Abstract
Objectives
To examine whether interaction terms improve predictive accuracy in EQ-5D valuation models using fully saturated datasets, and to identify parsimonious interaction specifications that balance model performance and feasibility.
Methods
A secondary analysis of three saturated EQ-5D visual analogue scale (VAS) datasets was conducted: EQ-5D-3 L from the Netherlands and the United Kingdom, and EQ-5D-5 L from China. The analysis followed three steps. First, main-effects models were estimated as a reference and compared with models including all second-order interaction terms. Second, the contribution of interaction terms was assessed by examining how model performance metrics (RMSE, R², AIC, and BIC) changed with the number of interaction terms included, and by ranking individual interaction terms according to their impact on RMSE across datasets. Third, parsimonious models using summary interaction terms were evaluated to capture key non-additive effects.
Results
Including interaction terms consistently improved predictive performance across all three datasets, although absolute improvements were modest. Interaction effects were strongest for combinations of severe health problems, and all interaction coefficients were positive, consistent with diminishing marginal utility. Interactions involving pain/discomfort and anxiety/depression were among the most influential across datasets. Parsimonious specifications incorporating summary interaction terms—N3D1 for EQ-5D-3 L and N45D1 for EQ-5D-5 L—captured much of the performance gain achieved by fully interacted models.
Conclusions
Interaction effects are empirically detectable and systematically improve model performance in EQ-5D valuation when saturated data are available. Summary interaction terms provide a practical approach to capturing key non-additive effects while maintaining model parsimony.
Supplementary Information
The online version contains supplementary material available at 10.1186/s12955-026-02548-2.
Keywords: EQ-5D, Valuation, Interaction effects, Visual analogue scale
Highlights
Evidence on interaction effects in EQ-5D valuation has been limited by non-saturated study designs.
Using three fully saturated EQ-VAS datasets, interaction terms improved model performance, particularly for severe health states.
Parsimonious summary interaction terms capture key non-additive effects while maintaining feasibility for health state valuation and HEOR practice.
Supplementary Information
The online version contains supplementary material available at 10.1186/s12955-026-02548-2.
Introduction
The EuroQol Valuation Technology (EQ-VT) design has provided a standardized framework for EQ-5D-5 L valuation studies since its inception, enabling comparability across countries and facilitating widespread adoption of EQ-5D-5 L in health economic evaluations [1–3]. The EQ-VT design—based on a relatively small number of health states with approximately 100 observations per state—represented a major improvement over earlier approaches and was considered sufficient for estimating main-effects (ME) models [4–6].
However, the reliance on ME models assumes that disutilities associated with problems in each EQ-5D dimension are independent and additive. This assumption has been increasingly questioned. Individuals experiencing multiple severe problems may perceive their combined impact differently than the sum of individual impairments [7–9]. The absence of strong evidence for interaction terms in early valuation studies may have reflected design limitations rather than a true lack of interactions: efficient designs optimized for ME estimation provided limited coverage of severe and implausible states, constraining the detectability of interactions [10, 11]. For example, concerns were raised in the review of the first English EQ-5D-5 L value set about whether the design sufficiently captured severe health states to test for interactions [12].
More recently, empirical studies have begun to identify interactions in EQ-5D valuation models [13, 14]. The Singapore EQ-5D-5 L value set explicitly included two interaction terms, demonstrating that models incorporating interactions can improve predictive performance relative to ME-only specifications [13]. In addition, Jonker et al. [8] applied an optimal scaling approach to Dutch EQ-5D data and similarly reported significant interaction effects, highlighting their importance for capturing preference heterogeneity. These findings add to earlier simulation evidence suggesting that ME models may be mis-specified if interactions are ignored [7, 12]. In EQ-5D-3 L valuation studies, interaction effects have been incorporated through summary terms such as N3 and D1. The N3 term, introduced in the UK MVH study [15], captures the additional disutility associated with the presence of any severe problem, while D1-type specifications, used in studies such as the US valuation [16], account for the cumulative burden of problems across dimensions. In EQ-5D-5 L valuation studies, similar attempts have been made to account for interaction effects, although no single standard specification has been established. In the recent Singapore EQ-5D-5 L valuation study, the inclusion of two two-way interaction terms improved model performance [13]. These approaches reflect attempts to capture the compounding impact of multiple impairments and suggest that interaction effects may play an important role in modelling preferences, particularly for more severe health states.
However, most published valuation studies lack the empirical coverage required to investigate interactions comprehensively. For instance, only 43 of 243 health states (17.7%) were directly valued in the EQ-5D-3 L [15], and 86 of 3,125 states (2.75%) in the EQ-5D-5 L [1]. This limited coverage makes it difficult to assess whether interactions systematically influence valuations, particularly for combinations of severe problems.
The emergence of saturated datasets valued using the visual analogue scale (VAS) provides a unique opportunity to address this gap. A saturated dataset refers to a dataset in which all possible health states defined by the EQ-5D descriptive system are directly valued at least once, ensuring full coverage of the health state space. In these datasets, all EQ-5D health states are directly valued, eliminating the sampling restrictions of conventional valuation studies and enabling a comprehensive assessment of interaction effects across the entire descriptive system.
Such full coverage is not feasible in conventional TTO or DCE valuation studies due to practical constraints on study design and respondent burden. As a result, VAS-based saturated datasets provide a unique platform for systematically examining interaction effects. Importantly, previous methodological studies have used such datasets to identify efficient health state sampling strategies, which were subsequently shown to perform well in TTO-based valuation studies [6, 11]. This suggests that, while VAS and TTO differ in their elicitation mechanisms, VAS data can provide a useful platform for testing methodological questions. Nevertheless, findings derived from VAS data should be interpreted with caution when considering their generalisability to TTO-based valuation.
Understanding whether and how interaction effects influence valuation models has important practical implications. If such effects are present and meaningful, they may inform model specification, improve predictive performance, and guide the design of valuation studies. In particular, identifying when interaction effects become relevant may help optimize health state selection and improve the efficiency of valuation designs. Against this background, the present study aimed to: (i) investigate whether including interaction terms improves predictive accuracy in EQ-5D valuation models using saturated EQ VAS datasets from the Netherlands, UK, and China; and (ii) identify parsimonious specifications that capture key interactions while remaining feasible for future valuation studies.
Methods
Data sources
This study utilized three saturated datasets: one EQ-5D-5 L VAS dataset collected in China [6] and two EQ-5D-3 L VAS datasets collected in the Netherlands [5] and the United Kingdom. In these datasets, all possible health states defined by the respective EQ-5D descriptive systems were directly valued, providing full coverage of the health state space. The Dutch EQ-5D-3 L dataset was derived from a saturation study in which 126 university students valued all 243 EQ-5D-3 L health states using EQ VAS. Health states were presented in 10 random orders to reduce framing effects, the procedure was piloted in advance, and the average completion time was 1 h 16 min [5].
The China EQ-5D-5 L dataset was derived from a direct EQ VAS valuation study in which 1603 university students valued all 3125 EQ-5D-5 L health states. To achieve approximately 100 observations per health state, the health states were divided into 16 blocks, and respondents valued one block each in organized group interview sessions [6].
