Calculating willingness-to-pay with discrete cost and random coefficients in discrete choice experiments

Abstract

Objectives

This study provides step-by-step guidance to calculate willingness-to-pay (WTP) in discrete choice experiments that involve discrete cost. It highlights the limitations of assuming a linear disutility for cost in WTP calculation.

Methods

Five mixed-logit models were considered. Log-normal distributions were applied to cost parameters for four models under the assumption that utility (disutility) for cost should be negative (positive) or at least non-positive (non-negative) for all individuals. Piecewise linear utility in cost, using an iterative process, was proposed to calculate the WTP for the discrete cost models. Individual level simulations – considering individual random preference – were conducted to obtain the median WTP across all individuals and compared with the population mean WTP. A case study exploring preferences for colorectal cancer screening was used to demonstrate these models and methods.

Results

Models utilising discrete cost exhibited higher disutilities in cost at lower costs relative to models using continuous cost, but lower disutilities in cost at higher costs. Modelling using continuous cost tended to overestimate the WTP at low costs and underestimate the WTP at high costs. Adding a quadratic cost term only partially solved the problem, as the quadratic functional form may not capture the sharp change in preference for cost at low-cost levels. Divergent policy recommendations emerged when comparing results from continuous and discrete cost models. Although WTP was calculated using the population mean and the median across individuals, no systematic pattern was identified.

Conclusion

This study highlights the importance of incorporating discrete cost and selecting appropriate distribution assumptions for cost parameters to accurately derive the WTP.

Supplementary Information

The online version contains supplementary material available at 10.1186/s13561-025-00658-z.

Introduction

Discrete choice experiments (DCEs) are widely used to understand people’s preferences and decision-making processes. In the context of DCEs, participants are presented with a set of options, each defined by multiple attributes, and are tasked with selecting their preferred option [1, 2]. In health economics and public health research, DCEs are particularly valuable for eliciting individuals’ trade-offs amidst different health-related attributes, encompassing factors such as cost, effectiveness, and treatment side effects [3]. By analysing participants’ choices, researchers can derive the relative importance of each attribute and its impact on decision-making [4–6].

While DCEs offer valuable insights into preferences, policymakers and the general public may find regression results from DCEs less intuitive [7]. Therefore, willingness-to-pay (WTP), calculated by including cost attributes, often serves as an easily interpretable metric guiding policy-making decisions. For example, policymakers can leverage WTP values to identify the appropriate subsidies to enhance uptake and compliance with healthcare services [7–10].

However, the prevalent assumption in most DCE studies is that an individual’s disutility for cost is linear, with cost typically treated as a continuous attribute to facilitate the calculation of WTP [11]. This entails a constant disutility for cost as it rises, and the independence of WTP from a base cost. For example, the marginal disutility of a $1 increase is the same whether the cost rises from $5 to $6, or from $1000 to $1001. In situations where the cost difference between options is small, assuming linear disutility for cost may be acceptable. However, this assumption may not accurately reflect reality, particularly when dealing with alternatives exhibiting a wide range of costs. For instance, screening tests for colorectal cancer (CRC) can range from a few dollars for a stool-based or blood-based test to several thousand dollars for a colonoscopy [12, 13]. Thus, the assumption of constant disutility for the cost of CRC screening tests may be inappropriate.

Several methods of calculating WTP using discrete cost have been proposed in the literature, including (i) the best-worst method that estimates disutility by considering only the lowest and highest cost variables, (ii) taking the mean of the categorical slopes, and (iii) a piecewise linear method [14]. The first two methods are straightforward to implement, but share drawbacks identical to estimating WTP using continuous cost. In essence, even though the two methods assume non-constant disutility for cost in the model estimation, they assume a constant disutility for cost when calculating WTP. Using piecewise linear methods to calculate WTP could partially account for non-constant disutility, but no step-by-step guidance is available.

A rapid review of DCEs on health and healthcare published from January 2022 onward was conducted using PubMed on 5 December 2023. Ninety-five studies that calculated WTP were included in the review. Seventy-nine studies used continuous cost in their baseline model. Of the remaining 16 studies that used discrete cost, six studies calculated WTP using continuous cost with a separate model that was not presented [15–20]. Six studies either did not specify how WTP was calculated, or used unclear formulas to calculate WTP [21–26]. One study calculated WTP only at specific cost levels of interest [27]. The remaining three studies used either the best-worst method [28, 29] or the mean of the categorical slopes [30]. As observed, method clarity and the format of reporting WTP calculations need to be improved. Unfortunately, existing recommendations for the statistical analysis of DCEs do not provide detailed guidance on the calculation of WTP with discrete cost [31, 32]. Hence, practical guidance on calculating WTP using discrete cost is greatly required.

