Sara Kherad and Olivier Renaud

Transcription

An Exact Permutation Method
for Testing All Effects For Any
Design of ANOVA
Sara Kherad and Olivier Renaud
University of Geneva, Switzerland
Swiss Statistics Meeting, 29 October 2009
Introduction
R.A. Fisher:
ANOVA (1920)
Permutation test (1930)
S. Kherad & O. Renaud
Introduction
R.A. Fisher:
ANOVA (1920)
Permutation test (1930)
What if the parametric conditions do not hold?
parametric ANOVA cannot be used!
Permutation Tests
Permutation Tests
Permutation test or Randomization test.
Permutation Tests
Weaker assumption than parametric one
Permutation Tests
Exchangeability of the observations under null hypothesis
Permutation Tests
f (y1 , y2 , .., yn ) = f (yπ(1) , yπ(2) , ..., yπ(n) )
Exact permutation test
Permutation Tests
f (y1 , y2 , .., yn ) = f (yπ(1) , yπ(2) , ..., yπ(n) )
Some of the parametric tests have the corresponding
Permutation tests, using the same statistic as parametric one
Permutation Approach
U1
Original:
U2
U1
Original:
statistics
U2
Analyze
Statorig = Ū1 − Ū2
U1
Original:
Perm. 1:
statistics
U2
Analyze
U1
Original:
Perm. 1:
statistics
U2
Analyze
Statperm(1) = Ū1∗ − Ū2∗
U1
Original:
Perm. 1:
Perm. 2:
statistics
U2
Analyze
U1
Original:
statistics
U2
Analyze
Perm. 1:
Perm. 2:
U1
Original:
statistics
U2
Analyze
Perm. 1:
Perm. 2:
Perm. B:
Statperm(B) = Ū1∗ − Ū2∗
histogram of permuted statistic distribution
Statperm(i) = Ū1∗ − Ū2∗
Q0.95
histogram of permuted statistic distribution
Statperm(i) = Ū1∗ − Ū2∗
Q0.95
p − value = Pr(statperm ≥ statorig )
!
# i : statperm(i) ≥ statorig
=
B
"
Two Kinds of Permutation Method in ANOVA
Anderson & Ter Braak:
Permutation of raw data or observations
Permutation of raw data or observations
Permutation of some form of residuals
[1] M. Anderson & C. Ter Braak, ``Permutation tests for multi-factorial analysis of variance,''
Journal of Statistical Computation and Simulation T-73(2), 2003.
Permutation of Raw Data
permutation of raw data or observations.
Unrestricted Permutation of raw data. (Manly 1997)
Approximation test
Approximation test
Restricted permutation test. (Good, 1994)
is exact for testing some cases of ANOVA.
there are not enough suitable observations to permute when the number of
sample in each cell is small.
Approximation test
Restricted permutation test. (Good, 1994)
is exact for testing some cases of ANOVA.
there are not enough suitable observations to permute when the number of
sample in each cell is small.
Synchronized permutation test. (Salmaso, 2003)
is exact for some parameter and some designs.
not used for unbalanced designs or small number of observations in each cell.
Permutation of Some Forms of Residulas
Permutation of residuals
