under the principle that similar states may be merged if the consequences of occupying them are the same. Here, the ‘consequences’ for a patient consist of the potential exit states, the probabilities of transition to these exit states, the costs and the utilities. We show that smaller models can be reformulated into practically equivalent models on the larger state space by constraining the outward transition probabilities, costs and utilities to be the same for the ‘merged’ states. The model choice is then a matter of assessing whether each of these constraints is reasonable, based on the fit to data. We also consider using ‘partially merged’ models with different state structures for transition probabilities, costs and utilities, depending on the most appropriate choice for each consequence; for example, we may assume the costs in high-risk CAD (\(c_{\text{H}}\)) to be the same as those in low-risk CAD (\(c_{\text{L}}\)) but that the transition probabilities to the dead state (\(P_{\text{HD}}\) and \(P_{\text{LD}}\)) are different. We illustrate our approach in models comparing treatment strategies for the management of depression, and diagnostic tests for CAD.

## 2 Methods for Comparing State Structures by Assessing Parameter Constraints

Suppose we have data consisting of the numbers of individuals who are observed to make a transition between each pair of states over a particular time interval, and corresponding denominators of the total number of patients at risk. The models are fitted to these data by maximum likelihood or Bayesian estimation, giving estimates of transition probabilities between states over one cycle of a discrete-time model [18]. If such data are not explicitly available, they might be derived from related data (such as published relative risks of death) under weak assumptions, and we discuss an example in Sect. 4. Costs and utilities for states are estimated from samples of individual-level costs and utilities, or from published unit costs combined with assumptions, expert beliefs or data on individual resource use.

### 2.1 Merging Two States with One Common Exit State

Thus, the uncertainty regarding state structure has been parameterised, as a choice of whether these three constraints are reasonable. If all three are supported by the data, the merged model can be used because it is equivalent to the constrained split model. If all constraints are invalid, then the fully split model is most appropriate. A ‘partially merged’ model can also be recommended, for example, if the transition probabilities but not the costs are found to be equivalent.

### 2.2 Merging Any Number of States with Any Number of Exit States

*i,*whether the probability of death from cause

*i*, the cost and the utility depends on the disease status being A or B.

*n*states \(A_{1} , \ldots ,A_{n}\) with transitions to

*m*states \(E_{1} , \ldots ,E_{m}\). The necessary constraints are

*A*

_{ j }. The constrained likelihood of the split-state model is proportional to that of the merged-state model, as proven formally in Appendix 2 in ESM.

The states *A* _{ j } may represent severities of disease, and the *E* _{ i } different causes of death, but this result is entirely general to problems of whether to split or combine a set of states \(A_{1} , \ldots ,A_{n}\) for which the potential destination states after leaving the set are the same for each \(i = 1, \ldots , n\). The ‘split’ and ‘merged’ models shown in Fig. 3 may both be part of a common larger state structure, for example, there may also be transitions into the *A* _{ j }, or into and out of the *E* _{ i }. However, only the constraints (2) on the outward transition probabilities, costs and utilities are required to effectively ‘merge’ the states. Thus, the choice of structures is parameterised as a choice of whether the outward transition probabilities, costs and utilities are common between \(A_{1} , \ldots ,A_{n}\).

### 2.3 Merging States with Different Exit Transitions

However, we can extend model (**a**) by including a non-zero transition between states 1 and 3 (\(P_{13} \ne 0\)) to obtain model (c) in Fig. 4. This model can be constrained to model (a) by setting \(P_{13} = 0\) or to model (b) by setting \(P_{13} = P_{23}\). A comparison between models (a) and (b) is then possible by assessing these constraints on model (c).

## 3 Application to a Markov Model with Individual Patient Data: PANDA

In this section, we present an application to a health economic model for patients with symptoms of depression for whom their general practitioner is considering prescribing anti-depressant medication. The model was used to compare the cost effectiveness of severity thresholds above which to treat patients with depression with anti-depressant medication, and to estimate the value of a proposed randomised controlled trial to compare severity thresholds. The severity of symptoms was measured on the Hamilton Depression Rating (HAMD) scale, and three alternative treatment thresholds (HAMD > 2, HAMD > 15 or HAMD > 25) are compared with a policy of no treatment.

