A Framework for the Estimation of Quality-Adjusted Life-Years Using Joint Models of Longitudinal and Survival Data

Abstract

Background:

Length and quality of life are frequently combined in health technology assessments to derive a single, generic measure of health improvement due to a treatment or medicine. One such measure is given by quality-adjusted life-years (QALYs), typically calculated assuming discrete health states and quality-of-life values. The accurate estimation of QALYs is crucial for informed decision making in health care policy and resource allocation; however, traditional methods often rely on assumptions that are sometimes biologically and statistically inappropriate. This study aims to develop a framework for estimating QALYs in populations where survival data are collected that addresses these limitations.

Methods:

A framework for estimating QALYs in continuous time is introduced, based on joint longitudinal-survival models fitted using maximum likelihood; the proposed framework requires patient-level data, including longitudinal health utility values and overall survival. In contrast to the conventional approach, which involves dichotomising health states and separate models, this method allows the estimation of QALYs from a single model while accounting for all the statistical and biological intricacies of the data, providing a more appropriate estimate for cost-effectiveness modeling. The joint modeling approach is validated using Monte Carlo simulation under realistic data-generating mechanisms.

Results:

Simulations showed that the joint longitudinal-survival modeling approach could recover the true QALYs without bias. Conversely, a comparison method based on calculating QALYs directly from the health utility trajectories, ignoring the survival process, was biased under most scenarios.

Conclusions:

A new methodology for estimating QALYs within a unified, flexible and extensible framework has been developed. This approach can improve QALY estimation and their use in cost-effectiveness analyses in practice, enabling more timely and robust health technology assessments. User-friendly Stata software is provided.

Highlights.

• Accurate estimation of quality-adjusted life-years (QALYs) is key for effective health technology assessments. Nonetheless, classical methods rely on modeling assumptions that are sometimes, if not always, biologically and statistically inappropriate.

• A new approach is introduced for the estimation of QALYs, based on multivariate joint models for longitudinal and survival data. The methodology provides a unified framework for modeling all components of QALYs jointly, thus incorporating all sources of heterogeneity and accounting for all the statistical and biological intricacies of the data.

• The new framework has the potential to enable more timely, accurate robust decision making, therefore significantly affecting clinical practice and health policy. Several extensions to this framework are possible, thus providing a platform for future research as well.

• User-friendly Stata software and example statistical code are provided to apply the methodology in practice.

The primary goal of new therapeutic options for cancer patients is to lengthen and improve patient survival, enabling affected individuals to live longer than they would without treatment. ¹ Nonetheless, over the past decades, health-related quality-of-life (HRQL) investigations have become increasingly important not only to quantify improvements in survival but also to understand what these improvements mean for patients. HRQL refers to the subjective assessment of the effect of health status on quality of life, encompassing physical, mental and social well-being. One commonly used measure that combines survival and quality of life is quality-adjusted life-years (QALYs), providing a single metric for evaluating health outcomes.

The significance of HRQL in cancer treatment is evidenced by several studies highlighting the number of trials collecting and reporting 1 or more patient-reported quality-of-life measures, or using such data to support regulatory submissions.^{2 –4} HRQL-related and patient-reported outcomes are increasingly measured throughout cancer trials and at regularly scheduled intervals to ensure appropriate data are captured on patient status. These HRQL scores not only are used to assess various trial endpoints but also play a crucial role in subsequent cost-effectiveness analysis, quantifying how much time patients spend at different levels of severity.

Despite the recent development of international standards, ⁵ statistical challenges persist in modeling and utilising HRQL data appropriately. HRQL data are often treated as a time-varying covariate, changing value only when a new measurement is taken—an assumption that is often (if not always) biologically implausible. Such data inherently suffer from measurement error, with observations that are correlated within patients. Longitudinal modeling of HRQL via mixed-effects models can yield biased estimates when trajectories are informatively truncated (eg, by death). ⁶ Moreover, mixed models implicitly impute values beyond the time of dropout, which may be unrealistic when dropout is due to death (and thus no HRQL exists after that time, in principle). Given the complexity of the interaction between survival and HRQL, there is a strong rationale for their joint analysis when evaluating a new product (or medicine) as ignoring one can bias the estimation of the other. ⁷

Joint models for longitudinal and survival data can effectively integrate postrandomisation data (such as HRQL measures of other longitudinal biomarkers recorded during follow-up of a trial) and survival data within a unified framework. ⁸ In health technology assessment, length of life and quality of life are commonly combined into a single, generic measure of health improvement via the QALY framework. ⁹ This process usually involves 3 steps: 1) describing the health states experienced by patients, 2) converting each health state into a value (or coefficient) and 3) combining the time spent in each health state and the corresponding value to obtain the total value (eg, QALYs). This article focuses on the estimation of QALYs and introduces a new approach based on joint longitudinal-survival modeling. Specifically, the proposed methodology involves jointly modeling quality of life and survival in continuous time, integrating their correlation and all sources of uncertainty within a single, unified framework. Compared with the usual approach based on dichotomising health states and using separate models for the various steps, the proposed approach enables the estimation of quality-adjusted life from a single model while accounting for all the statistical and biological intricacies of the data, providing a more appropriate estimate for use in cost-effectiveness modeling.