Residuals under full model. (Ter Braak, 1992)
Approximation test
Approximation test
Residuals under reduced model. (Still and White, 1981)
Approximation test
Approximation test
Residuals under reduced model. (Still and White, 1981)
Approximation test
Anderson and Ter Braak (2003) and our simulations, showed that in most of
the cases is more powerful method.
How to modify this Approximation test to get an Exact one?
Permutation of Residuals Under Reduced Model
Consider the general model for regression and ANOVA:
y = Xβ + ε
y = Xβ + ε
X = [X1 |X2 ]
β=
!
β1
β2
"
ε ∼ (0, σ 2 IN )
y = Xβ + ε
X = [X1 |X2 ]
β=
!
β1
β2
"
ε ∼ (0, σ 2 IN )
y = X1 β1 + X2 β2 + ε
y = Xβ + ε
X = [X1 |X2 ]
β=
!
β1
β2
"
ε ∼ (0, σ 2 IN )
y = X1 β1 + X2 β2 + ε
H0 : β2 = 0 vs.
H1 : β2 != 0
y = X1 β1 + X2 β2 + ε
(I − H1 )
y = X1 β1 + X2 β2 + ε
(I − H1 )
y = X1 β1 + X2 β2 + ε
H1 = X1 (X1! X1 )−1 X1!
(I − H1 )
y = X1 β1 + X2 β2 + ε
H1 = X1 (X1! X1 )−1 X1!
(I − H1 )
y = X1 β1 + X2 β2 + ε
H1 = X1 (X1! X1 )−1 X1!
yrr = (I − H1 )y, Xrr = (I − H1 )X2 , εrr = (I − H1 )ε
(I − H1 )
y = X1 β1 + X2 β2 + ε
H1 = X1 (X1! X1 )−1 X1!
yrr = Xrr β2 + εrr
(I − H1 )
y = X1 β1 + X2 β2 + ε
H1 = X1 (X1! X1 )−1 X1!
yrr = εrr ∼ (0, σ 2 (I − H1 ))
How Can We Have an Exact Permutation?
yrr
The aim is to make yrr exchangeable to obtain an exact
permutation test.
yrr
permutation test.
yrr
VN ×(N −q)
permutation test.
yrr
VN ×(N −q)
V V ! = IN − H1
and V ! V = IN −q
permutation test.
yrr
VN ×(N −q)
V V ! = IN − H1
V!
and V ! V = IN −q
permutation test.
yrr
VN ×(N −q)
V V ! = IN − H1
V!
and V ! V = IN −q
ymr = Xmr β2 + εmr
permutation test.
yrr
VN ×(N −q)
V V ! = IN − H1
V!
and V ! V = IN −q
ymr = Xmr β2 + εmr
E[ε2mr ]
=V
!
E[ε2rr ]V
= σ IN −q
2
2
y
=
ε
∼
(0,
σ
IN −q )
=⇒ mr
mr
Permutation of Residuals Under Modified Model
εij
iid
or
exchangeable
!
spherical distribution
ymr is weakly exchangeable
ymr is strongly exchangeable
A Geometric View of Three Methods
For testing β2
Fss =
y
yrr
RSS(β̂1 )
RSS(β̂)
X1
RSSrr (β1 )
SS(X2 )
SS(Xrr )
Frr
(RSS(βˆ1 ) − RSS(β̂))/(p − q)
RSS(β̂)/(N − p)
!
!
!
yrr
(Xrr (Xrr
Xrr )−1 Xrr
)yrr /(p − q)
=
! (I − X(X ! X)X ! )y /(N − p)
yrr
rr
ŷX1 ,X2
X2
ŷrr
Fmr
Xrr
!
!
!
ymr
(Xmr (Xmr
Xmr )−1 Xmr
)ymr /(p − q)
= !
! X
−1 X ! )y
ymr (IN −q − Xmr (Xmr
mr )
mr mr /(N − p)
Fmr
!
!
!
ymr
(Xmr (Xmr
Xmr )−1 Xmr
)ymr /(p − q)
= !
! X
−1 X ! )y
mr )
mr mr /(N − p)
Fmr
!
!
!
ymr
(Xmr (Xmr
Xmr )−1 Xmr
)ymr /(p − q)
= !
! X
−1 X ! )y
mr )
mr mr /(N − p)
∗
Fmr
∗ "
"
"
∗
ymr
(Xmr (Xmr
Xmr )−1 Xmr
)ymr
/(p − q)
= ∗ "
" X
−1 X " )y ∗ /(N − p)
mr )
mr mr
Fmr
!
!
!
ymr
(Xmr (Xmr
Xmr )−1 Xmr
)ymr /(p − q)
= !
! X
−1 X ! )y
mr )
mr mr /(N − p)
∗
Fmr
∗ "
"
"
∗
ymr
(Xmr (Xmr
Xmr )−1 Xmr
)ymr
/(p − q)
= ∗ "
" X
