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The NPAR1WAY Procedure |
If you specify the EDF option, PROC NPAR1WAY computes tests based on the empirical distribution function. These include the Kolmogorov-Smirnov and Cramer-von Mises tests, and also the Kuiper test for two-sample data. This section gives formulas for these test statistics. For further information about the formulas and the interpretation of EDF statistics, see Hollander and Wolfe (1999) and Gibbons and Chakraborti (1992). For details about the -sample analogs of the Kolmogorov-Smirnov and Cramer-von Mises statistics, see Kiefer (1959).
The empirical distribution function (EDF) of a sample ,
, is defined as
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where is an indicator function. PROC NPAR1WAY uses the subsample of values within the
th class level to generate an EDF for the class,
. The EDF for the overall sample, pooled over classes, can also be expressed as
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where is the number of observations in the
th class level, and
is the total number of observations.
The Kolmogorov-Smirnov statistic measures the maximum deviation of the EDF within the classes from the pooled EDF. PROC NPAR1WAY computes the Kolmogorov-Smirnov statistic as
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The asymptotic Kolmogorov-Smirnov statistic is computed as
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For each class level and overall, PROC NPAR1WAY displays the value of
at the maximum deviation from
and the value
at the maximum deviation from
. PROC NPAR1WAY also gives the observation where the maximum deviation occurs.
If there are only two class levels, PROC NPAR1WAY computes the two-sample Kolmogorov-Smirnov test statistic as
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The p-value for this test is the probability that is greater than the observed value
under the null hypothesis of no difference between class levels (samples). PROC NPAR1WAY computes the asymptotic p-value for
with the approximation
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where
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The quality of this approximation has been studied by Hodges (1957).
If you specify the D option, or if you request exact Kolmogorov-Smirnov p-values with the KS option in the EXACT statement, PROC NPAR1WAY also computes the one-sided Kolmogorov-Smirnov statistics and
for two-sample data as
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The asymptotic probability that is greater than the observed value
, under the null hypothesis of no difference between the two class levels, is computed as
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Similarly, the asymptotic probability that is greater than the observed value
is computed as
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To request exact p-values for the Kolmogorov-Smirnov statistics, you can specify the KS option in the EXACT statement. See the section Exact Tests for more information.
The Cramer-von Mises statistic is defined as
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where is the number of ties at the
th distinct value and
is the number of distinct values. The asymptotic value is computed as
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PROC NPAR1WAY displays the contribution of each class level to the sum .
For data with two class levels, PROC NPAR1WAY computes the Kuiper statistic, its scaled value for the asymptotic distribution, and the asymptotic p-value. The Kuiper statistic is computed as
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The asymptotic value is
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PROC NPAR1WAY displays the value of for each class level.
The p-value for the Kuiper test is the probability of observing a larger value of under the null hypothesis of no difference between the two classes. PROC NPAR1WAY computes this p-value according to Owen (1962, p. 441).
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