Mapping Function

Mapping Function for Predicting Health State Utility Values

A mapping function is the equation an analyst takes from a published mapping study and applies inside a cost-utility model, so that a trial recording only a clinical or disease-specific score can still yield health state utility values and QALYs. Given a patient's score and covariates such as age, it returns the expected utility on the target measure, usually the EQ-5D, for people with those characteristics. Because the output is an expected value and not an observed patient value, the way the function is applied matters as much as how it was estimated. Feeding in an arm's mean score can shift the ICER, and ignoring correlated coefficients misstates decision uncertainty, even when the function is sound.

What a mapping function predicts

The NICE Decision Support Unit's Technical Support Document 22 (TSD 22) describes the mapping model as an estimate of the conditional distribution of health utility given a vector of conditioning variables that includes the clinical outcome. For most decision models the quantity used is the mean of that distribution.

$$\hat{u}(x) = \hat{E}[\,u \mid x\,] = g(x; \hat{\beta})$$

where $u$ is the utility a patient would report on the target preference-based measure, $x$ is the vector of inputs (the clinical or disease-specific score plus covariates), $\hat{E}[\,u \mid x\,]$ is the estimated expected utility for patients with inputs $x$, $g$ is the functional form chosen in the mapping study, $\hat{\beta}$ is the vector of estimated coefficients and $\hat{u}(x)$ is the predicted utility.

TSD 22 points out that predictions always vary less than observed values, which does not matter for a cost-utility analysis of a whole population because the analysis averages over that variation.

Parts of a published mapping function

A mapping function is usable only if it is fully reported. The ISPOR Good Practices for Outcomes Research Task Force, reported by Wailoo and colleagues in 2017, asks for an algebraic description of the model, an example predicted value for a stated set of covariates, coefficients to enough decimal places, the variance-covariance matrix and details of the error terms.

Inputs from the source measure and covariates

The inputs are scores from a patient-reported outcome measure or a clinical measure, plus covariates. The task force notes that item or dimension scores usually explain utility better than a summary score, but decision models synthesising several trials typically rely on published results that report only the summary score, so an item-level function may be unusable there.

TSD 22 recommends that age feature as a conditioning variable in mappings that use adult data, and the task force says age should be retained even if not statistically significant.

Direct and response mapping outputs

A direct mapping function predicts the utility index itself, and TSD 22 notes that the relationship it estimates is relevant only to the value set in which the utilities were scored. A response mapping function predicts the probability of each response level on each dimension of the target instrument, and a value set is applied afterwards.

$$\hat{u}(x) = \sum_{h} \Pr(h \mid x)\, V(h)$$

where $h$ runs over the health states of the target instrument's descriptive system, $\Pr(h \mid x)$ is the predicted probability that a patient with inputs $x$ reports state $h$, $V(h)$ is the utility of state $h$ in the chosen value set, and $\hat{u}(x)$ is the expected utility. The task force notes that this separation lets an analyst apply any tariff, but that the model's fit is specific to the tariff in which it was tested.

Applying the function in cohort and patient-level models

TSD 22 states that a cohort model should apply the conditional mean for each health state. In a Markov cohort model with states defined by bands of a disease score, the state utility is the mean of the function's predictions for the patients in that band, which matches the function evaluated at one representative score only when the band is narrow or the function is close to linear.

In an individual patient simulation or a trial-based evaluation, TSD 22 notes that utility may instead be imputed for each patient by simulation from the reported error distribution. The task force warns that imputing the same conditional mean for identical patients ignores unexplained variability and misrepresents the clinical study and the mapping results.

When the function is non-linear, the mean of individual predictions differs from the prediction at the mean input. TSD 22 makes this point for mapping between the EQ-5D-5L and EQ-5D-3L, stating that with individual patient data the correct procedure maps each patient and averages the predictions, and that a mean value needs a distance-weighted average of nearby values controlled by a bandwidth. The same logic applies to any non-linear function.

Feasible range and extrapolation

TSD 22 gives the feasible range of the EQ-5D-3L with the UK value set as minus 0.594 to 1, and notes that model types reflecting the distribution of EQ-5D data will not predict outside it by design. Where a model type can, TSD 22 asks that the frequency of out-of-range predictions or sampled values, and how they were handled, be reported.

