Possible alternatives if your data violate one-sample t test assumptions
If the data to be analyzed by a one-sample t test
come from a
population
whose distribution
violates the assumption
of normality,
or outliers are present,
then the t test on the original data may provide misleading
results, or may not be the most powerful test available.
In such cases, transforming
the data or using a nonparametric test
may provide a better analysis.
Alternative procedures:
Transformations: correcting nonnormality
by transforming all the data values
Nonparametric tests: dealing with
nonnormality by employing a test that does not make the normality
assumption of the t test
Transformations (a single function applied to each
data value) are applied to correct problems of
nonnormality.
For example, taking logarithms of sample values
can reduce
skewness
to the right.
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.)
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.
Nonparametric tests
are tests that do not make the usual
distributional assumptions of the
normal-theory-based tests.
For the one-sample t test, the most common
nonparametric alternative tests are the one-sample Wilcoxon
one-sample signed rank test
and the one-sample sign test.
Although the signed rank test does
not assume
normality
of the distribution for the sample population,
it does assume that they come from the same, symmetric
distribution.
Thus the signed rank
test will not address the problem of
skewness.
Also, as with the one-sample t test, it is assumed that the
values in the sample are
independent of each other.
Although the Wilcoxon one-sample signed rank test is the most commonly used
nonparametric
alternative to the one-sample t test,
it is not the only one. However, all tests assume that the
data values are independent.
The one-sample sign test can be calculated
for data values even when only the direction (+ or -) of
the difference from the hypothesized mean value is known.
This means that it can be applied in
situations when the one-sample signed rank test, which requires at
least knowledge of the relative ranks and directions (signs)
of the differences between each data value and the hypothesized value,
can not be used.
Unlike the signed rank test, the sign test does not assume
symmetry of the population
distribution for the sample,
but is likely to be less powerful
than the one-sample signed 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 one-sample signed rank test or the one-sample t test.
If the data values do indeed come from a population
with a normal distribution,
then the t test is the most
powerful test of the equality
of the population mean and the hypothesized value,
meaning that no other test is more likely
to detect an actual departure.
(If a distribution is symmetric, its mean and median
are both equal to the center of symmetry. Since the
normal distribution is symmetric, the t test can also
be viewed as testing whether the population
median is different from the hypothesized value, if the normality assumption holds.)
If the population distribution of the sample is not normal, however,
the signed rank test may be more powerful at detecting
differences between the population median and the hypothesized value.
Because the signed rank test is nearly as powerful
as the one-sample t test in the case of data from a normal
distribution, and may be substantially more powerful
in the case of nonormality, the one-sample rank test is well suited
to analyzing data when outliers are suspected, even
if the underlying distribution is close to normal.
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