Possible alternatives if your data violate normality test assumptions
If the populations from which
data for a normality test were sampled
violate one or more of
the assumptions for the test, the results of the analysis may be
incorrect or misleading.
If there are factors
unaccounted for in the analysis,
then the
normality test may not give useful results.
In such cases, stratification
may provide a better analysis.
Although the various normality tests available in Prophet
are similar, they are not identical. In some
situations,
one of the tests may be preferable to the others.
There are also other normality tests.
Alternative procedures:
Stratification: Dividing
the sample into homogeneous subsamples
Stratification involves dividing a sample into
subsamples based on one or more characteristics
of the population. For example, a sample may
be stratified by gender.
If the distribution function is different for
the different strata, then the characteristic
used for stratification may be an
implicit factor,
and a separate analysis
for each individual subsample may be more
informative than an analysis of the entire sample.
A potential drawback with stratification is that one or
more of the subsamples may be small in size, leading to
problems
with the reliability of the test results.
Also, the results for each subsample are generalizable
to only a part of the sample population.
The Kolmogorov-Smirnov test is generally less
powerful
than the tests specifically designed to test for normality.
This is particularly the case when the mean and variance
are not specified in advance for the Kolmogorov-Smirnov test,
which then becomes conservative.
The Shapiro-Wilk test and the D'Agostino-Pearson omnibus tests are
both robust for efficiency,
having good power across a range of nonnormal distributions.
D'Agostino's test for skewness and the Anscombe-Glynn test for kurtosis
are good at detecting nonnormality caused by asymmetry or nonnormal
tail heaviness, respectively. If a distribution is symmetric but
heavy-tailed (positive kurtosis), the test for kurtosis may
be more powerful than the Shapiro-Wilk test, especially if the
heavy-tailedness is not extreme. If a distribution has normal
kurtosis but is skewed, the test for skewness may be more powerful
than the Shapiro-Wilk test, especially if the skewness is not
extreme.
Generally speaking, either the Shapiro-Wilk or D'Agostino-Pearson
test is a powerful overall test for normality.
D'Agostino's skewness test is particularly powerful for
detecting normality due to asymmetry, and the Anscombe-Glynn
test is particularly powerful for detecting normality due
to nonnormal kurtosis.
A number of normality tests have been proposed over the years.
Some tests are better than others in some situations, in that
no single test is uniformly most powerful. These tests include:
The Lilliefors test for normality
adjusts the Kolmogorov-Smirnov test specifically for
testing for normality when the mean and variance are unknown.
D'Agostino's D
is a powerful overall test for normality
like the Shapiro-Wilk or D'Agostino-Pearson tests, and may
be more powerful in detecting heavy-tailedness. It is not
as powerful as the Shapiro-Wilk test at detecting skewness
when the population distribution has normal kurtosis.
Spiegelhalter's T' is designed to test for normality
against other symmetric alternative distributions. Like
the Anscombe-Glynn test, it is powerful for detecting
nonnormal kurtosis (although not as powerful as the Anscombe-Glynn test),
but has little power in detecting nonnormality when the
population distribution is skewed.
The Martin-Iglewicz I is designed to test for normality
against other heavy-tailed alternative distributions. Like
the Anscombe-Glynn test, it is powerful for detecting
nonnormal kurtosis (although not as powerful as the Anscombe-Glynn test),
but has little power in detecting nonnormality when the
population distribution is skewed.
The chi-square goodness-of-fit test
can be used to test whether the population distribution
matches the hypothesized distribution, but it is not
a very powerful test for normality. Like the Kolmogorov-Smirnov
test, it requires that the mean and variance of the hypothesized
distribution be specified in advance. Moreover, the test
requires that the data be divided into categories. While this
may be appropriate with discrete data, which can take on only
a small number of values, it is at best an arbitrary process
when the values come from a continuous distribution. Since
the results of the chi-square test can vary with how the
data are divided, this test is not a good alternative
when dealing with continuous population distributions.
Gupta's test is a nonparametric test for symmetry
(as opposed to normality, which includes symmetry).
The Wilcoxon one-sample signed rank test
is sometimes
described as a test for symmetry, but actually assumes
the symmetry of the population distribution.
One alternative to testing against the null hypothesis of normality
is to test against the null hypothesis that the population distribution
is some other, nonnormal distribution, such as the uniform distribution.
The Kolmogorov-Smirnov test is
commonly used to test whether the population distribution follows a
specified continuous distribution.
A transformation of the data may create
a data set that more closely approximates that
from a normal distribution.
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.
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