Extrapolation beyond the severity seen in the estimation sample is best avoided, and using a function from one condition in another, such as a breast cancer EORTC mapping for another tumour, needs a judgement that the relationship generalises.

Coefficient uncertainty and probabilistic analysis

TSD 22 treats a mapping function like any other regression input to a cost-effectiveness model, so its coefficients are sampled jointly in probabilistic sensitivity analysis using the reported variance-covariance matrix. Coefficients from one regression are usually correlated, so independent draws give the wrong spread, as the variance of a linear index shows.

$$\operatorname{Var}(\hat{\eta}) = \operatorname{Var}(\hat{\beta}_0) + S^2 \operatorname{Var}(\hat{\beta}_1) + 2S \operatorname{Cov}(\hat{\beta}_0, \hat{\beta}_1)$$

where $\hat{\eta} = \hat{\beta}_0 + \hat{\beta}_1 S$ is the predicted linear index at score $S$, $\hat{\beta}_0$ is the estimated intercept, $\hat{\beta}_1$ is the estimated coefficient on the score, $\operatorname{Var}$ denotes a sampling variance and $\operatorname{Cov}(\hat{\beta}_0, \hat{\beta}_1)$ is their sampling covariance. Joint draws are usually made with a Cholesky decomposition of the matrix.

Illustrative worked example with a bounded mapping function

All numbers in this example are illustrative and do not come from any published mapping. A function predicts EQ-5D utility from a disease score $S$ on a scale of 0 (no symptoms) up to 50 (worst), with a form that keeps predictions below 1. Age is left out to keep the arithmetic short, although TSD 22 recommends it for adult mappings.

$$\hat{u}(S) = 1 - \exp(\hat{\beta}_0 + \hat{\beta}_1 S)$$

where $\hat{u}(S)$ is the predicted utility at score $S$, $\hat{\beta}_0 = -3.0$ is the intercept and $\hat{\beta}_1 = 0.04$ is the coefficient on the score, so the exponential term is the predicted loss from full health.

Step 1 (predictions for three scores). At $S = 10$ the index is $-3.0 + 0.04 \times 10 = -2.6$ and $\exp(-2.6) = 0.0743$, so $\hat{u} = 0.9257$. At $S = 30$ the index is $-1.8$ and $\exp(-1.8) = 0.1653$, so $\hat{u} = 0.8347$. At $S = 50$ the index is $-1.0$ and $\exp(-1.0) = 0.3679$, so $\hat{u} = 0.6321$.

Step 2 (mean of predictions against prediction at the mean). A comparator arm has three equally sized groups with scores 10, 30 and 50, so the mean score is 30.

$$\bar{u} = \frac{0.9257 + 0.8347 + 0.6321}{3} = \frac{2.3925}{3} = 0.7975$$

where $\bar{u}$ is the mean of the patient-level predictions. Evaluating the function at the mean score gives 0.8347, which is 0.8347 minus 0.7975, or 0.0372, higher.

Step 3 (effect on a one-year QALY gain). In the treatment arm every patient has a score of 20, with an index of $-2.2$, $\exp(-2.2) = 0.1108$ and $\hat{u} = 0.8892$. Over one year the gain is 0.8892 minus 0.7975, or 0.0917 QALYs, when the comparator is mapped patient by patient, and 0.8892 minus 0.8347, or 0.0545 QALYs, when it is mapped at its mean score. With an illustrative incremental cost of GBP 2,000 the ICER with patient-level mapping is $ICER_{P} = 2{,}000 / 0.0917 = 21{,}810$, or GBP 21,810 per QALY, against $ICER_{M} = 2{,}000 / 0.0545 = 36{,}697$, or GBP 36,697 per QALY, with mapping at the mean. Mapping at the mean understates the gain by about 41% because it hides the low utility of the severe group.

Step 4 (uncertainty in the prediction at a score of 30). Suppose the study reports $\operatorname{Var}(\hat{\beta}_0) = 0.0100$, $\operatorname{Var}(\hat{\beta}_1) = 0.000016$ and $\operatorname{Cov}(\hat{\beta}_0, \hat{\beta}_1) = -0.00036$, a correlation of minus 0.9.

$$\operatorname{Var}(\hat{\eta}) = 0.0100 + 900 \times 0.000016 - 60 \times 0.00036 = 0.0100 + 0.0144 - 0.0216 = 0.0028$$

where 900 is the square of the score of 30 and 60 is twice the score. The standard error is $\sqrt{0.0028} = 0.0529$, so the 95% interval for the index is $-1.8 \pm 1.96 \times 0.0529$, or $-1.9037$ to $-1.6963$. Transforming each limit gives a predicted utility between 0.8166 and 0.8510, a width of 0.0344. Ignoring the covariance gives a variance of 0.0244, a standard error of 0.1562 and a utility interval between 0.7755 and 0.8783, a width of 0.1028.