The UK EQ-5D-3 L dataset was derived from a postal saturation study conducted in England and Wales in 2003. Twenty-one questionnaire variants were developed, each containing anchor states and a subset of EQ-5D-3 L health states, and 1,100 questionnaires were mailed to individuals sampled from electoral-register sources. The responding sample comprised 685 individuals, with approximately 25 to 38 respondents contributing to each questionnaire variant.
These three datasets were selected because they represent available saturated EQ-VAS datasets across both the EQ-5D-3 L and EQ-5D-5 L systems, allowing the investigation of interaction effects under full health-state coverage. However, the original studies were not designed to produce nationally representative samples for the present purpose. In particular, the Dutch and Chinese datasets were based on university student samples, whereas the UK dataset used a postal general-population sample that was older and better educated than the population benchmark. Accordingly, the present analysis should be interpreted as methodological rather than population-representative. The use of saturated datasets allowed a comprehensive examination of interaction effects across the full EQ-5D descriptive system, without the sampling limitations inherent in conventional valuation designs. Key characteristics of the datasets are summarized in Table 1.
Table 1.
Characteristics of the saturated EQ-5D valuation datasets
| The Netherlands | UK | China | |
|---|---|---|---|
| EQ-5D version | 3 L | 3 L | 5 L |
| Mean number of observation per health state | 123.84 | 44.63 | 101.02 |
| Range of observation number per health state | From 122 to 124 | From 25 to 1364* | From 98 to 1601# |
| Lowest value, state | 8.88, 33333 | 7.64, 33333 | 14.50, 55555 |
| Highest value, state | 91.81, 11111 | 96.94, 11111 | 93.29, 11111 |
| Sample population | University students | General public | University students |
* 1364 observations for 11111 and 33333 only, for other states, it is between 25 and 70
#1601 observations for 55555 only, for other states, it is between 98 and 101
Data analysis
Step 1: Baseline and fully interacted model estimation
As a reference point, we first estimated conventional main-effects (ME) models for each dataset. In these models, the mean VAS value for each health state was expressed as a function of dimension-specific dummy variables and a constant. For the EQ-5D-3 L datasets, ten dummy variables represented deviations from full health (level 1) to levels 2 and 3 across the five EQ-5D dimensions (Formula 1). For the EQ-5D-5 L dataset, twenty dummy variables represented deviations from level 1 to levels 2–5.
 |
1 |
To examine the potential contribution of interaction effects, we then extended the ME models by including all possible second-order (two-way) interaction terms between dimension-level dummy variables. This resulted in 40 interaction terms for the EQ-5D-3 L datasets and 160 interaction terms for the EQ-5D-5 L dataset. These fully interacted models were estimated to assess whether interaction terms improved model performance relative to the ME specification and to explore the direction and magnitude of interaction effects across severity levels and dimensions.
Model performance for both ME and fully interacted models was evaluated using root mean squared error (RMSE), with additional goodness-of-fit statistics including R², Akaike information criterion (AIC), and Bayesian information criterion (BIC).
Step 2: Assessing the contribution of secondary interaction terms across datasets
To understand how interaction terms collectively influence model performance, we first examined how goodness-of-fit metrics changed as interaction terms were incrementally added to the model. Given the large number of possible interaction terms, it was not feasible to evaluate the effect of each term individually through sequential inclusion. Instead, we estimated a series of models in which interaction terms were added in increasing numbers, and plotted RMSE as a function of the number of interaction terms included. Interaction terms were added in arbitrary order to assess the general relationship between model complexity and performance, rather than to evaluate the marginal effect of any specific term. Corresponding analyses were conducted for R², AIC, and BIC. These analyses provide a general assessment of how increasing model complexity through additional interaction terms affects predictive performance and model fit.
These analyses provide a general assessment of how increasing model complexity through additional interaction terms affects predictive performance and model fit. RMSE results are presented in Figs. 1, 2 and 3, while corresponding figures for R², AIC, and BIC are reported in the supplementary material.
Fig. 1.
RMSE over number of interaction terms, UK 3 L
Fig. 2.
RMSE over number of interaction terms, Netherlands 3 L
Fig. 3.
RMSE over number of interaction terms, China 5 L
While these plots illustrate the overall relationship between the number of interaction terms and model performance, they do not identify which specific interaction terms are most influential. Therefore, to further disentangle the contribution of individual secondary interaction terms, we conducted an auxiliary regression analysis in which RMSE was regressed on the inclusion of each interaction term. This approach allowed us to quantify the marginal contribution of individual interaction terms to RMSE reduction and to rank interaction terms according to their importance within each dataset. The results of this ranking exercise are presented in Table 4 and were used to identify interaction patterns that were consistent across the three datasets.
Table 4.
Effect of selected secondary interaction terms on RMSE values, with aligned comparison of the Netherlands and UK EQ-5D-3 L datasets
| UK rank | 3 L term | UK coef | Netherlands coef | Netherlands rank | 5 L term | China coef |
|---|---|---|---|---|---|---|
| 1 | pd3ad3 | -0.02359 | -0.15480 | 1 | pd4ad5 | -0.00308 |
| 2 | mo3pd3 | -0.02009 | -0.06912 | 2 | pd5ad4 | -0.00270 |
| 3 | mo3ua3 | -0.01773 | -0.01895 | 25 | sc5ua4 | -0.00270 |
| 4 | mo3ad3 | -0.01473 | -0.05901 | 4 | mo4sc5 | -0.00263 |
| 5 | ua3pd3 | -0.01375 | -0.03490 | 10 | mo5sc4 | -0.00201 |
| 6 | ua3ad3 | -0.01143 | -0.05046 | 6 | mo4ua5 | -0.00193 |
