THE MOST APPROPRIATE METHODOLOGY FOR DEALING WITH MONOTONICITY IN THE CONTEXT OF SUMMARY DATA

Purpose: To evaluate the accuracy of sampling methodologies when monotonicity between two variables is required, but the parameter distributions allow the violation of this relationship when independently sampled.

Method: A dummy data set was constructed simulating the responses from 30 patients who provided utility values for a disease stage (d1) and for a more severe disease stage (d2). Monotonicity was required, with the utility for d1 (Ud1) being greater than d2 (Ud2). The mean and (95% confidence intervals) were as follows: d1 0.60 ( 0.56 – 0.64); d2  0.55 ( 0.51 – 0.59). Ten methodologies were evaluated. The mean, standard deviation, maximum and minimum values of Ud1 – Ud2 were recorded for each, having performed 1000 Monte Carlo simulations. These results were compared with samples where the covariance between d1 and d2 (cv12) was estimated from the dummy data set. The methodologies were: 1) Independent sampling from the distributions, 2) Using the same random number when sampling Ud1 and Ud2, 3) Increasing Ud1 so that it equalled Ud2, 4) Decreasing Ud2 so that it equalled Ud1,  5) Resampling Ud1 until Ud1>Ud2 6) Resampling Ud2 until Ud2<Ud1 7) assuming that cv12 was equal to the average of the individual variances of the means (aivm), 8) assuming a value for cv12 that produced plausible confidence intervals, with cv12 > aivm, 9) as methodology 8 but with cv12< aivm  and 10) assuming a plausible Beta distribution to represent the difference between Ud1 and Ud2.

Result: Methodologies 3 to 6 were biased with the mean difference being 3-6% greater than expected; the remaining methodologies had minimal error. The standard deviation of Ud1 – Ud2 varied between methods, being twenty-fold less than the expected value for methodologies 2 and 7, but conversely up to two-fold greater for methodologies 8 to 10, and four-fold higher for the remaining methodologies.  Methodology 1 violated the monotonicity assumption.

Conclusion: When presented with summary data and with a belief that monotonicity must apply, a judicious selection of the covariance parameter or of the distribution for the difference appears appropriate. The former strategy is likely to be preferential if the monotonicity assumption is required for more than two parameters.