When using random-coefficient models such as mixed-logit (MXL) models [33] to analyse discrete cost data, researchers may encounter the issue of maintaining monotonicity during simulations, such as when examining the population mean confidence intervals or utility at the individual level. The conventional expectation is that an individual’s disutility should rise as cost increases. However, instances where participants report lower disutility for a $100 cost compared to a $50 cost warrant a closer examination of potential bias such as cost signalling quality and evaluation of the suitability of the DCE design [34]. On the other hand, if the distributional assumptions are not carefully specified, the property of non-decreasing disutility as cost increases may not be upheld for utility at the individual level within random-coefficient models when simulations are conducted [35]. In this case, the issue resides not in the DCE study design, but with the choice of modelling.

In light of this gap, the objective of this study is to propose and compare multiple methodologies for computing WTP using discrete cost within random-coefficient DCE models. Using a case study centred on the population’s preferences for CRC screening tests in Singapore, we first present different modelling choices and methods of calculating the WTP using MXL models. Second, we show inconsistency in policy recommendations comparing the WTP calculated using continuous and discrete cost.

Methods

Theoretical foundation

DCEs build upon several theoretical areas including Lancaster’s characteristics theory of demand [36], and welfare and consumer theory [37, 38]. Consistent with the choice-based approach to consumer theory, it explicitly assumes that the choices observed in DCEs reveal the preferences of individuals. The choices made in DCEs are typically analysed using random utility theory as shown in Eq. (1) [39]. The utility for individual i, conditional on choice , can be decomposed into an explainable component , and a random component .

1

The random component accounts for unobserved attributes, specification and measurement errors, unobserved preference heterogeneity, and inherent variability within and between individuals [40, 41]. The explainable component can be modelled as a function of attributes, for example as shown in Eq. (2), where is the vector of attributes of good or service j as viewed by individual i. is the vector of coefficients to be estimated, which represents the strength of preference or preference weights [31].

2

Statistical method

A MXL model is a commonly used random-coefficient model in analysing DCEs [33]. The vector of preference weights is assumed to be different for each individual, denoted as . Let represent the preference weight of individual i towards a level k– one element from vector . As show in Eq. (3), represents the mean preference weight for a level k at the population level. represents the random component of the preference weight at the individual level, which follows a pre-specified distribution (e.g. normal distribution).

3

We define level k as a level from a non-cost attribute. We demonstrate the calculation of WTP for level k. When the cost is modelled as continuous, the parameters for the utility of cost are expressed as . When cost is modelled as discrete, the utility of cost are expressed as , , , , , , , , …, , where cost_1 represents the first level of cost (e.g. 0 dollar), cost_2 represents the second level, and cost_m represents the last level, assuming there are m levels in total. Further, assume the cost levels are ordered from smallest (level 1) to largest (level m). The parameters follow the relationship from Eq. 3.

We propose two methods to calculate WTP: (1) using only the population mean preference weight (i.e. ), and (2) combining the population mean preference weight (i.e. ) with individual-specific random preference weight (i.e. ) to calculate the median WTP through simulation. Both methods consider continuous and discrete cost. Method 1 includes stepwise guidance on calculating WTP using continuous and discrete cost. Method 2 maintains the same foundational approach but highlights the additional steps involving simulation components. Numerical examples and R code are provided in the Supplementary Material.

WTP using continuous cost: population mean

The WTP for level k using a continuous cost variable and considering the population mean is calculated as follows: . A negative sign is added because the coefficient for cost is usually negative. A non-linear functional form on cost can be used to relax the assumption on linear disutility of cost. However, WTP cannot be calculated directly using the regression coefficients. A similar algorithm for calculating WTP using discrete cost can be applied, which is presented in the next section.

WTP using discrete cost: population mean

We propose to calculate WTP assuming piecewise linear utility in cost. For any adjacent cost levels, for example, at level 1 and level 2, the linear utility per dollar is defined as = (. For cost beyond the highest cost level, the current method assumes the utility per dollar to be the same as the utility per dollar between the highest and second highest level (). Additional parametric assumption can be used to overcome this.

An iterative approach is applied to calculate WTP when discrete cost is used. For the ease of explanation, let us assume is positive. The same approach can be used if is negative by reversing the sign. The steps for calculating the WTP for the level k is presented below. and should be defined as . Base cost refers to the underlying cost of the product or service being examined - for example, the current market cost or prevailing cost on the ground.

Step 1: Calculate .

1. If the absolute value of () is less than or equal to (cost_2 – cost_1),

   • i.
     WTP is ;
   • ii.
     End the calculation.

2. If the absolute value of () is greater than (cost_2 – cost_1),

   • i.
     Add (cost_2 – cost_1) to the preliminary WTP estimate;
   • ii.
     Calculate
     
   • iii.
     Go to Step 2.

Step 2: Calculate .

1. If the absolute value of () is less than or equal to (cost_3 – cost_2),

   • i.
     Update the preliminary WTP estimate by adding the value of  ; 
   • ii.
     End the calculation.

