help qenvgamma 
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Generate quantile envelopes for gamma quantile-quantile plot


Syntax

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


Description

    qenvgamma 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 gamma
    distribution with the same number of values, shape and scale parameters
    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.


Remarks 

    The precise recipe is as follows.

    1. Fit gamma distribution using findit gammafit, which must be installed
    separately.

    2. Take reps samples of the same size from a gamma distribution with the
    same parameter values.

    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.  They are not aligned with the values of varname in the
    dataset, but they are assigned to the same set of observations, which is
    sufficient for qplot.

    In principle, the envelope variables could also be plotted in conjunction
    with findit qgamma, 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.


Options

    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.


Examples

    . sysuse auto, clear
    . qenvgamma weight, gen(lower upper)
    . qplot weight lower upper, ms(oh none ..) c(. l l) lc(gs10 ..)
        legend(off) ytitle("Weight, lb")
    . gammafit weight
    . local alpha = e(alpha)
    . local beta = e(beta)
    . qplot weight lower upper, ms(oh none ..) c(. l l) lc(gs10 ..)
        legend(off) ytitle("Weight, lb") trscale(`beta' * invgammap(`alpha',
        @)) xtitle(Gamma quantiles)
    . qenvgamma weight, overall reps(5000) gen(lower2 upper2)
    . qplot weight lower2 upper2, ms(oh none ..) c(. l l) lc(gs10 ..)
        legend(off) ytitle("Weight, lb") trscale(`beta' * invgammap(`alpha',
        @)) xtitle(Gamma quantiles)
 


Authors 

    Nicholas J. Cox, Durham University
    n.j.cox@durham.ac.uk

    Maarten Buis, WZB
    maarten.buis@wzb.eu

 
Reference         

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


Also see

    Manual:  [R] diagnostic plots