### 3.1 Model for Cost Effectiveness of Anti-Depressant Treatment by Depression Severity

### 3.2 Alternative Model Structures and Results

The transition probabilities between the four states are informed only by the individual transition history data, and there is no prior clinical belief regarding, for example, how the transition probability to well differs between mild, moderate and severe. Therefore, it is possible that, for these data, a more parsimonious structure that merges two or more of these states could give more precise estimates of cost effectiveness. Thus, we consider a ‘two-state’ model merging all depression states, a ‘Mod-Severe’ model merging the moderate and severe states, and a ‘Mild-Mod’ model merging the mild and moderate states. Merged IPCRESS, THREAD and TREAD data were re-analysed to estimate these transition probabilities for each structure. Costs for merged health states are estimated as weighted averages of their constituent costs, with weights defined by the baseline prevalence of the four depression states. The same prevalence was assumed for each cycle as the available prevalence estimate was representative of an average distribution over time. Utilities were mapped from incremental gains in HAMD. These models with ‘fully merged’ states ignore any prior clinical belief that costs or utilities are different between the states (Table 3). Finally, we consider ‘partially merged’ models, where outward transition probabilities across states are assumed to be equal but costs and HAMD, and therefore utilities, associated with the states are assumed to be different.

Results of cost-effectiveness value of information analyses for PANDA based on possible models for depression

Model | Optimal strategy | INB (£) of optimal strategy at willingness to pay £20,000 | P(CE) of optimal strategy at willingness to pay £20,000 | EVPI (£million) | EVPPI short term (£million) | EVPPI long term (£million) |
---|---|---|---|---|---|---|

Four-state (full) | HAMD > 2 | 223 (−217 to 798) | 0.64 | 80.04 | 67.29 | 0 |

Two-state | No treatment | NA | 0.61 | 95.61 | 103.62 | 4.11 |

Three-state (Mod-Severe) | HAMD > 2 | 224 (−213 to 805) | 0.67 | 74.88 | 62.26 | 0 |

Three-state (Mild-Mod) | HAMD > 2 | 234 (−205 to 830) | 0.68 | 70.70 | 60.53 | 0 |

Two-state unconstrained costs | HAMD > 2 | 225 (−214 to 812) | 0.65 | 77.95 | 65.45 | 0 |

Three-state (Mod-Severe) unconstrained costs | HAMD > 2 | 224 (−212, 813) | 0.65 | 77.41 | 64.88 | 0 |

Three-state (Mild-Mod) unconstrained costs | HAMD > 2 | 228 (−205, 830) | 0.65 | 77.06 | 64.61 | 0 |

Comparison of transition probabilities and costs for the four Markov cost-effectiveness depression models

### 3.3 Comparison of State Structures Using Constraints

We compare models by constraining parameters in the full (four-state) model to produce models that are equivalent to those with two or three states. We label the four health states as 1 (well), 2 (mild), 3 (moderate) and 4 (severe). The multi-state models being compared are illustrated in Fig. 5.

The four-state model is equivalent to the two-state model if the ‘recovery rates’ are constrained to be independent of depression severity, thus \(P_{21} = P_{31} = P_{41}\), and if the costs and HAMD/utilities of the mild, moderate and severe states are assumed to be equal to those of the single depressed state in the two-state model. To constrain the four-state model to be equivalent to the three-state ‘Mod-Severe’ model, we constrain the recovery rates to ‘well’ and the rates to ‘mild’ to be the same, \(P_{31} = P_{41}\) and \(P_{32} = P_{42}\), respectively, and constrain the costs and HAMD/utilities of the states to be equal. Likewise, the four-state model is constrained to the three-state ‘Mild-Mod’ model by constraining the recovery rates to ‘well’ and the progression rates to ‘severe’, \(P_{21} = P_{31}\), \(P_{24} = P_{34}\), along with the costs and HAMD/utilities for the mild and moderate states. Other transition probabilities, such as the probabilities of relapse (\(P_{12}\), \(P_{13} ,P_{14}\)), are unaffected by the constraints.

Each constraint is assessed by comparing the likelihood and AIC contributions, describing how well the resulting model fits when estimated using corresponding observed transitions between states. Full details of this method are given in Appendix 3 in ESM. The log-likelihood and AIC for each potential constraint are given in Table 2. An example code to conduct the comparisons in the R statistical software [32] is presented in Appendix 6 in ESM.

Under the unconstrained four-state model, the estimated recovery rates to well are substantially different for a patient with severe depression, thus \(P_{41} \ne P_{31}\) and \(P_{41} \ne P_{21}\). This is shown formally by the lower AIC for \(P_{21} \ne P_{31} \ne P_{41}\) compared with the constraints where \(P_{31} = P_{41}\) or \(P_{21} = P_{41}\). However, the recovery rates are similar between mild and moderate, thus the AIC is not changed substantially when moving between \(P_{21} = P_{31}\) and \(P_{21} \ne P_{31}\). The differences between \(P_{32}\) and \(P_{42}\) and between \(P_{24}\) and \(P_{34}\) under the four-state model are less striking. This is confirmed by the small difference in AIC between \(P_{32} = P_{42}\)and \(P_{32} \ne P_{42}\), and between \(P_{24} = P_{34}\) and \(P_{24} \ne P_{34}\). Thus, on the basis of transition probabilities, there is a negligible difference between the three-state Mild-Mod and four-state models, and these are both preferred over the two- and three-state Mod-Severe models, as expected.

