Economic modelling

Introduction

The aim of this chapter is to report on the model-based extrapolation to extend the economic analysis horizon beyond the 24-month trial follow-up, up to 10 years. While a clear answer for the cost-effectiveness of observation/conservative management versus laparoscopic cholecystectomy was obtained from the within-trial analysis, it was anticipated that further participants would need to undergo surgery for their gallstone disease after 24 months12,60 and that this could have an impact on the cost-effectiveness results. Moreover, current method guidelines for the conduct of economic evaluations61 recommend using a time horizon long enough to capture all relevant differences in costs and consequences between the strategies being compared. Therefore, a simple Markov model was developed to estimate longer-term economic differences beyond the 24-month trial follow-up.

Methods

Model structure

The Markov model was constructed in TreeAge Pro software (TreeAge Software, Inc., Williamstown, MA) and was informed by reviewing decision models identified in the literature.35 A similar structure was used for both trial groups and is illustrated in Figure 9. A cohort of individuals referred to secondary care and with confirmed symptomatic gallstones enter the model in the ‘No surgery’ health state and are assigned to either laparoscopic cholecystectomy or observation/conservative management. Individuals receive treatment as observed in the C-GALL trial on an ITT basis. As such, the model considers the waiting time which occurred in the trial follow-up for the first 24 months, with survival analysis used to extrapolate time from randomisation to surgical procedures. AEs (and costs) related to gallstone disease such as biliary colic or cholecystitis are experienced by a proportion of individuals in the ‘No surgery’ health state. Individuals undergoing surgery (cholecystectomy) accrue the cost of the surgical episode and move to the tunnel state ‘Recovery from surgery’ where a reduction in quality of life associated with the surgical intervention is accounted for. Markov tunnel states are a series of temporary states, lasting only one cycle, that must be visited in a fixed sequence.62 On exit from the tunnel Markov state, individuals either have their symptoms resolved or not, moving to the corresponding Markov state (i.e. ‘Symptoms solved’ or ‘Symptoms persist’). Finally, Markov models need at least one state that cannot be left, named absorbing states, for the Markov process to end. In the C-GALL economic evaluation model, the absorbing state ‘Death’ is included, which can be entered from any other state based on age-specific mortality rates reported in the UK life tables.63

FIGURE 9 Simplified schematic for the C-GALL Markov model. (Figure not reproduced here; see the original article.)

The state occupancy and estimated payoffs are updated on a constant monthly Markov cycle.

Population

The model analysis was conducted for a cohort of individuals with characteristics matching those of the participants in the C-GALL trial cohort (baseline mean age and gender proportion 50.5 years and 71% women, respectively). The ITT principle was followed to estimate input parameters for the economic model; therefore, the model reflects the fact that a number of participants in the C-GALL trial did not receive their intended treatment.

Time horizon and discounting

Based on the available literature,60 and upon discussion within the PMG, a time horizon of 10 years was adopted for this analysis. This decision was based on the expectation that most of the individuals needing surgery would go through this procedure in the first few years after the end of their 24-month C-GALL trial follow-up. Survival analyses of time to surgery (see Clinical input parameters) also show that the hazard transition to surgery declines through time with the Kaplan–Meier (KM) curves levelling off before the 10-year mark. The impact of adopting a medium-term time horizon of 5 years was assessed in a sensitivity analysis. The model starts at baseline/randomisation time with costs and QALYs accrued beyond year 1 discounted at an annual discount rate of 3.5%.61

Clinical input parameters

Surgery

The crucial clinical parameter in the model is the time-dependent risk of undergoing cholecystectomy. All participants in the C-GALL trial were followed for a minimum of 24 months. In addition, PQs were sent every 6 months for those completing the trial follow-up at 24 months. Therefore, information about surgical procedures beyond 24 months was available for some participants for up to 48 months post randomisation. Figure 10 shows the KM curve for cholecystectomies for all participants in the C-GALL trial.

FIGURE 10 Kaplan–Meier plot of time in years for cholecystectomy by treatment allocation. Lap. cholecystectomy, laparoscopic cholecystectomy; O/C management, observation/conservative management. (Figure not reproduced here; see the original article.)