| 7 | mo3sc3 | -0.00835 | -0.04800 | 7 | sc4ua5 | -0.00193 |
| 8 | sc3pd3 | -0.00657 | -0.04195 | 8 | mo3sc5 | -0.00175 |
| 9 | pd3ad2 | -0.00500 | -0.05297 | 5 | mo2sc2 | -0.00158 |
| 10 | sc2ua3 | -0.00467 | -0.02300 | 21 | sc5ua3 | -0.00152 |
| 11 | pd2ad3 | -0.00256 | -0.03099 | 13 | pd3ad5 | -0.00148 |
| 12 | ua2pd3 | -0.00255 | -0.02712 | 17 | pd3ad3 | -0.00146 |
| 13 | mo3sc2 | -0.00254 | -0.02039 | 24 | mo5sc5 | -0.00142 |
| 14 | mo2sc3 | -0.00250 | -0.02334 | 20 | pd5ad3 | -0.00134 |
| 15 | sc3ad3 | -0.00249 | -0.06308 | 3 | sc5ua5 | -0.00118 |
| 16 | sc3ua2 | -0.00243 | -0.00576 | 37 | mo3ua5 | -0.00108 |
| 17 | ua3ad2 | -0.00235 | -0.00885 | 35 | sc2ua2 | -0.00105 |
| 18 | mo2pd3 | -0.00233 | -0.02685 | 18 | mo4ad5 | -0.00104 |
| 19 | mo3ad2 | -0.00223 | -0.01397 | 29 | mo5sc3 | -0.00103 |
| 20 | mo2sc2 | -0.00213 | 0.000283 | 40 | mo4pd4 | -0.00102 |
| 21 | ua3pd2 | -0.00211 | -0.00364 | 38 | mo3sc4 | -0.00101 |
| 22 | sc3ua3 | -0.00211 | -0.01516 | 27 | mo4pd5 | -0.00097 |
| 23 | mo3ua2 | -0.00209 | -0.03057 | 14 | mo5ad4 | -0.00096 |
| 24 | sc2ua2 | -0.00199 | -0.01514 | 28 | mo5sc2 | -0.00090 |
| 25 | mo2ua3 | -0.00197 | -0.01690 | 26 | pd4ad4 | -0.00088 |
| 26 | mo3pd2 | -0.00192 | -0.00886 | 34 | mo5ua4 | -0.00087 |
| 27 | pd2ad2 | -0.00191 | -0.01396 | 31 | sc3ua5 | -0.00087 |
| 28 | ua2ad3 | -0.00181 | -0.02569 | 19 | mo3sc3 | -0.00080 |
| 29 | sc3pd2 | -0.00164 | -0.01372 | 32 | ua5pd5 | -0.00080 |
| 30 | mo2ad3 | -0.00154 | -0.00839 | 36 | ua5ad4 | -0.00077 |
| 31 | mo2ua2 | -0.00145 | -0.02910 | 16 | mo2sc3 | -0.00077 |
| 32 | sc2pd3 | -0.00140 | -0.02227 | 22 | pd2ad3 | -0.00075 |
| 33 | sc3ad2 | -0.00133 | -0.01396 | 30 | mo4ad4 | -0.00074 |
| 34 | ua2ad2 | -0.00117 | -0.03451 | 11 | mo3pd4 | -0.00074 |
| 35 | mo2pd2 | -0.00112 | -0.03028 | 15 | mo4sc4 | -0.00073 |
| 36 | sc2ad3 | -0.00109 | -0.03736 | 9 | mo2sc5 | -0.00070 |
| 37 | ua2pd2 | -0.00108 | -0.00933 | 33 | mo2ua2 | -0.00068 |
| 38 | sc2pd2 | -0.00100 | -0.02128 | 23 | mo3ad5 | -0.00066 |
| 39 | mo2ad2 | -0.00097 | -0.03154 | 12 | sc5pd4 | -0.00066 |
| 40 | sc2ad2 | -0.00092 | -0.00101 | 39 | pd2ad2 | -0.00065 |
| ua4ad4 | -0.00065 | |||||
| sc5ua2 | -0.00064 | |||||
| mo5ad3 | -0.00060 | |||||
| sc4pd4 | -0.00059 | |||||
| ua5pd4 | -0.00059 | |||||
| mo2sc4 | -0.00059 | |||||
| sc3ua3 | -0.00058 | |||||
| pd3ad4 | -0.00058 | |||||
| mo5pd4 | -0.00055 | |||||
| sc4ad3 | -0.00055 | |||||
| sc4ad4 | -0.00054 | |||||
| pd5ad5 | -0.00053 | |||||
| mo3ua3 | -0.00051 | |||||
| sc4ua2 | -0.00050 | |||||
| ua2pd5 | -0.00049 | |||||
| ua4pd5 | -0.00049 | |||||
| mo4sc2 | -0.00049 | |||||
| ua4ad3 | -0.00048 | |||||
| mo3sc2 | -0.00048 | |||||
| pd2ad5 | -0.00048 | |||||
| pd4ad2 | -0.00047 | |||||
| sc4ua3 | -0.00044 | |||||
| mo3ad3 | -0.00044 | |||||
| pd4ad3 | -0.00042 | |||||
| mo3ua4 | -0.00042 | |||||
| mo5ad5 | -0.00042 | |||||
| sc5pd5 | -0.00041 | |||||
| sc3pd2 | -0.00039 | |||||
| ua5ad3 | -0.00039 | |||||
| pd5ad2 | -0.00039 | |||||
| sc5ad4 | -0.00038 | |||||
| ua3ad4 | -0.00037 | |||||
| sc4ad5 | -0.00037 | |||||
| mo4ad3 | -0.00036 | |||||
| sc2ua3 | -0.00035 | |||||
| ua4pd4 | -0.00034 | |||||
| sc4pd5 | -0.00033 | |||||
| ua4pd3 | -0.00031 | |||||
| mo5ua5 | -0.00031 | |||||
| mo2ua5 | -0.00031 | |||||
| mo5ua3 | -0.00031 | |||||
| sc3ua4 | -0.00030 | |||||
| sc4pd2 | -0.00030 | |||||
| ua5ad5 | -0.00030 | |||||
| sc4pd3 | -0.00029 | |||||
| sc4ua4 | -0.00029 | |||||
| sc2ua5 | -0.00028 | |||||
| sc5ad3 | -0.00027 | |||||
| mo4sc3 | -0.00026 | |||||
| pd2ad4 | -0.00026 | |||||
| ua5pd3 | -0.00026 | |||||
| sc2ua4 | -0.00024 | |||||
| ua4pd2 | -0.00024 | |||||
| mo5pd3 | -0.00024 | |||||
| sc3ad3 | -0.00024 | |||||
| mo5pd5 | -0.00022 | |||||
| sc5ad5 | -0.00022 | |||||
| sc3pd4 | -0.00021 | |||||
| ua2pd3 | -0.00021 | |||||
| sc3ad5 | -0.00021 | |||||
| sc3pd3 | -0.00021 | |||||
| mo4pd2 | -0.00021 | |||||
| ua2pd2 | -0.00021 | |||||
| mo2ad4 | -0.00021 | |||||
| ua3pd5 | -0.00021 | |||||
| sc3ua2 | -0.00020 | |||||
| mo3pd2 | -0.00019 | |||||
| mo2ad3 | -0.00019 | |||||
| mo2ad2 | -0.00019 | |||||
| mo4ua4 | -0.00019 | |||||
| sc5pd3 | -0.00019 | |||||
| mo3pd3 | -0.00019 | |||||
| mo2pd5 | -0.00019 | |||||
| pd3ad2 | -0.00018 | |||||
| mo5ua2 | -0.00018 | |||||
| sc2pd5 | -0.00017 | |||||
| mo2pd4 | -0.00017 | |||||
| mo3ad2 | -0.00017 | |||||
| mo5pd2 | -0.00016 | |||||
| sc3ad4 | -0.00016 | |||||
| mo3ua2 | -0.00016 | |||||
| mo4ad2 | -0.00015 | |||||
| mo4ua3 | -0.00015 | |||||
| mo2ua4 | -0.00014 | |||||
| mo2ad5 | -0.00014 | |||||
| ua3pd2 | -0.00014 | |||||
| ua4ad5 | -0.00013 | |||||
| mo3ad4 | -0.00013 | |||||
| sc2ad4 | -0.00012 | |||||
| mo5ad2 | -0.00012 | |||||
| ua3ad5 | -0.00011 | |||||
| ua2ad5 | -0.00011 | |||||
| ua3pd4 | -0.00011 | |||||
| ua4ad2 | -0.00011 | |||||
| sc5ad2 | -9.4E-05 | |||||
| sc2ad5 | -8.8E-05 | |||||
| ua3ad3 | -8.6E-05 | |||||
| ua3pd3 | -8.6E-05 | |||||
| ua2pd4 | -8.0E-05 | |||||
| ua2ad4 | -7.5E-05 | |||||
| mo4ua2 | -7.0E-05 | |||||
| sc3pd5 | -6.7E-05 | |||||
| mo2pd2 | -6.5E-05 | |||||
| sc2pd4 | -5.1E-05 | |||||
| ua3ad2 | -4.6E-05 | |||||
| ua2ad2 | -4.5E-05 | |||||
| sc4ad2 | -4.3E-05 | |||||
| mo2ua3 | -3.2E-05 | |||||
| ua5ad2 | -3.2E-05 | |||||
| mo3pd5 | -2.5E-05 | |||||
| ua5pd2 | -1.5E-05 | |||||
| sc2ad3 | 2.57E-05 | |||||
| sc3ad2 | 2.58E-05 | |||||
| mo2pd3 | 4.67E-05 | |||||
| sc2ad2 | 5.14E-05 | |||||
| ua2ad3 | 5.60E-05 | |||||
| sc2pd2 | 6.27E-05 | |||||
| sc2pd3 | 9.21E-05 | |||||
| mo4pd3 | 0.000104 | |||||
| sc5pd2 | 0.000174 |
Note: UK and Netherlands results are both based on EQ-5D-3 L and are shown in the order of the UK ranking to facilitate comparison. The China results are based on EQ-5D-5 L and are therefore retained in their original within-dataset ranking. More negative coefficients indicate greater reduction in RMSE associated with inclusion of the interaction term. Coefficients shown in bold are not statistically significant at the 5% level
Step 3: Parsimonious modelling using summary interaction terms
Although models including all secondary interaction terms achieved the greatest improvements in model performance, such specifications involve a large number of parameters and are unlikely to be feasible in standard EQ-5D valuation studies. Therefore, in the final step, we explored whether parsimonious interaction structures could capture the key non-additive effects identified in Step 2.
