help qenvnormal 

Generate quantile envelopes for normal quantile-quantile plot


qenvnormal varname [if] [in] , generate(newvar1 newvar2) [ reps(#) level(#) overall ]


qenvnormal generates two numeric variables for later plotting on a quantile-quantile plot that define an envelope of low and high expectations for each quantile given repeated sampling from a normal distribution with the same number of values, mean and standard deviation as the data specified. The program is a helper program designed for use with qplot, which must be installed separately. Type findit qplot to get references and code sources.

When plotted, the envelopes are to be considered indicative, and not as implying a formal decision. In particular, the envelopes underline which quantiles are least reliable. With some experience, they should guide later analysis.


The precise recipe is as follows.

1. Calculate number of values, mean and standard deviation.

2. Take reps samples of the same size from a normal distribution with the same mean and standard deviation.

3. Order each sample from smallest to largest.

4. For each rank, calculate reference intervals containing level% of the sample quantiles. For example, with the default of 95%, the interval ranges from the 2.5% quantile to the 97.5% quantile, the quantiles being calculated using a procedure similar to that used for pctile.

5. The generated variables contain values defining the reference intervals.

In principle, the envelope variables could also be plotted in conjunction with qnorm, but that would require more work.

This envelope approximates a separate test for each rank. The probability that in a variable an observation falls outside this envelope is much larger than 100 - level. When specifying the overall option an envelope will be computed such that the overall error rate will be approximately 100 - level using an algorithm discussed by Davison and Hinkley (1997, Chapter 4).

1. Set L to ceil((100 - level)/200 * reps).

2. For each sample create an envelope using the remaining samples by storing for each rank the Lth and (reps - L)th smallest value, and determine whether the entire sample falls within this envelope. The proportion of samples for which this is not true is an estimate of the overall error rate of that envelope.

3. Decrease L until the overall error rate is less than 100 - level. If the nominal level has not been reached when L = 1, then the entire range is returned and a warning is displayed reporting the approximate overall error rate for that envelope.

4. Compute the envelope using all samples based on this L.


generate() specifies the names of two new variables to be generated. This is a required option.

reps() specifies the number of samples to be taken. The default is 100. This is likely to be adequate for informal exploration.

level() specifies the percent of sample values to be included within the envelope for each rank. level(100) is allowed and interpreted as the entire range.

overall specifies that an envelope with an approximate overall error rate of 100 - level is to be computed instead of a pointwise error rate of approximately 100 - level.


. sysuse auto, clear . qenvnormal weight, gen(lower upper) . qplot weight lower upper, ms(oh none ..) c(. l l) lc(gs10 ..) legend(off) ytitle("Weight, lb") . su weight . scalar mean = r(mean) . scalar sd = r(sd) . qplot weight lower upper, ms(oh none ..) c(. l l) lc(gs10 ..) legend(off) ytitle("Weight, lb") trscale(`=mean' + `=sd' * invnormal(@)) xtitle(Normal quantiles) . qenvnormal weight, overall reps(2000) gen(lower2 upper2) . qplot weight lower2 upper2, ms(oh none ..) c(. l l) lc(gs10 ..) legend(off) ytitle("Weight, lb") trscale(`=mean' + `=sd' * invnormal(@)) xtitle(Normal quantiles)


Nicholas J. Cox, Durham University

Maarten Buis, WZB


Davison, A.C. and Hinkley, D.V. 1997. Bootstrap methods and their application. Cambridge: Cambridge University Press.

Also see

Manual: [R] diagnostic plots