Simplified Methods for Modelling Dependent Parameters in Health Economic Evaluations: A Tutorial.

BACKGROUND: In health economic evaluations, model parameters are often dependent on other model parameters. Although methods exist to simulate multivariate normal (MVN) distribution data and estimate transition probabilities in Markov models while considering competing risks, they are technically challenging for health economic modellers to implement. This tutorial introduces easily implementable applications for handling dependent parameters in modelling. METHODS: Analytical proofs and proposed simplified methods for handling dependent parameters in typical health economic modelling scenarios are provided, and implementation of these methods are illustrated in seven examples along with the SAS and R code. RESULTS: Methods to quantify the covariance and correlation coefficients of correlated variables based on published summary statistics and generation of MVN distribution data are demonstrated using examples of physician visits data and cost component data. The use of univariate normal distribution data instead of MVN distribution data to capture population heterogeneity is illustrated based on the results from multiple regression models with linear predictors, and two examples are provided (linear fixed-effects model and Cox proportional hazards model). A conditional probability method is introduced to handle two or more state transitions in a single Markov model cycle and applied in examples of one- and two-way state transitions. CONCLUSIONS: This tutorial proposes an extension of routinely used methods along with several examples. These simplified methods may be easily applied by health economic modellers with varied statistical backgrounds.

Logarithmic confidence estimation of a ratio of binomial proportions for dependent populations.

This article investigates the logarithmic interval estimation of a ratio of two binomial proportions in dependent samples. Previous studies suggest that the confidence intervals of the difference between two correlated proportions and their ratio typically do not possess closed-form solutions. Moreover, the computation process is complex and often based on a maximum likelihood estimator, which is a biased estimator of the ratio. We look at the data from two dependent samples and explore the general problem of estimating the ratio of two proportions. Each sample is obtained in the framework of direct binomial sampling. Our goal is to demonstrate that the normal approximation for the estimation of the ratio is reliable for the construction of a confidence interval. The main characteristics of confidence estimators will be investigated by a Monte Carlo simulation. We also provide recommendations for applying the asymptotic logarithmic interval. The estimations of the coverage probability, average width, standard deviation of interval width, and index H are presented as the criteria of our judgment. The simulation studies indicate that the proposed interval performs well based on the aforementioned criteria. Finally, the confidence intervals are illustrated with three real data examples.

Evaluating the correlations of cost and utility parameters from summary statistics for probabilistic analysis in economic evaluations.

OBJECTIVES: The correlations between economic modeling input parameters directly impact the variance and may impact the expected values of model outputs. However, correlation coefficients are not often reported in the literature. We aim to understand the correlations between model inputs for probabilistic analysis from summary statistics. METHODS: We provide proof that for correlated random variables X and Y (e.g. inpatient visits and outpatient visits), the Pearson correlation coefficients of sample means and samples are equal to each other (corrX,Y=corrX-,Y-). Therefore, when studies report summary statistics of correlated parameters, we can quantify the correlation coefficient between parameters. RESULTS: We use examples to illustrate how to estimate the correlation coefficient between the incidence rates of non-severe and severe hypoglycemia events, and the common coefficient of five cost components for patients with diabetic foot ulcers. We further introduce three types of correlations for utilities and provide two examples to estimate the correlations for utilities based on published data. We also evaluate how correlations between cost parameters and utility parameters impact the cost-effectiveness results using a Markov model for major depression. CONCLUSION: Incorporation of the correlations can improve the precision of cost-effectiveness results and increase confidence in evidence-based decision-making. Further empirical evidence is warranted.

Age Differences in Thermal Pain Responses: A Direct Laboratory Comparison.

OBJECTIVES: According to a widely cited assertion, older adults are less likely than younger individuals to express pain complaints. Age-related differences in pain responses have been discussed in the literature despite a paucity of research involving direct comparisons of younger and older adults' pain reactions (i.e., verbal, nonverbal) in the context of a single experimental investigation. Our goal was to test the hypothesis that older adults are more stoic than younger adults in their expression of pain. METHODS: We measured trait stoicism as well as multiple responses to thermal pain. RESULTS: In contrast to suggestions in the literature, equivalence testing indicated that older and younger adults displayed similar verbal and nonverbal pain responses. Our results suggest that older adults are no more stoic about their pain than are younger persons. DISCUSSION: This is the first attempt to explore a wide array of age differences in pain expression within the context of a single experimental study.

Understanding bias in probabilistic analysis in model-based health economic evaluation.

