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Asymptotic Properties of Bayes Factor in One- Way Repeated
Measurements Model
Ameera Jaber Mohaisen and Khawla Abdul Razzaq Swadi
Mathematics Department College of Education for Pure Science AL-Basrah University-Iraq
E-mail: [email protected]
Abstract
In this paper, we consider the linear one- way repeated measurements model which has only one within units
factor and one between units factor incorporating univariate random effects as well as the experimental error
term. In this model, we investigate the consistency property of Bayes factor for testing the fixed effects linear
one- way repeated measurements model against the mixed one- way repeated measurements alternative model.
Under some conditions on the prior and design matrix, we identify the analytic form of the Bayes factor and
show that the Bayes factor is consistent.
Keywords: One- Way Repeated Measurements Model, ANOVA, Mixed model, Prior Distribution, Posterior
Distribution, covariance matrix, Bayes Factor, Consistent.
1. Introduction
Repeated measurements occur frequently in observational studies which are longitudinal in nature, and in
experimental studies incorporating repeated measures designs. For longitudinal studies, the underlying
metameter for the occasions at which measurements are taken is usually time. Here, interest often centers around
modeling the response as linear or nonlinear function of time, Repeated measures designs, on the other hand,
entail one or more response variables being measured repeatedly on each individual over arrange of conditions.
Here the metameter may be time or it may be a set of experimental conditions (e.g. ,dose levels of a drug).
Repeated measurements analysis is widely used in many fields, for example , in the health and life science,
epidemiology, biomedical, agricultural, industrial, psychological, educational researches and so on.[2],[3],[17]
A balanced repeated measurements design means that the p occasions of measurements are the same for all of
the experimental units. A complete repeated measurements design means that measurements are available each
time point for each experimental unit. A typical repeated measurements design consist of experimental units or
subjects randomized to a treatment (a between-units factor) and remaining on the assigned treatment throughout
the course of the experiment. [2],[3],[17]
A one โway repeated measurements analysis of variance refers to the situation with only one within-units
factor and a multi-way repeated measurements analysis of variance to the situation with more than one within
units factor. This is because the number of within factors, and not the number or between factor, dictates the
complexity of repeated measurements analysis of variance.[2],[3],[4],[12],[17]
In this paper, we consider the linear one- way repeated measurements model which has only one within units
factor and one between units factor incorporating univariate random effects as well as the experimental error
term, we can represent repeated measurements model as a mixed model.
The asymptotic properties of the Bayes factor have been studied mainly in nonparametric density estimation
problems related to goodness of fit testing [1],[5],[7],[8],[9],[15],[16]. Choi, Taeryon and et al in (2009) [6]
studied the semiparametric additive regression models as the encompassing model with algebraic smoothing and
obtained the closed form of the Bayes factor and studied the asymptotic behavior of the Bayes factor based on
the closed form. Mohaisen, A. J. and Abdulhussain , A. M. in (2013)[11] investigated large sample properties of
the Bayes factor for testing the pure polynomial component of spline null model whose mean function consists
of only the polynomial component against the fully spline semiparametric alternative model whose mean
function comprises both the pure polynomial and the component spline basis functions. In this paper, we
investigate the consistency property of Bayes factor for testing the fixed effects linear one- way repeated
measurements model against the mixed one- way repeated measurements alternative model. Under some
conditions on the prior and design matrix, we identify the analytic form of the Bayes factor and show that the
Bayes factor is consistent.
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2. One- Way Repeated Measurements Model
Consider the model
yijk = ฮผ+ ฯj + ฮดi(j) + ฮณk+ (ฯฮณ)jk + eijk (1)
Where
i=1,โฆ.,n is an index for experimental unit within group j ,
j=1,โฆ,q is an index for levels of the between-units factor (Group) ,
k=1,โฆ,p is an index for levels of the within-units factor (Time) ,
yijk is the response measurement at time k for unit i within group j,
ยต is the overall mean ,
ฯj is the added effect for treatment group j ,
ฮด i(j) is the random effect for due to experimental unit i within treatment group j ,
๐พ๐ is the added effect for time k ,
(ฯฮณ)jk is the added effect for the group j ร time k interaction ,
e ijk is the random error on time k for unit i within group j ,
For the parameterization to be of full rank, we imposed the
following set of conditions
โ ฯj=0qj=1 , โ ฮณ
k=0
pk=1 , โ (ฯฮณ)jk=0
qj=1 for each k=1,โฆ,p
โ (ฯฮณ)jk=0pk=1 for each j=1,โฆ,q .
And we assumed that the eijk and ฮดi(j) are indepndent with
eijk ~ i.i.d N (0,ฯe2) , ฮดij ~ i.i.d N (0,ฯฮด2) (2)
The model (1) is rewritten as follows
๐ = ๐๐ฝ + ๐๐ + ๐ (3)
Where
Y = [
y111y112โฎ
ynqp
]
nqpร1
, ฮฒ = [
ฮผ1ร1ฯqร1ฮณpร1
(ฯฮณ)qpร1
]
(qp+q+p+1)ร1
, Z = (1Pร1 โฏ 0Pร1โฎ โฑ โฎ
0Pร1 โฏ 1Pร1
)
nqpรnq
,
b = ฮด =
[ ฮด1(1)
ฮด1(2)โฎ
ฮดn(q)]
nqร1
, ๐ = [
e111e112โฎenqp
]
nqpร1
and ๐ = (๐1๐ , ๐2
๐ , ๐3๐ , ๐4
๐)๐is an
(๐๐๐ ร (1 + ๐ + ๐ + ๐๐)) design matrix of fixed effects.