Parametric survival functions (i.e. exponential, Weibull, Gompertz, log-logistic and generalised gamma) were fitted to 48-month data. However, visual inspection made clear that all curves provided a poor visual fit to observed data, underestimating the earlier proportion of surgeries and overestimating the proportion of surgeries at 48 months (and possibly after). An alternative piecewise approach was then taken, with the first 9 and 12 months of data dropped from the analysis, and parametric curves fitted to the remaining tails of the distributions. The statistical fit of the fitted curves, assessed using the Bayesian and Akaike information criteria, are reported in Appendix 4, Table 52. The estimated survival curves and the percentages of participants expected to have surgery at 10 years were discussed within the C-GALL PMG (Table 16). Furthermore, Schmidt et al.60 stated that almost no operations took place beyond 5 years, when reporting on the 14-year follow-up of a study that randomised 137 participants to observation or cholecystectomy. Given the proportion of C-GALL participants known to have had surgery at 24 months, coupled with those declining surgery in the laparoscopic cholecystectomy group, and based on the advice of the clinical experts, it was agreed that the proportion of participants having cholecystectomy at 10 years would be around 50% and 80% for the observation/conservative management and laparoscopic cholecystectomy groups, respectively. Therefore, it was agreed to use the generalised gamma function fitted to the 9-month time-to-surgery data for the base-case analysis and the Gompertz and log-logistic functions (also fitted to the 9-month data) for sensitivity analyses.

Monthly transition probabilities were obtained directly from the KM curves for the first 24 months of the model to mirror the survival observed in the C-GALL trial, and from the fitted survival curves thereafter.

Adverse events related to gallstone disease

Individuals not going to surgery can experience AEs related to gallstone disease such as biliary colic or cholelithiasis. C-GALL trial data were used to estimate the rate and mean cost for ‘presurgical’ events. Presurgical events were defined as events needing a hospital admission (e.g. A&E admission) with an onset date earlier than the date of surgery or end of follow-up, whichever happened first. The rate of presurgical events for the model was estimated using time-to-event data, calculating the rate of events per patient year. This was transformed into a monthly rate of 0.0068 events per patient month.

Health state utilities

The health state utilities applied in the model were estimated from SF-36 data collected in the C-GALL trial for the first 24 months (Table 17) and retrieved from the literature thereafter.64 Based on completed trial data, mean SF-6D scores and SDs by treatment allocation were used for the first 24 months of the model with beta distributions associated with these data for the probabilistic sensitivity analysis (PSA). Van den Berg (2012)64 derived SF-6D population norms for the UK using a sample of 22,166 respondents to a national representative sample of British citizens were used. These population norms were adjusted according to age and gender in the C-GALL trial. Following the methods proposed by Ara and Brazier,65 a multiplier was calculated as the ratio between the mean SF-6D score at 24 months and the corresponding age-specific population norm SF-6D score (weighted by the C-GALL gender proportions). The 95% CI reported in Van den Berg 201264 was used to estimate standard errors and build up beta distributions to reflect the parameter uncertainty associated with the mean utility scores (see Table 17).

Table 17 also reports the utility decrement related to the need for cholecystectomy surgery estimated using data from the C-GALL trial. SF-6D utility scores for the time points before and after the date of surgery were selected. A paired t-test was used to test the difference between SF-6D mean scores before and after surgery. The statistically significant mean score difference was then used to calculate the percentage reduction from the postsurgical utility score. The utility decrement was applied to (half of) the mean time between the utility scores before and after surgery obtained from C-GALL data. Applying the utility decrement to half of the mean time assumes a linear extrapolation between the lower (before surgery) and higher (after surgery) utility scores.

Health service resource use and costs

Costs for the cholecystectomy episode for the model were obtained from NHS reference costs31 (Table 18). Unit costs for elective laparoscopic cholecystectomy (HRG code GA10K) were assumed for all cholecystectomies in the model. Unit costs for day-case and non-elective long-stay (emergency) cholecystectomies were used to define alternative scenarios (see Table 18). The emergency cholecystectomy scenario is assumed to incorporate perioperative and postsurgery AEs associated with the index cholecystectomy procedure.

Additional health service costs’ input data for the model were informed by the analysis of C-GALL trial data. A total of 55 AEs associated with gallstone disease such as biliary colic or abdominal pain needing hospital admission were counted for participants who were either waiting for surgery or for whom surgery was not yet being considered. Mean cost and SD for these events were used to attach a cost to the rate of presurgical events that was applied to those participants staying in the No surgery Markov health state (see Adverse events related to gallstone disease).

As no differences in the number of outpatient or GP contacts were observed by study group in the C-GALL trial, a simplifying assumption was made and no cost for these contacts was incorporated in the model.

Finally, gamma distributions were used for all unit costs to explore parameter uncertainty by conducting PSA.