Specifically, we evaluated summary interaction terms that aggregate the effects of multiple secondary interactions. For EQ-5D-3 L, we examined the N3 term, which captures interactions among severe (level 3) problems, and the D1 term, which reflects overall marginal disutility beyond the first problem, as well as their combination (N3D1). For EQ-5D-5 L, we examined analogous specifications including N5, N45, and D1.
The performance of these parsimonious models was compared with that of the ME and fully interacted models using RMSE as the primary criterion, with results summarised in Table 5.
Table 5.
RMSE values of main effects model and models with different interaction terms
| Model name | RMSE | ||
|---|---|---|---|
| Netherlands | UK | China | |
| rmse_main | 2.496783 | 5.54918 | 1.81246 |
| rmse_all_interaction_terms | 1.213041 | 3.502183 | 1.507143 |
| rmse_d1 | 2.079927 | 3.99069 | 1.789281 |
| rmse_n3 | 1.96092 | 4.740939 | / |
| rmse_n3d1 | 1.578983 | 3.640426 | / |
| rmse_ad3 | 2.399362 | 5.157859 | / |
| rmse_ad3d1 | 1.982546 | 3.731316 | / |
| rmse_n5 | / | / | 1.813585 |
| rmse_n5d1 | / | / | 1.795188 |
| rmse_n45 | / | / | 1.720708 |
| rmse_n45d1 | / | / | 1.698956 |
Results
There are 30,092, 10,845 and 315,693 observations respectively for the Netherlands, UK and China. Table 1 shows some characteristics of these three saturated datasets. The number of observations per health state vary across three datasets, with the dataset from the Netherlands and China met the recommended sample size of 100 observations per state. The dataset from UK has a lower number of observations for all health states other than 11,111 and 33,333. This is because the UK dataset was collected using a block design, and states 11,111 and 33,333 were included in all blocks.
For modelling, we first estimated a main effects specification and then extended it by including all second-order interaction terms. The extended models yielded lower RMSE values and improved overall fit indices compared to the main effects models. The estimated coefficients for interactions were generally larger when both dimensions were at higher severity levels, and all coefficients were positive across datasets.
Table 2 shows the modelling coefficients and the model performance indicators for both 3 L datasets and Table 3 shows the modelling coefficients and model performance indication for the 5 L dataset. For 3 L datasets, all secondary interaction terms were significant at 0.01 level for the saturated dataset of the Netherlands, except for the terms sc2ad2 and sc2ad3, which were only significant at 0.05 level. In comparison, there were 5 interaction terms in the UK dataset that were not significant at 0.05 level (sc3ua2, sc3ua3, sc2ad2, ua3pd2, ua2pd3) and 2 terms were significant at 0.05 level (sc2ua2, mo3ua3, mo3ad3), with all other terms being significant at 0.01 level. The overall model performance indicators of the Netherlands dataset were better than the UK dataset. A similar pattern was observed for the 5 L dataset of China and only two interaction terms were significant at 0.05 (sc4ad2) and 0.1(sc2ad2) level respectively with all other 158 interaction terms being significant at 0.01 level.
Table 2.
Model performance and coefficients for both 3 L saturated datasets
| The Netherlands | UK | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Main effects | Interaction model | Main effects | Interaction model | ||||||
| Parameter | Coef | SE | Coef | SE. | Parameter | Coef | SE | Coef | SE |
| mo2 | -4.751 | 0.2 | -10.931 | 0.59 | mo2 | -10.417 | 0.443 | -19.525 | 1.082 |
| mo3 | -13.728 | 0.2 | -24.008 | 0.59 | mo3 | -18.234 | 0.437 | -29.856 | 1.162 |
| sc2 | -3.728 | 0.2 | -9.435 | 0.59 | sc2 | -10.417 | 0.444 | -17.308 | 1.074 |
| sc3 | -7.74 | 0.2 | -14.755 | 0.59 | sc3 | -14.232 | 0.437 | -22.904 | 1.161 |
| ua2 | -5.037 | 0.2 | -11.258 | 0.59 | ua2 | -10.014 | 0.443 | -16.973 | 1.079 |
| ua3 | -11.705 | 0.2 | -21.364 | 0.59 | ua3 | -13.163 | 0.436 | -18.935 | 1.164 |
| pd2 | -6.079 | 0.2 | -12.218 | 0.59 | pd2 | -12.031 | 0.443 | -17.977 | 1.084 |
| pd3 | -16.136 | 0.2 | -27.03 | 0.59 | pd3 | -20.075 | 0.436 | -36.094 | 1.132 |
| ad2 | -5.538 | 0.2 | -11.618 | 0.59 | ad2 | -10.188 | 0.444 | -17.929 | 1.082 |
| ad3 | -14.217 | 0.2 | -23.153 | 0.59 | ad3 | -17.819 | 0.435 | -34.071 | 1.161 |
| mo2sc2 | 2.687 | 0.482 | mo2sc2 | 5.674 | 1.035 | ||||
| mo3sc2 | 3.17 | 0.482 | mo3sc2 | 5.965 | 1.052 | ||||
| mo2sc3 | 3.135 | 0.482 | mo2sc3 | 6.019 | 1.042 | ||||
| mo3sc3 | 4.266 | 0.482 | mo3sc3 | 6.121 | 1.044 | ||||
| mo2ua2 | 1.971 | 0.482 | mo2ua2 | 5.488 | 1.029 | ||||
| mo3ua2 | 2.862 | 0.482 | mo3ua2 | 6.244 | 1.048 | ||||
| mo2ua3 | 2.914 | 0.482 | mo2ua3 | 4.296 | 1.048 | ||||
| mo3ua3 | 5.294 | 0.482 | mo3ua3 | 2.645 | 1.042 | ||||
| mo2pd2 | 1.666 | 0.482 | mo2pd2 | 5.297 | 1.037 | ||||
| mo3pd2 | 2.491 | 0.482 | mo3pd2 | 3.281 | 1.042 | ||||
| mo2pd3 | 2.937 | 0.482 | mo2pd3 | 6.142 | 1.043 | ||||
