In a study of health care expenditures attributable to smoking, we seek to compare the distribution of medical costs for persons with lung cancer or chronic obstructive pulmonary disease (cases) to those without (controls) using a national survey which includes hundreds of cases and thousands of controls. The distribution of costs is highly skewed toward larger values, making estimates of the mean from the smaller sample dependent on a small fraction of the biggest values. One approach to deal with the smaller sample is to rely on a simple parametric model such as the log-normal, but this makes the undesirable assumption that the distribution of the log-expenditures is symmetric.

We propose a novel approach to estimate the mean difference of two highly skewed distributions (Delta), which we call Smooth Quantile Ratio Estimation (SQUARE). SQUARE is obtained by smoothing, over percentiles, the ratio of the cost quantiles of the cases and controls. SQUARE defines a large class of estimators of Delta including: 1) the sample mean difference, 2) the maximum likelihood estimate under log-normal samples, and 3) L-estimates. We detail asymptotic properties of SQUARE such as consistency and asymptotic normality, and also provide a closed form expression for the asymptotic variance.

Through a simulation study, we show that SQUARE has lower mean squared error than several competitors including the sample mean difference, and log-normal parametric estimates in several realistic situations. We apply SQUARE to the 1987 National Medicare Expenditure Survey to estimate the difference in medical expenditures between persons suffering from the smoking attributable diseases, lung cancer and chronic obstructive pulmonary disease, and persons without these diseases. Software in R (Ihaka and Gentleman, 1996) for the implementation of SQUARE and of all its special cases, and the cost data used in this paper are available at http://biostat.jhsph.edu/~fdominic/square.html.


Health Services Research | Statistical Methodology | Statistical Models | Statistical Theory