## Abstract

### Objectives

### Methods

### Results

### Conclusion

## Keywords

## Introduction

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^{2}

^{3}

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^{4}

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^{5}

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### Data and Methods

^{7}

Mobility | No | Some | Confined to bed |
---|---|---|---|

No | 1651 (1782) | 0 (29) | 0 (1) |

Slight | 103 (119) | 507 (552) | 0 (1) |

Moderate | 10 (16) | 525 (586) | 4 (4) |

Severe | 0 (1) | 355 (386) | 25 (30) |

Unable | 0 (4) | 0 (23) | 105 (112) |

Self-care | No | Some | Unable |

No | 2280 (2468) | 0 (43) | 0 (3) |

Slight | 73 (82) | 382 (408) | 0 (5) |

Moderate | 10 (13) | 288 (313) | 6 (6) |

Severe | 0 (5) | 90 (109) | 30 (35) |

Unable | 0 (0) | 0 (6) | 126 (140) |

Usual activities | No | Some | Unable |

No | 1308 (1382) | 0 (42) | 0 (5) |

Slight | 146 (163) | 601 (661) | 0 (70) |

Moderate | 15 (20) | 608 (656) | 19 (23) |

Severe | 0 (9) | 254 (274) | 122 (134) |

Unable | 0 (0) | 0 (15) | 212 (239) |

Pain/discomfort | No | Moderate | Extreme |

No | 1061 (1126) | 0 (65) | 0 (1) |

Slight | 199 (211) | 787 (850) | 0 (4) |

Moderate | 16 (21) | 761 (837) | 18 (19) |

Severe | 0 (6) | 221 (239) | 149 (159) |

Extreme | 0 (2) | 0 (8) | 73 (82) |

Anxiety/depression | No | Moderate | Extreme |

No | 1275 (1352) | 0 (45) | 0 (1) |

Slight | 207 (219) | 752 (841) | 0 (3) |

Moderate | 26 (30) | 631 (692) | 14 (17) |

Severe | 0 (10) | 147 (164) | 150 (158) |

Extreme | 0 (3) | 0 (6) | 83 (93) |

*k*= 1,…, 5 index the EQ-5D dimension (mobility, usual activities, self-care, pain/discomfort, and anxiety/depression),

*i*= 1,…, 5 index problem levels as measured by the EQ-5D-5L, and

*j =*1,…, 3 index problem levels as measured by the EQ-5D-3L. Then

*i*

^{k}and

*j*

^{k}represent the observed problem levels for the

*k*th dimension of the EQ-5D-5L and EQ-5D-3L, respectively. For the

*k*th dimension, the probability of observing the

*i*th EQ-5D-5L problem level conditional on the

*j*th EQ-5D-3L problem level can be denoted as ${p}_{i,j}^{k}$. The probability of being in a given EQ-5D-5L health state contingent on an EQ-5D-3L health state can be derived as the product of the

*k*dimension specific probabilities. This is represented in Eq. 1.

^{7}

*k*th dimension with the first group having more extreme problems in other dimensions than the second group. If offered the more graded set of response options included in the EQ-5D-5L, members of the first group would be expected to rate their problem level for the

*k*th dimension as being worse than would members of the second group. Similarly, among persons having the same level of problems for a given dimension as measured by the EQ-5D-3L, one might expect older individuals to grade their problem level for that dimension as more extreme than would younger individuals if presented with the EQ-5D-5L.

*i*th row and

*j*th column of Table 1 for dimension

*k*, and all other terms are as previously defined. For instance, ${p}_{\mathrm{2,1}}^{1}$ was estimated by the number of respondents who categorized themselves as having slight mobility problems using the EQ-5D-5L and no mobility problems using the EQ-5D-3L divided by the total number of respondents classified as having no mobility problems using the EQ-5D-3L, that is, [103/(1651 + 103 + 10)] = 0.0584.

