## Abstract

### Background

### Objectives

### Methods

### Results

### Conclusions

## Keywords

## Introduction

## Methods

### Data

### Preference-Based Measures

Devlin NJ, Shah KK, Feng Y, et al. Valuing health-related quality of life: an EQ-5D-5L value set for England. Health Econ (published online ahead of print August 27, 2017). 〈doi:10.1002/hec.3564〉.

### Asthma Quality of Life Questionnaire

### Statistical Methods

*i*defined at point 1 and the interval $[a,\tau ]$, where $a<\tau <1$ and can be written as follows:

with probabilities:

where ${x}_{i3}$ is a vector of variables influencing the probabilities, ${\gamma}_{k}$ is a vector of coefficients, and

*s*refers to each section of the distribution. For identification, the coefficients corresponding to the continuous part of the distribution are set to 0. The probability density function for the continuous part of the distribution has probability density function $h(\cdot )$ made up of a mixture of C-components each representing a beta distribution, with mean ${\mu}_{{c}_{i}}$ and precision parameter ${\varphi}_{c}$, where $c=1,\dots ,C$, such that:

where $f(\cdot )$ is a beta density with alternative parameterization and $C$ is the number of components included in the analysis. Component membership is determined using a second multinomial logit model, such that:

where ${x}_{i2}$ is a vector of variables influencing the probability of component membership and ${\delta}_{c}$ is a vector of corresponding coefficients. Again, one set of coefficients is set to 0 for identification.

## Results

Mean ± SD | Minimum | Maximum | |
---|---|---|---|

AQLQ-S | 0.7085 ± 0.7766 | 0 | 4 |

EQ-5D-5L | 0.8425 ± 0.1693 | −0.073 | 1 |

HUI3 | 0.7560 ± 0.2408 | −0.1958 | 1 |

Age (y) | 43.03 ± 15.00 | 18 | 89 |

Country | No. of observations (%) | ||

Australia | 141 (16.55) | – | – |

USA | 150 (17.61) | – | – |

UK | 150 (17.61) | – | – |

Canada | 138 (16.20) | – | – |

Norway | 126 (14.79) | – | – |

Germany | 147 (17.25) | – | – |

### Five-Level EuroQol Five-Dimensional Questionnaire

No. of components | Specification | Log likelihood | No. of parameters | RMSE | MAE | ME | AIC | BIC |
---|---|---|---|---|---|---|---|---|

EQ-5D-5L betamix | ||||||||

3 | Probability mass at full health | 527.68 | 33 | 0.1430 | 0.1001 | −0.0003 | −989.36 | −832.69 |

3 | Probability mass at full health and truncation point | 436.00 | 38 | 0.1425 | 0.1003 | 0.0005 | −796.00 | −615.59 |

4 | Probability mass at full health | 538.87 | 44 | 0.1429 | 0.1002 | 0.0003 | −989.75 | −780.86 |

4 | Probability mass at full health and truncation point | 401.81 | 45 | 0.1474 | 0.1018 | −0.0012 | −713.62 | −499.98 |

EQ-5D-5L ALDVMM | ||||||||

3 | Bounded | 322.42 | 28 | 0.1439 | 0.1006 | 0.0003 | −588.84 | −455.90 |

4 | Bounded | 336.64 | 39 | 0.1439 | 0.1004 | 0.00003 | −595.28 | −410.13 |

HUI3 betamix | ||||||||

3 | Probability mass at full health | 708.05 | 33 | 0.2081 | 0.1566 | 0.0018 | −1350.09 | −1193.42 |

3 | Probability mass at full health and truncation point | 207.64 | 38 | 0.2081 | 0.1563 | 0.0024 | −339.28 | −158.88 |

4 | Probability mass at full health | 727.92 | 44 | 0.2076 | 0.1562 | 0.0017 | −1367.80 | −1158.95 |

4 | Probability mass at full health and truncation point | 224.48 | 49 | 0.2067 | 0.1548 | 0.0009 | −350.96 | −118.33 |

HUI3 ALDVMM | ||||||||

3 | Age included | 192.51 | 28 | 0.2076 | 0.1556 | 0.00041 | −329.03 | −196.10 |

4 | Age included | 212.42 | 39 | 0.2071 | 0.1550 | 0.00033 | −346.85 | −161.69 |

3 | No age in probability variables ${x}_{2}$ | 189.87 | 24 | 0.2082 | 0.1563 | 0.00054 | −331.75 | −217.80 |

4 | No age in probability variables ${x}_{2}$ | 201.09 | 33 | 0.2082 | 0.1563 | 0.00026 | −336.18 | −179.51 |

Linear model | ||||||||

-– | OLS-EQ-5D | 424.21 | 5 | 0.1471 | 0.1023 | −6.47 × 10^{−17} | −838.42 | −814.69 |

– | OLS-HUI3 | 112.35 | 5 | 0.2121 | 0.1598 | 3.23 × 10^{−17} | −214.70 | −190.96 |

*Note*. All models outlined here include a truncation at the best possible health state other than full health.

### Health Utilities Index Mark 3

*P*values all in excess of 0.03. We therefore investigated the use of different variables to predict component membership. Results are presented in Table 2 for models that do not include age to predict component probabilities as well as those that do. Although AIC and BIC generally favor the exclusion of age, other measures of error are worse. Figures 4B and 5B show a marked difference between these models and suggest that age should remain an explanatory variable for the probabilities because they considerably improve the fit of the model. The evidence indicates that the observed statistical insignificance associated with age may be related to the limited sample size.

### Comparison with Traditional Models

Specification | RMSE | MAE |
---|---|---|

EQ-5D | ||

OLS | 0.1468 | 0.1027 |

CLAD | 0.1491 | 0.1001 |

GLM | 0.1463 | 0.1025 |

BB | 0.1491 | 0.1051 |

HUI3 | ||

OLS | 0.2130 | 0.1608 |

CLAD | 0.2188 | 0.1545 |

GLM | 0.2120 | 0.1605 |

BB | 0.2154 | 0.1643 |

## Discussion

*any*point in the distribution without some theoretical rather than empirical justification for doing so. The increased number of parameters required by the beta-based mixture model means that for smaller data sets there is a danger that it might be more difficult to identify. For example, we attempted to estimate the EQ-5D-5L using a four-component model with probability masses at full health and the truncation point, but this model would not converge when the AQLQ-S was included in the component membership probabilities. To reduce the number of parameters, we estimated a model without the AQLQ-S in the probabilities and achieved a much worse fit as a result.

### Study Limitations

## Conclusions

## Acknowledgments

## Supplementary material

Supplementary material

Supplementary material

Supplementary material

Supplementary material

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