The use of joint longitudinal-survival models to simultaneously analyse quality-of-life data and survival data is not new. For instance, Billingham et al ¹⁰ discussed how joint longitudinal-survival models can account for informative dropout in longitudinal HRQL data analysis or simultaneously model longitudinal HRQL data and survival via shared random effects models. The latter topic is further discussed by Billingham and Abrams. ¹¹ Spertus et al ¹² used a joint model in a cardiovascular clinical trial, comparing it with separate longitudinal and time-to-event models. Bruno et al ¹³ discussed the use of joint models to study treatment effects on secondary, alternative endpoints, such as changes in longitudinal biomarkers (eg, HRQL), and how these can complement traditional Response Evaluation Criteria in Solid Tumors (RECIST) endpoints. ¹⁴ Touraine et al ¹⁵ compared linear mixed-effects and joint models for the analysis of HRQL data in cancer clinical trials, showing that biased estimated longitudinal trajectories arise if the survival outcome is ignored. Severe bias in some model parameters was also reported by Henderson et al ¹⁶ when the association between the processes is ignored, further motivating the use of joint models.

Approaches other than joint models have been proposed in the literature to integrate repeated quality-of-life measures and survival, including extensions of the Q-TWiST method, partitioned survival models, multistate models and methods based on time-to-death utilities.^{17 –24} Despite the popularity of these alternative approaches, there is a substantial interest in incorporating survival outcomes and surrogates (such as tumour burden and quality-of-life measures). Joint longitudinal-survival modeling has been recognised in this context as an important technique to leverage all the available information collected throughout clinical trials, addressing some of the limitations of alternative approaches. By providing a more accurate and comprehensive estimate of QALYs, this article aims to fill existing gaps in the literature and improve the assessment of health technologies.

The rest of the article is organised as follows. The next section introduces the joint modeling framework and QALY estimators. The third section (‘Monte Carlo Simulation Study’) reports simulation results, including a misspecification assessment in the fourth section. The fifth section (‘Discussion’) discusses the methodology, explores its limitations and concludes the article with opportunities for future research.

Methodology

Joint Modeling of Longitudinal and Survival Data

The joint longitudinal-survival modeling framework is now introduced in the context of randomised trials with a time-to-event outcome and where longitudinal information (such as quality-of-life data or other biomarkers) is collected throughout the trial.

Longitudinal submodels

Let $n$ be the number of study subjects on which 2 longitudinal outcomes, $Q$ and $Y$ , are measured over time. $Q$ denotes quality of life and $Y$ a certain other biomarker, such as prostate-specific antigen levels. The values of $Q$ and $Y$ at time $t$ for the ith subject, with $i=1,\dots ,n$ , are modelled via the following linear mixed-effects submodels:

$$q_i(t)=X_i^q(t){\beta }^q+Z_i^q(t)b_i^q+{\epsilon }_i^q(t)=m_i^q(t)+{\epsilon }_i^q(t)$$ (1)

$$y_i(t)=X_i^y(t){\beta }^y+Z_i^y(t)b_i^y+{\epsilon }_i^y(t)=m_i^y(t)+{\epsilon }_i^y(t)$$ (2)

where $X^q$ , $X^y$ denote outcome-specific and possibly time-varying covariates with associated fixed effects ${\beta }^q$ , ${\beta }^y$ , respectively, and $Z^q$ , $Z^y$ denote covariates with subject-specific random effects $b^q$ , $b^y$ and residual error terms ${\epsilon }^q$ , ${\epsilon }^y$ . Note that a covariate denoting time is usually included in $X^q$ and $X^y$ and that covariates $X^q$ , $X^y$ and $Z^q$ , $Z^y$ may overlap between the 2 models, but not necessarily; analogously, there could be some overlap between $X^q,Z^q$ and between $X^y$ , $Z^y$ . The error terms are assumed to follow independent normal distributions, ${\epsilon }^q\sim N(0,{\sigma }_{{\epsilon }^q}^2)$ and ${\epsilon }^y\sim N(0,{\sigma }_{{\epsilon }^y}^2)$ . The correlation between the longitudinal outcomes is captured by assuming a multivariate normal distribution for all the random effects, centred at zero and with variance-covariance matrix $\mathrm{\Sigma }$ :

$$b_i=\left[\begin{array}{c}{b}_i^q\\ b_i^y\end{array}\right]\sim N(0,\mathrm{\Sigma })$$

A common choice for the structure of $\mathrm{\Sigma }$ is the unstructured form, which allows full flexibility in estimating pairwise correlations between all random effects at the cost of additional computational complexity; alternative structures (such as diagonal, exchangeable or block-exchangeable) are possible as well. ²⁵

Note that we assume $Q$ and $Y$ to be continuous outcomes, for simplicity; the software implementation currently supports other distributions (such as binary or count) for the biomarker $Y$ ²⁶ but requires a continuous model for $Q$ . The marker submodel $Y$ is, however, not strictly required by the framework and could be ignored if such data are not available. Moreover, we recommend rescaling and modeling HRQL on a transformed scale (eg, after applying a logit transformation) to ensure predicted values are in a plausible range. Alternatively, more complex distributions (eg, Beta) for $Q$ could be accommodated in future work.

Survival submodel

For the time-to-event outcome, let $T_i$ be the observed event time for the ith participant, taken as $T_i=\min \left(\underset{i}{\stackrel{*}{T}},C_i\right)$ , with $T_i^{*}$ representing the true event time and $C_i$ the censoring time. ${\delta }_i=\{0,1\}$ is the event indicator variable, with 1 denoting events and 0 denoting censored observations. The survival submodel for the time-to-event outcome is defined, assuming a proportional hazards parametrisation, as

$$\begin{array}{l}{h}_i(t)=h_0(t)\exp [X_i^h(t)\gamma +g^q({\mathcal{Q}}_i(t),b_i^q,{\alpha }^q)\\ +g^y({\mathcal{Y}}_i(t),b_i^y,{\alpha }^y)]\end{array}$$ (3)

where ${\mathcal{Q}}_i(t)=\{q_i(s),0\le s\le t\}$ and ${\mathcal{Y}}_i(t)=\{y_i(s),0\le s\le t\}$ denote the history of the longitudinal processes for quality of life $Q$ and the biomarker $Y$ , up until time $t$ . The parameters ${\alpha }^q$ , ${\alpha }^y$ quantify the association between a certain function of $Q$ , $Y$ and the hazard of event; more details on the form of $g^q(\cdot )$ , $g^y(\cdot )$ are discussed in the ‘Association Structures’ subsection. $h_0(t)$ denotes the baseline hazard function, which could follow parametric, semiparametric or flexible parametric formulations. ²⁷ $X_i^h(t)$ denotes a set of exogenous (external, ie, not affected by earlier events) and possibly time-varying covariates with corresponding regression coefficients $\gamma$ . For instance, an exogenous time-varying covariate could be a deterministic treatment modality that changes over time, for example, with a second-line treatment administered after a fixed period.

The joint longitudinal-survival model for the ith subject is defined by combining equations (1), (2) and (3), which are jointly estimated within a unified framework.

Estimation

Estimation of the joint model relies on the assumption of full conditional independence, which implies that, given the random effects, 1) repeated measurements in each outcome are independent, 2) longitudinal outcomes are independent of one another and 3) longitudinal outcomes and the time-to-event outcome are independent. Moreover, the methodology relies on the common assumption that the observation processes (driving the times at which each longitudinal outcome was recorded) are noninformative: given the trial context with scheduled observations, this assumption seems appropriate. Finally, estimation of the parameters of the joint model is performed via either a frequentist or Bayesian framework and is a computationally intensive task given 1) the (potentially) high dimensional random effects and 2) that there is no closed form for the likelihood of the joint model. More details on the estimation of joint longitudinal-survival models are available elsewhere and omitted for conciseness^8,28; this work relies on the frequentist implementation from the user-written merlin command in Stata.^26,29

Association structures

The longitudinal and time-to-event submodels are linked in equation (3) via general functions $g^q(\cdot )$ and $g^y(\cdot )$ . The most common form of $g(\cdot )$ yields the so-called expected value association structure, for example, for the quality-of-life submodel, $g^q({\mathcal{Q}}_i(t),b_i^q,{\alpha }^q)={\alpha }^q{m}_i^q(t)$ . The association parameter ${\alpha }^q$ can be interpreted as the log hazard ratio for a unit change in $m_i^q(t)=X_i^q(t){\beta }^q+Z_i^q(t)b_i^q$ , that is, the expected value of the quality-of-life measure $Q$ from equation (1).