*k*and each respondent $r=1,\dots ,\phantom{\rule{0.25em}{0ex}}R$ were modeled as

for

*i*= 2, … , 4 and

*j*= 1, …, 3, where ${\alpha}^{k}-{{x}_{j}}^{\text{\u2032}}{\beta}^{k}$ was a linear function of the regressors, and ${\mu}_{r}$ was a latent factor capturing unobserved heterogeneity due to within-respondent correlation. The latter was assumed to be normally distributed with mean zero and variance ${\sigma}^{2}$. The effects of the latent factor on dimensions were measured by ${\theta}^{k}$ with ${\theta}^{1}$ normalized to 1. When interpreting parameter estimates for this model, the $\stackrel{\u02c6}{\beta}$’s reflect the degree to which the probability of being in a more severe state increases as a function of the regressors, while the $\stackrel{\u02c6}{\alpha}$’s represent thresholds (points on the latent outcome) used to differentiate adjacent levels of the response variable.

### Comparison With the DSU Approach

*EQGcopula*option.

## Results

*k*th dimension, the probability of exhibiting the

*i*th EQ-5D-5L problem level conditional on the

*j*th EQ-5D-3L problem level is reflected by the corresponding

*ij*th cell count divided by the

*jk*th column-sum. Table 2 presents the parameter estimates for ordinal logistic regressions excluding age and gender but including the latent factor. The first 4 estimates are the ancillary cut points (see Eq. 3). The parameter estimates for all other models are presented in Appendix B in Supplemental Materials found at https://doi.org/10.1016/j.jval.2021.03.009.

Parameter | Mobility | Self-care | Usual activities | Pain/Discomfort | Anxiety/Depression | |||||
---|---|---|---|---|---|---|---|---|---|---|

Estimate | SE | Estimate | SE | Estimate | SE | Estimate | SE | Estimate | SE | |

1|2 $\left({\alpha}_{1}\right)$ | 3.933 | 0.188 | 6.091 | 0.347 | 3.291 | 0.172 | 2.243 | 0.101 | 2.027 | 0.094 |

2|3 $\left({\alpha}_{2}\right)$ | 9.472 | 0.437 | 11.673 | 0.579 | 8.822 | 0.405 | 7.442 | 0.288 | 6.428 | 0.227 |

3|4 $\left({\alpha}_{3}\right)$ | 12.135 | 0.493 | 14.608 | 0.658 | 11.940 | 0.495 | 10.216 | 0.313 | 9.001 | 0.253 |

4|5 $\left({\alpha}_{4}\right)$ | 18.118 | 0.779 | 18.481 | 0.844 | 16.474 | 0.658 | 14.473 | 0.425 | 13.131 | 0.395 |

${\beta}_{2}^{mob}$ | 8.019 | 0.400 | 2.221 | 0.276 | 0.906 | 0.154 | 0.771 | 0.118 | –0.225 | 0.124 |

${\beta}_{3}^{mob}$ | 14.996 | 0.793 | 4.076 | 0.477 | 2.901 | 0.457 | 0.953 | 0.315 | 0.052 | 0.315 |

${\beta}_{2}^{sc}$ | 1.499 | 0.147 | 7.957 | 0.433 | 1.421 | 0.152 | 0.818 | 0.118 | 0.240 | 0.123 |

${\beta}_{3}^{sc}$ | 2.165 | 0.370 | 13.817 | 0.748 | 2.513 | 0.402 | 1.170 | 0.297 | 0.188 | 0.297 |

${\beta}_{2}^{ua}$ | 0.904 | 0.171 | 0.996 | 0.301 | 7.361 | 0.369 | 0.791 | 0.118 | 0.541 | 0.117 |

${\beta}_{3}^{ua}$ | 2.063 | 0.255 | 2.607 | 0.359 | 12.812 | 0.568 | 1.056 | 0.195 | 1.351 | 0.192 |

${\beta}_{2}^{pd}$ | 0.472 | 0.159 | –0.063 | 0.231 | 0.250 | 0.143 | 6.239 | 0.274 | 0.031 | 0.107 |

