Let Y be the variable of interest in a complex survey. Denote as the cumulative distribution for . For , the th quantile of the population cumulative distribution function is
Let be the observed values for variable associated with sampling weights, where are the stratum index, cluster index, and member index, respectively, as shown in the section Definitions and Notation. Let denote the sample order statistics for variable .
An estimate of quantile is
where is the estimated cumulative distribution for :
and is the indicator function.
When you use VARMETHOD=TAYLOR, or by default if you do not specify the VARMETHOD= option, PROC SURVEYMEANS uses Woodruff’s method (Dorfman and Valliant 1993; Särndal, Swensson, and Wretman 1992; and Francisco and Fuller 1991) to estimate the variances of quantiles. This method first constructs a confidence interval on a quantile. Then it uses the width of the confidence interval to estimate the standard error of a quantile.
In order to estimate the variance for , first the procedure estimates the variance of the estimated distribution function by
where
Then % confidence limits of can be constructed by
where is the th percentile of the t distribution with df degrees of freedom, described in the section Degrees of Freedom.
When is out of the range of [0,1], the procedure does not compute the standard error.
The th quantile is defined as
and the th quantile is defined as
The standard error of then is estimated by
where is the th percentile of the t distribution with df degrees of freedom.
When you use the replication method, PROC SURVEYMEANS uses the usual variance estimates for a quantile as described in the section Replication Methods for Variance Estimation. However, you should proceed cautiously because this variance estimator can have poor properties (Dorfman and Valliant 1993).
Symmetric % confidence limits are computed as
If you specify the NONSYMCL option in the SURVEYMEANS statement when you use VARMETHOD=TAYLOR option, the procedure computes % nonsymmetric confidence limits: