CIRCULAR ONE-SAMPLE TESTS |
Rao’s Spacing |
Continuous |
Uniform |
Any alternative |
θ1 ≤ θ2 ≤ … ≤ θn
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n ≥ 5 |
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Tn−1 = θn − θn−1
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where Ti is the arclength between consecutive angles |
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Tn = 360 + θ1 − θn
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Kuiper’s |
Continuous |
Uniform |
Any alternative |
Vn = D+ + D−
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where D is the deviation from (uniform) theoretical distribution |
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for small n, more powerful than χ2 test |
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circular goodness-of-fit test corresponding to Kolmogorov-Smirnov |
Rayleigh’s |
Continuous |
Uniform |
Unimodal |
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unimodal von Mises alternative |
Watson’s U2
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Continuous |
Uniform |
Unimodal |
θ1 ≤ θ2 ≤ … ≤ θn
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where F(θ) is the function of (uniform) theoretical distribution |
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suitable for unimodal and multimodal distributions |
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V-test |
Continuous |
Uniform |
Unimodal |
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external direction (µ0) specified a priori |
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CIRCULAR MULTISAMPLE TESTS |
Mardia-Watson-Wheeler |
Continuous |
Two samples have identical distributions |
Two samples have different distributions |
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data is ungrouped |
Watson’s U2
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Continuous |
Two samples have same distributions |
Two samples have different distributions |
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where N = n1 + n2
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no ties between samples |
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LINEAR TESTS |
χ2
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Grouped |
Uniform |
Non-uniform |
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where ei = expected value for ith term |
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based on approximation, need large n
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suitable for both unimodal and multimodal distributions |
Kolmogorov-Smirnov |
Grouped |
Uniform |
Non-uniform |
Dmax = max(|di|) |
where F̄i is the function of (uniform) theoretical distribution |
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where |di| = |Fi − F̄i| |
more powerful than χ2 when n is small |
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dependent on starting point, more sensitive near median than at tails |
Student’s t-test |
Continuous |
Two samples have same means |
Two samples have different means |
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where x̄ is the sample mean and s is the sample standard deviation |
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where
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assuming equal variances |
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