| mo3pd3 | 5.421 | 0.482 | mo3pd3 | 6.899 | 1.048 | ||||
| mo2ad2 | 1.244 | 0.482 | mo2ad2 | 4.747 | 1.044 | ||||
| mo3ad2 | 2.681 | 0.482 | mo3ad2 | 4.072 | 1.048 | ||||
| mo2ad3 | 1.986 | 0.482 | mo2ad3 | 2.302 | 1.047 | ||||
| mo3ad3 | 4.646 | 0.482 | mo3ad3 | 5.368 | 1.048 | ||||
| sc2ua2 | 2.581 | 0.482 | sc2ua2 | 2.609 | 1.033 | ||||
| sc3ua2 | 2.554 | 0.482 | sc3ua2 | 1.026 | 1.05 | ||||
| sc2ua3 | 3.257 | 0.482 | sc2ua3 | 3.831 | 1.055 | ||||
| sc3ua3 | 2.66 | 0.482 | sc3ua3 | 0.725 | 1.04 | ||||
| sc2pd2 | 1.363 | 0.482 | sc2pd2 | 4.259 | 1.038 | ||||
| sc3pd2 | 1.926 | 0.482 | sc3pd2 | 3.921 | 1.046 | ||||
| sc2pd3 | 1.985 | 0.482 | sc2pd3 | 5.305 | 1.051 | ||||
| sc3pd3 | 3.25 | 0.481 | sc3pd3 | 4.632 | 1.042 | ||||
| sc2ad2 | 0.946 | 0.482 | sc2ad2 | 0.045 | 1.033 | ||||
| sc3ad2 | 1.384 | 0.482 | sc3ad2 | 3.314 | 1.052 | ||||
| sc2ad3 | 1.129 | 0.482 | sc2ad3 | 5.993 | 1.054 | ||||
| sc3ad3 | 1.871 | 0.482 | sc3ad3 | 6.702 | 1.04 | ||||
| ua2pd2 | 1.654 | 0.482 | ua2pd2 | 1.563 | 1.037 | ||||
| ua3pd2 | 2.697 | 0.482 | ua3pd2 | 0.197 | 1.048 | ||||
| ua2pd3 | 2.992 | 0.482 | ua2pd3 | 4.391 | 1.052 | ||||
| ua3pd3 | 4.871 | 0.482 | ua3pd3 | 2.698 | 1.044 | ||||
| ua2ad2 | 1.612 | 0.482 | ua2ad2 | 6.013 | 1.04 | ||||
| ua3ad2 | 2.838 | 0.482 | ua3ad2 | 3.383 | 1.05 | ||||
| ua2ad3 | 2.43 | 0.482 | ua2ad3 | 6.327 | 1.046 | ||||
| ua3ad3 | 4.439 | 0.482 | ua3ad3 | 5.484 | 1.039 | ||||
| pd2ad2 | 2.947 | 0.482 | pd2ad2 | 4.808 | 1.038 | ||||
| pd3ad2 | 4.586 | 0.482 | pd3ad2 | 9.801 | 1.047 | ||||
| pd2ad3 | 3.667 | 0.482 | pd2ad3 | 7.811 | 1.05 | ||||
| pd3ad3 | 6.635 | 0.482 | pd3ad3 | 13.57 | 1.04 | ||||
| Constant | 69.675 | 0.27 | 82.528 | 0.573 | Constant | 87 | 0.37 | 95.408 | 0.413 |
| Model performance indicators | |||||||||
| RMSE | 2.497 | 1.213 | 5.549 | 3.502 | |||||
| R2 | 0.419 | 0.437 | 0.680 | 0.723 | |||||
| AIC | 244,819 | 243,955 | 91,696 | 90,225 | |||||
| BIC | 244,910 | 244,379 | 91,776 | 90,596 | |||||
* Coefficients shown in bold are not statistically significant at the 5% level, whereas coefficients shown in italics are statistically significant at the 5% level
Table 3.
Model performance and coefficients for China 5 L dataset
| China | ||||||||
|---|---|---|---|---|---|---|---|---|
| Main effects | Interaction model | |||||||
| Parameter | Coef | SE | Coef | SE | ||||
| mo2 | -3.84 | 0.092 | -10.698 | 0.357 | ||||
| mo3 | -5.981 | 0.092 | -13.979 | 0.357 | ||||
| mo4 | -9.12 | 0.092 | -17.346 | 0.357 | ||||
| mo5 | -11.257 | 0.091 | -18.689 | 0.357 | ||||
| sc2 | -3.417 | 0.092 | -10.12 | 0.358 | ||||
| sc3 | -5.595 | 0.092 | -12.73 | 0.358 | ||||
| sc4 | -8.069 | 0.092 | -16.237 | 0.357 | ||||
| sc5 | -10.122 | 0.091 | -17.621 | 0.357 | ||||
| ua2 | -3.359 | 0.092 | -8.649 | 0.357 | ||||
| ua3 | -5.568 | 0.092 | -11.85 | 0.358 | ||||
| ua4 | -8.379 | 0.092 | -15.058 | 0.358 | ||||
| ua5 | -10.393 | 0.091 | -17.317 | 0.357 | ||||
| pd2 | -2.897 | 0.092 | -8.614 | 0.357 | ||||
| pd3 | -5.581 | 0.092 | -11.959 | 0.357 | ||||
| pd4 | -9.593 | 0.092 | -17.235 | 0.358 | ||||
| pd5 | -11.125 | 0.091 | -18.738 | 0.357 | ||||
| ad2 | -3.32 | 0.092 | -8.241 | 0.358 | ||||
| ad3 | -6.386 | 0.092 | -13.648 | 0.358 | ||||
| ad4 | -10.74 | 0.092 | -18.18 | 0.357 | ||||
| ad5 | -12.757 | 0.091 | -20.013 | 0.357 | ||||
| mo2sc2 | 4.306 | 0.288 | ||||||
| mo2sc3 | 3.691 | 0.288 | ||||||
| mo2sc4 | 3.89 | 0.288 | ||||||
| mo2sc5 | 3.647 | 0.288 | ||||||
| mo2ua2 | 2.045 | 0.288 | ||||||
| mo2ua3 | 1.72 | 0.288 | ||||||
| mo2ua4 | 1.881 | 0.288 | ||||||
| mo2ua5 | 2.195 | 0.288 | ||||||
| mo2pd2 | 1.387 | 0.288 | ||||||
| mo2pd3 | 1.238 | 0.288 | ||||||
| mo2pd4 | 1.354 | 0.288 | ||||||
| mo2pd5 | 1.741 | 0.288 | ||||||
| mo2ad2 | 1.424 | 0.288 | ||||||
| mo2ad3 | 1.758 | 0.288 | ||||||
| mo2ad4 | 1.946 | 0.288 | ||||||
| mo2ad5 | 1.658 | 0.288 | ||||||
| mo3sc2 | 3.549 | 0.288 | ||||||
| mo3sc3 | 3.906 | 0.288 | ||||||
| mo3sc4 | 4.161 | 0.288 | ||||||
| mo3sc5 | 4.386 | 0.288 | ||||||
| mo3ua2 | 1.895 | 0.288 | ||||||
| mo3ua3 | 2.166 | 0.288 | ||||||
| mo3ua4 | 2.142 | 0.288 | ||||||
| mo3ua5 | 2.879 | 0.288 | ||||||
| mo3pd2 | 1.725 | 0.288 | ||||||
| mo3pd3 | 1.634 | 0.288 | ||||||
| mo3pd4 | 2.31 | 0.288 | ||||||
| mo3pd5 | 2.007 | 0.288 | ||||||
| mo3ad2 | 1.661 | 0.288 | ||||||
| mo3ad3 | 2.361 | 0.288 | ||||||
| mo3ad4 | 2.167 | 0.288 | ||||||
| mo3ad5 | 2.622 | 0.288 | ||||||
| mo4sc2 | 3.483 | 0.288 | ||||||
| mo4sc3 | 3.561 | 0.288 | ||||||
| mo4sc4 | 3.901 | 0.288 | ||||||
| mo4sc5 | 4.573 | 0.288 | ||||||
| mo4ua2 | 1.536 | 0.288 | ||||||
| mo4ua3 | 2.035 | 0.288 | ||||||
| mo4ua4 | 2.191 | 0.288 | ||||||
| mo4ua5 | 3.337 | 0.288 | ||||||
| mo4pd2 | 1.627 | 0.288 | ||||||
| mo4pd3 | 1.599 | 0.288 | ||||||
| mo4pd4 | 2.623 | 0.288 | ||||||
| mo4pd5 | 2.504 | 0.288 | ||||||
| mo4ad2 | 1.568 | 0.288 | ||||||
| mo4ad3 | 2.49 | 0.288 | ||||||
| mo4ad4 | 2.783 | 0.288 | ||||||
| mo4ad5 | 2.907 | 0.288 | ||||||
| mo5sc2 | 3.67 | 0.288 | ||||||