2. If the absolute value of () is greater than (cost_3 – cost_2),

   • i.
     Update the preliminary WTP estimate by adding the value of (cost_3 – cost_2);
   • ii.
     Calculate
     
   • iii.
     Go to Step 3.

Step 3: Apply the same sequence of actions as stated in Step 2 iteratively by moving to the next cost level. If the calculation reaches the highest cost level, the calculation will end as the upper bound is infinity, and condition (a) will always be satisfied.

To apply the algorithm to non-linear continuous cost, levels need to be first defined, which may be different from the levels used in the DCE questionnaire. The disutility of cost can then be calculated for each selected level. The same aforementioned algorithm can be used to calculate WTP by going through the smallest to the largest cost level.

WTP using continuous cost and discrete cost: median across individuals

The approaches above use the mean parameters at the population level to calculate WTP. However, another approach is to calculate WTP at the individual level by simulation, and then generate the median WTP across all individuals [32]. As WTP is derived from a ratio of two values, the median WTP across all individuals will most likely be different from the WTP that uses the mean parameters. To calculate the median WTP across all individuals, the aforementioned approach can be applied to calculate WTP when either the continuous or discrete cost is used, by replacing the parameters of the population mean with the parameters of individual-level values. The individual level parameters can be simulated using and .

However, one technical issue often countered is the requirement that the coefficients for cost should be negative, or at the very least, non-positive for all individuals. This can be addressed by assuming that the negative of the cost parameters follows a log-normal distribution [32].

However, when discrete cost is considered, it is also essential to maintain the principle of non-decreasing disutility, or monotonicity, as cost increases - for example, disutility should be higher for higher costs. To ensure this monotonicity, we propose to model the incremental disutility of cost in the DCE analysis as shown in Eq. 4, with j ranging from 1 to the number of levels minus 1:

4

where is the incremental disutility as cost increases from the level j to the next level j + 1. Hence, the monotonicity in the disutility of cost can be maintained at the individual level. The mean incremental disutility at the population level is .

Case study

A DCE examining the population’s preference for CRC screening tests in Singapore is used to demonstrate the method [42]. Detailed description of the study can be found from the previous publication. Six attributes were identified and ultimately used in the DCE: (i) procedure, (ii) pain level, (iii) sensitivity, (iv) recommendation, (v) out-of-pocket cost, and (vi) risk of test. The cost levels included S$0 (0 Singapore Dollar or SGD), S$5, S$30, S$400, and S$1000 to cover the cost for stool- and visual-based tests. An “Opt-Out” option was included in the second stage for participants to choose whether they would take the preferred option from the first stage in real life. This study was conducted as a web-based survey hosted on REDCap from 19 May 2022 to 28 May 2022.

Overview of the analyses

We compared the results from five models in the Results section. Model I and Model II used linear continuous cost; Model III used quadratic continuous cost; Model IV and Model V used discrete cost. Specifically, we considered: Model I - a model with a continuous cost variable and a normal distribution for the cost parameter; Model II - a model with a continuous cost variable and a log-normal distribution for the cost parameter; Model III – a model with continuous cost and cost-square variables and a log-normal distribution for both parameters; Model IV – a model with discrete cost variables and log-normal distributions for the cost parameters, and Model V - a model with discrete incremental cost variables and log-normal distributions for the incremental-cost parameters. The WTP using population mean parameters and the median WTP across all individuals were calculated for the five models. Normal distributions were assumed for all other parameters to isolate the impact of different assumptions on the cost parameters.

For simulating individual parameters and calculating 95% confidence intervals, 10,000 iterations were used. All quantitative data analyses were carried out using the statistical software R 4.2.1 [43]. Statistical significance was set at p < 0.05.

Results

The summary statistics of the case study are presented in the Supplementary Material. Responses from 1,021 participants were used for analysis. The attributes and level used, and a sample DCE question are also presented in the Supplementary Material.

Table 1 presents the regression results for all five models. Model II exhibited a higher log-likelihood and lower Bayesian Information Criteria (BIC) compared to model I, suggesting that a log-normal distribution for the cost parameter is more appropriate than a normal distribution. Model V provided the best fit to the data among all five models. The mean non-cost parameters were similar across five models except for “Stool-Based (1 Day)”. The parameter was not statistically significant in model I (Coefficient = 0.14, CI =[− 0.03, 0.31]) and II (Coefficient = 0.16, CI=[− 0.02, 0.33]), but statistically significant in model III (Coefficient = 0.22, CI=[0.04, 0.39]), IV (Coefficient = 0.32, CI=[0.15, 0.49]) and V (Coefficient = 0.27, CI =[0.09, 0.44]).

Table 1.