Guidelines of economic evaluations suggest that probabilistic analysis (using probability distributions as inputs) provides less biased estimates than deterministic analysis (using point estimates) owing to the non-linear relationship of model inputs and model outputs. However, other factors can also impact the magnitude of bias for model results. We evaluate bias in probabilistic analysis and deterministic analysis through three simulation studies. The simulation studies illustrate that in some cases, compared with deterministic analyses, probabilistic analyses may be associated with greater biases in model inputs (risk ratios and mean cost estimates using the smearing estimator), as well as model outputs (life-years in a Markov model). Point estimates often represent the most likely value of the parameter in the population, given the observed data. When model parameters have wide, asymmetric confidence intervals, model inputs with larger likelihoods (e.g., point estimates) may result in less bias in model outputs (e.g., costs and life-years) than inputs with lower likelihoods (e.g., probability distributions). Further, when the variance of a parameter is large, simulations from probabilistic analyses may yield extreme values that tend to bias the results of some non-linear models. Deterministic analysis can avoid extreme values that probabilistic analysis may encounter. We conclude that there is no definitive answer on which analytical approach (probabilistic or deterministic) is associated with a less-biased estimate in non-linear models. Health economists should consider the bias of probabilistic analysis and select the most suitable approach for their analyses.

Examining the correlation between treatment effects in clinical trials and economic modeling.

INTRODUCTION: Many diseases have a sequential treatment pathway. Compared with patients without previous treatment, patients who fail initial treatment may have lower success rates with a second treatment. This phenomenon can be explained by a correlation between treatment effects. METHODS: We developed a statistical model of covariance for the underlying unobserved correlation between treatments and established a mathematical expression for the magnitude of the latent correlation term. We conducted a simulation study of clinical trials to investigate the correlation between two treatments and explored clinical examples based on published literature to illustrate the identification and evaluation of these correlations. RESULTS: Our simulation study confirmed that a treatment correlation reduces the probability of success for the second treatment, compared with no correlation. We found that treatment correlations may be observable in clinical trials, such as for depression and lung cancer, and the magnitude of correlation may be estimated. We illustrated that treatment correlations can be incorporated into an economic model, with possible impacts on cost-effectiveness results. Additional applications of correlation concepts are also discussed. CONCLUSIONS: We evaluated the correlation between treatment effects and our approach can be applied to clinical trial design and economic modeling of sequential clinical treatment pathways.

Review and estimation of disutility for joint health states of severe and nonsevere hypoglycemic events in diabetes.

Aim: Many economic evaluations used linear or log-transformed additive methods to estimate the disutility of hypoglycemic events in diabetes, both nonsevere (NSHEs) and severe (SHEs). Methods: We conducted a literature search for studies of disutility for hypoglycemia. We used additive, minimum and multiplicative methods, and the adjusted decrement estimator to estimate the disutilities of joint health states with both NSHEs and SHEs in six scenarios. Results: Twenty-four studies reported disutilities for hypoglycemia in diabetes. Based on construct validity, the adjusted decrement estimator method likely provides less biased estimates, predicting that when SHEs occur, the additional impact from NSHEs is marginal. Conclusion: Our proposed new method provides a different perspective on the estimation of quality-adjusted life-years in economic evaluations of hypoglycemic treatments.

Comparison of the expected rewards between probabilistic and deterministic analyses in a Markov model.

Objectives: In Markov models that evaluate the cost-effectiveness of health-care technologies, it is generally recommended to use probabilistic analysis instead of deterministic analysis. We sought to compare the performance of probabilistic and deterministic analysis in estimating the expected rewards in a Markov model.Methods: We applied Jensen's inequality to compare the expected Markov rewards between probabilistic and deterministic analysis and conducted a simulation study to compare the bias and accuracy between the two approaches.Results: We provided mathematical justification why probabilistic analysis is associated with greater Markov rewards (life-years and quality-adjusted life-years) compared with deterministic analysis. In our simulations, probabilistic analyses tended to generate greater life-years, bias, and mean square error for the estimated rewards compared with deterministic analyses. When the expected values of transition probabilities were the same, weaker evidence derived from smaller sample sizes resulted in larger Markov rewards compared with stronger evidence derived from larger sample sizes. When longer time horizons were applied in cases of weak evidence, there was a substantial increase in bias where the rewards in both probabilistic and deterministic analysis were overestimated.Conclusion: Authors should be aware that probabilistic analysis may lead to increased bias when the evidence is weak.