Several alternatives are possible: the expected rate of change association structure, time-independent association structures that link the submodels by sharing the random effects and association structures that denote cumulative effects, lagged effects or combinations (such as expected value and rate of change, together), among others. These are discussed and described in more detail elsewhere.^28,30 The choice of association structure may be data driven (eg, by assessing fit of the joint model with the observed data) or motivated by the clinical question, the literature or computational reasons.

QALYs

An overall measure that combines quality of life and survival probability is that of QALYs. ³¹ Assuming a clearly defined sequence of distinct health states, this can be defined as

$$QALY=\underset{i=1}{\overset{p}{\mathrm{\Sigma }}}{Q}_i^{*}{S}_i^{*},$$ (4)

where $Q_1^{*},\dots ,Q_p^{*}$ is the quality of life assigned to the $p$ distinct health states and $S_1^{*},\dots ,S_p^{*}$ is the time spent in each state. This method is often used to provide information on how much of the survival benefit of a certain treatment is valuable, for example, taking into account quality of life, symptoms of the disease, drug toxicity or side effects.^32,33

For this work, we define QALYs in continuous time as

$$QALY={\int }_0^{\infty }Q(u)S(u)du$$ (5)

where $Q(u)$ denotes the quality of life at time $u$ with corresponding survival probability $S(u)$ . Equation (4) is a discrete-time approximation of equation (5), where the former relies on the assumptions that quality-of-life values are constant between measurements (states) and that they are free of measurement error. By directly modeling quality of life and survival via the joint modeling approach, a complete, continuous time model for quality of life and survival at arbitrary time points is defined, which we can then use to directly estimate QALYs in continuous time.

In practice, the integral in equation (5) does not have a closed form and is approximated numerically, as discussed in the ‘Estimation’ subsection.

Restricted QALYs

The QALY measure in continuous time introduced in the ‘QALYs’ subsection relies on extrapolation of both the quality-of-life measure $Q$ and the survival curve $S$ to a lifetime horizon. To avoid such extrapolation, a second measure named ‘restricted QALYs’ can be defined:

$$QALY(t)={\int }_0^{t}Q(u)S(u)du$$ (6)

Unlike equation (5), this requires no extrapolation assumptions as long as $t$ is within the range of observed data; the choice of $t$ is left to the analyst and/or key stakeholders. ³⁴

Marginal and conditional measures

QALYs and restricted QALYs, as introduced in the ‘QALYs’ and ‘Restricted QALYs’ subsections, are defined as marginal measures that refer to the entire study population; this is now extended to conditional measures (eg, for a specific treatment modality). The focus of this section is on restricted QALY measures, but this generalises to lifetime settings as well.

Let $X$ be a binary treatment indicator variable denoting, for example, the experimental treatment of interest (X = 1) or placebo (X = 0). A measure of conditional restricted QALYs can be defined as follows:

$$QALY(t|X=x)={\int }_0^{t}Q(u|X=x)S(u|X=x)du$$ (7)

for exposure level $X=x$ . A schematic representation of this measure is included in Figure 1, for illustration purposes.

Figure 1.

Schematic representation of the conditional restricted quality-adjusted life-year (QALY) measure, equation (7).

If $X$ consists of every covariate included in the joint model, fully conditional measures are then obtained. Conversely, if $X$ denotes only a certain covariate (or a combination thereof) to condition on while adjusting for other factors in the joint model (such as sex or a certain prognostic factor, eg, to increase precision), it is necessary to marginalise (standardise) over these additional covariates for calculating (7).

Contrasts of QALY measures

Contrasts involving QALY measures can be readily defined using the potential outcomes, counterfactual framework. ³⁵

For instance, one may want to compare the difference in restricted QALYs had every study participant been exposed compared with the counterfactual scenario in which every participant was assigned to the control arm; this can be defined as $QALY(t|X=1)-QALY(t|X=0)$ , where $QALY(t|X=1)$ and $QALY(t|X=0)$ follow from Equation (7). This can be interpreted as the marginal, population-level difference in QALYs between the experimental and control (reference) arm, restricted at a certain time $t$ . Notably, this quantity could be computed for different values of $t$ , allowing for the comparison and examination of both short- and long-term differences.

Estimation

The estimation process for QALY measures starts by fitting the joint model introduced in the ‘Joint Modeling of Longitudinal and Survival Data’ subsection with the survival outcome, the longitudinal quality-of-life measures and (possibly) any other longitudinal biomarker.

