Compare Two Independent Samples

The command
compares two independent samples using the Mann-Whitney U test (nonparametric
alternative to the *t-test* for independent samples), Kolmogorov-Smirnov
test, Wald-Wolfowitz Runs test and Rosenbaum Q criterion.

# How To

ü Run: Statistics→Nonparametric Statistics → Compare Two Independent Samples...

ü Select two variables to compare.

# Results

Mann-Whitney U Test is a nonparametric counterpart to the t-test, it is also known as Mann–Whitney–Wilcoxon (MWW) or Wilcoxon rank-sum test. The test assumes that the variables are measured on at least an ordinal scale (rank order).

The test statistics is the smallest of the two *U* values,
defined as:

(when no ties),

where is
the size and the is
the sum of the ranks for the *i ^{th}* sample.

For small samples the significance level is calculated using the Dineen and Blakesley (1973) algorithm [DBA]. If the final p-level < 0.05 the decision is to reject the null hypothesis. Null hypothesis for Mann-Whitney U: there is no difference between the ranks of the two samples

Z, approximately distributed as a standard normal, is defined as:

Provided p-level is two-tailed.

Kolmogorov-Smirnov test
is a two-sample test of the null hypothesis that *x* and *y* were
drawn from the same continuous distribution is performed.

Wald-Wolfowitz Runs test is another alternative of t-test, that detects two independent samples from different populations with different cumulative distribution functions. The assumption for the test – data collected from two independent groups is identical. The test can detect differences in averages or spread or any other important aspect between the two populations. It is efficient when each sample size is moderately large [BNT].

H0 for Wald-Wolfowitz Runs – there is no statistically significant difference between the two continuous cumulative distribution functions

If Runs Count R ≤
R_{c } we reject H0.

# References

[KBS] Ken Black Business Statistics: For Contemporary Decision Making, Sixth Edition, 2010, 906 p.

[DBA] Dineen, L. C., and B. C. Blakesley. 1973. Algorithm AS 62: Generator for the sampling distribution of the Mann-Whitney U statistic. Applied Statistics, 22, 269-273.

[WKS] Daniel, Wayne W. (1990). "Kolmogorov–Smirnov one-sample test". Applied Nonparametric Statistics (2nd ed.). Boston: PWS-Kent. pp. 319–330

[LTM] L.S. Titkova (2002) Mathematical methods in psychology