${\beta}_{3}^{pd}$ | 2.235 | 0.250 | 0.089 | 0.291 | 1.096 | 0.238 | 10.930 | 0.394 | 0.019 | 0.185 |

${\beta}_{2}^{ad}$ | –0.019 | 0.123 | 0.480 | 0.163 | 0.678 | 0.121 | 0.091 | 0.094 | 6.046 | 0.220 |

${\beta}_{3}^{ad}$ | –0.287 | 0.224 | 0.773 | 0.270 | 1.767 | 0.220 | 0.547 | 0.167 | 11.517 | 0.379 |

$\theta $ | 1.000 | 0.000 | 1.055 | 0.133 | 1.146 | 0.158 | 0.523 | 0.075 | 0.460 | 0.072 |

${\sigma}^{2}$ | 1.121 | 0.049 |

*k*th dimension, a histogram is presented for the distribution of ${x}^{\prime}{\stackrel{\u02c6}{\beta}}^{k}$ along with the predicted probability of each of the 5 problem levels conditional on ${x}^{\prime}{\stackrel{\u02c6}{\beta}}^{k}$. At very low values of ${x}^{\prime}{\stackrel{\u02c6}{\beta}}^{k}$, respondents were most likely to exhibit slight problems as measured by the EQ-5D-5L. Respondents pivoted to the right (exhibited increasing problem extremity) with increases equal to the values of parameter estimates for regressors. The degree to which this occurred is illustrated at the top of each panel. The points at which the curves cross relate to the estimated ancillary cut points.

^{11}

Shaw JW, Bennett B, Trigg A, DeRosa M, Taylor F, Cocks K. Associations between the EQ-5D-3L and QLU-C10D descriptive systems: use of correlation networks to explore preference differences in solid tumor trials. 25th Annual International Meeting of the International Society for Pharmacoeconomics and Outcomes Research, May 18-20, 2020. Abstract PCN317.

^{,}

Non-parametric | Ordered logistic regression + complementary dimensions | Ordered logistic regression + complementary dimensions + latent factor | |||||
---|---|---|---|---|---|---|---|

Age and gender | Age, age^{2} and gender | Age and gender | Age, age^{2} and gender | ||||

Mean absolute error | |||||||

All | 0.0811 | 0.0706 | 0.0708 | 0.0707 | 0.0756 | 0.0772 | 0.0753 |

COPD/asthma | 0.1010 | 0.0874 | 0.0881 | 0.0882 | 0.0815 | 0.0873 | 0.0876 |

Diabetes | 0.0688 | 0.0554 | 0.0550 | 0.0549 | 0.0500 | 0.0512 | 0.0510 |

Liver disease | 0.0667 | 0.0535 | 0.0523 | 0.0523 | 0.0434 | 0.0469 | 0.0467 |

RA/arthritis | 0.0984 | 0.0838 | 0.0837 | 0.0837 | 0.0781 | 0.0824 | 0.0826 |

CVD | 0.1042 | 0.0981 | 0.0989 | 0.0990 | 0.0912 | 0.0986 | 0.0989 |

Stroke | 0.1206 | 0.1047 | 0.1048 | 0.1048 | 0.0954 | 0.1020 | 0.1021 |

Depression | 0.0848 | 0.0720 | 0.0716 | 0.0712 | 0.0613 | 0.0687 | 0.0684 |

Personality disorders | 0.0718 | 0.0656 | 0.0657 | 0.0658 | 0.0601 | 0.0644 | 0.0644 |

