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

ABSTRACT

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 ($\mathrm{c}orr\left(X,Y\right)=\mathrm{c}orr\left(\bar{X},\bar{Y}\right)$). 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.

KEYWORDS:

• probabilistic analysis
• correlation
• parameter uncertainty
• modeling
• simulation

Article highlights

• The correlations between economic modeling input parameters directly impact the variance and may impact the expected values of model outputs. Economic evaluation guidelines recommend incorporating correlations in modeling. However, published studies do not often report the correlations of health outcomes and cost inputs.

• We provide proof that for correlated random variables X and Y (e.g. inpatient visits and outpatient visits), the correlation coefficients of sample means and samples are equal to each other ($\mathrm{c}orr\left(X,Y\right)=\mathrm{c}orr\left(\bar{X},\bar{Y}\right)$). This allows us to quantify the covariance and the correlation coefficient between parameters from summary statistics using a simple formula, $Var\left(X+Y\right)=Var\left(X\right)+Var\left(Y\right)+2Cov\left(X,Y\right)$.

• In four examples, we illustrate methods to detect and quantify the correlation coefficients for costs and utility parameters from published summary statistics. Using correlation coefficients or covariance, we can simulate multivariate normal distribution data for probabilistic analysis, and probably correlated Poisson distributions for count data (e.g. numbers of general practitioner and specialist visits) in microsimulation. We provide both SAS and R code for these examples.

• We demonstrate how correlations between parameters impact the variance of cost-effectiveness results in a simplified Markov model. When cost and utility parameters are positively correlated, omitting these correlations in modeling may lead to underestimate the parameter uncertainty of total costs, QALYs and incremental cost, and overestimate the parameter uncertainty of incremental QALYs. However, it is difficult to predict how correlations impact the ICERs.

• When a model includes many parameters with various unknown correlations, it is not unreasonable to assume that different correlations would influence the cost-effectiveness results in different directions and that the impact of unknown correlations may be partially balanced. Under these assumptions, ignoring unknown correlations likely would not substantially influence decision-making.

Acknowledgments

Dr Wendy J. Ungar holds a Canada Research Chair in Economic Evaluation and Technology Assessment in Child Health.

Declaration of interest

The authors have no relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript. This includes employment, consultancies, honoraria, stock ownership or options, expert testimony, grants or patents received or pending, or royalties.

Reviewer disclosures

Peer reviewers on this manuscript have no relevant financial or other relationships to disclose.

Disclaimer

The opinions expressed in this publication do not necessarily represent the opinions of Ontario Health. No endorsement is intended or should be inferred.

Author contributions

X Xie, A Schaink and A Volodin conceived the study idea, designed the study, and drafted the manuscript. C Gao, O Gajic-Veljanoski and W Ungar provided important intellectual content and revised the draft manuscript. A Volodin provided the proof for sample means of correlated random variables. X Xie provided the mathematical justification of how correlations between model parameters impact cost-effectiveness results. X Xie and C Gao simulated the data and conducted the analyses. All authors agreed for the final version of the manuscript to be published.

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/14737167.2023.2221436.

Additional information

Funding

This paper was not funded.