Regression results of the five models

	Model I		Model II		Model III		Model IV		Model V
	Coefficient	95%−CI	Coefficient	95%−CI	Coefficient	95%−CI	Coefficient	95%−CI	Coefficient	95%−CI
MEAN COEFFICIENT
Left	0.19	(0.12, 0.25)	0.20	(0.13, 0.26)	0.20	(0.13, 0.27)	0.15	(0.08, 0.22)	0.16	(0.09, 0.23)
None	−1.22	(− 1.40, − 1.05)	−1.54	(− 1.72, − 1.36)	-1.55	(− 1.73, − 1.37)	−1.67	(− 1.86, − 1.48)	−1.63	(− 1.82, − 1.44)
Procedure
Colonoscopy	−0.68	(− 0.85, − 0.52)	−0.74	(− 0.91, − 0.57)	-0.69	(− 0.86, − 0.52)	−0.74	(− 0.91, − 0.58)	−0.82	(− 0.99, − 0.65)
CT Colonoscopy	−0.82	(− 0.98, − 0.67)	−0.87	(− 1.02, − 0.71)	-0.77	(-0.93, -0.61)	−0.77	(− 0.93, − 0.62)	−0.81	(− 0.97, − 0.65)
Stool-Based (2 Days)	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference
Stool-Based (1 Day)	0.14	(− 0.03, 0.31)	0.16	(− 0.02, 0.33)	0.22	(0.04, 0.392)	0.32	(0.15, 0.49)	0.27	(0.09, 0.44)
Blood-Based	0.29	(0.14, 0.44)	0.26	(0.11, 0.41)	0.31	(0.15, 0.46)	0.38	(0.23, 0.53)	0.35	(0.19, 0.50)
Pain Level
No Pain	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference
Mild Pain	−0.59	(− 0.68, − 0.49)	−0.58	(− 0.67, − 0.48)	-0.59	(-0.69, -0.50)	−0.56	(− 0.66, − 0.48)	−0.58	(− 0.68, − 0.48)
Sensitivity
100%	1.68	(1.57, 1.80)	1.73	(1.61, 1.85)	1.73	(1.61, 1.85)	1.64	(1.52, 1.76)	1.75	(1.63, 1.88)
95%	0.59	(0.48, 0.71)	0.64	(0.53, 0.76)	0.65	(0.54, 0.77)	0.71	(0.60, 0.83)	0.76	(0.64, 0.88)
80%	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference
60%	−1.05	(− 1.23, − 0.89)	−1.15	(− 1.32, − 0.98)	-1.18	(-1.35, -1.01)	−1.29	(− 1.47, − 1.11)	−1.18	(− 1.35, − 1.02)
Recommendation
Health Promotion Board	0.83	(0.72, 0.94)	0.85	(0.74, 0.97)	0.89	(0.77, 1.00)	0.89	(0.77, 1.01)	0.90	(0.79, 1.02)
Doctors	0.70	(0.58, 0.82)	0.69	(0.58, 0.81)	0.74	(0.62, 0.85)	0.69	(0.58, 0.81)	0.69	(0.57, 0.80)
Family & Friend	0.36	(0.24, 0.49)	0.32	(0.20, 0.44)	0.36	(0.24, 0.49)	0.34	(0.22, 0.46)	0.33	(0.21, 0.46)
Neither	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference
Cost	−0.0040	(− 0.0042, − 0.0038)	−5.64	(− 5.69, − 5.59)	-5.07	(-5.15, -4.98)
SGD0							0.00	Reference	0.00	Reference
SGD5							−1.12	(− 1.66, − 0.59)	−1.97	(− 1.34, − 0.58)
SGD30							−0.19	(− 0.33, − 0.04)	−0.96	(− 1.34, − 0.58)
SGD400							0.88	(0.82, 0.94)	0.27	(0.15, 0.40)
SGD1000							1.31	(1.26, 1.36)	0.11	(− 0.03, 0.25)
Cost 2					-12.82	(-13.00, -12.65)
Risk of Test
No Risk	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference
Positive Risk of 1%	−0.84	(− 0.95, − 0.72)	−0.83	(− 0.72, − 0.95)	-0.87	(-0.99, -0.75)	−0.77	(− 0.89, − 0.65)	−0.88	(− 0.95, − 0.71)
STANDARD DEVIATION
None	3.89	(3.70, 4.08)	3.76	(3.58, 3.95)	3.80	(3.62, 3.99)	3.77	(3.58, 3.95)	3.78	(3.59, 3.96)
Procedure
Colonoscopy	1.03	(0.90, 1.16)	1.13	(1.00, 1.27)	1.23	(1.10, 1.37)	1.22	(1.08, 1.36)	1.32	(1.18, 1.46)
CT Colonoscopy	0.63	(0.49, 0.78)	0.73	(0.57, 0.88)	0.82	(0.67, 0.97)	0.82	(0.67, 0.96)	0.99	(0.85, 1.14)
Stool-Based (2 Days)	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference
Stool-Based (1 Day)	0.38	(0.14, 0.62)	0.60	(0.36, 0.84)	0.58	(0.35, 0.82)	0.62	(0.40, 0.85)	0.23	(0.01, 0.45)
Blood-Based	0.78	(0.63, 0.94)	0.71	(0.55, 0.86)	0.63	(0.47, 0.79)	0.76	(0.61, 0.91)	0.80	(0.20, 0.48)
Pain Level
No Pain	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference
Mild Pain	0.28	(0.15, 0.42)	0.13	(− 0.01, 0.27)	0.07	(-0.07, 0.21)	0.03	(− 0.12, 0.18)	0.34	(0.20, 0.48)
Sensitivity
100%	0.04	(− 0.11, 0.18)	0.00	(− 0.13, 0.14)	0.02	(-0.12, 0.15)	0.28	(0.14, 0.42)	0.20	(0.07, 0.33)
95%	0.03	(− 0.12, 0.18)	0.03	(− 0.12, 0.19)	0.03	(-0.13, 0.19)	0.04	(− 0.12, 0.21)	0.03	(− 0.12, 0.18)
80%	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference
60%	1.77	(1.55, 1.99)	1.58	(1.37, 1.80)	1.66	(1.44, 1.87)	1.89	(1.66, 2.21)	1.49	(1.28, 1.69)
Recommendation
Health Promotion Board	0.10	(− 0.06, 0.25)	0.10	(− 0.05, 0.25)	0.05	(-0.10, 0.20)	0.06	(− 0.10, 0.22)	0.02	(− 0.13, 0.17)
Doctors	0.08	(− 0.07, 0.23)	0.07	(− 0.07, 0.21)	0.00	(-0.14, 0.15)	0.19	(0.04, 0.34)	0.06	(− 0.08, 0.20)
Family & Friend	0.07	(− 0.08, 0.22)	0.02	(− 0.12, 0.17)	0.05	(-0.10, 0.20)	0.16	(− 0.004, 0.32)	0.05	(− 0.09, 0.19)
Neither	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference	0.00	Reference
Cost	0.0026	(0.0024, 0.0028)	0.97	(0.90, 1.04)	0.43	(0.38, 0.48)
SGD0						Reference	0.00	Reference	0.00	Reference
SGS5							0.70	(0.32, 1.07)	1.40	(1.17, 1.64)
SGD30							0.05	(− 0.15, 0.26)	0.58	(0.32, 0.85)
SGD400							0.04	(− 0.04, 0.11)	1.04	(0.51, 0.81)
SGD1000							0.40	(0.34, 0.45)	0.66	(0.51, 0.81)
Cost 2					0.04	(-0.03, 0.11)
Risk of Test
No Risk	0.00	Reference	0.00	Reference		Reference	0.00	Reference	0.00	Reference
Positive Risk of 1%	0.04	(− 0.12, 0.20)	0.05	(− 0.10, 0.21)	0.21	(0.05, 0.37)	0.15	(− 0.01, 0.32)	0.16	(0.01, 0.30)
Log-likelihood	−8414		−8366		−8308		−8353		−8270
AIC	16,887		16,790		16,678		16,775		16,609
BIC	17,117		17,020		16,924		17,053		16,887