Students | 0.1075 | 0.1040 | 0.1039 | 0.1038 | 0.0804 | 0.1012 | 0.1010 |

Other | 0.0639 | 0.0529 | 0.0547 | 0.0555 | 0.0417 | 0.0521 | 0.0530 |

Root mean squared error | |||||||

All | 0.1101 | 0.1016 | 0.1019 | 0.1018 | 0.1145 | 0.1163 | 0.1151 |

COPD/asthma | 0.1356 | 0.1223 | 0.1236 | 0.1237 | 0.1208 | 0.1259 | 0.1261 |

Diabetes | 0.0934 | 0.0836 | 0.0833 | 0.0833 | 0.0794 | 0.0827 | 0.0825 |

Liver disease | 0.0834 | 0.0761 | 0.0757 | 0.0757 | 0.0672 | 0.0725 | 0.0725 |

RA/arthritis | 0.1296 | 0.1185 | 0.1185 | 0.1186 | 0.1109 | 0.1200 | 0.1204 |

CVD | 0.1479 | 0.1447 | 0.1457 | 0.1460 | 0.1328 | 0.1479 | 0.1483 |

Stroke | 0.1595 | 0.1392 | 0.1388 | 0.1389 | 0.1337 | 0.1390 | 0.1390 |

Depression | 0.1137 | 0.1052 | 0.1051 | 0.1052 | 0.0949 | 0.1056 | 0.1059 |

Personality disorders | 0.0924 | 0.0941 | 0.0938 | 0.0937 | 0.0909 | 0.0954 | 0.0953 |

Students | 0.1550 | 0.1579 | 0.1580 | 0.1579 | 0.1118 | 0.1586 | 0.1582 |

Other | 0.0826 | 0.0773 | 0.0779 | 0.0781 | 0.0648 | 0.0778 | 0.0778 |

AIC | |||||||

All | 21315 | 19950 | 19647 | 19639 | 19459 | 19138 | 19127 |

COPD/asthma | 18836 | 17689 | 17435 | 17428 | 17262 | 16997 | 16987 |

Diabetes | 20114 | 18825 | 18555 | 18549 | 18367 | 18077 | 18068 |

Liver disease | 19848 | 18597 | 18308 | 18300 | 18157 | 17850 | 17839 |

RA/arthritis | 18533 | 17436 | 17171 | 17167 | 17024 | 16745 | 16740 |

CVD | 19505 | 18186 | 17907 | 17902 | 17752 | 17459 | 17452 |

Stroke | 16866 | 15919 | 15648 | 15643 | 15522 | 15232 | 15222 |

Depression | 19853 | 18611 | 18323 | 18316 | 18154 | 17851 | 17841 |

Personality disorders | 19260 | 17810 | 17540 | 17535 | 17299 | 17017 | 17009 |

Students | 19241 | 17881 | 17594 | 17583 | 17457 | 17150 | 17135 |

Other | 19778 | 18501 | 18260 | 18248 | 18017 | 17758 | 17740 |

### Comparison With the DSU Approach

- 1.
*Treat the 3L and 5L responses symmetrically.*The best road to work is not necessarily the best road home. Symmetry may well be an unnecessary restriction that is likely to lead to suboptimal predictions. - 2.
*Avoid the assumption that the 5L response scale is simply a more detailed categorization than the 3L scale of the same underlying concept.*The EQ-5D-5L was developed to provide a more graded, sensitive, and responsive descriptive system than the EQ-5D-3L. Aside from the expansion of the number of problem levels for each dimension, deviations from the EQ-5D-3L in wording and formatting are minimal. Accordingly, the aforementioned assumption would seem to be justified. - 3.
*Allow for the effects of covariates.*The approaches applied in this research accommodated for the effects of various regressors, including problems in other EQ-5D-3L dimensions, age, and gender. Conversely, the DSU approach does not allow for the generation of predictions without conditioning on age and gender. - 4.
*Capture the strong association between 3L and 5L responses within each health domain, without necessarily assuming that the strength of the association is the same in all parts of the health distribution.*The nonparametric approach described in this article does not assume that the strength of the association between EQ-5D-3L and EQ-5D-5L responses is constant across the health continuum. All of the ordinal logistic regression models captured the association between domain-specific problems as measured by the EQ-5D-3L and EQ-5D-5L via the inclusion of dummy variables. However, the models did assume proportionality, which means that for any split of response variable categories (eg, no problems vs slight or more, extreme problems vs severe or less), the parameter estimates would remain unchanged. - 5.
*Be sufficiently flexible to fit the diverse response patterns... so we generalize the usual assumption of normally distributed**errors by allowing for a 2-part normal mixture distribution.*Appendix 1 in the DSU report shows that 86.5% of cases fell into a category with a mean of 0.151 and a variance of 0.373, whereas 13.5% of cases belonged to a category with a mean of -0.976 and a variance of 3.947. Respondents with a high likelihood of belonging to this latter group affected the central estimate less than respondents belonging to the first group. This may be a concern for the respondents who were excluded from estimation in the current study due to inconsistent data because these individuals likely resembled those in the second group. Aside from this, each dimension may have its own mixture allowing for a respondent’s data to contribute differentially to different dimensions. This reads like the econometric version of pairwise deletion, which the DSU researchers once labeled as a dangerous practice. - 6.
*Allow dependence across the five domains of EQ-5D ... incorporating a random latent factor influencing responses in all domains.*Ordinal logistic regression can be adapted to allow for dependence in the problem levels for different dimensions, as has been done in this research.