Estimators for the QALY measures introduced in the previous sections are now introduced, starting with the marginal unrestricted QALYs of equation (5):

$$\begin{array}{r}\hfill \begin{array}{l}\widehat{QALY}={\int }_0^{\infty }\widehat{Q}(u)\widehat{S}(u)du\hfill \\ ={\int }_0^{\infty }\left\{\frac{1}{n}\sum _{i=1}^n\left[{\int }_{-\infty }^{+\infty }\widehat{Q}(u|X_i^q(u),b)\widehat{S}(u|X_i^h(u),m_i^q(u),m_i^y(u),b)\varphi (b|\widehat{\mathrm{\Sigma }})db\right]\right\}du\hfill \end{array}\end{array}$$ (8)

Note that $\widehat{Q}(\cdot )$ and $\widehat{S}(\cdot )$ depend on the estimated joint model parameters as well, omitted to simplify the notation; the summation is over the number of study subjects $n$ . The inner integral in (8) is used to obtain marginal (with respect to the random effects) predictions for survival and quality of life; this does not have a closed-form solution and is approximated using multivariate Gauss–Hermite quadrature. ³⁶ Then, regression standardisation is used to marginalise (standardise) over the observed distribution of every possibly time-varying model covariate ( $X^q,X^h,m^q,m^y$ ), across all $n$ study subjects. This approach is analogous to Stata’s margins command or the marginaleffects R package.^29,37 Note that, if time-varying, exogenous covariates were present, they need to either be measured at all time points or be predictable to estimate QALYs via equation (8); in practice, time-fixed covariates and time-varying endogenous (internal, ie, measured on the patient) covariates are much more common, so this is unlikely to be a significant issue in practice. A comparison of exogenous and endogenous covariates is available elsewhere. ³⁸ Moreover, regression standardisation relies on the positivity assumption, which implies that there must be both exposed and unexposed subjects for every possible combination of values of all model covariates. ³⁹ This is unlikely to be an issue when analysing data originating from clinical trials. Finally, the outer integral is used to approximate the QALY measure to a lifetime horizon (ie, from time zero and onward): this does not have a closed-form solution either and is approximated using Gauss–Laguerre quadrature. ⁴⁰ More details on these approximations are included in the supplementary material available online.

For the restricted QALY measures of equation (6), the estimator of equation (8) is updated by simply modifying the bounds of the outer integral:

$$\begin{array}{r}\hfill \begin{array}{l}\widehat{QALY}(t)={\int }_0^t\widehat{Q}(u)\widehat{S}(u)du\hfill \\ ={\int }_0^t\left\{\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}\left[{\int }_{-\infty }^{+\infty }\widehat{Q}(u|X_i^q(u),b)\widehat{S}(u|X_i^h(u),m_i^q(u),m_i^y(u),b)\varphi (b|\widehat{\mathrm{\Sigma }})db\right]\right\}du\hfill \end{array}\end{array}$$ (9)

As in equation (8), the outer integral does not have a closed form and needs to be approximated numerically, eg, using Gauss–Legendre quadrature ⁴¹ ; more details are included in the supplementary material as well.

For the conditional (eg, on treatment modality $X=x$ ) restricted QALY measures of equation (7), the model covariates are partitioned as $X^q(u)=\{X,X^{*q}(u)\}$ and $X^h(u)=\{X,X^{*h}(u)\}$ , where $X$ denotes the covariate of interest (or combinations thereof) and $X^{*q}(u)$ , $X^{*h}(u)$ denote every other covariate. An estimator for the conditional restricted QALYs is then obtained by replacing $\widehat{Q}(u),\widehat{S}(u)$ in equation (9) for their conditional analogues $\widehat{Q}(u|X=x),\widehat{S}(u|X=x)$ , and therefore, $\widehat{Q}(u|X_i^q(u),b),\widehat{S}(u|X_i^h(u),m_i^q(u),m_i^y(u),b)$ for $\widehat{Q}(u|X_i=x,X_i^{*q}(u),b),\widehat{S}(u|X_i=x,X_i^{*h}(u),m_i^q(u),m_i^y(u),b)$ .

For contrasts of QALY measures (as in the ‘Contrasts of QALY Measures’ subsection), estimators previously introduced in this section can be combined, as needed. Finally, the variance of estimated QALY measures (or contrasts thereof) can be obtained using the numerical delta method, as in our implementation. ⁴² A nonparametric bootstrap procedure is theoretically possible, but the computational burden is likely prohibitive in practice. ⁴²

These estimators are implemented in jmqaly, a user-written postestimation command for merlin in Stata that is openly available on GitHub. An R implementation is also in development.

Monte Carlo Simulation Study

The performance and validity of the methodology introduced in the ‘Methodology’ section is studied using Monte Carlo simulation. In the interest of space, we report only a summary of the simulation study protocol (‘Short Simulation Protocol’ subsection), with results in the ‘Results’ subsection. A comprehensive protocol document, reported according to the ADEMP framework, ⁴³ is included in the supplementary material available online.

