COMPARING THE ACCURACY OF THE CONSTANT PROPORTIONAL RISK POSTURE VERSUS DELTA MODEL: MODELING THE STANDARD-GAMBLE - TIME TRADE-OFF RELATIONSHIP

Purpose: Standard gamble utilities as a function of time trade-off utilities represent a utility (risk attitude) curve. This study aims to evaluate the difference between Howard’s Delta Model (HDM) and Constant Proportional Risk Posture (CPRP) as methods to model the relationship between the Standard Gamble (SG) and Time Trade-Off (TTO) utility estimation methods.

Method: Utility data were obtained from 1986 individuals attending clinics or public events, for a study of health state valuation for themselves or others. Utilities were obtained by SG and TTO for four health states (diabetes, severe bilateral vision loss, severe seizure disorder, and severe mental impairment), perfect health and death; for the individual’s self, a child and/or an elderly adult. HDM suggests an exponential relationship between the two utility metrics:

SG =		1−e−γ⋅TTO
		1−e−γ

CPRP suggests a power-law relationship between the two utility metrics:

SG = TTO^γ

To select an algorithm for the fit of both of the models (HDM and CPRP), a global fit was performed for all subjects with a nonlinear weighted least-squares estimator, using the Gauss-Newton and Golub-Pereyra algorithms.  The two models’ predicted standard gamble distributions were compared using an ensemble of statistical tests.

Result: The plots of the fit using both algorithms can be found in the model fit section of the appendix below. The ensemble of statistical tests demonstrate that the predicted standard gambles from the two model fits do not differ with a p-value <0.05, given the null hypothesis that the cumulative distributions, means and location shifts are equivalent.

Conclusion: The Gauss-Newton algorithm more accurately fits the necessary anchor point hypothesis of 0,0 and 1,1. In addition, it exhibits a higher convergence tolerance for the fit of the standard gamble utility estimates.

We can further conclude that the HDM and CPRP models are statistically similar to each other. This is further suggested by the correlation between the two models’ residual sum of squares distributions being from a range of 0.82 to 0.96, and the covariance between the two error distributions of an order of magnitude 10⁻². Theoretical and application considerations may be more important than the empirical fit to the data one model over another to assess risk posture.

Appendix:

Model Fit