Each paired difference comes from a continuous
distribution that is symmetric, with the same center of symmetry.
Strictly speaking, the population distributions need not be
the same for all the paired differences.
However, if we want a
consistent test,
we assume that the paired differences
all come from the same continuous, symmetric distribution.
(The Wilcoxon signed rank test
is a nonparametric
test. We need not specify or know what the
distribution is,
only that all the paired difference follow the same one.)
The paired differences all have the same median.
(This median will also be the center of symmetry for
the population distribution associated with each
paired difference. Moreover, since the mean of a
continuous symmetric distribution is equal to its median,
this means that the paired differences will also
all have the same mean.)
Because the test statistic for the Wilcoxon signed rank
is based only on the
ranks of the paired differences, the test can
be performed when the only data available are those relative ranks
for the paired differences.
Note that it is not assumed that the two samples are
independent of each other.
In fact, they should be related to each other such that
they create pairs of data points, such as the measurements
on two matched people in a case/control study, or
before- and after-treatment measurements on the same person.
The two-sample paired signed rank test is equivalent to performing
a one-sample signed rank test on the
paired differences.
Guidance:
Ways to detect
before performing the signed rank test whether your data violate any
assumptions.
Ways to examine
signed rank test results to detect assumption violations.
Possible alternatives if
your data or signed rank test results indicate assumption violations.
To properly analyze and interpret results of
the two-sample paired signed rank test, you should be familiar with the
following terms and concepts:
If you are not familiar with these terms and concepts, you are advised to
consult with a statistician. Failure to understand and properly apply the
two-sample paired signed rank test may result in drawing erroneous conclusions from your data.
Additionally, you may want to consult the following references:
Brownlee, K. A. 1965. Statistical Theory and Methodology
in Science and Engineering. New York: John Wiley & Sons.
Conover, W. J. 1980. Practical Nonparametric Statistics. 2nd ed.
New York: John Wiley & Sons.
Daniel, Wayne W. 1978. Applied Nonparametric Statistics.
Boston: Houghton Mifflin.
Daniel, Wayne W. 1995. Biostatistics. 6th ed.
New York: John Wiley & Sons.
Hollander, M. and Wolfe, D. A. 1973. Nonparametric Statistical Methods.
New York: John Wiley & Sons.
Lehmann, E. L. 1975. Nonparametrics: Statistical Methods Based on
Ranks. San Francisco: Holden-Day.
Miller, Rupert G. Jr. 1996. Beyond ANOVA, Basics of Applied
Statistics. 2nd. ed.
London: Chapman & Hall.
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