Short Simulation Protocol

The simulation study consisted of 24 distinct data-generating mechanisms (DGMs), varying study sample size, measurement pattern for longitudinal utilities, shape of the utility trajectories and treatment effect on the longitudinal trajectories, in a factorial fashion. Briefly, each DGM is based on a joint model that mimics the settings of a clinical trial with participants randomised 1:1 to a binary treatment (or placebo) at baseline. The joint model includes a longitudinal submodel for the health utility values and a time-to-event submodel for overall survival. The longitudinal submodel for HRQL assumes linear or nonlinear trajectories (depending on the DGM), a random intercept and slope and a possible treatment effect on the longitudinal trajectories. The survival submodel assumes a treatment effect on overall survival, a Weibull baseline hazard function and links with the longitudinal submodel via the shared random effects association structure. Finally, participants are followed up for a maximum of 10 y. Each DGM is described in more detail in the full protocol, included in the online supplementary material.

The estimands for this simulation study are restricted QALYs at time $t=2,5,10$  y, by treatment arm and difference thereof. These are denoted by $Q^1(t)$ for the treatment arm, $Q^0(t)$ for the control arm and $Q^{1-0}(t)=Q^1(t)-Q^0(t)$ for the QALYs difference at time $t$ , respectively.

These estimands are estimated using 3 methods. First, a method that estimates QALYs as the area under the HRQL trajectory, denoted as the AUC method. This effectively ignores the survival process, assuming that quality of life is minimised at the time of death and beyond. Second, a method that uses the QALYs estimators introduced in the ‘Estimation’ subsection and that is based on a multivariate joint model assuming linear trajectories for both the marker and the HRQL trajectories with time-by-treatment statistical interactions, a random intercept in the marker submodel and a random intercept and slope in the HRQL submodel. This is denoted as JM^Lin and is fully well-specified under all simulation scenarios assuming linear HRQL trajectories. Third, a joint model with nonlinear HRQL trajectories, modelled using restricted cubic splines with 3 degrees of freedom. This is denoted as JM^NonLin and is expected to accommodate both linear and nonlinear trajectories (eg, according to every DGM of this study).

Key performance measures are bias, relative bias and coverage probability, including Monte Carlo standard errors to quantify uncertainty in their estimation. ⁴³ Computational time and convergence rates are reported as well.

Finally, the number of repetitions was determined by running a preliminary batch of 10 repetitions per DGM. Based on those repetitions, we estimated that 250 additional repetitions are sufficient to control Monte Carlo error, as described in the full simulation protocol, for a total of $10+250=260$ repetitions.

Results

Median survival times, across all simulated datasets, were between 6 and 7 y; by the end of follow-up, approximately 70% to 80% of simulated study participants died. The median number of measurements per participants was 7 and 26 for scenarios with HRQL measurements every 12 and 3 mo, respectively. Detailed values are included in Supplementary Appendix B.1.

Convergence rates were satisfactory: 100% for the AUC method, greater than 94% for JM^Lin and greater than 91% for JM^NonLin (Supplementary Appendix B.2). The median computational time, across simulation scenarios, was 0.3 min for the AUC method, 2.5 min for JM^Lin and 3.2 min for JM^NonLin (Supplementary Appendix B.3).

The results for bias are depicted in Figures 2, 3 and 4 and tabulated in Supplementary Appendix B.4. JM^NonLin performed best overall, despite small bias under certain scenarios at prediction time $t=2$ (up to 0.023 QALYs, in absolute terms); nonetheless, under the same scenarios, JM^NonLin still outperformed both the AUC and JM^Lin methods. Otherwise, JM^NonLin consistently yielded unbiased results across estimands, simulation scenarios and prediction times. JM^Lin performed as well as JM^NonLin when correctly specified (ie, under DGMs with linear HRQL trajectories); when misspecified, JM^Lin yielded biased results (up to 0.174 QALYs for arm-specific estimates and 0.063 QALYs for the difference between arms), performing worse than the AUC method for $t=2$ and better for $t=5,10$ . The AUC method performed worst overall, with positive bias for arm-specific estimates under all scenarios (up to 0.256 QALYs) and negative bias for the difference (up to −0.107 QALYs). Despite biased results, the AUC and JM^Lin methods performed better at estimating the QALYs difference as compared with estimating arm-specific QALYs; the AUC method, specifically, performed as well as the joint models under all scenarios with no treatment effect on the longitudinal HRQL trajectories. Relative bias patterns were similar to absolute bias (Supplementary Appendix B.5). Sample size and observation patterns did not affect bias significantly.

Figure 2.

