The Theoretical Relationship between Sample Size and Expected Predictive Precision for EQ-5D Valuation Studies: A Mathematical Exploration and Simulation Study

Background. Scoring algorithms of multi-attribute utility instruments (MAUI) are developed in valuation studies and are hence estimated subject to uncertainty. Valuation studies need to be designed to achieve reasonable accuracy. We aim to provide the first closed-form mathematical formula for the mean square error (MSE) of an additive MAUI as a function of sample size that acknowledges that the MAUI model for the mean utility is not a perfect fit. Methods. Based on the design of the EQ-5D valuation study, we derived our closed-form formula in terms of sample size and number of directly valued health states overall and per subject. We validated our formula by conducting a simulation study using the US EQ-5D-3L valuation data set and examined the effect of using a random-effects versus an ordinary least-squares model and the effect of heteroscedasticity. We explored the effect of sample size and number of valued health states. Results. The simulation study validated our MSE-based closed-form formula regardless of whether assuming a random-effects model versus an ordinary least squares model or heteroscedasticity versus homoscedasticity. As the sample size approaches infinity, the MSE does not approach zero but levels off asymptotically. The improvement based on increasing sample is more prominent when the sample is small. When the sample size is greater than 300 to 500, further increases do not meaningfully improve the MSE, while increasing the number of health states can further improve the MSE. Conclusion. We have derived a closed-form formula to calculate the MSE of an additive MAUI scoring algorithm based on sample size and number of health states, which will enable the developers of MAUI valuation studies to calculate the required sample size for their desired predictive precision.