*EQGcopula*option. In their analyses, the DSU researchers excluded eight 15-year-old respondents and two 14-year-old respondents. When comparing the observed and predicted values from the DSU approach with those of the ordinal logistic regression excluding age and gender, the former yielded a mean error of 0.0006, MAE of 0.0804, and RMSE of 0.1168, while the latter yielded a mean error of 0.001, MAE of 0.0706, and RMSE of 0.1016.

^{8}

Mean absolute error | Root mean squared error | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Ordered logistic regression | Ordered logistic regression | |||||||||

No age and gender | Age, age^{2} and gender | No age and gender | Age, age^{2} and gender | |||||||

Males | -Latent | +Latent | -Latent | +Latent | Copula | -Latent | +Latent | -Latent | +Latent | Copula |

<26 | 0.0518 | 0.0491 | 0.0526 | 0.0495 | 0.0528 | 0.0749 | 0.0752 | 0.0742 | 0.0742 | 0.0745 |

26-35 | 0.1026 | 0.1014 | 0.1029 | 0.1015 | 0.1091 | 0.1812 | 0.1835 | 0.1826 | 0.1855 | 0.1872 |

36-45 | 0.0654 | 0.0613 | 0.0640 | 0.0594 | 0.0669 | 0.1083 | 0.1068 | 0.1078 | 0.1070 | 0.1092 |

46-55 | 0.0803 | 0.0768 | 0.0786 | 0.0748 | 0.0821 | 0.1210 | 0.1211 | 0.1207 | 0.1215 | 0.1248 |

56-65 | 0.0820 | 0.0786 | 0.0807 | 0.0768 | 0.0833 | 0.1130 | 0.1125 | 0.1128 | 0.1129 | 0.1144 |

65-75 | 0.0733 | 0.0714 | 0.0729 | 0.0706 | 0.0745 | 0.1049 | 0.1056 | 0.1049 | 0.1060 | 0.1051 |

>75 | 0.0991 | 0.0979 | 0.0994 | 0.0980 | 0.1031 | 0.1360 | 0.1370 | 0.1359 | 0.1367 | 0.1381 |

Females | ||||||||||

<26 | 0.0596 | 0.0573 | 0.0613 | 0.0587 | 0.0632 | 0.0861 | 0.0864 | 0.0858 | 0.0859 | 0.0881 |

26-35 | 0.0711 | 0.0678 | 0.0711 | 0.0676 | 0.0754 | 0.1040 | 0.1037 | 0.1039 | 0.1041 | 0.1130 |

36-45 | 0.0638 | 0.0596 | 0.0627 | 0.0578 | 0.0638 | 0.0897 | 0.0881 | 0.0886 | 0.0871 | 0.0889 |

46-55 | 0.0882 | 0.0863 | 0.0878 | 0.0859 | 0.0897 | 0.1238 | 0.1253 | 0.1243 | 0.1270 | 0.1269 |