Step 5 (interpretation). Dropping the covariance makes the predicted utility look about three times as uncertain as it is, which would inflate decision uncertainty in the probabilistic analysis. The intervals describe the conditional mean, and individual patients with a score of 30 would still vary more widely.

Model choices that move the predictions

The form of the function matters most at the extremes of severity. The task force reports evidence that linear regression applied to the EQ-5D-3L, and also Tobit, two-part and censored least absolute deviations models, tend to underestimate utility in mild states and overestimate it in severe states, so that clinically effective therapies appear less cost-effective than they truly are. TSD 22 concludes that linear regression is not appropriate for mapping and reports very good performance from mixture model approaches, such as the finite mixture model, when they are carefully specified.

Input level, covariates, target value set and estimation dataset also change the function. Longworth and Rowen, reviewing mapping at NICE, recommended explicit consideration of whether the mapping function generalises to the target sample, and called mapping usually a second-best solution to collecting the preferred instrument.

Mapping functions in NICE appraisals

The NICE manual for technology appraisal and highly specialised technologies guidance (PMG36), section 4.3.9, allows EQ-5D data to be estimated by mapping when they are not available and treats this as a departure from the reference case. It asks that the mapping function be fully described, its choice justified and its fit to the data shown, with sensitivity analyses exploring variation from using different mapping algorithms.

For technology appraisal topics with an invitation to participate issued after 27 August 2026, NICE's interim methods statement (PMG51) states that mapped utilities should be based on the UK EQ-5D-5L value set, with EQ-5D-3L data mapped using the model developed by the DSU and the Economic Evaluation Policy Research Unit.

Mapping function, mapping study and crosswalk

The mapping study is the research that estimates and tests the relationship, and the mapping function is its usable output. TSD 22 notes that mapping, cross-walking and transfer to utility have all named the process, and PMG36 uses both mapping function and mapping algorithm. A crosswalk is the special case linking versions of one instrument, such as the EQ-5D-3L and EQ-5D-5L. When a trial collected the preference-based measure itself, its value set scores the responses directly, as in indirect utility assessment, and no mapping function is needed.

Common misreadings of mapped utilities

Several errors come from treating predicted conditional means as if they were observed data. Each concerns how the fit of a function is judged.

1. Reading a low R-squared as failure. The task force notes that high between-patient variability in quality of life data gives relatively low R-squared values, and that this measure says little about validity for cost-effectiveness use.
2. Comparing the spread of predictions with the spread of data. TSD 22 calls this misleading, because predictions always vary less than observations. Fit is better judged by comparing mean predicted and mean observed utility across bands of the clinical score.
3. Relying on average error alone. TSD 22 calls the mean absolute error and root mean square error relatively insensitive and able to conceal systematic prediction errors.

Sources

• Longworth L, Rowen D. Mapping to obtain EQ-5D utility values for use in NICE health technology assessments. Value in Health. 2013;16(1):202-210. https://doi.org/10.1016/j.jval.2012.10.010
• Wailoo AJ, Hernandez-Alava M, Manca A, Mejia A, Ray J, Crawford B, Botteman M, Busschbach J. Mapping to estimate health-state utility from non-preference-based outcome measures: an ISPOR Good Practices for Outcomes Research Task Force report. Value in Health. 2017;20(1):18-27. https://doi.org/10.1016/j.jval.2016.11.006
• Wailoo A, Hernández Alava M, Pudney S. NICE DSU Technical Support Document 22: Mapping to estimate health state utilities. Decision Support Unit, ScHARR, University of Sheffield; June 2023, updated September 2026. https://doi.org/10.15131/shef.data.33849340
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. Section 4.3.9. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• National Institute for Health and Care Excellence. Interim methods statement: implementing the EQ-5D-5L value set (PMG51). Published 27 August 2026. Sections 1.10 and 1.11, and process considerations. https://www.nice.org.uk/process/pmg51