Notes

Abbreviations: CI, Confidence Interval; AIC, Akaike Information Criteria; BIC, Bayesian Information Criteria

Model I: Model with continuous cost parameter and a normal distribution in the cost parameter

Model II: Model with continuous cost parameter and a log-normal distribution in the cost parameter

Model III: Model with continuous cost and cost-square parameters and log-normal distributions in the cost and cost-square parameters

Model III: Model with discrete cost parameters and log-normal distributions in the cost parameters

Model IV: Model with discrete incremental cost parameters and log-normal distributions in the incremental cost parameters

Figure 1 illustrates the utility weights at different cost levels across the five models using the population mean parameter. Compared to model I, model II, which assumed a log-normal distribution, exhibited higher disutility (negative of utility weight) for cost. Model III had similar disutility levels as model II at lower cost levels, but exhibited lower disutility at higher cost levels. Compared to model I, II and III, there was a steeper increase in the disutility for cost in model IV and V at lower costs, and a gentler increase in the disutility for cost in model III and IV at higher costs.

Fig. 1.

Utility weight at different cost levels of the five models. Notes. Model I: Model with a continuous cost variable and a normal distribution for the cost parameter. Model II: Model with a continuous cost variable and a log-normal distribution for the cost parameter. Model III: Model with continuous cost and cost-square parameters and log-normal distributions in the cost and cost-square parameters. Model IV: Model with discrete cost variables and log-normal distributions for the cost parameters. Model V: Model with discrete incremental cost variables and log-normal distributions for the cost parameters

In model I, where a normal distribution was specified for the continuous cost parameter, the cost parameter represents the mean disutility value. However, for models II, III, IV and V, a log-normal distribution was specified for the continuous and discrete cost parameters, respectively. Thus, the cost parameters in these models represent the location parameter of a log-normal distribution and not the mean disutility value. Following the formula in the Methods section, this corresponds to a mean utility value per dollar of − 0.0057 for model II at any cost level, − 0.0069, − 0.0829 and − 0.0745 per dollar for models III, IV and V between S$0 and S$5, respectively. Considering a test with a base cost of S$0 to S$5, models with continuous cost underestimated the disutility of cost compared to models with discrete costs. Figure 2 visually presents the increment disutility per incremental cost at different cost levels across all five models.