Second, we compare the costs informally because these were based on expert belief. The treated costs are the same, though the untreated costs are slightly different, between mild and moderate. Thus, there is some evidence that a model with unconstrained costs is more appropriate. The costs for severe depression are substantially different from mild and moderate depression, arguing against the two-state and ‘Mod-Severe’ models. Prior judgement deemed that utilities are primarily determined by severity of depression, which broadly favours models that have finer classifications of HAMD.

Based on the chosen model, ‘treat if HAMD > 2’ is the optimal strategy. Because the model extrapolates beyond HAMD-D scores included in trials, we conclude that anti-depressant medications are cost effective over the range of HAMD scores included in the trials. We also conclude that there is likely to be value in a short-term trial that recruits patients with milder disease (lower HAMD scores); however, a long-term follow-up is not likely to be cost effective.

## 4 Application to a Model Informed by Published Parameters: CECaT

In this application, there are no individual-level data. Instead, the transition probabilities out of the states being considered for merging are obtained from published estimates. To formally compare the state structures, we have to derive the implicit transition counts underlying the published data.

The Cost-Effectiveness of non-invasive Cardiac Testing (CECaT) study [33] was a randomised trial of diagnostic strategies for CAD, comparing angiography alone with three non-invasive functional tests (followed by confirmatory angiography if positive). Following the trial, a Markov multi-state health economic model was developed, based on previous models by Mowatt et al. [34, 35] and Kuntz et al. [36]. The full structure and assumptions are detailed by Thom [17]. Briefly, a patient with suspected CAD receives one of five alternative diagnostic test strategies and is assigned a diagnosed severity, as a result of which they may receive either medical management or revascularisation. The diagnosed severity may be incorrect because the tests are not perfect and vary in their sensitivity and specificity. The model then proceeds with an annual cycle for 30 years, and at each cycle, a patient may have a myocardial infarction and/or die from any cause.

In these models, CAD severity is categorised into discrete states, representing the increasing risk of myocardial infarction and death, and the increasing need for revascularisation. Mowatt et al. [34, 35] used three risk states: low (no CAD), medium (CAD in one or two vessels excluding the left main stem) or high (CAD in three or more vessels and poor left-ventricular function, or disease in the left main stem). We compare the three-state categorisation with a model where medium- and high-risk states are merged, giving two states representing no CAD or CAD. While the three-state representation is typically used in the literature, it relies on having sufficient information about the differences between medium and high risk to justify separating them. Under the two models, the optimal diagnostic strategy at conventional cost-effectiveness thresholds, and extent of decision uncertainty, are different [17].

Published and derived data on parameters of the CECaT model of coronary artery disease, and AIC difference assessing the constraint that the corresponding parameters are equal between medium- and high-risk states (positive AIC difference favours different parameters)

Medium risk | High risk | AIC (medium = high) − AIC (medium ≠ high) | |
---|---|---|---|

Published parameter estimates (with 95% CI) | |||

Relative risk of death (vs. no CAD) | 2.3 (1.9–2.8) | 3.6 (3.1–4.1) | |

Annual risk of non-fatal MI | 0.022 (0.016–0.029) | 0.028 (0.021–0.035) | |

Derived event count data | |||

Number/denominator of deaths in 1 year (%) | 126/571 (22) | 259/754 (34) | 22.2 |

Number/denominator of non-fatal MIs (%) | 39/1717 (2.2) | 62/2159 (2.8) | −0.6 |

Summary of individual-level data (mean, SD, sample size) | |||

Costs | 1530, 880, | 1930, 1070, | 1.4 |

Utilities | 0.81, 0.12, | 0.78, 0.21, | −1.4 |

A similar analysis is performed for the risk of non-fatal myocardial infarction, which has overlapping confidence intervals between medium and high risk, though this does not necessarily imply a non-significant difference. An AIC difference of −0.6, however, mildly favours a common risk between the ‘medium’ and ‘high’ states.

The costs and utilities used for the medium- and high-risk states in the economic model (excluding the costs of revascularisation) were estimated from the subset of patients in the CECaT trial whose CAD severity was known. With only 19 of these patients in high risk and 59 in medium risk, it is not clear from the data in Table 3 whether we can assume that expected cost and utility are different between medium and high risk. To assess this formally, generalised linear regression models were fitted to the individual-level cost and utility outcomes by maximum likelihood in R [32], using a gamma distribution for the costs, and a truncated normal distribution for the utilities. The AIC marginally favours a model with different mean costs (AIC difference 1.4) and a model with common mean utility (AIC difference −1.4) between medium and high risk.

Thus, in this model, separating medium- and high-risk states is strongly justified based on their different mortality rates. Though within this structure, there is some evidence that constraining the myocardial infarction rates and utilities to be common between the states will lead to a better trade-off between model fit and model complexity, or bias and precision. Appendix 7 in ESM provides an example R code for all the likelihood and AIC calculations in this example.

