APPENDIX I: Detailed steps for assessing the external validity of existing mapping
algorithms
1. Estimate predicted EQ-5D utility values using a mapping algorithm for the main analysis
(pooled) sample. The ‘preferred’ algorithms identified were either estimated using
ordinary least squares (OLS) regression or a multinomial logit model. The functional form
for both estimation procedures are briefly described below.
Example of the functional form of an OLS mapping algorithm:
𝑦𝑖 = 𝛽0 + 𝑥𝑖 𝛽,
𝑖 = 1,2, … , N
where the dependent variable y is the EQ-5D utility value and xi is a 1 x K vector and β is a
K x 1 vector of various QLQ-C30 functioning, symptom and overall health scale or item
scores treated as either continuous or dummy variables.
Example of the functional form of a multinomial logit mapping algorithm:
𝐽
Pr(𝑦𝑖 = 𝑚|𝑥𝑖 ) = 𝑒 (𝑥𝑖 𝛽𝑚) /[1 + ∑ 𝑒 (𝑥𝑖 𝛽𝑗) ],
𝑖 = 1,2, … , N
𝑗=2
where y is the dependent variable representing one of the five EQ-5D domains (e.g.,
mobility domain) with J unordered categories (e.g., a level 1, 2 or 3 response); Pr(y=m|x)
is the probability of observing outcome m given x; and βm a vector of QLQ-C30
functioning, symptom and overall health scale or item scores treated as continuous
variables that can differ for each outcome.
2. Plot the observed EQ-5D utility values (estimated using the country-specific tariff applied
in the original mapping study) and the predicted EQ-5D utility values estimated using the
main analysis (pooled) sample; include a line of best fit (solid red diagonal line in Figures
3A-J) and a line of perfect fit (dash green diagonal line in Figures 3A-J). In addition, the
predicted EQ-5D utility values for the best and worst QLQ-C30 health states were added
to the plot (horizontal red lines labelled ‘Best score’ and ‘Worst score’ in Figures 3A-J).
3. The differences (i.e., errors) between the observed and the predicted EQ-5D utility values
were calculated for the main analysis (pooled) sample. Errors were also reported for three
subsets of the EQ-5D range when both observed and predicted values fell within the stated
range (e.g., EQ-5D < 0.5, 0.5 ≤ EQ-5D < 0.75, 0.75 ≤ EQ-5D ≤ 1). For example,
calculation of errors for the lowest values (EQ-5D<0.5) only included patients that had
observed and predicted values within the stated range. Therefore, patients with observed
values in the lowest range but predicted values in a higher range or vice versa were not
included in the error calculation.
4. The mean absolute error (MAE) and the root mean squared error (RMSE) were estimated
for the main analysis (pooled) sample and for the three subsets of the EQ-5D range. The
RMSE was standardized to a percentage by dividing the RMSE by the maximal tariff EQ5D range to account for the differences in ranges of the applied EQ-5D country-specific
tariffs.
5. Using the main analysis (pooled) sample observed and predicted EQ-5D utility values
along with their respective 95% confidence intervals were estimated for ten patient
samples that differed by tumour type (e.g., breast, prostate, head and neck, colorectal, lung,
oesophagogastric, renal, bone and soft tissue, cancer of the unknown primary, and
cervical). The MAEs for each of the ten patient samples were also estimated.
6. Quality-adjusted life-years (QALYs) were calculated for two disease severity patient
samples. QALYs were estimated at six months and one year for the less severe patients
(N=496 and N=192 respectively) and at three and six months for the more severe patients
(N=93 and N=25 respectively). QALYs were calculated by summing the estimated area
under the curve for each subinterval using the mid-point between each observation
multiplied by the difference in the number of days between each observation. The values
were then standardized by the length of follow up for each patient. The differences (i.e.,
errors) between the observed and the predicted QALYs were calculated and a paired t-test
was used to assess whether the means were different.
7. Using the main analysis (pooled) sample, observed EQ-5D utility values estimated using
the EQ-5D country-specific tariff applied in the original mapping study were regressed on
the predicted EQ-5D utility values using OLS and quantile regression (QR). QR was used
to test whether the association between observed and predicted EQ-5D utility values
varied across the distribution. With this method, for each conditional quantile we observe
the incremental change in observed values that would be expected following a unit change
in predicted values. The quantiles were specified as (0.10; 0.25; 0.50; 0.75; 0.90).
8. Step 7 was repeated, however this time controlling for a number of demographic and
patient characteristics variables within the Cancer 2015 dataset. These covariates included
age at consent, sex (male=1 or female=0), and four additional dummy variables. To
account for the rurality of patients’ home addresses, postcodes were converted to the
Accessibility/Remoteness Index of Australia (ARIA). Five ARIA classes were then
generated based on the following thresholds: ARIA ≤0.2 representing class 1 or a ‘highly
accessible (metro/major city)’; >0.2 ARIA ≤2.4 representing class 2 or ‘accessible
(provincial/inner regional)’; >2.4 ARIA ≤5.92 representing class 3 or ‘moderately
accessible (other provincial/outer regional)’; >5.92 ARIA ≤10.53 representing class 4 or
‘remote’; and >10.53 ARIA ≤15 representing class 5 or ‘very remote’. Note that there
were no observations for ARIA class 5 in the Cancer 2015 dataset. Dummy variables were
then created for ARIA classes 2-4 with ARIA class 1 as the reference group. To account
for the stage of patients’ cancer (i.e., the stage of cancer when first diagnosed), tumour
stream-specific stage definitions were broadly grouped into six stage classifications
(‘Stage 0’ ‘in situ’ cancer, ‘Stage 1’ localized cancer, ‘Stage 2’ regional spread in the
general region it first began including nearest lymph nodes, ‘Stage 3’ regional spread and
more extensive lymph node involvement, ‘Stage 4’ distant spread, and ‘Other’ that could
not be classified into any other stage). Dummy variables were then created for ‘Stages 1-4’
and the ‘Other’ group, with ‘Stage 0’ as the reference group. To account for the primary
site of patients’ tumours dummy variables for breast, head and neck, colorectal, lung, bone
and soft tissue, cervical, cancer of the unknown primary, renal, oesophagogastric, and all
other sites were created with prostate as the reference group. Differences in patients’
Eastern Cooperative Oncology Group (ECOG) performance status was accounted for by
creating a dummy variables for ECOG status 1 “Limited in normal activity”, 2 “Self-care
capable but not work”, 3 “Limited self-care, 50% disabled”, 4 “No self-care, 100%
disabled” and 5 “Death” with ECOG status 0 “Normal activity” as the reference group.
9. Steps 1 through 8 were repeated for all identified ‘preferred’ mapping algorithms.