Estimated bias (with 95% confidence intervals based on Monte Carlo standard errors) for the $Q^1(t)$ estimand (restricted quality-adjusted life-years in the active arm), across models, prediction times and simulation scenarios.

Figure 3.

Estimated bias (with 95% confidence intervals based on Monte Carlo standard errors) for the $Q^0(t)$ estimand (restricted quality-adjusted life-years in the control arm), across models, prediction times and simulation scenarios.

Figure 4.

Estimated bias (with 95% confidence intervals based on Monte Carlo standard errors) for the $Q^{1-0}(t)$ estimand (difference in restricted quality-adjusted life-years between active and control arm), across models, prediction times and simulation scenarios.

Coverage probabilities followed the same patterns as bias and relative bias (Supplementary Appendix B.6), with JM^NonLin performing best and JM^Lin second best. The AUC method yielded suboptimal coverage probabilities under most scenarios; nonetheless, performance for the difference estimator was close to that of joint models here as well.

In conclusion, the simulation study highlighted that well-specified joint models yield unbiased estimates when longitudinal HRQL is modelled flexibly enough. Some small sample bias may be present, especially at early prediction times, but the more flexible approach still outperformed simpler approaches, highlighting the importance of effective model-building strategies when using the QALY estimators introduced in this article. For instance, JM^NonLin modelled HRQL trajectories using restricted cubic splines with 3 degrees of freedom; alternative knots locations or nonlinear formulation should be assessed in practice, as a sensitivity analysis. Later prediction times and QALYs differences were estimated more robustly under all DGMs of this simulation study.

Assessment of Model Misspecification

The results of the previous section highlight that the proposed estimator has satisfactory finite-sample performance when a joint model is either well-specified or flexible enough to capture nonlinear patterns. However, as with any model-based estimator (including other methods routinely used to estimate QALYs), misspecification of the joint model can lead to biased estimates (as in the case of the JM^Lin method). We further expand on the potential impact of model misspecification in this section by 1) reviewing the literature on model misspecification in joint models for longitudinal and survival data, with a focus on the implications for QALYs estimation, and 2) providing additional simulation results for misspecified estimators.

In the context of QALYs estimation, bias in any of the regression coefficients would result in biased estimates of the QALY measures as well, as these rely on predictions from the joint model, although the magnitude of this bias may be reduced due to the cumulative nature of the QALY estimand. Song et al ⁴⁴ suggested remarkable robustness to the assumption of normal random effects for the estimation of most model parameters. These results were noted by Hsieh et al ⁴⁵ as well, citing that the posterior density of the random effects can be well approximated by a normal density when a reasonably large number of longitudinal measurements per study subject is available. Similar results were obtained by Rizopoulos et al, ⁴⁶ further noting that estimators for parameters of the longitudinal and survival submodels are rather robust to random effects misspecification. Crowther et al ²⁷ showed that parameter estimates were insensitive to the specification of the baseline hazard, as long as a sufficiently flexible distribution is assumed (eg, flexible parametric with a sufficient number of degrees of freedom). Crowther et al ⁴⁷ specifically tackled the issue of model misspecification, showing that misspecification of the longitudinal trajectory minimally affected the survival parameters when using the expected value association structure. Conversely, bias was observed in the association parameters when assuming a linear trajectory (in place of a true nonlinear trajectory) under the rate of change association structure. Overall, the authors noted that the current value association structure was quite robust to model misspecification, with the rate of change association structure being more sensitive to oversimplified model specifications.

Next, we derive some additional simulation results to further study model misspecification in our context. Specifically, we extend simulation scenario 11 (400 participants, HRQL measurements every 3 mo, nonlinear HRQL trajectories, significant treatment effect on HRQL trajectories) by defining 2 additional scenarios in which all joint models compared in the simulation study of the ‘Monte Carlo Simulation Study’ section are misspecified: a scenario in which the nonlinear HRQL trajectory is simulated from a Beta distribution, and a scenario in which the submodels are linked via the expected value association structure. The models compared in these settings are the same: the AUC method, JM^Lin and JM^NonLin. Both scenarios are described in more detail in the simulation study protocol, available online.

The results of this additional simulation study are included in Appendix C (supplementary material available online). Under both scenarios, previous results from the literature were confirmed, with biased QALY estimates when relying on a misspecified joint model estimator. Bias was larger in the scenario assuming the expected value association structure as compared with the scenario assuming Beta-distributed HRQL trajectories, and the magnitude of this bias varied, with smaller bias at early prediction times and larger bias at later prediction times. Moreover, the more flexibly specified joint model (JM^NonLin) performed generally better than the less flexible joint model (JM^Lin) did, and the AUC method performed slightly better than the joint model under the expected value association structure for the difference estimand, likely due to not assuming proportional hazards for the assumed time-dependent treatment effect.