| mo5sc3 | 3.863 | 0.288 | ||||||
| mo5sc4 | 4.426 | 0.288 | ||||||
| mo5sc5 | 1.953 | 0.286 | ||||||
| mo5ua2 | 1.773 | 0.288 | ||||||
| mo5ua3 | 2.114 | 0.288 | ||||||
| mo5ua4 | 2.485 | 0.288 | ||||||
| mo5ua5 | 1.842 | 0.286 | ||||||
| mo5pd2 | 1.309 | 0.288 | ||||||
| mo5pd3 | 1.75 | 0.288 | ||||||
| mo5pd4 | 2.247 | 0.288 | ||||||
| mo5pd5 | 1.759 | 0.286 | ||||||
| mo5ad2 | 1.73 | 0.288 | ||||||
| mo5ad3 | 2.59 | 0.288 | ||||||
| mo5ad4 | 2.994 | 0.288 | ||||||
| mo5ad5 | 2.509 | 0.286 | ||||||
| sc2ua2 | 3.089 | 0.288 | ||||||
| sc2ua3 | 2.697 | 0.288 | ||||||
| sc2ua4 | 2.277 | 0.288 | ||||||
| sc2ua5 | 2.567 | 0.288 | ||||||
| sc2pd2 | 1.203 | 0.288 | ||||||
| sc2pd3 | 1.476 | 0.288 | ||||||
| sc2pd4 | 1.671 | 0.288 | ||||||
| sc2pd5 | 1.421 | 0.288 | ||||||
| sc2ad2 | 0.552 | 0.288 | ||||||
| sc2ad3 | 0.921 | 0.288 | ||||||
| sc2ad4 | 1.202 | 0.288 | ||||||
| sc2ad5 | 1.022 | 0.288 | ||||||
| sc3ua2 | 2.318 | 0.288 | ||||||
| sc3ua3 | 2.927 | 0.288 | ||||||
| sc3ua4 | 2.401 | 0.288 | ||||||
| sc3ua5 | 2.996 | 0.288 | ||||||
| sc3pd2 | 1.842 | 0.288 | ||||||
| sc3pd3 | 1.607 | 0.288 | ||||||
| sc3pd4 | 1.84 | 0.288 | ||||||
| sc3pd5 | 1.373 | 0.288 | ||||||
| sc3ad2 | 0.743 | 0.288 | ||||||
| sc3ad3 | 1.518 | 0.288 | ||||||
| sc3ad4 | 1.326 | 0.288 | ||||||
| sc3ad5 | 1.351 | 0.288 | ||||||
| sc4ua2 | 2.802 | 0.288 | ||||||
| sc4ua3 | 2.953 | 0.288 | ||||||
| sc4ua4 | 2.324 | 0.288 | ||||||
| sc4ua5 | 3.547 | 0.288 | ||||||
| sc4pd2 | 1.872 | 0.288 | ||||||
| sc4pd3 | 2.046 | 0.288 | ||||||
| sc4pd4 | 2.385 | 0.288 | ||||||
| sc4pd5 | 2.06 | 0.288 | ||||||
| sc4ad2 | 0.649 | 0.288 | ||||||
| sc4ad3 | 1.993 | 0.288 | ||||||
| sc4ad4 | 1.801 | 0.288 | ||||||
| sc4ad5 | 1.62 | 0.288 | ||||||
| sc5ua2 | 2.928 | 0.288 | ||||||
| sc5ua3 | 3.75 | 0.288 | ||||||
| sc5ua4 | 3.876 | 0.288 | ||||||
| sc5ua5 | 1.48 | 0.286 | ||||||
| sc5pd2 | 1.24 | 0.288 | ||||||
| sc5pd3 | 1.792 | 0.288 | ||||||
| sc5pd4 | 2.313 | 0.288 | ||||||
| sc5pd5 | 1.997 | 0.286 | ||||||
| sc5ad2 | 0.899 | 0.288 | ||||||
| sc5ad3 | 1.388 | 0.288 | ||||||
| sc5ad4 | 1.652 | 0.288 | ||||||
| sc5ad5 | 1.474 | 0.286 | ||||||
| ua2pd2 | 1.118 | 0.288 | ||||||
| ua2pd3 | 0.826 | 0.288 | ||||||
| ua2pd4 | 1.179 | 0.288 | ||||||
| ua2pd5 | 1.902 | 0.288 | ||||||
| ua2ad2 | 0.984 | 0.288 | ||||||
| ua2ad3 | 1.245 | 0.288 | ||||||
| ua2ad4 | 1.065 | 0.288 | ||||||
| ua2ad5 | 1.331 | 0.288 | ||||||
| ua3pd2 | 1.384 | 0.288 | ||||||
| ua3pd3 | 1.406 | 0.288 | ||||||
| ua3pd4 | 1.664 | 0.288 | ||||||
| ua3pd5 | 1.928 | 0.288 | ||||||
| ua3ad2 | 1.233 | 0.288 | ||||||
| ua3ad3 | 1.569 | 0.288 | ||||||
| ua3ad4 | 1.809 | 0.288 | ||||||
| ua3ad5 | 1.643 | 0.288 | ||||||
| ua4pd2 | 1.447 | 0.288 | ||||||
| ua4pd3 | 1.779 | 0.288 | ||||||
| ua4pd4 | 2.216 | 0.288 | ||||||
| ua4pd5 | 2.299 | 0.288 | ||||||
| ua4ad2 | 1.477 | 0.288 | ||||||
| ua4ad3 | 2.148 | 0.288 | ||||||
| ua4ad4 | 2.25 | 0.288 | ||||||
| ua4ad5 | 1.792 | 0.288 | ||||||
| ua5pd2 | 1.232 | 0.288 | ||||||
| ua5pd3 | 1.751 | 0.288 | ||||||
| ua5pd4 | 2.266 | 0.288 | ||||||
| ua5pd5 | 2.601 | 0.286 | ||||||
| ua5ad2 | 1.419 | 0.288 | ||||||
| ua5ad3 | 2.171 | 0.288 | ||||||
| ua5ad4 | 2.397 | 0.288 | ||||||
| ua5ad5 | 1.791 | 0.286 | ||||||
| pd2ad2 | 3.097 | 0.288 | ||||||
| pd2ad3 | 3.479 | 0.288 | ||||||
| pd2ad4 | 2.949 | 0.288 | ||||||
| pd2ad5 | 3.26 | 0.288 | ||||||
| pd3ad2 | 2.464 | 0.288 | ||||||
| pd3ad3 | 4.196 | 0.288 | ||||||
| pd3ad4 | 3.657 | 0.288 | ||||||
| pd3ad5 | 4.254 | 0.288 | ||||||
| pd4ad2 | 3.052 | 0.288 | ||||||
| pd4ad3 | 3.707 | 0.288 | ||||||
| pd4ad4 | 3.934 | 0.288 | ||||||
| pd4ad5 | 5.039 | 0.288 | ||||||
| pd5ad2 | 3.239 | 0.288 | ||||||
| pd5ad3 | 4.368 | 0.288 | ||||||
| pd5ad4 | 4.859 | 0.288 | ||||||
| pd5ad5 | 3.861 | 0.286 | ||||||
| Constant | 75.215 | 0.127 | 88.443 | 0.286 | ||||
| Model performance indicators | ||||||||
| RMSE | 1.812 | 1.507 | ||||||
| R2 | 0.246 | 0.255 | ||||||
| AIC | 2,657,028 | 2,653,469 | ||||||
| BIC | 2,657,252 | 2,655,399 | ||||||
* Coefficients shown in bold are not statistically significant at the 5% level, whereas coefficients shown in italics are statistically significant at the 5% level
From Tables 2 and 3, the parameters with the highest weights are pd3ad3, mo3pd3, mo3ua3 for the Netherlands and pd3ad3, pd3ad2, pd2ad3 for the UK. In comparison, the parameters with the highest weights are pd4ad5, pd5ad4, mo4sc5 for China.
Figures 1, 2 and 3 present the relationship between RMSE and the number of interaction terms included in the model for the UK, Netherlands and China. Across all three datasets, RMSE declines monotonically as interaction terms are added, while the overall magnitude of change remains limited. Corresponding figures for R², AIC, and BIC are presented in the supplementary material and show similar trends.