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We assume ๐๐๐ < 1 + ๐ + ๐ + ๐๐ + ๐๐, ๐๐๐ + ๐ = 1 + ๐ + ๐ + ๐๐ + ๐๐, (๐ is integer number and greater
than 1 ), and we assume that the design matrix as the following way:
๐๐๐๐๐ ๐๐๐๐ = ๐๐๐๐ผ๐๐๐ โ๐๐๐ โฅ 1 (4)
where:
๐๐๐๐ = [๐ ๐๐] and ๐๐ be the ๐๐๐ ร [๐๐๐ โ (1 + ๐ + ๐ + ๐๐)] matrix, and let ๐ = [๐๐ ๐๐ ].
We would like to choose between a Bayesian mixed one- way repeated measurements model and itโs one- way
repeated measurements model (the model without random effects) by the criterion of the Bayes factor for two
hypotheses,
๐ป0: ๐ = ๐๐ฝ + ๐ i . e ๐ป0: yijk = ฮผ+ ฯj + ฮณk+ (ฯฮณ)jk + eijk versus
๐ป1: ๐ = ๐๐ฝ + ๐๐ + ๐ i . e ๐ป1: yijk = ฮผ+ ฯj + ฮดi(j) + ฮณk+ (ฯฮณ)jk + eijk . (5)
We assume ๐ฝ and ๐ are independent and the prior distribution for the parameters are
๐ฝ ~ ๐(0, โ0) , where
โ0 = ๐๐ฝ2๐ผ1+๐+๐+๐๐ =
[
๐๐2 01ร๐ 01ร๐
0๐ร1 ๐๐2๐ผ๐ร๐ 0๐ร๐
0๐ร1 0๐ร๐ ๐๐พ2๐ผ๐ร๐
01ร๐๐ 0๐ร๐๐ 0๐ร๐๐
0๐๐ร1 0๐๐ร๐ 0๐๐ร๐ ๐(๐๐พ)2 ๐ผ๐๐ร๐๐]
(1+๐+๐+๐๐)ร(1+๐+๐+๐๐)
, (6)
and ๐ ~ ๐(0, โ1) , where
โ1 = ๐๐2๐ผ๐๐ = ๐๐ฟ
2๐ผ๐๐ =
[ ๐๐ฟ112 0 โฏ 0
0 ๐๐ฟ122 โฏ 0
โฎ0
โฎ0
โฑ โฎโฏ ๐๐ฟ๐๐
2]
๐๐ร๐๐
. (7)
2.Posterior distribution
From the model (1), we have the following:
๐๐๐๐|๐๐๐๐ ~ ๐๐๐๐(๐๐๐๐, ๐๐2๐ผ๐๐๐) , ๐๐๐๐ ~ ๐๐๐๐(0๐๐๐, ๐โ0๐
๐ + ๐โ1๐๐) (8)
where
๐๐๐๐ = ฮผ+ ฯj + ฮดi(j) + ฮณk+ (ฯฮณ)jk + eijk,for, ๐ = 1,2, โฆ , ๐, ๐ = 1,โฆ , ๐ ๐๐๐ ๐ = 1,โฆ , ๐.
Then, the posterior distribution ๐1(๐๐๐๐|๐๐๐๐) is the multivariate normal with
๐ธ(๐๐๐๐|๐๐๐๐ , ๐๐2) = โ2(โ2 + ๐๐
2๐ผ๐)โ1๐๐๐๐ , where N=1+q+p+qp+nq and
๐๐๐(๐๐๐๐|๐๐๐๐, ๐๐2) = (โ2
โ1 + (๐๐2๐ผ๐)
โ1)โ1 = โ2(โ2 + ๐๐2๐ผ๐ )
โ1๐๐2๐ผ๐
where
โ2 = ๐โ0๐๐ + ๐โ1๐
๐ = ๐
[
๐๐2 01ร๐ 01ร๐
0๐ร1 ๐๐2๐ผ๐ร๐ 0๐ร๐
0๐ร1 0๐ร๐ ๐๐พ2๐ผ๐ร๐
01ร๐๐ 0๐ร๐๐ 0๐ร๐๐
0๐๐ร1 0๐๐ร๐ 0๐๐ร๐ ๐(๐๐พ)2 ๐ผ๐๐ร๐๐]
๐๐ + ๐
[ ๐๐ฟ112 0 โฏ 0
0 ๐๐ฟ122 โฏ 0
โฎ0
โฎ0
โฑ โฎโฏ ๐๐ฟ๐๐
2]
๐๐
Then, ๐๐๐๐~ ๐๐๐(0, ๐๐2๐ผ๐๐๐ + โ2). (9)
Hence, the distribution of ๐๐๐๐ under ๐ป0 and ๐ป1 are respectively ๐๐๐๐(0, ๐๐2๐ผ๐๐๐ + ๐โ0๐
๐) and
๐๐๐๐(0, ๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐). (10)
We investigate the consistency of the Bayes factor, under the one- way repeated measurements model with
design matrix of the random effects (๐) to the first ๐๐๐ terms. We get
yijk = ฮผ + ฯj + ฮดKโ + ฮณk+ (ฯฮณ)jk + eijk , K
โ = 1,โฆ . , (๐๐๐ โ (1 + ๐ + ๐ + ๐๐)).The covariance matrix of the
distribution of ๐๐๐๐ under ๐ป1 as:
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๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐ = ๐๐
2๐ผ๐๐๐ +๐๐๐๐๐ท๐๐๐๐๐๐๐๐ (11)
where โ1,๐=๐๐2๐ผ๐๐๐โ(1+๐+๐+๐๐) =
[ ๐๐ฟ112 0 โฏ 0
0 ๐๐ฟ122 โฏ 0
โฎ0
โฎ0
โฑ โฎโฏ ๐๐ฟ๐๐๐โ(1+๐+๐+๐๐)
2]
๐๐๐โ(1+๐+๐+๐๐)ร๐๐๐โ(1+๐+๐+๐๐)
= ๐๐ฟ2๐ผ๐๐๐โ(1+๐+๐+๐๐) and ๐ท๐๐๐ = [
โ0 0๐101๐ โ1,๐
] , where 0๐1, 01๐ are the matrices for zeros element with size
(1 + ๐ + ๐ + ๐๐) ร (๐๐๐ โ (1 + ๐ + ๐ + ๐๐)) and (๐๐๐ โ (1 + ๐ + ๐ + ๐๐)) ร (1 + ๐ + ๐ + ๐๐) ,
respectively.
