A Computationally Efficient Method for Probabilistic Parameter Threshold Analysis for Health Economic Evaluations

Abstract

<jats:p> Background. Threshold analysis is used to determine the threshold value of an input parameter at which a health care strategy becomes cost-effective. Typically, it is performed in a deterministic manner, in which inputs are varied one at a time while the remaining inputs are each fixed at their mean value. This approach will result in incorrect threshold values if the cost-effectiveness model is nonlinear or if inputs are correlated. Objective. To propose a probabilistic method for performing threshold analysis, which accounts for the joint uncertainty in all input parameters and makes no assumption about the linearity of the cost-effectiveness model. Methods. Three methods are compared: 1) deterministic threshold analysis (DTA); 2) a 2-level Monte Carlo approach, which is considered the gold standard; and 3) a regression-based method using a generalized additive model (GAM), which identifies threshold values directly from a probabilistic sensitivity analysis sample. Results. We applied the 3 methods to estimate the minimum probability of hospitalization for typhoid fever at which 3 different vaccination strategies become cost-effective in Uganda. The threshold probability of hospitalization at which routine vaccination at 9 months with catchup campaign to 5 years becomes cost-effective is estimated to be 0.060 and 0.061 (95% confidence interval [CI], 0.058–0.064), respectively, for 2-level and GAM. According to DTA, routine vaccination at 9 months with catchup campaign to 5 years would never become cost-effective. The threshold probability at which routine vaccination at 9 months with catchup campaign to 15 years becomes cost-effective is estimated to be 0.092 (DTA), 0.074 (2-level), and 0.072 (95% CI, 0.069–0.075) (GAM). GAM is 430 times faster than the 2-level approach. Conclusions. When the cost-effectiveness model is nonlinear, GAM provides similar threshold values to the 2-level Monte Carlo approach and is computationally more efficient. DTA provides incorrect results and should not be used. </jats:p>

Probabilistic threshold analysis by pairwise stochastic approximation for decision-making under uncertainty

“…Interestingly, as clearly seen from the result for the pair (d 1 , d 2 ) for the variable θ 7 , the stochastic approximation method can find the pairwise threshold successfully even if it is located far from the initial estimate. It seems quite hard to get similar results if we only generate θ 7 randomly from its marginal distribution as done in the existing approaches 1,10 .…”

“…for any sample size M ≥ 1 in (1). Hence, by formulating the probabilistic threshold analysis as a stochastic root-finding problem, we can avoid the difficulty inherent to the nested structure considered in the literature 1,10 . Moreover, in order to improve the stability of the algorithm, we can apply some of variance reduction techniques including Latin hypercube sampling 19 or (randomized) quasi-Monte Carlo sampling 20 , as long as the resulting estimator is unbiased as with the standard one (1).…”

“…This is exactly what the probabilistic parameter threshold analysis aims to do. Following the closely-related literature 1,10 , let us focus on the case where θ θ θ 1 consists only of a single individual variable θ j for some 1 ≤ j ≤ s. Then the probabilistic parameter threshold for θ j , denoted by K j , is simply defined as follows.…”

“…)/2. In order to reduce the necessary computational cost, a regression-based approach has been proposed 10 , which first approximates the inner conditional expectation E θ θ θ − j |θ j [ f d (θ θ θ )] by a regression model (as a function of θ j ) and then applies a single-loop Monte Carlo sampling for θ j to estimate K j . However, these existing approaches rely upon accurate estimation (with large M) or approximation of inner conditional expectations, and both lack a theoretical support on convergence and computational complexity.…”

“…Then we develop a pairwise stochastic approximation approach to search for K j efficiently. The key difference from the existing approaches 1,10 is that our proposed approach only requires an unbiased, but rough estimator of the inner conditional expectations and that the parameter threshold estimate is generated randomly only at the initial step and then updated iteratively. In fact, in our numerical experiments below, we use only one Monte Carlo sample to estimate the inner conditional expectations at each iteration step.…”