Handling missing data in patient-level cost-effectiveness analysis alongside randomised clinical trials

Abstract

Background

Missing data are potentially an extensive problem in cost-effectiveness analyses conducted alongside randomised clinical trials, where prospective collection of both resource use and health outcome information is required. There are several possible reasons for the presence of incomplete records, and the validity of the analysis in the presence of data with missing values is dependent upon the mechanism generating the missing data phenomenon. In the past, the most commonly used methods for analysing datasets with incomplete observations were relatively ad hoc (e.g. case deletion, mean imputation) and suffered from potential limitations. Recently, several alternative and more sophisticated approaches (e.g. multiple imputation) have been proposed that attempt to correct the flaws of the simple imputation methods.

Objectives

The objectives are to provide a concise and accessible description of the quantitative methods most commonly used in trial-based cost-effectiveness analysis for handling missing data, and also to demonstrate the potential impact of these alternative approaches on the cost-effectiveness results reported in two case studies.

Methods

Data from two recently conducted, trial-based economic evaluations are used to explore the sensitivity of the study results to the technique used to deal with incomplete observations. A statistical framework for representing the uncertainty in the alternative methods is outlined using an approach based on net benefits and cost-effectiveness acceptability curves.

Results

The case studies demonstrate the potential importance of the approach used to handle missing data. Although the analytical strategy did not appear to alter the results of one of the studies, the other case study showed that that the results of the cost-effectiveness analysis were sensitive to both the decision to impute and also the imputation strategy adopted.

Conclusions

Analysts should be more explicit in reporting the analytical strategies applied in the presence of missing data. The use of a multiple imputation approach is recommended in the majority of cases, so as to adequately reflect the uncertainty in the study results due to the presence of missing data.

Appendix

Using the notation provided in Schafer,[23] suppose the analyst is interested in analysing the variable Y (e.g. patient-level net benefit), part of which is observed (Yobs) and part is missing (Y_mis).

Let the parameter ^Q be the statistic of interest (i.e. mean costs, mean effects, mean net benefits) and Û its estimated variance, and suppose the researcher has created m completed datasets (equation 1):

$$Y^{(1)}=(Y_{obs},Y^{(1)}_{mis}),Y^{(2)}=(Y_{obs},Y^{(2)}_{mis}),....,Y^{(m)}=(Y_{obs},Y^{(m)}_{mis})$$

(1)

S/he can now calculate m plausible estimates of the statistic of interest ^Q⁽¹⁾,…, ^Q^(m) together with their estimated variances, Û⁽¹⁾,Û⁽²⁾, …,Û^(m).

In the univariate case, it is possible to combine these quantities to obtain an MI estimate of the statistic of interest, together with its variance, by first calculating the average value of -Q across the m datasets as (equation 2):

$${\overline Q}=m^{-1}\sum^{m}_{k=1}{\hat Q^{(k)}}$$

(2)

The total variance of this estimate can be obtained as (equation 3):

$$T=\overline U+(1+{1\over m})B$$

(3)

Which combines the ‘within-imputation variance’ (U) with the ‘between-imputation variance’ (B), where $\overline U={1\over m}\sum^{m}_{k=1}U^{(l)}$ is the average of the variances across the m dataset, and $B={1\over m-1}\sum^{m}_{k=1}(\hat Q^{(l)}-\overline Q)^2$ is the sample variance among the m estimates.

The analyst can then construct confidence intervals around the Q estimate using the Student’s t-test, with a number of degrees of freedom that are a function of m, B and U.