Then, we can rewrite (11) by using ๐ท๐๐๐ as:
๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]๐๐๐๐
๐ . (12)
Then, the covariance matrix of the distribution of ๐๐๐๐ under ๐ป0, can be represented as:
๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐0๐๐๐๐๐ = ๐๐
2๐ผ๐๐๐ +๐๐๐๐๐ถ๐๐๐๐๐๐๐๐ , where:
๐ถ๐๐๐ = [โ0 0๐101๐ 0๐๐
]
and 0๐๐ is the matrix for zeros element with size (๐๐๐ โ (1 + ๐ + ๐ + ๐๐)) ร (๐๐๐ โ (1 + ๐ + ๐ + ๐๐)), then
we can rewrite the covariance matrix of the distribution of ๐๐๐๐ under ๐ป0, the following:
๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]๐๐๐๐
๐ (13)
and the covariance matrix of the distribution of ๐๐๐๐ under ๐ป1, can be represented as:
๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐ = ๐๐
2๐ผ๐๐๐ + [๐ ๐]๐บ๐๐๐[๐ ๐]๐, where:
๐บ๐๐๐ = [โ0 0๐101๐ โ1
]
then we can rewrite the covariance matrix of the distribution of ๐๐๐๐ under ๐ป1, the following:
[๐ ๐] [๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐] [๐ ๐]
๐ (14)
Lemma(1):Let the covariance matrices of the distribution of ๐๐๐๐ under ๐ป๐ and ๐ป1 be defined as in (12) and
(13) respectively, and suppose ๐๐๐๐satisfy (4). Then, the following hold.
1 โ๐๐๐๐๐๐๐๐๐ = ๐๐๐๐
๐ ๐๐๐๐ = ๐๐๐๐ผ๐๐๐
2 โ det(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐) = ๐๐๐1+๐+๐+๐๐(๐๐2
๐๐๐+ ๐๐
2)(๐๐2
๐๐๐+ ๐๐
2)๐(๐๐2
๐๐๐+ ๐๐พ
2)๐(๐๐2
๐๐๐+ ๐(๐๐พ)
2 )๐๐
(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
3 โ det(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐) =
(๐๐๐)๐๐๐ (๐๐2
๐๐๐+ ๐๐
2) (๐๐2
๐๐๐+ ๐๐
2)๐
(๐๐2
๐๐๐+ ๐๐พ
2)๐
(๐๐2
๐๐๐+ ๐(๐๐พ)
2 )๐๐
(๐๐2
๐๐๐+ ๐๐ฟ
2)๐๐๐โ(1+๐+๐+๐๐)
4 โ (๐๐2๐ผ๐๐๐ + ๐๐ด0๐
๐)โ1 = 1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐๐
5 โ (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1 =
1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐
Proof:-
1 โ By multiplying ๐๐๐๐ and ๐๐๐๐๐ to equation (4) from right and left side, we have:-
๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐
๐ = ๐๐๐๐๐๐๐๐ผ๐๐๐๐๐๐๐๐
= ๐๐๐๐๐๐๐๐๐๐๐๐
= ๐๐๐๐๐๐๐๐๐๐๐๐
Multiplying (๐๐๐๐๐๐๐๐๐ )โ1 to the above equation from left side, we get:-
๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐
๐ (๐๐๐๐๐๐๐๐๐ )โ1 = ๐๐๐๐๐๐๐๐๐๐๐
๐ (๐๐๐๐๐๐๐๐๐ )โ1
๐๐๐๐๐๐๐๐๐ = ๐๐๐๐ผ๐๐๐
2 โ ๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ = ๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]๐๐๐๐
๐ , and by using (4), we get
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= [๐ , ๐๐]
[ (
๐๐2
๐๐๐+ ๐๐
2) 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 (๐๐2๐ผ๐
๐๐๐+ ๐๐
2๐ผ๐) 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
(๐๐2๐ผ๐
๐๐๐+ ๐๐พ
2๐ผ๐) 0๐ร๐๐ 0๐ร๐๐
0๐๐ร๐ (๐๐2๐ผ๐๐
๐๐๐+ ๐(๐๐พ)
2 ๐ผ๐๐) 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐๐๐2
๐๐๐๐ผ๐๐๐โ(1+๐+๐+๐๐)]
[๐ , ๐๐]๐
=
[ ๐๐๐(
๐๐2
๐๐๐+ ๐๐
2) 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 ๐๐๐(๐๐2๐ผ๐
๐๐๐+ ๐๐
2๐ผ๐) 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
๐๐๐(๐๐2๐ผ๐
๐๐๐+ ๐๐พ
2๐ผ๐) 0๐ร๐๐ 0๐ร๐๐
0๐๐ร๐ ๐๐๐(๐๐2๐ผ๐๐
๐๐๐+ ๐(๐๐พ)
2 ๐ผ๐๐) 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ๐๐2๐ผ๐๐๐โ(1+๐+๐+๐๐)]
Then
det(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐) = ๐๐๐1+๐+๐+๐๐(๐๐2
