Directional tests for vector parameters
Nancy Reid
September 29, 2012
with Don Fraser, Nicola Sartori, Anthony Davison
Directional tests
A Symposium in Honor of Jack Kalbfleisch
Parametric models and likelihood
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model f (y; θ),
data y = (y1 , . . . , yn )
log-likelihood function
parameter of interest
θ ∈ Rp
independent observations
`(θ; y ) = log f (y; θ)
θ = (ψ, λ), ψ ∈ Rd
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likelihood inference
w(ψ) = 2{`(ψ̂, λ̂) − `(ψ, λ̂ψ )}
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(1234a)
(a)
M
m
-4
-3
--2
-1
2
1
M
0
Directional tests
A Symposium in Honor of Jack Kalbfleisch
(a)
Example: 2 × 3 contingency table
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contingency table on activity amongst psychiatric patients
(Everitt, 1992 CH)
Affective disorders
12
18
Retarded
Not retarded
Schizophrenics
13
17
Neurotics
5
25
log{E(y )} = X θ,
θ ∈ R6
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model: log-linear y ∼ Poisson,
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log-likelihood `(θ; y) = θ0 X 0 y − 10 eX θ = θ0 s − c(θ)
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θ = (ψ, λ)
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(ψ1 , ψ2 )
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H0 : ψ = ψ0 = (0, 0)
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Directional tests
ψ ∈ R2 , λ ∈ R4
interaction parameters
log-likelihood `(ψ, λ; y) =
A Symposium in Honor of Jack Kalbfleisch
independence
ψ0s
1
+ λ0 s2 − c(ψ, λ)
Testing ψ = 0
(1234a)
(a)
M
m
-4
-3
--2
-1
2
1
M
0
-4
-3
-2
-1
0
1
2
(a)
-4
-3
-2
-1
0
1
2
.
w(ψ0 ) ∼ χ22
0.0
0.5
p-value 0.047
1.0
t
1.5
2.0
directional p-value 0.050
‘exact’ conditional p-value 0.051
Directional tests
A Symposium in Honor of Jack Kalbfleisch
Directional tests
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log-likelihood
`(ψ, λ) = ψ 0 s1 + λ0 s2 − c(ψ, λ)
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Step 1: get rid of nuisance parameter
use conditional density of s1 , given s2
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Step 2: measure the directed departure from H0
sψ : expected value of s1 , under H0
s0 : observed value of s1
line through these two points
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Directional tests
L∗ = ts0 + (1 − t)(sψ − s0 ),
A Symposium in Honor of Jack Kalbfleisch
t ∈R
L∗ = ts0 + (1 − t)sψ
... directional tests
S(t)
2
0
S2
−2
2
−2
−4
0
psi2
4
4
6
Relative log likelihood
−3
−2
−1
0
1
2
3
psi1
−3
−2
−1
0
null hypothesis of independence
observed value of s
R∞
g(t)dt
p-value = R1∞
0 g(t)dt
x
1
2
3
S1
t =0
t =1
like a 2-sided p-value
Pr ( response > observed | response > 0)
Directional tests
A Symposium in Honor of Jack Kalbfleisch
... directional p-value
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R∞
R ∞ d−1
g(t)dt
t
f {s(t); ψ}dt
1
p-value = R ∞
= R1∞ d−1
f {s(t); ψ}dt
0 g(t)dt
0 t
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sample space, dimension p (= 6),
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reduced space Lψ , dimension d (= 2),
L∗ ,
dimension 1,
f (s; θ)
f (s1 ; ψ)
f {s1 (t); ψ}
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line
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directed departure – magnitude, given direction
polar coordinates −→ t d−1
Simplifications
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1: L∗ ⊂ Lψ ⊂ Rp
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2: ratio of two integrals – drop any terms that
don’t depend on t
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3: saddle point approximation to f (s; θ) gives a very
accurate starting point
Directional tests
A Symposium in Honor of Jack Kalbfleisch
Aside: From full likelihood to density along the line
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suppose log-likelihood function `(ϕ; s) = ϕ0 s + `(ϕ; s0 )
such as in exponential families
ϕ = ϕ(θ); θ = (ψ, λ) = (ψ(ϕ), λ(ϕ))
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double saddle point approximation gives
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|jϕϕ {ϕ̂(s); s}|−1/2
.
