One-way nonparametric ANOVA with trigonometric scores

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One-way nonparametric ANOVA with trigonometric
scores

• by Kravchuk, O.Y.
• School of Land and Food Sciences,
• University of Queensland

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Inspired by the simplicity of the Kruskal-Wallis
k-sample procedure, we introduce a new rank test
of the ?2 type that allows one to work with data
that violates the normality assumption, being
unimodal and symmetric but more heavier tailed
than the normal. This type of non-normality is
common in biometrical applications and also
describes the distribution of the log-transformed
Cauchy data. The distribution of the test
statistic corresponds to the distribution of the
first component of the well-known Cramer-von
Mises test statistic.
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The test is asymptotically most efficient for the
hyperbolic secant distribution that is compared
to the normal and logistic distributions in the
diagram below.
Fig1 Standardised normal, hyperbolic secant and
logistic densities
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The test is a one way rank based ANOVA, where we
assume that within the k treatments the
populations are continuous, belong to the same
location family and may differ in the location
parameter only. There are N experimental units,
where the jth treatment accumulates nj units.The
test statistic is built on k bridges
corresponding to k linear contrasts of type
T1(A1ltA2,A3). The asymptotic distribution of the
test statistic is the ?2 with k-1 degrees of
freedom. Computationally, the exact distribution
is easy to construct on the basis of k-1
orthogonal contrasts (for example, for k 3,
T1, 2U3 and T2,3).
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For small samples (max(nj)lt6), the chi-square
approximation is more conservative than the exact
null distribution. The diagram and table below
provide the exact distribution for n13, n2n32
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We illustrate the method by an artificial example
of three normal populations different in location
only. The populations are, correspondingly,
N(0,1), N(-1,1), N(2,1). Random samples of size
8 are taken from these populations.
One-way ANOVA F 18.07, p 0.000 Kruskal-Wallis
KW 14.11, p 0.001 Trigonometric
ANOVA Q14.32, p 0.001
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Illustrating the procedure When there is a
certain linear trend among the treatments, the
corresponding bridge tends to have the U-shape.
We measure the strength of such a tendency by the
first coefficient of the Fourier
sine-decomposition of the bridge.
S11.17 S22.58 S3-3.75
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The larger the sample sizes, the smoother the
bridge. The actual shape of the bridge depends on
the difference in location as well as on the
distributions of the underlying populations. If
the difference is large, the shape is strictly
triangular regardless of the underlying
distribution and the median k-sample test works
well. If the difference in location is small,
for symmetric, unimodal distributions, the shape
of the bridges is determined by the tails of the
distributions.
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The difference in scale among several Cauchy
distributions may be analysed by means of the
current test. To illustrate such an application,
we perform the following ANOVA on the
log-transformed Cauchy populations Cauchy(0,1),
Cauchy(0,5) and Cauchy(0,2). The
log-transformation of the absolute values of the
data makes it more normal-like. However, the
analysis of the residuals of one-way ANOVA shows
a departure from normality.The test allows us to
perform the formal analysis and detect the
difference in scale. The Kruskal-Wallis test
gives a similar conclusion.
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The trigonometric ANOVA on log-transformed
Cauchy Random samples of size 8 were taken from
the parent populations.
One-way ANOVA Kruskal-Wallis F 5.78, p
0.01 KW 11.26, p 0.004 Trigonometric
ANOVA Q11.33, p 0.003
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Olena Kravchuk, LAFS, UQo.kravchuk_at_uq.edu.au(07)
33652171
The multiple comparisons and contrasts are to be
further developed for this test. The two-way
test with trigonometric scores is to be
investigated.The test performances are to be
compared to the k-sample Cramer-von Mises test.