๐๐๐+ ๐๐
2)(๐๐2
๐๐๐+ ๐๐
2)๐(๐๐2
๐๐๐+ ๐๐พ
2)๐(๐๐2
๐๐๐+ ๐(๐๐พ)
2 )๐๐
(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
3 โ ๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐ = ๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]๐๐๐๐
๐
= [๐ ๐๐]
[ (๐๐2
๐๐๐+ ๐๐
2) 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 (๐๐2๐ผ๐๐๐๐
+ ๐๐2๐ผ๐) 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
(๐๐2๐ผ๐๐๐๐
+ ๐๐พ2๐ผ๐) 0๐ร๐๐ 0๐ร๐๐
0๐๐ร๐ (๐๐2๐ผ๐๐๐๐๐
+ ๐(๐๐พ)2 ๐ผ๐๐) 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ (๐๐2
๐๐๐๐ผ๐๐๐โ(1+๐+๐+๐๐) + ๐๐ฟ
2๐ผ๐๐๐โ(1+๐+๐+๐๐))]
[๐ ๐๐]๐
=
[ ๐๐๐(
๐๐2
๐๐๐+ ๐๐
2) 01ร๐ 01ร๐ 01ร๐ 01ร๐
0๐ร1 ๐๐๐(๐๐2๐ผ๐๐๐๐
+ ๐๐2๐ผ๐) 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
๐๐๐(๐๐2๐ผ๐๐๐๐
+ ๐๐พ2๐ผ๐) 0๐ร๐๐ 0๐ร๐๐
0๐๐ร๐ ๐๐๐(๐๐2๐ผ๐๐๐๐๐
+ ๐(๐๐พ)2 ๐ผ๐๐) 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ๐๐๐(๐๐2
๐๐๐๐ผ๐๐๐โ(1+๐+๐+๐๐) + ๐๐ฟ
2๐ผ๐๐๐โ(1+๐+๐+๐๐))]
.
Then
det(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐) = (๐๐๐)๐๐๐(
๐๐2
๐๐๐+ ๐๐
2)(๐๐2
๐๐๐+ ๐๐
2)๐(๐๐2
๐๐๐+ ๐๐พ
2)๐(๐๐2
๐๐๐+ ๐(๐๐พ)
2 )๐๐(๐๐2
๐๐๐+
๐๐ฟ2)๐๐๐โ(1+๐+๐+๐๐)
4 โ (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐)โ1 = {๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]๐๐๐๐
๐ }โ1
= ๐๐๐๐๐ โ1
[๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐โ1
=๐ผ๐๐๐โ1
๐๐๐๐๐๐๐ผ๐๐๐๐๐๐๐
๐ โ1[๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐โ1๐๐๐๐ผ๐๐๐
๐ผ๐๐๐โ1
๐๐๐
=1
(๐๐๐)2๐๐๐๐๐๐๐๐
๐ ๐๐๐๐๐ โ1
[๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐โ1๐๐๐๐๐๐๐๐
๐
= 1
(๐๐๐)2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐๐ (by lemma 1)
5 โ (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1 = {๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]๐๐๐๐
๐ }โ1
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= ๐๐๐๐๐ โ1
[๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐โ1
=๐ผ๐๐๐โ1
๐๐๐๐๐๐๐ผ๐๐๐๐๐๐๐
๐ โ1[๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐โ1๐๐๐๐ผ๐๐๐
๐ผ๐๐๐โ1
๐๐๐
=1
(๐๐๐)2๐๐๐๐๐๐๐๐
๐ ๐๐๐๐๐ โ1
[๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐โ1๐๐๐๐๐๐๐๐
๐
= 1
(๐๐๐)2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ . โ
3- The Bayes factor
The Bayes factor for testing problem (5) is given by:
๐ต01(yijk) = m(yijk|๐ป๐)
m(yijk|๐ป1)
=
1
โ2๐๐๐๐1+๐+๐+๐๐(๐๐2
๐๐๐+๐๐2 )(
๐๐2
๐๐๐+๐๐2)๐(
๐๐2
๐๐๐+๐๐พ2)๐(
๐๐2
๐๐๐+๐(๐๐พ)2 )๐๐(๐๐
2)๐๐๐โ(1+๐+๐+๐)
exp{โ1
2๐๐๐๐๐ (
1
๐๐๐2๐๐๐๐[
๐๐2
๐๐๐๐ผ๐๐๐+ ๐ถ๐๐๐]
โ1
๐๐๐๐๐ )๐๐๐๐}
1
โ2๐(๐๐๐)๐๐๐(๐๐2
๐๐๐+๐๐2)(
๐๐2
๐๐๐+๐๐2)
๐
(๐๐2
๐๐๐+๐๐พ2)
๐
(๐๐2
๐๐๐+๐(๐๐พ)2 )
๐๐
(๐๐2
๐๐๐+๐๐ฟ2)๐๐
exp{โ1
2๐๐๐๐๐ (
1
๐๐๐2[๐ ๐][
๐๐2
๐๐๐๐ผ๐๐๐+ ๐บ๐๐๐]
โ1
[๐ ๐]๐)๐๐๐๐}
=
โ(๐๐๐)๐๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐
(๐๐2
๐๐๐+๐๐พ
2)๐
(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐
(๐๐2
๐๐๐+๐๐ฟ
2)๐๐
โ๐๐๐1+๐+๐+๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
exp{ โ1
2๐๐๐2๐๐๐๐๐
(๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐๐ โ[๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐]
โ1
[๐ ๐]๐)๐๐๐๐}. (15)
Now let the approximation of ๐ต01 with design matrix of the random effects (๐) to the first ๐๐๐ terms as
following
๏ฟฝฬ๏ฟฝ01 =
โ(๐๐๐)๐๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2