f̂ (s; ψ) = c exp `{ϕ(θ̂ψ0 ); s}−`{ϕ̂(s); s}
|j(λλ) {ϕ(θ̂ψ0 ; s)}|−1/2
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Directional tests
θ̂ψ0 = (ψ, λ̂0ψ )
ϕ̂(s) from `ϕ (ϕ; s) = 0
uses only likelihood for the full model, plus constrained
maximum likelihood estimator
density along the line: use f {s(t); ψ}, t ∈ R
f̂ (s; ψ) approximates the marginal likelihood
(Kalbfleisch & Sprott, 1970, 1974)
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2 × 3 table
h(t)
10
20
11.0
19.0
0
0.5
1
2
td 1h(t)
0
t
Directional tests
A Symposium in Honor of Jack Kalbfleisch
0.5
1
2
t=0
10
20
10
20
t = 0.5
11.5
7.5
18.5 22.5
12
18
t=1
13
17
5
25
14
16
t=2
16
14
0
30
Simulations, 2 × 3 table
Nominal
LRT
Directional
Skovgaard, 2001
0.010
0.011
0.010
0.010
0.025
0.028
0.024
0.025
0.050
0.055
0.050
0.050
0.100
0.107
0.100
0.101
0.250
0.260
0.250
0.251
Nominal
LRT
Directional
Skovgaard, 2001
0.750
0.757
0.752
0.752
0.900
0.905
0.902
0.900
0.950
0.952
0.950
0.950
0.975
0.974
0.973
0.973
0.990
0.992
0.992
0.991
Directional tests
A Symposium in Honor of Jack Kalbfleisch
0.500
0.510
0.501
0.501
1.00
0.98
0.99
diameter
1.01
Example: comparison of normal variances
1
2
3
4
5
6
batch
F -test
Directional tests
A Symposium in Honor of Jack Kalbfleisch
7
8
9
10
Likelihood ratio statistic
Directional
Skovgaard, 2001
Bartlett’s test
0.0042
0.0389
0.0622
0.0136
Aside: Bartlett correction of likelihood ratio test
(1234a)
(a)
M
m
-4
-3
--2
-1
2
1
M
0
(1234c)
(c)
M
m
-4
-3
--2
-1
2
1
M
0
1
0
-1
-2
-3
-4
-4
-3
-2
-1
0
1
2
(c)
2
(a)
-4
-3
-2
-1
0
1
2
-4
-3
-2
0
1
2
.
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w(ψ) = 2{`(θ̂) − `(θ̂ψ )} ∼ χ2d
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E{w(ψ)} = d{1 +
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w̃(ψ) =
Directional tests
-1
B(ψ)
+ O(n−2 )
n
w(ψ)
.
∼ χ2d
1 + B(ψ)/n
A Symposium in Honor of Jack Kalbfleisch
O(n−3 )
Aside: compare directional test
4
−1
0
1
2
3
−3
−1
0
psi1
t=1
t=2
1
2
3
1
2
3
4
−2
0
2
psi2
4
2
−2
0
psi2
−2
psi1
6
−2
6
−3
Directional tests
2
psi2
−2
0
2
−2
0
psi2
4
6
t=0.5
6
t=0
−3
−2
−1
0
1
A Symposium in Honor of Jack Kalbfleisch
2
3
−3
−2
−1
0
Other examples
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binary regression
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equality of exponential rates
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covariance selection models
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comparison of normal variances
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symmetry restrictions in contingency tables
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...
Directional tests
A Symposium in Honor of Jack Kalbfleisch
Binary regression
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binary response
– presence/absence of bacteria
MASS, §10.4
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covariate of interest
– week: factor variable with levels 0, 2, 4, 6, 11
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nuisance parameter
– subject effects: factor with 50 levels
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108 observations
– 24 nuisance parameters, 4 parameters of interest
Likelihood ratio statistic
Directional
Skovgaard, 2001
Directional tests
A Symposium in Honor of Jack Kalbfleisch
0.0005
0.0053
0.0043
Equality of exponential rates
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300
0
100
200
time
400
500
600
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yij ∼ exp(θi ), j = 1, . . . , ni ; i = 1, . . . , g
H0 : θ1 = · · · = θg
Example: air-conditioning data, Cox & Snell (1981)
1
Directional tests
2
3
4
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5
6
7
8
9
10
... exponential rates
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R ∞ d−1
t
h(t)dt
p-value = R1∞ d−1
h(t)dt
0 t
g
Y
t(ȳ − ȳi ) ni −1
h(t) ∝
1−
ȳ
i=1
Nominal
LRT
Directional
Skovgaard, 2001
0.010
0.013
0.010
0.008
0.025
0.031
0.025
0.020
0.050
0.059
0.049
0.039
0.100
0.115
0.099
0.080
0.250
0.275
0.248
0.210
Nominal
LRT
Directional
Skovgaard, 2001
0.750
0.770
0.750
0.688
0.900
0.910
0.899
0.855
0.950
0.955
0.949
0.918
0.975
0.977
0.975
0.953
0.990
0.990
0.989
0.977
Directional tests
A Symposium in Honor of Jack Kalbfleisch
0.500
0.529
0.500
0.441
Conclusion
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new way to assess vector parameters
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incorporates information in the direction of departure
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easy to compute: two model fits, plus 1-d numerical
integration
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accurate conditionally, by construction, and unconditionally
(simulations)
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can be used in models of practical interest – discrete and
continuous data
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exponential family model not necessary – easily
generalized using approximate exponential family model
Directional tests
A Symposium in Honor of Jack Kalbfleisch
Thank you!
Directional tests
A Symposium in Honor of Jack Kalbfleisch
References
Fraser, D.A.S. and Massam, H. (1985). Conical tests. Stat. Hefte
Skovgaard, I.M. (1988). Saddlepoint expansions for directional test
probabilities. JRSS B
Skovgaard, I.M. (2001). Likelihood asymptotics. Scand. J. Statist. 28,
3–32.
Cheah, P.K., Fraser, D.A.S., Reid, N. (1994). Multiparameter testing in
exponential models. Biometrika
Davison, A.C., Fraser, D.A.S., Reid, N. (2006). Improved likelihood
inference for discrete data. JRSS B
Davison, A.C., Fraser, D.A.S., Reid, N., Sartori, N. (2012). Directional
inference for vector parameters.
Directional tests
A Symposium in Honor of Jack Kalbfleisch