Model validation

A number of steps were taken in order to secure the internal validity of the model. Testing was conducted throughout the model implementation such as verification of the model formulae using Microsoft Excel^® (Microsoft Corporation, Redmond, WA, USA) (white-box testing). The final model results were checked when varying the model input parameter values (e.g. defining all utility scores equal to one and zero discount rate to check whether total QALYs equal total life years) to assess the consistency of the model performance given specific variation in the model input values (black-box testing).

The validity of the projected estimates of cholecystectomies was assessed by plotting the model Markov traces with the Kaplan–Maier curves for the C-GALL trial (Figure 11). Visual inspection shows the Markov traces follow closely the C-GALL Kaplan–Maier curves. In addition, the rate of cholecystectomies at 10 years extracted from the model base case are 48% and 79% for observation/conservative management and cholecystectomy groups, respectively. This is in line with the expected rates discussed in the PMG (i.e. 50% and 80%, respectively).

FIGURE 11 Kaplan–Meier and model Markov trace plot of time for cholecystectomy by treatment allocation. GGamma, generalised gamma; KM, Kaplan–Meier; Lap. Chole, laparoscopic cholecystectomy; O/C M, observation/conservative management. (Figure not reproduced here; see the original article.)

Finally, within-trial analysis and model results were compared. For this, the model was run for a 24-month time horizon using the unit cost for the day-case cholecystectomy as this unit cost was similar to the cost of the cholecystectomy episode resulting from the within-trial analysis. The cost-effectiveness results for this comparison are reported in Appendix 4, Table 53. While the model expected costs and expected QALYs up to 24 months are lower than the corresponding values for the within-trial analysis, the difference in costs and QALYs between the two analyses produce consistent but substantially different ICERs (i.e. £55,235 and £88,930 for the within-trial and model-based analyses, respectively). The difference in costs reflects the fewer cost categories incorporated in the modelling analysis. The difference in the incremental QALYs is the result of the alternative approach used to obtain the AUC (linear interpolation for the within-trial analysis vs. simple extrapolation for the model) and the fact that neither cost or QALYs are adjusted for minimisation factors in the model-based analysis.

Model analysis

The model was run deterministically (using expected costs and outcomes) and probabilistically (using second-order Monte Carlo simulations) to characterise the joint uncertainty in the incremental costs and QALYs between observation/conservative management and cholecystectomy, arising from the model input parameter values. Probability distributions were assigned to model input mean parameter values reflecting uncertainty due to sampling variation. Beta distributions were defined for probabilities (risks) and utilities, and gamma for costs. Mean and standard errors, used to define these distributions using TreeAge^® (TreeAge Software, Inc., Williamstown, MA, USA) software, are provided in Tables 17 and 18. Probabilistic analyses were conducted by running 10,000 random draws from the allocated, producing 10,000 estimates of incremental costs and QALYs. The point estimate was calculated by averaging across the 10,000 estimates for costs and QALYs. Results tables also show the probability of observation/conservative management and laparoscopic cholecystectomy being cost-effective at £13,000, £20,000 and £30,000 per QALY thresholds.

Further deterministic analysis was conducted to assess the sensitivity of the model results to changes in key input parameters and structural assumptions. The following scenario and one-way sensitivity analyses were conducted:

1. Gompertz distribution for the survival analysis for the laparoscopic cholecystectomy and observation/conservative management groups. This results in a lower proportion of individuals going to cholecystectomy compared with the base case that used a generalised gamma survival model.
2. Log-logistic distribution for the survival analysis for the laparoscopic cholecystectomy and observation/conservative management groups. This shows a higher proportion of individuals going to cholecystectomy compared with the base case that used a generalised gamma survival model.
3. Gompertz distribution for laparoscopic cholecystectomy group and log-logistic for the observation/conservative management group. This scenario assumes a relatively higher proportion of individuals going to surgery in the observation/conservative management group and a lower proportion in the laparoscopic cholecystectomy group compared with the base case.
4. Day-case cholecystectomy cost (£2693) assumed for all cholecystectomies.
5. Emergency cholecystectomy cost (£4897) assumed for all cholecystectomies.
6. Day-case (£2693) and emergency (£4897) cholecystectomy cost assumed for laparoscopic cholecystectomy and observation/conservative management groups, respectively.
7. One-way sensitivity analysis: additional cost for surgery in the observation/conservative management group reflecting the possible higher costs associated with unexpected cholecystectomies in the observation/conservative management group, that are not necessarily associated with emergency procedures.
8. One-way sensitivity analysis. The utility reduction due to the need for cholecystectomy after 24 months is varied from 0 to 20% (base case, 5.5%).
9. One-way sensitivity analysis. Additional cost for surgery in the observation/conservative management group and scenario 3 for survival analysis.
10. Using C-GALL within-trial analysis results into the model for the first 24 months as the base case thereafter.

