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The CAPABILITY Procedure |
The CAPABILITY procedure automatically computes the 1st, 5th, 10th, 25th, 50th, 75th, 90th, 95th, and 99th percentiles (quantiles), as well as the minimum and maximum of each analysis variable. To compute percentiles other than these default percentiles, use the PCTLPTS= and PCTLPRE= options in the OUTPUT statement.
You can specify one of five definitions for computing the percentiles with the PCTLDEF= option. Let be the number of nonmissing values for a variable, and let
represent the ordered values of the variable. Let the
th percentile be
, set
, and let
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where is the integer part of np, and
is the fractional part of np. Then the PCTLDEF= option defines the
th percentile,
, as described in the following table:
PCTLDEF= |
Description |
Formula |
---|---|---|
1 |
weighted average at |
|
where |
||
2 |
observation numbered closest to np |
|
where |
||
3 |
empirical distribution function |
|
4 |
weighted average aimed |
|
at |
where |
|
5 |
empirical distribution function with averaging |
|
When you use a WEIGHT statement, the percentiles are computed differently. The 100th weighted percentile
is computed from the empirical distribution function with averaging
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where is the weight associated with
, and where
is the sum of the weights.
Note that the PCTLDEF= option is not applicable when a WEIGHT statement is used. However, in this case, if all the weights are identical, the weighted percentiles are the same as the percentiles that would be computed without a WEIGHT statement and with PCTLDEF=5.
You can use the CIPCTLNORMAL option to request confidence limits for percentiles which assume the data are normally distributed. These limits are described in Section 4.4.1 of Hahn and Meeker (1991). When , the two-sided
% confidence limits for the
-th percentile are
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where is the sample size. When
, the two-sided
% confidence limits for the
-th percentile are
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One-sided % confidence bounds are computed by replacing
by
in the appropriate preceding equation. The factor
is related to the noncentral
distribution and is described in Owen and Hua (1977) and Odeh and Owen (1980).
You can use the CIPCTLDF option to request confidence limits for percentiles which are distribution free (in particular, it is not necessary to assume that the data are normally distributed). These limits are described in Section 5.2 of Hahn and Meeker (1991). The two-sided % confidence limits for the
-th percentile are
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where is the jth order statistic when the data values are arranged in increasing order:
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The lower rank and upper rank
are integers that are symmetric (or nearly symmetric) around
where
is the integer part of
, and where
is the sample size. Furthermore,
and
are chosen so that
and
are as close to
as possible while satisfying the coverage probability requirement
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where is the cumulative binomial probability
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In some cases, the coverage requirement cannot be met, particularly when is small and
is near 0 or 1. To relax the requirement of symmetry, you can specify CIPCTLDF( TYPE = ASYMMETRIC ). This option requests symmetric limits when the coverage requirement can be met, and asymmetric limits otherwise.
If you specify CIPCTLDF( TYPE = LOWER ), a one-sided % lower confidence bound is computed as
, where
is the largest integer that satisfies the inequality
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with . Likewise, if you specify CIPCTLDF( TYPE = UPPER ), a one-sided
% lower confidence bound is computed as
, where
is the largest integer that satisfies the inequality
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where .
Note that confidence limits for percentiles are not computed when a WEIGHT statement is specified.
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