Statistics in Medicine: Calculating confidence intervals for some non-parametric analyses

Abstract

Gardner and Altman1 described the rationale behind the use of confidence intervals and gave methods for their calculation for a population mean and for differences between two population means for paired and unpaired samples. These methods are based on sample means, standard errors, and the t distribution and should strictly be used only for continuous data from Normal distributions (although small deviations from Normality are not important2). For non-Normal continuous data the median of the population or the sample is preferable to the mean as a measure of location. Medians are also appropriate in other situations?for example, when measurements are on an ordinal scale. This paper describes methods of calculating confidence intervals for a population median or for other population quantiles from a sample of observations. Calculations of confidence intervals for the difference between two population medians or means (a non-parametric approach rather than the parametric approach mentioned above) for both unpaired and paired samples are described. Worked examples are given for each situation. Because of the discrete nature of some of the sampling distribu? tions involved in non-parametric analyses it is not usually possible to calculate confidence intervals with exactly the desired level of confidence. Hence, if a 95% confidence interval is wanted the choice is between the lowest possible level of confidence over 95% (a interval) and the highest possible under 95%. There is no firm policy on which of these is preferred, but we will mainly describe conservative intervals in this paper. The exact level of confidence associated with any particular approximate level can be calculated from the distribution of the statistic being used. The methods outlined for obtaining confidence intervals are described in more detail in textbooks on non-parametric statistics.3 The calculations can be carried out using the statistical computer package MINITAB.4 A method for calculating confidence intervals for Spearman's rank correlation coefficient is given in an accom? panying paper.5 A confidence interval indicates the precision of the sample statistic as an estimate of the overall population value. Confidence intervals convey the effects of sampling variation but cannot control for non-sampling errors in study design or conduct. They should not be used for basic description of the sample data but only for indicating the uncertainty in sample estimates for population values of medians or other statistics.