Although the C-GALL trial follow-up was 24 months, participants continued to be sent questionnaires every 6 months at 24 months. SF-36 data were collected and SF-6D scores obtained and analysed. A mixed-effects regression model for repeat measures with adjustment for the minimisation covariates (gender, age) and including centre as a random effect was used to obtain the SF-6D utility score difference between trial groups by data collection time point. The results of this analysis are reported in Appendix 4, Table 54. Consistent with the trial’s main statistical analysis, non-significant utility score differences between randomised groups were observed for up to 24 months. However, statistically significant differences are seen for 30 and 36 months post randomisation. Therefore, a final sensitivity analysis (scenario 11) was conducted using the utility data reported in Appendix 4, Table 54 over the initial 48 months of the 10-year time horizon.

Results

Base-case analysis

The results for the base-case cost–utility analysis are reported in Table 19. As anticipated, the expected costs and QALYs are higher for the 10-year time horizon compared to the 24-month follow-up for the within-trial analysis. The higher costs are explained by the higher proportion of individuals going to surgery and the longer time period considered. Differences in costs and QALYs decrease compared with the within-trial base-case analysis (see Table 14). However, the ICER rises, indicating that the expected saving per QALY loss is increased.

The expected costs and QALY differences from the PSA are reported in Figure 12. Cost and QALY differences are measured in the y- and x-axis, respectively. All PSA results show observation/conservative management is less costly than laparoscopic cholecystectomy. Only 23% of the results show observation/conservative management is more effective; however, the QALY difference in favour of laparoscopic cholecystectomy is not large enough for this strategy to be considered cost-effective in the base case. Moreover, the red dotted line represents the £20,000 cost-effectiveness threshold. Just 1.6% of the PSA iterations cross over the left of this line; that is, the probability of laparoscopic cholecystectomy being cost-effective at a £20,000 threshold is just 1.6% for the base-case analysis (see Table 19).

FIGURE 12 Model-based incremental cost-effectiveness scatterplot for observation/conservative management vs. laparoscopic cholecystectomy (base case; probabilistic sensitivity analysis – 1000 iterations). (Figure not reproduced here; see the original article.)

The CEACs showing the probability of observation/conservative management or laparoscopic cholecystectomy being cost-effective at alternative cost-effectiveness threshold values are reported in Figure 13. The CEAC for observation/conservative management shows a 98% probability of being cost-effective at £20,000 threshold value. This probability decreases at higher threshold levels reflecting the higher mean cost and QALYs resulting from laparoscopic cholecystectomy, but observation/conservative management has the higher probability of cost-effectiveness up to a cost-effectiveness threshold of £78,000 (data not shown).

FIGURE 13 Cost-effectiveness acceptability curves for observation/conservative management and laparoscopic cholecystectomy groups (model-based analysis; base case; probabilistic sensitivity analysis – 1000 iterations). (Figure not reproduced here; see the original article.)

Scenario analyses

In Table 20 the results of the first six scenario analyses are presented. Assuming alternative distributions such as Gompertz or log-logistic for the survival analysis, or lower (day-case) or higher (emergency procedures) unit cost for all cholecystectomies, have an impact on the ICER. However, ICERs are well above the usual cost-effectiveness threshold used in the UK for decision-making (i.e. £20,000).66 Moreover, the probability of observation/conservative management being considered cost-effective remains very high for scenarios 1 to 5. Assuming all cholecystectomies being conducted as emergencies for the observation/conservative management and as elective procedures for the laparoscopic cholecystectomy group (scenario 6) reverse the result seen in scenarios 1 to 5. Laparoscopic cholecystectomy becomes less costly and more effective (dominating) than observation/conservative management with a high probability of being cost-effective. This result was expected as scenario 6 is an extreme case scenario.

Table 21 shows the results for the one-way sensitivity analysis where alternative differences in the unit cost for the surgical episode between the trial groups were defined. Differences over £1700 are needed for a drop in the expected savings per QALY lost for the £20,000 threshold. As a reference, the difference in the NHS reference cost for laparoscopic cholecystectomy episode between a day-case and an elective procedure with an overnight stay and a non-elective procedure with long stay are £886 and £2205, respectively. That is to say, a large proportion of individuals in the observation/conservative management group will need to undergo emergency cholecystectomies for the laparoscopic cholecystectomy strategy to be cost-effective.

