Publication date: 11/10/2021

The Kolmogorov-Smirnov test is available only when X has exactly two levels. The report shows descriptive statistics followed by test results. The descriptive statistics are the following:

Level

The two levels of X.

Count

The frequencies of each level.

EDF at Maximum

For a level of X, gives the value of the empirical cumulative distribution function (EDF) for that level at the value of X for which the difference between the two EDFs is a maximum. For the row named Total, gives the value of the pooled EDF (the EDF for the entire data set) at the value of X for which the difference between the two EDFs is a maximum.

Deviation from Mean at Maximum

For each level, gives the value obtained by the following steps:

– Compute the difference between the EDF at Maximum for the given level and the EDF at maximum for the pooled data set (Total).

– Multiply this difference by the square root of the number of observations in that level, given as Count.

This report gives the details for the test.

KS

A Kolmogorov-Smirnov statistic computed as follows:

The formula uses the following notation:

– xj, j = 1,..., n are the observations

– ni is the number of observations in the ith level of X

– F is the pooled cumulative empirical distribution function

– Fi is the cumulative empirical distribution function for the ith level of X

This version of the Kolmogorov-Smirnov statistic applies even when there are more than two levels of X. Note, however, that JMP performs the Kolmogorov-Smirnov analysis only when X has only two levels of X.

KSa

An asymptotic Kolmogorov-Smirnov statistic computed as , where n is the total number of observations.

D=max|F1-F2|

The maximum absolute deviation between the EDFs for the two levels. This is the version of the Kolmogorov-Smirnov statistic typically used to compare two samples.

Prob > D

The p-value for the test. This is the probability that D exceeds the computed value under the null hypothesis of no difference between the levels.

D+ = max(F1-F2)

A one-sided test statistic for the alternative hypothesis that the level of the first group exceeds the level of the second group.

Prob > D+

The p-value for the test of D+.

D- = max(F2-F1)

A one-sided test statistic for the alternative hypothesis that the level of the second group exceeds the level of the first group

Prob > D-

The p-value for the test of D-.

For the Kolmogorov-Smirnov exact test, the report gives the same statistics as does the asymptotic test, but the p-values are computed to be exact.

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