## Introduction

## Methods and materials

### Data

### Patients

### Statistical analysis

_{5}score (the average of the five KOOS subscales scores, ranged from 0 to 100 in our sample) and the KOOS

_{4}score (the average of the KOOS subscales scores excluding the ADL subscale, as previously used in ACL injured populations [18], ranged from 1.25 to 100 in our sample). We also used the KOOS individual items but it caused convergence problem in beta-mixture model and we decided to not include them in our final analysis to ensure the models were comparable. For each of these alternatives, we applied a series of model specifications based on main terms, and main terms plus squared and square root terms (likelihood ratio test was used for exclusion of squared and square root terms). The models estimated for linear regression and response mapping are presented in Supplementary Tables 1 and 2. For beta-mixture model, we estimated different specifications with different numbers of components (starting with a one-component model equivalent to a beta regression model), with and without inclusion of the gap between full health and the next feasible value (UK EQ-5D-3L = 0.883), and with and without probability masses at full health and truncation point of the EQ-5D-3L distribution. An example of models estimated for a single specification is presented in Supplementary Table 3.

### Assessment of model performance

_{5}and the KOOS

_{4}scores) and each class of models (linear, response mapping, beta-mixture), we selected one model with the smallest BIC, and the lowest ME, MAE and RMSE in the whole sample and across the distribution of disease severity measured by the KOOS

_{5}/KOOS

_{4}scores as preferred model (Supplementary Tables 4–12). Then, we selected one model as the optimal model for each class of models. In our decision to select the preferred models, we gave higher priority to the models with smallest BIC, while in selecting the optimal models higher priority was given to models with better predictive ability (in our study optimal models had both lower BIC and better predictive ability compared to other models).

## Results

Pre-operation | Post-operation | Total | |
---|---|---|---|

Number of patients | 12,582 | 16,983 | 21,854 |

Number of observations | 12,582 | 27,877 | 40,459 |

Mean (SD) age at operation, years | 30.0 (9.4) | 29.5 (10.3) | 29.1 (10.0) |

EQ-5D mobility | |||

No problems (%) | 66.8 | 88.5 | 81.7 |

Some problems (%) | 32.8 | 11.5 | 18.1 |

Extreme problems (%) | 0.4 | 0.0 | 0.2 |

EQ-5D self-care | |||

No problems (%) | 97.0 | 99.0 | 98.4 |

Some problems (%) | 2.4 | 0.7 | 1.2 |

Extreme problems (%) | 0.6 | 0.3 | 0.4 |

EQ-5D usual activities | |||

No problems (%) | 54.1 | 82.5 | 73.6 |

Some problems (%) | 35.9 | 16.6 | 22.6 |

Extreme problems (%) | 10.0 | 0.9 | 3.8 |

EQ-5D pain | |||

No problems (%) | 15.5 | 39.2 | 31.9 |

Some problems (%) | 79.1 | 58.1 | 64.6 |

Extreme problems (%) | 5.4 | 2.7 | 3.5 |

EQ-5D anxiety/depression | |||

No problems (%) | 50.6 | 71.8 | 65.2 |

Some problems (%) | 43.9 | 25.8 | 31.4 |

Extreme problems (%) | 5.5 | 2.4 | 3.4 |

Proportion in full health (EQ-5D-3L = 1), % | 9.0 | 35.5 | 27.3 |

Proportion reporting EQ-5D-3L < 0 | 1.8 | 0.9 | 1.2 |

Mean (SD) EQ-5D-3L index | 0.66 (0.24) | 0.81 (0.20) | 0.77 (0.22) |

Mean (SD) KOOS _{5} | 58.9 (16.5) | 75.7 (17.8) | 70.5 (19.1) |

Mean (SD) KOOS _{4} | 53.1 (16.9) | 71.9 (19.5) | 66.1 (20.6) |

Mean (SD) KOOS-pain | 73.4 (17.8) | 84.6 (15.9) | 81.1 (17.3) |

Mean (SD) KOOS-symptoms | 68.7 (18.3) | 77.9 (18.1) | 75.1 (18.7) |

Mean (SD) KOOS-activity of daily living | 82.1 (17.6) | 91.1 (13.5) | 88.3 (15.5) |

Mean (SD) KOOS-sports/recreation | 38.3 (26.7) | 64.5 (27.7) | 56.3 (30.0) |

Mean (SD) KOOS-quality of life | 32.0 (17.3) | 60.6 (24.0) | 51.7 (25.8) |

_{5}and KOOS

_{4}for three classes of models (linear regression, response mapping and beta-mixture) are reported in Supplementary Tables 4–12. In all three classes of models, the optimal models were those based on individual KOOS subscales.

_{5}scores, the response mapping and mixture model outperformed linear regression in terms of all summary measures and importantly this was more profound (the highest proportional improvements in the MAE and RMSE) at the extremes of the distribution of disease severity. Compared with the response mapping model, the beta-mixture model estimated closer mean to the observed mean in overall and across the range of the KOOS

_{5}score except those < 25 (most severe). For all models the magnitude of MAE and RMSE rose with the severity of the disease. The results were generally similar when we measured disease severity by the KOOS

_{4}scores (Table 3).

