Possible alternatives if your data violate paired sign
test assumptions
If
outliers are present,
then the paired sign test on the original data may provide misleading
results, or may not be the most powerful test available.
Transforming
the data to promote normality and then performing a
paired t test,
or using another nonparametric test
may provide a better analysis.
Alternative procedures:
Transformations: correcting
skewness by transforming the paired differences
Transformations (a single function applied to each
data value) can be applied to correct problems of
nonnormality.
For example, taking logarithms of sample values
can reduce
skewness
to the right.
If such a transformation can be found, the transformed
data may be suitable for use with a
paired t test.
The resulting test might be more powerful than
the original sign test.
The same transformation
should be applied to both samples.
Unless scientific
theory suggests a specific transformation a priori,
transformations are usually chosen from the "power family"
of transformations, where each value is replaced by
x**p, where p is an integer or half-integer, usually
one of:
-2 (reciprocal square)
-1 (reciprocal)
-0.5 (reciprocal square root)
0 (log transformation)
0.5 (square root)
1 (leaving the data untransformed)
2 (square)
For p = -0.5 (reciprocal square root),
0, or 0.5 (square root), the data values must all be
positive. To use these transformations when there
are negative and positive values,
a constant can be added to all the data values
such that the smallest is greater than 0 (say,
such that the smallest value is 1). (If all
the data values are negative, the data can
instead be multiplied by -1, but note that
in this situation, data suggesting
skewness
to the right
would now become data suggesting skewness to the left.)
Note that if you transform the paired differences so that
those that originally had value 0 no longer do,
the effective sample size of the data set will be
changed. You can, of course, preserve the same sample
size by only including in the transformation the non-zero
paired differences, and making sure that none of the
transformed paired differences become 0.
To preserve the order of the original data
in the transformed data, if the value of p is
negative, the transformed data are
multiplied by -1.0; e.g., for p = -1,
the data are transformed as x --> -1.0/x.
Taking logs or square roots tends to "pull in"
values greater than 1 relative to values less
than 1, which is useful in correcting skewness
to the right. Transformation involves changing
the metric in which the data are analyzed, which
may make interpretation of the results difficult if the
transformation is complicated. If you are unfamiliar
with transformations, you may wish to consult a
statistician before proceeding.
Although the paired sign test is a frequently used
nonparametric
alternative to the paired two-sample t test,
it is not the only one. However, all tests assume that the
paired differences are independent.
The Wilcoxon paired signed rank test
is generally more powerful than the sign test, but it requires
more information about the paired differences, at the very least
the difference is known. This means that it can be applied in
situations when the paired signed rank test, which requires at
least knowledge of their relative ranks as well as their directions (signs).
The signed rank test assumes
symmetry of the population
distribution for the paired differences,
and is likely to be more powerful
than the paired sign rank test when that distribution is
in fact symmetric. If the distribution
is extremely heavy-tailed,
the sign test may be more powerful than either
the paired signed rank test or the paired t test.
The sign test is the most powerful of these three tests for
paired differences from a double exponential or Cauchy distribution.
If the sampled paired differences do indeed come a population
with a normal distribution,
then the paired
two-sample t test is the most
powerful test of the equality
of the two means, meaning that no other test is more likely
to detect an actual difference between the two means.
If the population distribution for the paired differences is not normal,
but is symmetric,
the signed rank test may be more powerful than either the
t test or the sign test at detecting
differences between the sample medians of the
paired differences.
If applying a transformation
promotes normality, the paired two-sample t test
may be a more powerful test than the paired signed rank test for
the transformed data.
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