Nonparametric Dispersion and Equality Tests

40 Pages·2017·0.21 MB·English

by  Nathaniel E. Helwig

Nonparametric Dispersion and Equality Tests
Nathaniel E. Helwig
AssistantProfessorofPsychologyandStatistics
UniversityofMinnesota(TwinCities)
Updated04-Jan-2017
NathanielE.Helwig (UofMinnesota) NonparametricDispersionandEqualityTests Updated04-Jan-2017 : Slide1
Copyright
Copyright (cid:13)c 2017 by Nathaniel E. Helwig
NathanielE.Helwig (UofMinnesota) NonparametricDispersionandEqualityTests Updated04-Jan-2017 : Slide2
Outline of Notes
1) Dispersion (Ansari-Bradley): 3) Equality (Kolmogorov-Smirnov):
Overview Overview
Procedure Procedure
Example 1 Example 3
2) Dispersion/Location (Lepage):
Overview
Procedure
Example 2
NathanielE.Helwig (UofMinnesota) NonparametricDispersionandEqualityTests Updated04-Jan-2017 : Slide3
DispersionTest(Ansari-Bradley)
Dispersion Test
(Ansari-Bradley)
NathanielE.Helwig (UofMinnesota) NonparametricDispersionandEqualityTests Updated04-Jan-2017 : Slide4
DispersionTest(Ansari-Bradley) Overview
Problem(s) of Interest
Like two-sample location problem, we have N = m+n observations
X ,...,X are iid random sample from population 1
1 m
Y ,...,Y are iid random sample from population 2
1 n
We want to make inferences about difference in distributions
Let F and F denote distributions of populations 1 and 2
1 2
Null hypothesis is same distribution (F (z) = F (z) for all z)
1 2
Using the location-scale parameter model, we have
F (z) = G([z −θ ]/η ) and F (z) = G([z −θ ]/η )
1 1 1 2 2 2
θ and η are median and scale parameters for population j
j j
NathanielE.Helwig (UofMinnesota) NonparametricDispersionandEqualityTests Updated04-Jan-2017 : Slide5
DispersionTest(Ansari-Bradley) Overview
Assumptions
Within sample independence assumption
X ,...,X are iid random sample from population 1
1 m
Y ,...,Y are iid random sample from population 2
1 n
Between sample independence assumption
Samples {X}m and {Y}n are mutually independent
i i=1 i i=1
Continuity assumption: both F and F are continuous distributions
1 2
Location assumption: θ = θ or θ and θ are known
1 2 1 2
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DispersionTest(Ansari-Bradley) Procedure
Parameter of Interest and Hypothesis
Parameter of interest is the ratio of the variances:
V(X)
γ2 =
V(Y)
so that γ2 = 1 whenever V(X) = V(Y).
The null hypothesis about γ2 is
H : γ2 = 1
0
and we could have one of three alternative hypotheses:
One-Sided Upper-Tail: H : γ2 > 1
1
One-Sided Lower-Tail: H : γ2 < 1
1
Two-Sided: H : γ2 (cid:54)= 1
1
NathanielE.Helwig (UofMinnesota) NonparametricDispersionandEqualityTests Updated04-Jan-2017 : Slide7
DispersionTest(Ansari-Bradley) Procedure
Test Statistic
Let {Z }N denote the order statistics for the combined sample, and
(k) k=1
assign rank scores
(cid:26) 1,2,3,..., N, N,...,3,2,1 if N is even
R∗ = 2 2
k 1,2,3,..., N−1, N+1, N−1,...,3,2,1 if N is odd
2 2 2
to the combined sample {Z }N .
(k) k=1
The Ansari-Bradley test statistic C is defined as
n
(cid:88)
C = R
j
j=1
where R is the assigned rank score of Y for j = 1,...,n
j j
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DispersionTest(Ansari-Bradley) Procedure
Distribution of Test Statistic under H
0
Under H all (cid:0)N(cid:1) arrangements of Y-ranks occur with equal probability
0 n
Given (N,n), calculate C for all (cid:0)N(cid:1) possible outcomes
n
Each outcome has probability 1/(cid:0)N(cid:1) under H
n 0
Example null distribution with m = 3 and n = 2:
Y-ranks C ProbabilityunderH
0
1,2 3 1/10
1,3 4 1/10
1,4 3 1/10
1,5 2 1/10
2,3 5 1/10 Note: thereare(cid:0)5(cid:1)=10possibilities
2
2,4 4 1/10
2,5 3 1/10
3,4 5 1/10
3,5 4 1/10
4,5 3 1/10
NathanielE.Helwig (UofMinnesota) NonparametricDispersionandEqualityTests Updated04-Jan-2017 : Slide9
DispersionTest(Ansari-Bradley) Procedure
Hypothesis Testing
One-Sided Upper Tail Test:
H : γ2 = 1 versus H : γ2 > 1
0 1
Reject H if C ≥ c where P(C > c ) = α
0 α α
One-Sided Lower Tail Test:
H : γ2 = 1 versus H : γ2 < 1
0 1
Reject H if C ≤ [c −1]
0 1−α
Two-Sided Test:
H : γ2 = 1 versus H : γ2 (cid:54)= 1
0 1
Reject H if C ≥ c or C ≤ [c −1]
0 α/2 1−α/2
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