Fig. 2.

Incremental disutility per incremental cost of the five models. Notes. Model I: Model with a continuous cost variable and a normal distribution for the cost parameter. Model II: Model with a continuous cost variable and a log-normal distribution for the cost parameter. Model III: Model with continuous cost and cost-square parameters and log-normal distributions in the cost and cost-square parameters. Model IV: Model with discrete cost variables and log-normal distributions for the cost parameters. Model V: Model with discrete incremental cost variables and log-normal distributions for the cost parameters

The graphs present the absolute value of disutility per unit of incremental cost. The values are the absolute values of the slopes of the lines in Fig. 1.

Table 2 summarises the WTP based on two scenarios: (i) switching from a two-day stool-based test to a blood-based test, and (ii) improving the sensitivity of a two-day stool-based test from 80% to 95%. At a population level, the assumption of a log-normal distribution for the cost parameter (model II) resulted in a lower WTP compared to the assumption of a normal distribution for the cost parameter (model I). At lower base cost levels of S$0 and S$5, the WTP estimates from model III were similar to those of model II, but models IV and V had much lower WTP estimates. However, at higher base cost levels of S$30 and S$400, the WTP estimates from models III, IV and V were higher than those from models II. Among models III, IV and V, the range of WTP estimates according to model III was smaller than models IV and V, with relatively higher WTP estimates at lower costs, but relatively lower WTP estimates at higher costs. For model IV, wide confidence intervals of the mean population WTP were observed when the base cost was S$5, for scenarios (i) (WTP = S$22.72, CI=[− S$57.91, S$104.54]) and scenario (ii) (WTP = S$93.92, CI=[S$−107.21, S$178.55]).

Table 2.

WTP for blood-based test and 95% sensitivity

	Model I		Model II		Model III		Model IV			Model V
	WTP	95%-CI	WTP	95%-CI	WTP	95%-CI	WTP	95%-CI		WTP	95%-CI
Blood-Based Test
Population Level Mean	S$72.23	(S$34.08, S$111.03)	S$45.24	(S$17.62, S$72.93)
S$0					S$47.43	(S$22.20, S$74.33)	S$4.56	(S$1.80, S$14.72)	S$4.68		(S$2.84, S$11.88)
S$5					S$48.43	(S$22.26, $75.30)	S$22.72	(-S$57.91, S$104.54)	S$19.13		(S$13.28, S$35.60)
S$30					S$53.05	(S$26.46, S$79.79)	S$88.54	(S$52.53, S$125.58)	S$57.21		(S$36.92, S$88.17)
S$400					S$97.14	(S$48.11, S$148.96)	S$142.51	(S$86.16, S$203.08)	S$150.82		(S$96.32, S$234.94)
Individual Level Median	S61.52	(S$28.77, S$96.29)	S$48.45	(S$17.80, S$83.69)
S$0					S$43.28	(S$21.87, S$67.75)	S$4.69	(S$2.00, S$9.82)	S$6.94		(S$2.96, S$12.86)
S$5					S$43.59	(S$22.00, S$68.23	S$16.51	(S$5.00, S$23.33)	S$19.22		(S$8.95, S$35.29)
S$30					S$52.56	(S$26.22, S$81.65)	S$86.47	(S$52.16, S$125.53)	S$64.69		(S$33.85, S$101.67)
S$400					S$74.61	(S$34.71, S$111.16)	S$93.18	(S$55.93, S$142.88)	S$146.68		(S$83.47, S$231.54)
95% Sensitivity
Population Level Mean	S$147.17	(S$119.53, S$175.40)	S$112.53	(S$92.39, S$133.66)
S$0					S$107.30	(S$87.46, S$127.52)	S$22.80	(S$3.77, S$36.44)		S$26.13	(S$15.86, S$49.71)
S$5					S$108.30	(S$88.49, S$128.56)	S$93.92	(-S$107.21, S$178.55)		S$74.52	(S$23.59, S$109.83)
S$30					S$112.92	(S$93.24, S$133.07)	S$166.35	(S$138.24, S$197.19)		S$124.28	(S$103.52, S$145.92)
S$400					S$206.75	(S$168.42, S$252.93)	S$267.75	(S$218.12, S$323.04)		S$327.62	(S$268.08, S$390.37)
Individual Level Median	S$135.97	(S$109.42, S$160.43)	S$179.39	(S$146.34, S$212.53)
S$0					S$103.41	(S$84.13, S$122.80)	S$10.90	(S$6.28, S$18.76)		S$27.07	(S$14.82, S$49.13)
S$5					S$104.19	(S$84.83, S$123.72)	S$32.55	(S$23.71, S$39.75)		S$49.42	(S$32.00, S$74.40)
S$30					S$126.34	(S$103.46, S$148.28)	S$166.51	(S$138.14, S$196.66)		S$213.92	(S$173.08, S$256.91)
S$400					S$189.48	(S$154.02, S$225.72)	S$226.73	(S$181.25, S$276.82)		S$406.38	(S$325.27, S$495.48)