Table 4 presents the ranking of interaction terms based on their impact on RMSE. In both 3 L datasets, the terms pd3ad3 and mo3pd3 produced the largest reductions in RMSE, with pd3ad3 showing the strongest effect in the Netherlands dataset. In the 5 L dataset from China, the terms pd4ad5 and pd5ad4 had the largest influence, whereas pd5ad5 showed a comparatively smaller effect. The mo3pd3 term, which was important in the 3 L datasets, did not emerge as influential in the 5 L dataset. Across all three datasets, the interaction involving high levels of pain/discomfort and anxiety/depression was consistently among the top-ranked terms.
From Table 4, it is evident that the effect of diminishing marginal utility differs between mild and severe health states, with stronger interactions observed at higher severity levels. To capture these effects efficiently, we selected N3 for the EQ-5D-3 L models and N45 for the EQ-5D-5 L model, representing interactions among severe levels, and incorporated D1 to account for overall marginal disutility effects. This targeted approach ensures that key interaction effects are reflected in the model while maintaining feasibility within realistic sample size constraints.
Table 5 demonstrates that incorporating interaction terms significantly improves model performance across all datasets, as indicated by lower RMSE values compared to the main effects model. Although the absolute reduction is marginal, the relative reduction of RMSE is considerable. Specifically, in the Netherlands dataset, RMSE decreases from 2.50 to 1.21, representing a 51% reduction, while in the UK dataset, RMSE drops from 5.55 to 3.50 (37% reduction). The China dataset shows a more modest 17% reduction (1.81 to 1.51). Notably, while models including all interaction terms achieve the greatest reduction in RMSE, more parsimonious models, such as N3D1 in the Netherlands and UK and N45D1 in China, also yield substantial improvements, suggesting that focusing on key interactions rather than including all possible terms may provide a balance between model parsimony and predictive accuracy.
Discussion
This study investigated the impact of incorporating interaction terms in EQ-5D valuation models using three saturated datasets: EQ-5D-3 L datasets from the Netherlands and the UK, and an EQ-5D-5 L dataset from China. The findings reveal three key insights. First, models that included interaction terms demonstrated improved predictive performance, as reflected in reduced RMSE values and enhanced model fit indicators (AIC, BIC, and R²). Second, all interaction terms were positive, supporting the presence of diminishing marginal utility, where individuals perceive smaller additional losses when multiple health problems are combined. Third, interaction terms involving higher severity levels exhibited the largest coefficients, suggesting that the diminishing marginal utility effect is more sensitive among health states with severe/extreme problems. Importantly, while the results demonstrate the superiority of models with interaction terms, the main effects model also provide sufficient prediction accuracy. This suggests the validity of published studies using main effects model, especially for those studies using efficient design optimized for main effects model [6].
A further finding is that no single universal interaction term emerged across all three datasets, except for interaction terms between high levels between anxiety/depression and pain/discomfort, which captures the effect of experiencing severe or extreme problems in both pain/discomfort and anxiety/depression simultaneously. This interaction was consistently identified as significant in both the 3 L and 5 L datasets, highlighting its potential importance in capturing the compounding effect of severe problems in these two dimensions. However, incorporating secondary interaction terms between these two dimensions alone did not outperform the N3 term in model performance, suggesting that broader interactions involving multiple severe dimensions provide a better representation of valuation patterns [15, 17]. Moreover, in the 5 L dataset, the N5-only term did not outperform the N45 term, indicating that interactions among severe and extreme health problems (N45) better account for valuation responses than focusing solely on extreme levels. These findings suggest that while certain interactions may be consistently significant across datasets, a single specific interaction term like pd3ad3 for EQ-5D-3 L is insufficient to fully capture the complexities of preference-based valuation models.
Based on the findings of this study, we conclude that for EQ-5D-3 L, the inclusion of N3 (capturing interactions among severe levels) and D1 (representing overall marginal disutility effects) appears to be the optimal model specification [15–18]. For EQ-5D-5 L, the combination of N45 and D1 interactions provides the best model fit and predictive accuracy. These interaction terms effectively capture the compounding effects of severe health impairments across multiple dimensions.