๐๐๐+๐๐ฟ
2)๐๐๐โ(1+๐+๐+๐๐)
โ๐๐๐1+๐+๐+๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
exp{โ1
2๐๐๐2๐๐๐๐๐
(๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐๐ โ๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ )๐๐๐๐}. (16)
Then,
log๏ฟฝฬ๏ฟฝ01 = 1
2log
(๐๐๐)๐๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2
๐๐๐+๐๐ฟ
2)
๐๐๐โ(1+๐+๐+๐๐)
๐๐๐1+๐+๐+๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
โ 1
2๐๐๐2๐๐๐๐๐
(๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐๐ โ๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ )๐๐๐๐
2 log๏ฟฝฬ๏ฟฝ01 = log(๐๐๐)๐๐๐(
๐๐2
๐๐๐+๐๐ฟ
2)
๐๐๐โ(1+๐+๐+๐๐)
๐๐๐1+๐+๐+๐๐(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
โ 1
๐๐๐2๐๐๐๐๐ (๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐๐
โ๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ )๐๐๐๐
โ 2 log๏ฟฝฬ๏ฟฝ01 = log(๐๐๐)๐๐๐(
๐๐2
๐๐๐+๐๐ฟ
2)๐๐๐โ(1+๐+๐+๐๐)
๐๐๐1+๐+๐+๐๐(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
โ1
(๐๐๐)2 {๐๐๐๐
๐ ๐๐๐๐ {[๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
โ [๐๐2
๐๐๐๐ผ๐๐๐ +
๐ท๐๐๐]โ1
}๐๐๐๐๐ ๐๐๐๐}
= log(๐๐๐)๐๐๐โ(1+๐+๐+๐๐)(
๐๐2
๐๐๐+๐๐ฟ
2)
๐๐๐โ(1+๐+๐+๐๐)
(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
โ1
(๐๐๐)2{๐๐๐๐๐ ๐๐๐๐
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[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐ 0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)2๐๐ฟ
2
(๐๐2+๐๐๐๐๐ฟ
2)๐๐2 )๐ผ๐๐๐โ(1+๐+๐+๐๐)]
๐๐๐ร๐๐๐
๐๐๐๐๐ ๐๐๐๐}
= log(๐๐2+๐๐๐๐๐ฟ
2
๐๐2 )๐๐๐โ(1+๐+๐+๐๐) โ {๐๐๐๐
๐ ๐๐๐๐๐๐๐๐}
where
๐๐๐๐ = 1
(๐๐๐)2๐๐๐๐
[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐ 0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)2๐๐ฟ
2
(๐๐2+๐๐๐๐๐ฟ
2)๐๐2 )๐ผ๐๐๐โ(1+๐+๐+๐๐)]
๐๐๐ร๐๐๐
๐๐๐๐๐ .
The Bayes factor for testing the model in (5) satisfy is consistent if [6]
๐๐๐๐๐๐โโ ๐ต01 = โ , ๐๐๐๐๐ ๐ก ๐ ๐ข๐๐๐๐ฆ ๐๐ ๐๐๐๐๐๐๐๐๐ก๐ฆ ๐0โ under ๐ป0, (17)
๐๐๐๐๐๐โโ ๐ต01 = 0 , ๐๐๐๐๐ ๐ก ๐ ๐ข๐๐๐๐ฆ ๐๐ ๐๐๐๐๐๐๐๐๐ก๐ฆ ๐1โ under ๐ป1, (18)
Lemma(2):let ๐๐๐๐ = (๐ฆ111 , โฆ , ๐ฆ๐๐๐)๐ where ๐ฆ๐๐๐ โฒ๐ are independent normal random variable
with mean ฮผ + ฯj + ฮณk+ (ฯฮณ)jk and variance ๐๐
2 = 1, then there exist a positive constant ๐ถ1 such that 1
๐๐๐๐,1[โ log(1 + ๐๐๐๐๐ฟ
2) โ ๐๐๐โ(1+๐+๐+๐๐)๐=1 ๐๐๐๐
๐ ๐๐๐๐๐๐๐๐] > ๐ถ1 , with probability tending to 1, i.e. log๏ฟฝฬ๏ฟฝ01
๐๐๐โโโ โ, in probability, where Snqp,1 = โ
nqpฯฮด2
(1+nqpฯฮด2)
nqpi=1 .
Proof:
Let Xnqp = ( X111, X121, โฆ , Xnqp) are independent standard normal random variables, Note that Ynqp Xnqp=d +
๐๐ฝ and
YnqpT QnqpYnqp = (Xnqp + ๐๐ฝ)
TQnqp (Xnqp + ๐๐ฝ)
= XnqpT QnqpXnqp + (๐๐ฝ)
TQnqp(๐๐ฝ) + XnqpT Qnqp(๐๐ฝ) + (๐๐ฝ)
TQnqpXnqp.
By the property of the design matrix (4) get
(๐๐ฝ)TQnqp = (๐๐ฝ)T 1
(nqp)2 Mnqp
[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐ 0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)2๐๐ฟ
2
(๐๐2+๐๐๐๐๐ฟ
2)๐๐2 )๐ผ๐๐๐โ(1+๐+๐+๐๐)]
MnqpT =
1
(๐๐๐)2[๐, ๐, ๐พ, (๐๐พ)]๐๐๐ (๐ผ1+๐+๐+๐๐ , 0๐๐๐โ(1+๐+๐+๐๐)
[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐ 0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)
2๐๐ฟ2
(๐๐2+๐๐๐๐
๐ฟ
2)๐๐2 )๐ผ๐๐๐โ(1+๐+๐+๐๐)
]
๐๐๐(๐ผ1+๐+๐+๐๐ , 0๐๐โ(1+๐+๐+๐๐) [๐, ๐, ๐พ, (๐๐พ)]๐ = 0.