Notes:

Abbreviations: WTP, Willingness-to-pay; CI, Confidence Interval

Model I: Model with continuous cost parameter and a normal distribution in the cost parameter

Model II: Model with continuous cost parameter and a log-normal distribution in the cost parameter

Model III: Model with continuous cost and cost-square parameters and log-normal distributions in the cost and cost-square parameters

Model IV: Model with discrete cost parameters and log-normal distributions in the cost parameters

Model V: Model with discrete incremental cost parameters and log-normal distributions in the incremental cost parameters

Comparing the population mean WTP and the median WTP across all individuals based on the two scenarios, model I had median WTP values across all individuals that were lower than the mean WTP values at a population level. However, the opposite was true in model II when a log-normal distribution was assumed for cost. For scenario (ii) in model II, the population mean WTP (WTP = S$112.53, CI=[S$92.39, S$133.66]) was statistically different from the median WTP across all individuals (WTP = S$179.39, CI=[S$146.34, S$212.53]). For models III, IV and V, there were no clear patterns of the differences between the population mean WTP values and the median WTP values across all individuals.

Discussion

We provided step-by-step guidance to calculate WTP when non-linear continuous and discrete cost are considered in the analysis of DCEs. We proposed a simple way to maintain the monotonicity of preference in cost in random coefficient models using discrete cost. Based on a case study of a DCE of the population’s preference for CRC screening tests in Singapore, we explored and highlighted the importance of calculating WTP considering non-linear disutility for cost. According to economic theory, modelling linear disutility of cost is unrealistic as it violates the concept of decreasing marginal disutility [44] and fails to capture the non-linear relationship between cost and choice accurately. Our findings on the decreasing marginal disutility as cost increases land support for the economic theory. While we demonstrated the method using a MXL model, this method can also be easily applied to other models for analysing DCE, such as latent class [45] and mixed-mixed logit models [46].

The advantage of models that use discrete cost is the ability to account for an appropriate base cost, without which wrong policy recommendations could potentially be prescribed. Consider the scenario of switching from a two-day stool-based test to a blood-based test: using a model with continuous cost suggested that the mean incremental WTP at a population level was S$72.23 (CI=[S$34.08, S$111.03]) based on model I and S$45.24 (CI=[S$17.62, S$72.93]) based on model II. Using a quadratic cost term, the WTP ranged from S$47.43 to S$97.14, based on the population mean estimate. However, in the context of Singapore, citizens and permanent residents aged 50 and above may receive stool-based test kits at a subsidised price of S$5, or have them completely free. Considering a base cost of S$0 for patients receiving free stool-based test kits, the WTP for a blood-based test is only S$4.56 (CI=[S$1.80, S$14.72]) and S$4.68 (CI=[S$2.84, S$11.88]) based on model IV and V, respectively. This means that models I, II and III had severely overestimated the patients’ WTP by a large magnitude. Hence, if policymakers were to change from a free stool-based test to a blood-based test that costs S$30, the observed screening rate is likely to decrease, unlike what was suggested by models I, II, and III.

This approach to selecting an appropriate base cost for policy decision-making is also applicable to programmes such as colonoscopy screening, where the base cost of testing is much higher. In this scenario, patients are more likely to have a higher WTP when procedural attributes such as pain level and risk of the test are improved. Based on our analysis of the scenario of moving from mild pain to no pain for a colonoscopy, the mean incremental WTP at a population level was S$145.80 (CI=[S$122.92, S$170.00]), S$100.87 (CI=[S$83.40, S$118.43]) and S$187.45 (CI=[S$155.09, S$226.13]) based on models I, II and III, respectively. However, considering a base cost of $400, the mean incremental WTP at a population level was S$212.10 (CI=[S$172.85, S$257.22]) and S$250.19 (CI=[S$202.36, S$302.02]) based on models IV and V, respectively. In this scenario, using continuous cost would underestimate the WTP for colonoscopy among the population.