Another key contribution of this study is to clarify the role and interpretation of summary interaction terms such as N3, N45, and D1 within EQ-5D valuation models. Conceptually, these terms are often described as capturing non-additive effects arising from the joint presence of multiple problems or severe problems across dimensions. Rather than representing specific pairwise interactions between dimensions [7, 8], summary terms provide a low-dimensional way to account for systematic deviations from strict additivity that arise under particular conditions of severity.
Empirically, evidence from saturated EQ VAS datasets suggests that these terms may be closely related to boundary effects on the valuation scale. In particular, the N3 and N45 terms indicate the presence of at least one dimension at the most severe (or near-most severe) level, a situation in which observed VAS values tend to decline more sharply than predicted by additive main-effects alone. This pattern is consistent with a boundary-induced compression effect, whereby respondents increasingly anchor their valuations toward the lower end of the scale as health states approach the worst imaginable condition.
Under this interpretation, summary interaction terms do not necessarily reflect complex psychological interactions between specific dimensions (e.g., mobility and pain), but instead capture a global adjustment associated with proximity to the lower bound of the valuation scale. Such boundary effects are particularly salient in VAS data, where the finite scale and anchoring at “worst imaginable health” may induce non-linearities as severity accumulates across dimensions. The consistent improvement in model performance associated with N3 and N45 across multiple datasets supports the view that these terms capture a systematic and reproducible feature of the valuation process. Importantly, the mechanisms underlying the valuation of severe health states may differ between VAS and other elicitation methods such as TTO, e.g., EQ VAS has a known “end aversion” effect [11, 19]. While the boundary-related compression observed here is consistent with properties of the VAS scale, TTO tasks involve explicit trade-offs with life duration and may therefore engage different cognitive and decision processes when valuing very poor health states [11]. As such, the interaction patterns identified in this study should be interpreted as reflecting VAS-based valuation behavior and may not directly translate to TTO or DCE data.
This interpretation has important implications for valuation study design. If summary interaction terms primarily reflect boundary effects rather than dimension-specific interactions, their estimation does not require valuation designs that are explicitly optimised to detect between-dimension interactions. Instead, design optimization may benefit from placing greater emphasis on the inclusion of more severe health states, while expanding the total number of health states or observations may not be necessary. Designs that under-represent such states—whether due to efficiency-driven selection or limited health-state coverage—may therefore provide insufficient empirical support for estimating boundary-related effects, even when interaction terms are included in the model [10].
From this perspective, the advantage of summary interaction terms lies precisely in their parsimony [14]. They allow boundary-related non-additivity to be accommodated within conventional modelling frameworks and efficiency-oriented designs, without requiring substantial expansion of the health-state set or a dramatic increase in sample size. This helps explain why these terms have been consistently beneficial in saturated datasets, while their empirical contribution may be difficult to detect in smaller or more restricted designs.
Our findings are consistent with previous research exploring the impact of interaction terms in EQ-5D valuation. Studies such as Jonker et al. [8] and Nicolet et al. [7] also reported that incorporating interactions improved model fit, particularly for severe health states. The present study extends this work by demonstrating that the significance of interaction effects is robust across different datasets and valuation settings. Furthermore, the observed pattern of diminishing marginal utility aligns with prior EQ-5D valuation studies, which have shown that additional losses in utility tend to shrink at higher severity levels. Although the consistent positivity of interaction terms is compatible with diminishing marginal utility, alternative explanations related to VAS scale properties cannot be fully excluded.
However, our results differ from some studies that have found only marginal gains from including interaction terms. This discrepancy may be due to differences in study design, particularly in health state sampling. Whereas earlier studies often relied on small or non-saturated datasets, our analysis utilized full-coverage datasets, allowing for a more comprehensive assessment of interaction effects. These findings suggest that the detectability of interactions is influenced by the breadth of health state sampling, reinforcing the importance of well-designed valuation studies that capture a wide range of health state interactions.
Despite its strengths, this study has several limitations. First, the datasets analyzed were derived from VAS valuation studies. While VAS allows full health state coverage and provides a useful platform for exploring interaction structures, it differs conceptually from TTO and DCE approaches, particularly in how respondents evaluate severe health states. As a result, the extent to which the observed interaction patterns generalize to TTO- or DCE-based valuation remains uncertain. Second, while this study explored interaction effects within existing datasets, it did not directly test alternative experimental designs that might be better optimized for capturing interactions. Future research should examine whether targeted oversampling of high-severity health states or adaptive experimental designs could further improve the estimation of interaction effects. Third, this study focused on two-way (second-order) interactions but did not investigate higher-order interactions (e.g., three-way or four-way interactions). While higher-order interactions may theoretically contribute to valuation models, their complexity increases the risk of overfitting. Future studies should explore whether higher-order interactions are empirically supported and whether their inclusion substantively improves model performance. Last but not least, the datasets used in this study were not designed to be nationally representative, and differences in sampling strategies and respondent characteristics across studies may limit the generalizability of the findings. However, the primary aim of this study is methodological rather than population-based, and the results should be interpreted in this context.
Conclusion
This study shows that interaction effects can improve model fit and predictive accuracy in EQ-5D valuation, particularly for health states with multiple severe problems. However, the gains were modest in absolute terms, suggesting that the main-effects model remains a robust baseline specification for standard valuation studies. Parsimonious summary terms such as N3 and D1 for EQ-5D-3 L and N45 and D1 for EQ-5D-5 L appear to offer a practical way to capture key non-additive effects.
These findings are mainly relevant in a methodological context and may help inform future model specification and valuation study design. However, because this study was based on secondary analysis of saturated EQ VAS datasets, the results should be interpreted cautiously. Further research is needed to examine whether similar interaction structures are observed in TTO-based valuation studies.
Supplementary Information
Below is the link to the electronic supplementary material.
Acknowledgements
We thank EuroQol Research Foundation for funding this study (Grant number: 92-RA).
Author contributions
Z.Y. conceived and designed the study, coordinated data acquisition, and drafted the initial manuscript. Z.Y. and J.M.R.-G. conducted the statistical analyses and contributed to the interpretation of the results. Z.Y., P.K., and J.B. collected the data. J.M.R.-G. provided methodological expertise on model specification and interaction modelling and critically revised the manuscript for methodological rigor. P.K. contributed to the conceptual framing of the study and interpretation of findings. J.B. contributed to study design and interpretation of results. All authors critically revised the manuscript, approved the final version, and agree to be accountable for all aspects of the work.
Funding
This work was supported by grants 92-RA from the EuroQol Research Foundation.
Data availability
No datasets were generated or analysed during the current study.
Declarations
Ethical approval
Not applicable.
Competing interests
Dr. Yang reports grants from The EuroQol Research Foundation, during the conduct of the study; grants from The EuroQol Research Foundation, outside the submitted work. Dr. Ramos-Goñi reports grants from The EuroQol Research Foundation, during the conduct of the study; grants from The EuroQol Research Foundation, outside the submitted work. Dr. Kind reports grants from The EuroQol Research Foundation, during the conduct of the study; grants from The EuroQol Research Foundation, outside the submitted work. Dr. Busschbach reports grants from The EuroQol Research Foundation, during the conduct of the study; grants from The EuroQol Research Foundation, outside the submitted work. All authors are members of the EuroQol Group.
Footnotes
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This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Data Availability Statement
No datasets were generated or analysed during the current study.