Then YnqpT QnqpYnqp = Xnqp
T QnqpXnqp.
Now by using the expectation formula of the quadratic form[4],[10],[12],[14], get
0 < ๐ธ(XnqpT QnqpXnqp) = tr(Qnqp) = โ
nqpฯฮด2
(1+nqpฯฮด2)
nqpi=1 = Snqp,1
Note that
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Qnqp2 =
1
(nqp)4Mnqp
[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐ 0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)2๐๐ฟ
2
(๐๐2+๐๐๐๐๐ฟ
2)๐๐2 )๐ผ๐๐๐โ(1+๐+๐+๐๐)]
MnqpT Mnqp
[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐ 0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)
2๐๐ฟ2
(๐๐2+๐๐๐๐
๐ฟ
2)๐๐2 )๐ผ๐๐๐โ(1+๐+๐+๐๐)
]
MnqpT
= 1
(nqp)3Mnqp
[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐ 0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)2๐๐ฟ
2
(๐๐2+๐๐๐๐๐ฟ
2)๐๐2 )
2๐ผ๐๐๐โ(1+๐+๐+๐๐) ]
MnqpT
and
tr(Qnqp2 ) =
1
(nqp)2โ (
(nqp)2ฯฮด2
1+nqpฯฮด2)2nqp
i=1 = โ (nqpฯฮด
2
1+nqpฯฮด2)2 = nqp3(
ฯฮด2
1+nqpฯฮด2)2 = Snqp,2
nqpi=1 .
Using the variance formula of the quadratic form[4],[10],[12],[14], get
var(XnqpT QnqpXnqp) = 2tr(Qnqp
2 ) = 2Snqp,2 = 2โ (nqpฯฮด
2
1+nqpฯฮด2)2nqp
i=1 .
Let cnqp = Snqp,3โSnqp,1
2Snqp,1 , where Snqp,3 = โ log(1 + ๐๐๐๐๐ฟ
2)๐๐๐๐=1 = ๐๐๐ log(1 + ๐๐๐๐๐ฟ
2) . Then,
Pr { 1
Snqp,1[โ log(1 + nqpฯฮด
2)2 โ nqpโ(1+q+p+qp)i=1 Ynqp
T QnqpYnqp] โค cnqp } = Pr [ {YnqpT QnqpYnqp}
Snqp,1 โฅ
Snqp,3
Snqp,1โ
cnqp ]
= Pr [ {YnqpT QnqpYnqp}
Snqp,1โ
E(YnqpT QnqpYnqp)
Snqp,1โฅSnqp,3
Snqp,1โ cnqp โ
E(YnqpT QnqpYnqp)
Snqp,1 ]
= Pr [ 1
Snqp,1{(Ynqp
T QnqpYnqp) โ E(YnqpT QnqpYnqp)} โฅ
Snqp,3โE(YnqpT QnqpYnqp)
Snqp,1โ cnqp ]
โค1
Snqp,12
var(YnqpT QnqpYnqp)
(Snqp,2โSnqp,1
2Snqp,1)2
= 8Snqp,2
(Snqp,3โSnqp,1)2
nqpโโโ 0, from the Chebyshev inequality[13] .
Since Snqp,3 โ Snqp,1 โฅ c1Snqp,1 for some c1 > 0 and let cnqp โฅ c1/2.
Then, there exists a positive constant C1 = c1/2 such as 1
Snqp,1[โ log(1 + nqpฯฮด
2)2 โ nqpโ(1+q+p+qp)i=1 Ynqp
T QnqpYnqp] > C1, with probability tending to 1.
Then,
2logBฬ01 = โ log(1 + nqpฯฮด2)2 โ
nqpโ(1+q+p+qp)i=1 Ynqp
T QnqpYnqpnqpโโโ โ in probability under H0 i.e.
logBฬ01nqpโโโ โ
Lemma 3: Suppose we have the one- way repeated measurements model, then
1
๐๐๐๐,1๐ก๐ (
1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ โ [๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐]
โ1
[๐ ๐]๐)๐๐๐โโโ 0
Proof:
Since[๐ ๐] [๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐] [๐ ๐]
๐ โ๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]๐๐๐๐
๐ is the covariance matrix of
๐โ = ๐ฟ๐พโ = ๐๐ ๐, ๐พโ = ๐๐๐ โ (1 + ๐ + ๐ + ๐๐), โฆ , ๐๐๐ + ๐ , it is a positive definite matrix. Thus
1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ โ
1
๐๐๐2[๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐]
โ1
[๐ ๐]๐is also a positive definite matrix.
Thus, ๐ก๐ (1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ ) โฅ ๐ก๐(
1
๐๐๐2[๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐]
โ1
[๐ ๐]๐ ), also
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151
๐ก๐ (๐๐ ๐ 1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ ) โค โ๐ก๐(๐๐
๐๐๐ )๐ก๐ (1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ )
= โ๐๐ โ โ1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ โ by the Cauchy-Schwarz inequality, where
โ1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ โ = { ๐ก๐ ((
1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]๐๐๐๐
๐ )โ2)}1
2
= [1
๐๐๐2{ ((
๐๐2
๐๐๐+ ๐๐
2) + โ (๐๐2
๐๐๐+ ๐๐
2) + โ (๐๐2
๐๐๐+ ๐๐พ
2) + โ (๐๐2
๐๐๐+ ๐(๐๐พ)
2 )๐๐๐=1
๐๐=1
๐๐=1 )
โ2
+
โ (๐๐2
๐๐๐+ ๐๐ฟ
2)โ2
๐๐๐โ(1+๐=๐+๐๐)๐=1 }]
1
2
.