In conclusion, these findings reinforce the need for careful model specification and the use of modern fit diagnostics (eg, posterior predictive checks as suggested by Rizopoulos et al ⁴⁸ ).

Discussion

With this article, a new method for estimating QALYs based on multivariate joint models for longitudinal and survival data is introduced. Compared to previous approaches, this method directly and jointly models survival and HRQL trajectories over time, with great flexibility, allowing for the inclusion of any number of other covariates, either time fixed or time varying, and for a unified quantification of uncertainty and heterogeneity across all processes. Intermittent noninformative missing data and informative dropout due to death are handled naturally within the joint modeling framework.^8,49

The availability of user-friendly, easy-to-use statistical software (in Stata) makes it accessible to practitioners and health care policy decision makers alike. Therefore, example data and annotated code to apply the methodology in practice are provided, under a permissive licence, on GitHub. The simulation study reported in the ‘Monte Carlo Simulation Study’ section highlighted that the method implementation can estimate QALYs without bias, significantly outperforming a simpler method that disregards the survival process completely. This framework has substantial potential to improve the estimation of QALYs in practice, thereby enhancing cost-effectiveness modeling efforts, as it allows for a comprehensive accounting of the statistical and biological intricacies of data within a single model.

While the joint modeling approach offers significant advantages, it also has limitations. First, given the inherent flexibility of the framework, analysts must rely on subject matter knowledge and modeling assumptions when defining each submodel and their association: this is a nontrivial task, which may result in biased estimates when some of the assumptions are violated or when fit to the observed data is poor, as highlighted by the results of the simulation study and the assessment of model misspecification in the ‘Assessment of Model Misspecification’ section. Effective model-building strategies, including graphical model diagnostics and validation techniques, ⁴⁸ are therefore essential to ensure adequate model fit and accurate estimates; this limitation is shared with every other modeling approach for the estimation of QALYs. Second, the estimation algorithm is complex and can be time-consuming with larger datasets. Methodological and software improvements, currently under development, and the increased availability of computational resources are likely to alleviate this issue in the future. Third, QALY prediction relies on nested numerical integration without closed-form solutions; users should assess accuracy by testing an increasing number of quadrature points. Fourth, lifetime QALY calculation requires extrapolation of survival and HRQL trajectories beyond the observed data, whose accuracy must be assessed case by case (eg, using prior literature or longer follow-up data). Finally, the simulation study covers a limited number of scenarios and comparison methods; alternative parameters and methods could be explored, where estimand comparability allows, and further methodological studies are warranted. ⁵⁰

The methodological framework can be further extended. For instance, extrapolation issues can be mitigated by allowing the inclusion of modeling constraints and alternative distributions to ensure biologically plausible extrapolations (eg, to prevent impossible HRQL values). A second natural extension is allowing the inclusion of discounting and costs within the joint modeling framework (eg, similar to the work of Achana et al ⁵¹ ); this would further bridge the gap between QALY estimation and trial-based economic evaluations. The addition of discounting (with rate $r$ ) is particularly straightforward, as it simply requires including the discounting factor $e^{-ru}$ within the integrands of equations (5) and (6). Then, further methodology to allow for the indirect comparison of marginal QALY measures could be developed, for example, with a weighting methodology similar to that of matching-adjusted indirect comparisons. ⁵² This would enable the indirect comparison of treatments and medicines in terms of QALYs as well, allowing for more effective and comprehensive health technology assessments. If sufficient data on disease progression were available, the methodology could be extended to accommodate dropout (and informative truncation of HRQL trajectories) due to other causes by fitting a competing risks joint model and updating the estimands in terms of cumulative incidence functions.^53,54 The methodology can also be reframed within a Bayesian framework, allowing the incorporation of prior evidence, more complex model specifications, and, in principle, external treatment effects at the prediction step. Finally, recent deep learning approaches to joint longitudinal-survival modeling relaxed parametric assumptions to capture complex nonlinear patterns.^{55 –57} However, for health technology assessment, the parametric framework remains advantageous because it is interpretable, supports established uncertainty quantification, is computationally efficient for typical trial sizes, and aligns with current regulatory guidance.

In conclusion, a unified and extensible framework for the estimation of QALYs has been developed, accompanied by user-friendly Stata software for practical applications. Despite some limitations, this methodology has the potential to significantly affect clinical practice, health policy and future research. The framework relies on the timely and accurate collection of quality-of-life data in clinical trials, beyond the usual intercurrent events of disease progression, treatment discontinuation and so on. Therefore, we advocate for the collection of such data to inform timely and robust health technology assessments, thus enabling more effective decisions when bringing new medicines to patients.