_{5}scores)

Sample and summary statistics | Linear regression ^{a} | Response mapping ^{b} | Beta-mixture model ^{c} |
---|---|---|---|

Full sample (n = 40,459) | |||

ME | 1.65 × 10 ^{−17} | 0.0026 | − 0.0003 |

MAE | 0.1037 | 0.0996 | 0.0988 |

RMSE | 0.1505 | 0.1489 | 0.1490 |

KOOS _{5} 0 to < 25 (n = 543) | |||

ME | − 0.0359 | − 0.0005 | − 0.0145 |

MAE | 0.2305 | 0.2140 | 0.2177 |

RMSE | 0.2776 | 0.2671 | 0.2691 |

KOOS _{5} 25 to < 50 (n = 6069) | |||

ME | 0.0067 | 0.0078 | − 0.0001 |

MAE | 0.1948 | 0.1897 | 0.1877 |

RMSE | 0.2358 | 0.2340 | 0.2340 |

KOOS _{5} 50 to < 70 (n = 11,370) | |||

ME | 0.0030 | 0.0007 | 0.0004 |

MAE | 0.0951 | 0.0923 | 0.0922 |

RMSE | 0.1504 | 0.1493 | 0.1495 |

KOOS _{5} 70 to < 85 (n = 11,535) | |||

ME | − 0.0107 | 0.0012 | 0.0007 |

MAE | 0.0874 | 0.0831 | 0.0830 |

RMSE | 0.1242 | 0.1229 | 0.1229 |

KOOS _{5} 85 to ≤ 100 (n = 10,942) | |||

ME | 0.0062 | 0.0032 | − 0.0015 |

MAE | 0.0729 | 0.0693 | 0.0671 |

RMSE | 0.0962 | 0.0947 | 0.0946 |

_{4}scores)

Sample and summary statistics | Linear regression ^{a} | Response mapping ^{b} | Beta-mixture model ^{c} |
---|---|---|---|

KOOS _{4} 0 to < 25 (n = 1136) | |||

ME | − 0.0246 | 0.0121 | − 0.0017 |

MAE | 0.2472 | 0.2364 | 0.2390 |

RMSE | 0.2821 | 0.2760 | 0.2768 |

KOOS _{4} 25 to < 50 (n = 8528) | |||

ME | 0.0088 | 0.0035 | − 0.0017 |

MAE | 0.1633 | 0.1582 | 0.1565 |

RMSE | 0.2114 | 0.2098 | 0.2098 |

KOOS _{4} 50 to < 70 (n = 11,892) | |||

ME | − 0.0025 | 0.0011 | 0.0012 |

MAE | 0.0826 | 0.0806 | 0.0806 |

RMSE | 0.1363 | 0.1354 | 0.1355 |

KOOS _{4} 70 to < 85 (n = 10,167) | |||

ME | − 0.0084 | 0.0011 | − 0.0002 |

MAE | 0.0957 | 0.0912 | 0.0910 |

RMSE | 0.1231 | 0.1218 | 0.1218 |

KOOS _{4} 85 to ≤ 100 (n = 8736) | |||

ME | 0.0078 | 0.0040 | − 0.0009 |

MAE | 0.0647 | 0.0607 | 0.0580 |

RMSE | 0.0889 | 0.0870 | 0.0870 |

_{5}score was associated with 0.785 change in EQ-5D-3L values in the observed data. The corresponding value was 0.743, 0.782 and 0.772 for linear, response mapping and beta-mixture model, respectively, indicating a difference of 0.038 between models.

Observed | Linear regression ^{a} | Response mapping ^{b} | Beta-mixture model ^{c} | |
---|---|---|---|---|

Mean | 0.766 | 0.767 | 0.781 | 0.767 |

Variance | 0.049 | 0.069 | 0.055 | 0.050 |

Skewness | − 1.633 | − 0.268 | − 2.289 | − 1.644 |

Kurtosis | 6.478 | 3.289 | 10.459 | 6.571 |

Minimum | − 0.594 | − 1.112 | − 0.594 | − 0.594 |

Maximum | 1.000 | 2.002 | 1.000 | 1.000 |

EQ-5D-3L = 1, % | 27.26 | 0.0 | 31.81 | 27.26 |

EQ-5D-EL > 1, % | 0.0 | 18.59 | 0.0 | 0.0 |

EQ-5D-EL < 0, % | 1.16 | 0.47 | 2.52 | 1.01 |

Percentiles | ||||

1% | − 0.016 | 0.097 | − 0.239 | − 0.002 |

5% | 0.228 | 0.320 | 0.293 | 0.223 |

10% | 0.620 | 0.429 | 0.620 | 0.592 |

25% | 0.725 | 0.599 | 0.689 | 0.711 |

50% | 0.796 | 0.777 | 0.796 | 0.777 |

75% | 1.000 | 0.946 | 1.000 | 1.000 |

90% | 1.000 | 1.093 | 1.000 | 1.000 |

95% | 1.000 | 1.179 | 1.000 | 1.000 |

99% | 1.000 | 1.337 | 1.000 | 1.000 |

## Discussion

_{5}and KOOS

_{4}). However, individual scores are not always available to map from and models using average scores are needed. It should be noted that previous studies suggested that the KOOS-ADL subscale might have poor content validity for young adults with ACL injury [20], however its inclusion in our study improved the predictive ability of our mapping models.