−1 X " )y ∗ /(N − p)
mr )
mr mr
∗
#(Fmr
≥ Fmr )
P − value =
B
Simulation Results
Simulation Results
Comparing four methods for a balanced two-way ANOVA with a=2, b=2, and
n=2 to test the interaction effect for different values of t.
Simulation Results
N (0, 1)
√
U (− 3,
√
3)
exp(1) − 1
t(4)
stat.
Ymr
Yrr
Y
F-test
Ymr
Yrr
Y
F-test
Ymr
Yrr
Y
F-test
Ymr
Yrr
Y
F-test
t=0
0.0493
0.0318
0.0615
0.0510
0.0500
0.0390
0.0737
0.0537
0.0507
0.0330
0.0647
0.0520
0.0450
0.0254
0.0618
0.0429
t = 0.5
0.1843
0.1373
0.2308
0.1908
0.0800
0.0647
0.1083
0.0870
0.2450
0.2217
0.3347
0.0520
0.1543
0.1170
0.1827
0.1372
t=1
0.5460
0.4590
0.6260
0.5770
0.2867
0.2483
0.3743
0.3160
0.6147
0.5717
0.6897
0.6007
0.7060
0.6080
0.7220
0.6400
t = 1.5
0.8413
0.7857
0.8942
0.8782
0.6623
0.6130
0.7828
0.7337
0.8270
0.5717
0.8870
0.8193
0.8600
0.7830
0.8540
0.8250
t=2
0.9533
0.9460
0.9815
0.9828
0.9060
0.8977
0.9733
0.9737
0.9019
0.9270
0.9573
0.8983
0.8833
0.8460
0.9115
0.8628
t = 2.5
0.9918
0.9910
0.9988
0.9998
0.9969
0.9917
0.9999
0.9902
0.9928
0.9910
0.9912
0.9902
0.9440
0.9050
0.9460
0.9390
Simulation Results
N (0, 1)
√
U (− 3,
√
3)
exp(1) − 1
t(4)
stat.
Ymr
Yrr
Y
F-test
Ymr
Yrr
Y
F-test
Ymr
Yrr
Y
F-test
Ymr
Yrr
Y
F-test
t=0
0.0493
0.0318
0.0615
0.0510
0.0500
0.0390
0.0737
0.0537
0.0507
0.0330
0.0647
0.0520
0.0450
0.0254
0.0618
0.0429
t = 0.5
0.1843
0.1373
0.2308
0.1908
0.0800
0.0647
0.1083
0.0870
0.2450
0.2217
0.3347
0.0520
0.1543
0.1170
0.1827
0.1372
t=1
0.5460
0.4590
0.6260
0.5770
0.2867
0.2483
0.3743
0.3160
0.6147
0.5717
0.6897
0.6007
0.7060
0.6080
0.7220
0.6400
t = 1.5
0.8413
0.7857
0.8942
0.8782
0.6623
0.6130
0.7828
0.7337
0.8270
0.5717
0.8870
0.8193
0.8600
0.7830
0.8540
0.8250
t=2
0.9533
0.9460
0.9815
0.9828
0.9060
0.8977
0.9733
0.9737
0.9019
0.9270
0.9573
0.8983
0.8833
0.8460
0.9115
0.8628
t = 2.5
0.9918
0.9910
0.9988
0.9998
0.9969
0.9917
0.9999
0.9902
0.9928
0.9910
0.9912
0.9902
0.9440
0.9050
0.9460
0.9390
Application in Analysis of Psychological Data
age<45
age>45
Postman
Prison Guard
Secretary
7
7
5
7
7
7
Ambiguity
Postman
Prison Guard
Secretary
7
7
5
7
7
age<45
7
age>45
Ambiguity
p-value
age
career
Ymr
Yrr
0.089
0.300
0.298
0.198
0.002
0.025
0.028
0.012
Y
F − test
Postman
Prison Guard
Secretary
7
7
5
7
7
age<45
7
age>45
Ambiguity
p-value
age
career
Ymr
Yrr
0.089
0.300
0.298
0.198
0.002
0.025
0.028
0.012
Y
F − test
* Thanks to Katia Iglesias for providing the data used in this analysis,
Summary
Advantages of the method:
An exact permutation test
Could be applied to all ANOVA designs
Suitable for unbalanced designs
Can be used to test all the factors in ANOVA
Thank you!

Sara Kherad and Olivier Renaud

Transcription

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