## 5 Discussion

Currently, state structure choices are made informally, based upon clinical opinion or availability of data, or compared through simple scenario analyses [2, 7, 8, 9]. In this article, we have developed a formal statistical basis to compare state structures in cost-effectiveness models. Specifically, two or more similar states in a transition model can be merged if they have the same consequences for a patient who enters them. The models are then compared by assessing a constraint on these consequences using standard statistical methods, if the parameters are estimated from data, or by expert belief. Thus, we can decide whether the risk of bias in a more parsimonious model outweighs the reduced uncertainty from such a model. While assessing constraints on parameters is common practice, we have shown that models with merged states and models with constrained parameters can be used interchangeably. We proved that this method works for comparing any pair of structures where the states to be merged have the same exit states (Sect. 2.2) and that the method can be adapted to work if they have different exit states (Sect. 2.3). We also showed this method to be valuable even if state structure uncertainties do not affect the current treatment decision as the value of further research, quantified by the EVPI, expected value of partial perfect information or the expected value of sample information, may be sensitive to structural choices [14].

Statistical methods to assess the equality of model parameters require that the data used to estimate those parameters are available, to form the likelihood. For transition probabilities, the number of individuals who are observed to move between each pair of states in a time period, and denominators are required. The Prescribing ANtiDepressants that will leAd to a clinical benefit (PANDA) study used randomised controlled trials but our methods apply to any source, including registries or cohort studies, which provide the data necessary to estimate transition probabilities. We recommend using the data to choose the appropriate state structure before building the full model. Individual patient data were not available for the CECaT model. W recreated the numerators and denominators by assuming that the split between risk groups was the same across randomised arms of the trial, which should approximately hold if randomisation was adequate. To aid such calculations, we recommend that data of this form are published routinely.

Constraints for state selection can also be applied to continuous time multi-state models, which have been advocated for use in health economic modelling [37, 38], as we show in Appendix 5 in ESM, and to the selection of structures for patient-level simulation and heterogeneity models through the inclusion of covariates on the transition probabilities and comparing their effects between states. The principle should also extend to non-Markov multi-state models but this needs to be investigated. In a Bayesian model comparison, expert belief can be used by placing prior probabilities on parameters or model structures and combining with data via Bayes theorem.

Conversely, our method deals only with comparing multi-state structures. Further research into formal statistical methods for other forms of structural uncertainty is required. The choice between continuous and discrete outcome models is difficult. A multi-state model for changes in disease severity is essentially a continuous outcome model, where ranges of the outcome are constrained to have equivalent costs, utilities and future disease progression. However, there is no routinely applicable method to constrain a multi-level regression model, for example, to be equivalent to a multi-state model. Consideration is also required for more complex models, such as dynamic transmission models in infectious diseases [6, 39].

## 6 Conclusion

We have developed a formal method to parameterise state structure uncertainty using constraints on the parameters of the most complex model and have illustrated its wide applicability through examples in depression and CAD. Further research is required for structural uncertainty in non-multi-state cost-effectiveness models.

## Notes

#### Author contributions

All authors conceived and designed the study. Howard Thom developed the general methods in conjunction with Chris Jackson, Nicky Welton and Linda Sharples. Howard Thom and Nicky Welton developed the PANDA depression model and Howard Thom conducted the model comparison. Chris Jackson applied the methods to CECaT. Howard Thom drafted the paper and all authors critically reviewed and agreed to it as submitted.

### Compliance with ethical standards

#### Funding

Chris Jackson was fully funded and Linda Sharples and Howard Thom were partly funded by the Medical Research Council (MRC) Programme Grant U015232027. Howard Thom and Nicky Welton also received support from the MRC ConDuCT-II Hub (Collaboration and innovation for Difficult and Complex randomised controlled Trials In Invasive procedures—MR/K025643/1). The CECaT study was funded by a Grant from the UK National Institute for Health Research (NIHR) Health Technology Assessment Program (Project No. 99/26/04). The PANDA depression study was funded by the NIHR Programme Grants for Applied Research Award (Grant Reference No. RP-PG-0610-10048). The views expressed are those of the authors and not necessarily those of the MRC, National Health Service, the NIHR or the Department of Health.

#### Conflict of interest

Howard Thom, Chris Jackson, Nicky Welton and Linda Sharples declare they have no financial or non-financial conflicts of interest that are directly relevant to the content of this article.

#### Data availability statement

The datasets used for model comparison during the current study are included in this published article (within Table 3 for the heart disease example and in the code provided in Appendix 6 in ESM for the depression example). Further code and data, to generate the cost-effectiveness results, are available from the corresponding author on reasonable request.

## Supplementary material

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