Since
((๐๐2
๐๐๐+ ๐๐
2) + โ (๐๐2
๐๐๐+ ๐๐
2) + โ (๐๐2
๐๐๐+ ๐๐พ
2) + โ (๐๐2
๐๐๐+ ๐(๐๐พ)
2 )๐๐๐=1
๐๐=1
๐๐=1 )
โ2
= 1
((๐๐2
๐๐๐+๐๐
2)+๐(๐๐2
๐๐๐+๐๐
2)+๐(๐๐2
๐๐๐+๐๐พ
2)+๐๐(๐๐2
๐๐๐+๐(๐๐พ)
2 ))
2 =
๐๐๐2
((๐๐2+๐๐๐๐๐
2)+๐(๐๐2+๐๐๐๐๐
2)+๐(๐๐2+๐๐๐๐๐พ
2)+๐๐(๐๐2+๐๐๐๐(๐๐พ)
2 ))2 โค
๐๐๐2
((๐๐๐๐๐2)+๐(๐๐๐๐๐
2)+๐(๐๐๐๐๐พ2)+๐๐(๐๐๐๐(๐๐พ)
2 ))2
= 1
((๐๐2)+๐(๐๐
2)+๐(๐๐พ2)+๐๐(๐(๐๐พ)
2 ))2 = ๐(1), and
โ (๐๐2
๐๐๐+ ๐๐ฟ
2)โ2
๐๐๐โ(1+๐+๐+๐๐)๐=1 =
๐๐๐โ(1+๐+๐+๐๐)
(๐๐2
๐๐๐+๐๐ฟ
2)2 โค
๐๐๐
(๐๐2
๐๐๐+๐๐ฟ
2)2 =
(๐๐๐)3
(๐๐2+๐๐๐๐๐ฟ
2)2 โค
(๐๐๐)3
(๐๐๐๐๐ฟ2)2 = ๐(๐๐๐).
Then, โ1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ โ = ๐{(๐๐๐)
1
2 }.
Since ๐ ๐ข๐โ๐๐ โ = 1 from the distances between the elements of design matrix ๐, then โ๐๐ โ2 โค ๐๐๐.
Thus , 1
๐๐๐๐,1๐ก๐ (
1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ โ
1
๐๐๐2[๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐]
โ1
[๐ ๐]๐) =
1
๐๐๐๐,1๐ก๐ (
1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ {
1
๐๐๐2[๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐] [๐ ๐]
๐ โ1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ +
๐ท๐๐๐]๐๐๐๐๐ } [๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐]
โ1
[๐ ๐]๐ )
= 1
๐๐๐๐,1โ ๐๐ฟ
2(๐๐พโ๐ 1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ (๐๐พโ
๐ 1
๐๐๐2[๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ +
๐๐๐+๐ ๐พโ=๐๐๐โ(1+๐+๐+๐๐)
๐บ๐๐๐]โ1
[๐ ๐]๐)๐) โค ๐๐ฟ2
๐๐๐๐,1โ
1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ โ
2
โ โ๐ง๐พโโ2๐๐๐+๐
๐พโ=๐๐๐โ(1+๐+๐+๐๐)
โค๐๐๐๐๐ฟ
2
๐๐๐๐๐๐๐,1โ 1๐๐๐+๐ ๐พโ=๐๐๐โ(1+๐+๐+๐๐) โค
๐พโ๐๐ฟ2
๐๐๐ ๐๐๐๐,1=
๐พโ๐๐ฟ2
(๐๐๐)2๐๐ฟ2
(1+๐๐๐๐๐ฟ2)
=๐พโ(1+๐๐๐๐๐ฟ
2)
(๐๐๐)2
Thus
1
๐๐๐๐,1๐ก๐ (
1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ โ
1
๐๐๐2[๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐]
โ1
[๐ ๐]๐) โค ๐พโ(1+๐๐๐๐๐ฟ
2)
(๐๐๐)2
๐๐๐โโโ 0
Theorem(1): Suppose that the true model is the one- way repeated measurements model with fixed effect
only (the model under ๐ป0 in 5), and let ๐0๐๐๐
denote the true distribution of the data, with probability density
function ๐0( yijk|ฮผ, ฯj, ฮณk, (ฯฮณ)jk) = ๐ท{(๐ฆ๐๐๐ โ ฮผ โ ฯj โ ฮณkโ (ฯฮณ)jk)/ ๐๐} , where ๐ท(โ) is the standard normal
density, then, the Bayes factor is consistent under the null hypothesis ๐ป0 i.e
lim๐๐๐โโ ๐ต01 = โ in probability ๐0๐๐๐
.