The results for the combined one-way analysis are reported in Table 21 and those for scenario 3 are reported in Table 22. Scenario 3 assumed a Gompertz survival model for the laparoscopic cholecystectomy group and log-logistic for the observation/conservative management group (see Table 20, scenario 3). Differences in the unit cost for the cholecystectomy episode over £800 would reduce the expected saving per QALY lost below £20,000.

Table 23 shows the results for the one-way sensitivity analysis where a higher reduction in QoL (utility decrement) due to cholecystectomy after 24 months was assumed. All ICERs remained above the usual cost-effectiveness threshold used in the UK, meaning observation/conservative management is still cost-effective for the utility decrements assumed.

The results for the last two scenarios are reported in Table 24. For scenario 10, the results from the C-GALL within-trial base-case analysis in Chapter 4 were incorporated in the model. The cost difference between the strategies reduces and the QALY difference increases slightly, resulting in a sharp reduction in the ICER compared with the base-case analysis (see Table 19). However, the ICER is still above the £20,000 cost-effectiveness threshold and the probability of observation/conservative management being cost-effective remains high.

In scenario 11, the adjusted SF-6D utility scores from the analysis of the C-GALL trial data were used for the initial 48 months of the 10-year model time horizon. The SF-6D scores show a significant difference in QoL at 30 and 36 months post randomisation between the trial groups, and non-significant utility scores favouring laparoscopic cholecystectomy for all but three and 48 months time points. As anticipated, there are no changes in costs or cost difference with respect to the base-case analysis as only input utility scores were varied. Expected QALYs are higher for the laparoscopic cholecystectomy group and slightly lower for the observation/conservative group, dropping the ICER below the £20,000 threshold. Laparoscopic cholecystectomy has 83% probability of being cost-effective at £20,000 cost-effectiveness threshold for this final scenario.

Discussion

The results from the 10-year extrapolation modelling exercise reported in this chapter suggest that laparoscopic cholecystectomy is more costly than observation/conservative management. Only under extreme assumptions, such as assuming all surgeries conducted as emergency procedures in the observation/conservative group and as day-cases for the laparoscopic cholecystectomy group, was the cost difference favouring observation/conservative management reversed. However, higher resource used should be related to individuals presenting with more severe clinical cases. Clinical cases at presentation may be more severe in the observation/conservative management group once individuals have waited several months without surgery. Moreover, a small but consistent QALY difference favouring laparoscopic cholecystectomy was observed for the base case and all sensitivity analyses. These cost and QALY differences translated into an ICER of £78,063 for the base case, meaning that important savings could be obtained for a relatively small QALY loss. A final scenario analysis used SF-6D-adjusted utility scores up to 48 months post randomisation. This produced a QALY difference large enough to reverse the base-case results, reducing the ICER to £14,698.

A strength of the modelling exercise is that it is supported by randomised data collected in a prospective, large multicentre and pragmatic RCT. Survival analysis was used to extrapolate time from randomisation to surgical procedures and time to event and cost data were used to estimate rate and costs associated with presurgical AEs. In addition, the reduction in QoL for individuals undergoing cholecystectomy was estimated using SF-6D scores collected in the trial covering a period of 6 months before and after cholecystectomy.

There are a number of limitations in our analysis. The estimated survival curve data were directly incorporated into the decision modelling software in a deterministic way and therefore there was no parameter uncertainty associated with the survival analysis in the PSA. As such, the probabilistic analysis might underestimate the overall parameter uncertainty in the model. Nevertheless, alternative survival functions were defined to cover the range of extreme but plausible scenarios with the base-case results being robust to the different survival distributions used. Furthermore, there was no difference in the utility score attached to the Symptoms resolved and Symptoms persist Markov health states after surgery. As the proportion of cholecystectomies are higher in the cholecystectomy group, a reduced quality of life due to symptoms after surgery would make cholecystectomy less cost-effective and therefore our assumtion is conservative. However, the long-term quality-of-life data are needed to assess if this assumption is valid beyond the 24-month trial follow-up.

In summary, the cost–utility analysis based on the C-GALL trial participant data suggests that observational/conservative management might be cost-effective at 24-month follow-up. However, the extrapolation over 10 years introduced uncertainty over this short-term result and could in some scenarios reverse the findings. The current decision uncertainty could be reduced with a long-term follow-up of the C-GALL trial participants.