While non-linear parameterisation for continuous cost can be used to incorporate non-linear preference, the functional form needs to be examined carefully to understand whether it can reflect the non-linear pattern. In our example, the model with quadratic cost term (model III) overestimated the WTP at low cost level compared to models with discrete cost, which could still lead to wrong policy recommendations. On the other hand, one advantage of using continuous cost with non-linear function form, is that it allows for the arbitrary selection of appropriate cost levels to construct a piecewise linear disutility of cost based on the estimated non-linear curve. Since DCEs may only support a limited number of levels for each attribute, this method allows for the chosen cost levels to reflect not only those used in the questionnaire, but also to capture regions where the disutility of cost exhibits non-linearity. This approach also allows for changing disutilities for costs exceeding the highest level in the DCE design, as the functional form assumption permits extrapolation beyond the highest cost level.

Models using discrete cost to estimate WTP are, however, not without its limitations. Model IV presented a case where monotonicity was violated, resulting in confidence intervals with negative WTP values, where people preferred higher over lower cost. This limitation stems from the method used to derive the piecewise linear disutility of cost, which depends on the distribution of values of the adjacent cost parameters. However, modelling the incremental disutility rather than the piecewise linear disutility allows for the maintenance of monotonicity, as seen in model V. In model V, the distribution of piecewise linear disutility from the simulation yielded only positive values.

This study further provided empirical evidence in estimating WTP using random coefficients, comparing the WTP values generated using the population mean values with those derived individually through simulation, such as the median preferences across individuals. The two methods have different policy implications: the former represents the population average, while the latter focuses on the median individual. Nevertheless, estimating the distribution of WTP values across all individuals can be computationally demanding, especially when estimating confidence intervals which involve second-order simulation. Issues may also arise when simulating a range of standard deviations from a multivariable normal distribution for calculating confidence intervals, if the simulation produces negative standard deviations. In such cases, we assumed that a negative standard deviation implied the absence of individual-level preference heterogeneity. In our case study, we also did not observe any consistent pattern, such as the WTP estimate from one method being consistently higher or lower than that from the other method. However, as shown in Table 2, the WTP for improved sensitivity differed statistically between the two methods for model II. Based on model IV, the differences were mainly driven by the difference in the WTP at higher base costs. Future studies should be conducted to understand the reasons.

However, we do not claim that all DCEs should model using discrete cost. WTP calculations derived from continuous cost coefficients have their advantages in terms of the ease of calculation and interpretation. Furthermore, in studies modelling a small range of cost levels, linear disutility of cost could hold. However, researchers ought to justify their choice of modelling. When discrete cost is used, researchers should clearly describe the method used for calculating WTP and the method used for incorporating the non-linear disutility of cost.

This study has several limitations. Firstly, the use of a piecewise linear assumption to address non-linear disutility of cost is itself a limitation. However, our method offers the potential for extension through the incorporation of parametric assumptions. In theory, any higher-order polynomial may be employed to model disutility of cost. However, in practice, the order of the polynomial should remain lower than the number of levels of the cost attribute in the DCE. Secondly, the choice of cost levels could affect the results and the WTP calculations. Even with additional parametric assumptions, the choices of cost for linear interpolation based on the parametric curve may influence the calculated WTP. Future research is required to investigate and address these considerations.

We also recognise that there remain other unaddressed concerns, such as the behavioural issues that may arise from the magnitude and imbalance of the cost levels. Apart from economic theory, the non-linearity of cost may potentially be related to respondent characteristics as well. These issues, while pertinent to the estimation of WTP, are beyond the scope of this paper, which attempts to address the aforementioned situations. Nevertheless, these methods of addressing non-linearity may be extended beyond cost to other inherently continuous attributes within DCEs, such as sensitivity, severity, and risk levels, among others.

Conclusion

Our study improves the accuracy of WTP estimation in DCEs by using discrete cost, accounting for non-linear cost disutility, and considering the appropriate base cost. We also explored variations in WTP estimates when considering population mean utility values versus individual utility value distributions, allowing policymakers to choose the values that best fit their context. By addressing non-linear cost disutility and utility value distribution, our study enhances WTP estimation methodologies, offering insights for research and policymaking.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Author contributions

Y.W., K.K.T., and A.R.C. were responsible for the concept and design of the study and obtaining the funding for this study. C.O., K.K.T., and A.R.C. were responsible for the recruitment of participants and data collection. C.O. and Y.W. were responsible for the analysis of data. Y.W., K.K.T., and A.R.C. provided supervision and administrative support. C.O. and Y.W. wrote the first draft of the manuscript. All authors were involved in the interpretation of the data and worked on critical revisions of the manuscript. All authors read and approved the final manuscript.

Funding

This research is supported by the Singapore Ministry of Health’s (MOH) National Medical Research Council (NMRC) under its Health Services Research New Investigator Grant (MOH-000740-00). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not reflect the views of MOH/NMRC.

Data availability

Data is available upon request, subject to the approval of the review boards.

Declarations

Ethics approval and consent to participate

The study was approved by the National Healthcare Group Domain Specific Review Board (2021/00753). Participants of the case study were from an online cohort, and their participation in the research was approved by the National University of Singapore institutional review board (NUS-IRB: H-18-011).

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.