One-way nonparametric One-way nonparametric ANOVA with trigonometric ANOVA with trigonometric

scoresscores

by Kravchuk, O.Y. by Kravchuk, O.Y.

School of Land and Food Sciences,School of Land and Food Sciences,

University of QueenslandUniversity of Queensland

Inspired by the simplicity of the Kruskal-Wallis Inspired by the simplicity of the Kruskal-Wallis k-sample procedure, we introduce a new rank k-sample procedure, we introduce a new rank test of the test of the χχ22 type that allows one to work with type that allows one to work with data that violates the normality assumption, data that violates the normality assumption, being unimodal and symmetric but more being unimodal and symmetric but more heavier tailed than the normal. This type of heavier tailed than the normal. This type of non-normality is common in biometrical non-normality is common in biometrical applications and also describes the distribution applications and also describes the distribution of the log-transformed Cauchy data. of the log-transformed Cauchy data. The distribution of the test statistic The distribution of the test statistic corresponds to the distribution of the first corresponds to the distribution of the first component of the well-known Cramer-von component of the well-known Cramer-von Mises test statistic. Mises test statistic.

The test is asymptotically most efficient for The test is asymptotically most efficient for the hyperbolic secant distribution that is the hyperbolic secant distribution that is compared to the normal and logistic compared to the normal and logistic distributions in the diagram below.distributions in the diagram below.

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Fig1: Standardised normal, hyperbolic secant and logistic densities

The test is a one way rank based ANOVA, The test is a one way rank based ANOVA, where we assume that within the k where we assume that within the k treatments the populations are continuous, treatments the populations are continuous, belong to the same location family and may belong to the same location family and may differ in the location parameter only. There differ in the location parameter only. There are N experimental units, where the jare N experimental units, where the jthth treatment accumulates ntreatment accumulates njj units. units.The test statistic is built on k “bridges” The test statistic is built on k “bridges” corresponding to k linear contrasts of type corresponding to k linear contrasts of type TT11=(A=(A11<A<A22,A,A33). The asymptotic distribution of ). The asymptotic distribution of the test statistic is the the test statistic is the χχ22 with k-1 degrees of with k-1 degrees of freedom. Computationally, the exact freedom. Computationally, the exact distribution is easy to construct on the basis distribution is easy to construct on the basis of k-1 orthogonal contrasts (for example, for of k-1 orthogonal contrasts (for example, for k = 3, Tk = 3, T1, 2U31, 2U3 and T and T2,32,3).).

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For small samples (max(nFor small samples (max(njj)<6), the chi-)<6), the chi-square approximation is more conservative square approximation is more conservative than the exact null distribution. The diagram than the exact null distribution. The diagram and table below provide the exact and table below provide the exact distribution for ndistribution for n11=3, n=3, n22=n=n33=2=2

Q=q P(Q<=q)

3.172 0.810

3.511 0.848

3.525 0.867

4.039 0.886

4.311 0.905

4.545 0.924

4.799 0.943

5.238 0.962

5.647 0.9817.359 1.000

We illustrate the method by an artificial We illustrate the method by an artificial example of three normal populations different example of three normal populations different in location only. The populations are, in location only. The populations are, correspondingly, N(0,1), correspondingly, N(0,1), N(-1,1), N(2,1). Random samples of size 8 are N(-1,1), N(2,1). Random samples of size 8 are taken from these populations.taken from these populations.

N(2,1)N(-1,1)N(0,1)

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One-way ANOVAF = 18.07, p = 0.000

Kruskal-WallisKW = 14.11, p = 0.001

Trigonometric ANOVAQ=14.32, p = 0.001

Illustrating the procedure… When there is a Illustrating the procedure… When there is a certain linear trend among the treatments, the certain linear trend among the treatments, the corresponding bridge tends to have the U-corresponding bridge tends to have the U-shape. We measure the strength of such a shape. We measure the strength of such a tendency by the first coefficient of the Fourier tendency by the first coefficient of the Fourier sine-decomposition of the bridge.sine-decomposition of the bridge.

S1=1.17S2=2.58S3=-3.75

The larger the sample sizes, the smoother The larger the sample sizes, the smoother the bridge. The actual shape of the bridge the bridge. The actual shape of the bridge depends on the difference in location as well depends on the difference in location as well as on the distributions of the underlying as on the distributions of the underlying populations. If the difference is large, the populations. If the difference is large, the shape is strictly triangular regardless of the shape is strictly triangular regardless of the underlying distribution and the median k-underlying distribution and the median k-sample test works well. sample test works well. If the difference in location is small, for If the difference in location is small, for symmetric, unimodal distributions, the symmetric, unimodal distributions, the shape of the bridges is determined by the shape of the bridges is determined by the tails of the distributions.tails of the distributions.

Normal Logistic Hyperbolic secantEfficiency 0.905 0.986 1.000

The difference in scale among several The difference in scale among several Cauchy distributions may be analysed by Cauchy distributions may be analysed by means of the current test. To illustrate such means of the current test. To illustrate such an application, we perform the following an application, we perform the following ANOVA on the log-transformed Cauchy ANOVA on the log-transformed Cauchy populations: Cauchy(0,1), Cauchy(0,5) and populations: Cauchy(0,1), Cauchy(0,5) and Cauchy(0,2). The log-transformation of the Cauchy(0,2). The log-transformation of the absolute values of the data makes it more absolute values of the data makes it more normal-like. However, the analysis of the normal-like. However, the analysis of the residuals of one-way ANOVA shows a residuals of one-way ANOVA shows a departure from normality.departure from normality.The test allows us to perform the formal The test allows us to perform the formal analysis and detect the difference in scale. analysis and detect the difference in scale. The Kruskal-Wallis test gives a similar The Kruskal-Wallis test gives a similar conclusion.conclusion.

The trigonometric ANOVA on log-The trigonometric ANOVA on log-transformed Cauchy… Random samples of transformed Cauchy… Random samples of size 8 were taken from the parent size 8 were taken from the parent populations.populations.

Log(C(0,2))Log(C(0,5))Log(C(0,1))

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P-Value: 0.088A-Squared: 0.631

Anderson-Darling Normality Test

N: 24StDev: 1.34377Average: -0.0000000

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Normal Probability Plot

One-way ANOVA Kruskal-WallisF = 5.78, p = 0.01 KW = 11.26, p = 0.004

Trigonometric ANOVAQ=11.33, p = 0.003

Olena Kravchuk, LAFS, UQOlena Kravchuk, LAFS, UQo.kravchuk@uq.edu.auo.kravchuk@uq.edu.au(07) 33652171(07) 33652171

The multiple comparisons and contrasts are The multiple comparisons and contrasts are to be further developed for this test. to be further developed for this test. The two-way test with trigonometric scores is The two-way test with trigonometric scores is to be investigated.to be investigated.The test performances are to be compared to The test performances are to be compared to the k-sample Cramer-von Mises test.the k-sample Cramer-von Mises test.