56-65 | 0.0834 | 0.0818 | 0.0830 | 0.0813 | 0.0837 | 0.1193 | 0.1202 | 0.1195 | 0.1210 | 0.1205 |

65-75 | 0.0886 | 0.0868 | 0.0892 | 0.0871 | 0.0891 | 0.1173 | 0.1182 | 0.1180 | 0.1189 | 0.1184 |

>75 | 0.1025 | 0.1008 | 0.1045 | 0.1024 | 0.1078 | 0.1404 | 0.1407 | 0.1411 | 0.1410 | 0.1435 |

## Discussion

## Conclusions

## Article and Author Information

**Author Contributions:**

*Concept and design*: van Hout

*Analysis and interpretation of data*: van Hout, Shaw

*Drafting of the manuscript*: van Hout, Shaw

*Critical revision of the paper for important intellectual content*: Shaw

*Statistical analysis*: van Hout

*Obtaining*

*funding*: van Hout

*Administrative, technical, or logistic*

*support:*van Hout

*Supervision:*van Hout

**Conflict of Interest Disclosures:**Dr Shaw is an employee and stockholder of Bristol-Myers Squibb. No other disclosures were reported.

**Funding/Support:**The work of Dr van Hout was supported by a grant from the EuroQol foundation. Dr Shaw did not receive any financial support for this research.

**Role of the**

**Funder/Sponsor:**The funder designed and organized the data collection which was used in an earlier study. It did not have any role in the management, analysis, and interpretation of the data; the preparation, review, or approval of the manuscript; and the decision to submit the manuscript for publication.

## Acknowledgment

## Supplemental Material

- Appendix A

- Appendix B

## References

- EuroQol—a new facility for the measurement of health-related quality of life.
*Health Policy.*1990; 16: 199-208 - Development and preliminary testing of the new five-level version of EQ-5D (EQ-5D-5L).
*Quality of Life Research.*2011; 20: 1727-1736 - A systematic review of studies comparing the measurement properties of the three-level and five-level versions of the EQ-5D.
*Pharmacoeconomics.*2018; 36: 645-661 - Estimation of minimally important differences in EQ-5D utility and VAS scores in cancer.
*Health and Quality of Life Outcomes.*2007; 5 - Comparing the mapping between EQ-5D-5L, EQ-5D-3L and the EORTC-QLQ-C30 in non-small cell lung cancer patients.
*Health and Quality of Life Outcomes.*2016; 14: 60 - A comparison of the EQ-5D-3L and EQ-5D-5L.
*Pharmacoeconomics.*2020; 38: 575-591 - Interim scoring for the EQ-5D-5L: mapping the EQ-5D-5L to EQ-5D-3L value sets.
*Value in Health.*2012; 15: 708-715 - Methods for mapping between the EQ-5D-5L and the 3L for technology appraisal. Report by the Decision Support Unit. Decision Support Unit, ScHARR, University of Sheffield, Sheffield, UK2017
- Valuing health-related quality of life: An EQ-5D-5L value set for England.
*Health Economics.*2018; 27: 7-22 - eq5dmap: a command for mapping from 3-level to 5-level EQ-5D.
*The Stata Journal.*2018; 18: 395-415 Shaw JW, Bennett B, Trigg A, DeRosa M, Taylor F, Cocks K. Associations between the EQ-5D-3L and QLU-C10D descriptive systems: use of correlation networks to explore preference differences in solid tumor trials. 25th Annual International Meeting of the International Society for Pharmacoeconomics and Outcomes Research, May 18-20, 2020. Abstract PCN317.

Shaw JW, Bennett B, Trigg A, et al. A comparison of generic and condition-specific preference-based measures using data from nivolumab trials: EQ-5D-3L, a mapping to the EQ-5D-5L, and QLU-C10D. Manuscript submitted for publication.

- Econometric modelling of multiple self-reports of health states: Theswitch from EQ-5D-3L to EQ-5D-5L in evaluating drug therapies forrheumatoid arthritis.
*Journal of Health Economics.*2017; 55: 139-152

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