Proof:
log ๐ต01 = 1
2log
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐)โ1
2๐๐๐๐๐ ( (๐๐
2๐ผ๐๐๐ + ๐โ0๐๐)โ1โ(๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐โ1๐
๐)โ1)๐๐๐๐
=1
2log
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)+
1
2log
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐)โ1
2๐๐๐๐๐ (๐๐
2๐ผ๐๐๐ + ๐โ0๐๐) โ1
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152
โ(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1)๐๐๐๐ โ
1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1โ(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ +
๐โ1๐๐)โ1)๐๐๐๐
= log๏ฟฝฬ๏ฟฝ01 +1
2log
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐) โ
1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1โ(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ +
๐โ1๐๐)โ1)๐๐๐๐
Since (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐) โ (๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐) is a positive definite matrix,
๐๐๐ก (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐) โฅ ๐๐๐ก (๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐), thus
1
2log
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐) โฅ 0
๐ธ( ๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1โ(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1)๐๐๐๐) = ๐๐
2 ๐ก๐((๐๐2๐ผ๐๐๐ +
๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1 โ (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1) + (๐๐ฝ)๐((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1 โ
(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1)(๐๐ฝ) ,
since (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1 โ (๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐โ1๐
๐)โ1 , (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1
and (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1 are all nonnegative definite matrices, it follows that
0 โค (๐๐ฝ)๐((๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1 โ (๐๐
2๐ผ๐๐๐ + (๐โ0๐๐ + ๐โ1๐
๐)โ1)(๐๐ฝ) โค (๐๐ฝ)๐
(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1 (๐๐ฝ).
By the property of design matrix (4), we get
๐๐๐๐๐๐ = ๐๐๐๐๐ ๐ = ๐๐๐ (๐ผ1+๐+๐+๐๐ , 0๐๐๐โ(1+๐+๐+๐๐).
Now as the proof of lemma (3), we get:
(๐๐ฝ)๐(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1(๐๐ฝ) =
1
(๐๐๐)2(๐๐ฝ)๐๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ (๐๐ฝ)
=1
(๐๐๐)2[๐, ๐, ๐พ, (๐๐พ)]๐๐๐ (๐ผ1+๐+๐+๐๐ , 0๐๐โ(1+๐+๐+๐๐)
[ (๐๐2
๐๐๐+ ๐๐
2) 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 (๐๐2
๐๐๐+ ๐๐
2)๐ผ๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
(๐๐2
๐๐๐+ ๐๐พ
2)๐ผ๐ 0๐ร๐๐ 0๐ร๐๐
0๐๐ร๐ (๐๐2
๐๐๐+ ๐(๐๐พ)
2 )๐ผ๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ (๐๐2
๐๐๐+ ๐๐ฟ
2)๐ผ๐๐๐โ(1+๐+๐+๐๐) ] โ1
๐๐๐
(๐ผ1+๐+๐+๐๐ , 0๐๐โ(1+๐+๐+๐๐) [๐, ๐, ๐พ, (๐๐พ)]๐
= ๐2 [๐๐2
๐๐๐+ ๐๐
2]โ1
+ โ ๐๐2๐
๐=1 [๐๐2
๐๐๐+ ๐๐
2]โ1
+ โ ๐พ๐2๐
๐=1 [๐๐2
๐๐๐+ ๐๐พ
2]โ1
+ โ (๐๐พ)๐2๐๐
๐=1 [๐๐2
๐๐๐+ ๐(๐๐พ)
2 ]โ1
= ๐(1).
Since ๐๐ { 1
๐๐๐๐,1๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1โ(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1)๐๐๐๐ > ํ } โค
1
๐๐๐๐,1 {
๐ธ( ๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐+๐โ0๐๐+๐๐โ1,๐๐๐
๐)โ1โ(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)โ1)๐๐๐๐)
}
โค 1
๐๐๐๐,1
๐ก๐((๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)โ1โ(๐๐
2๐ผ๐๐๐+๐โ0๐๐+๐โ1๐
๐)โ1
+ 1
๐๐๐๐,1
(๐๐ฝ)๐(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)โ1 (๐๐ฝ)
๐๐๐โโโ 0 (by the Markov inequality[13], where ํ > 0) .
Since (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1 โ (๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐โ1๐
๐)โ1 is nonnegative definite โ
๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1โ(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1)๐๐๐๐ is nonnegative .
Therefore,
โ1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1โ(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1)๐๐๐๐ = ๐(๐๐๐๐,1)
โ โ
1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1โ(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1)๐๐๐๐
๐๐๐๐,1
= ๐(1)
Then, as in lemma (2) there exists a constant ๐ถ1 such that
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1
๐๐๐๐,1log ๐ต01 โฅ
1
๐๐๐๐,1log ๏ฟฝฬ๏ฟฝ01 + ๐(1) > ๐ถ1 + ๐(1).
That is,
๐ต01 โฅ exp( ๐๐๐๐,1{ ๐ถ1 + ๐(1)}) ๐๐๐โโโ โ in probability ๐0
๐๐๐.โ
Lemma(4):
Assuming that the true model is the mixed one-way repeated measurements model (the model under ๐ป1in 5). Let
๐๐๐๐ = (๐ฆ1 , โฆ , ๐ฆ๐๐๐) where ๐ฆ๐๐๐ โฒ๐ are independent normal random variables with mean ฮผ+ ฯj + ฮดi(j) + ฮณk+
(ฯฮณ)jk and variance ๐๐2 = 1 and let ๐ค๐๐๐ = ๐ธ(๐๐๐๐)
๐๐๐๐๐ ๐ธ(๐๐๐๐).
Then, lim๐๐๐โโ1
๐ค๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ = 1, in probability ๐1
โ.
Proof:
From the moment formula of the quadratic form of normal random variables, the expectation and variance of
quadratic form ๐๐๐๐๐ ๐๐๐๐๐๐๐๐ are given by:
๐ธ(๐๐๐๐๐ ๐๐๐๐๐๐๐๐ ) = ๐ก๐ (๐๐๐๐) + ๐ธ(๐๐๐๐)
๐๐๐๐๐ ๐ธ(๐๐๐๐)
= โ๐๐๐๐๐ฟ
2
(1+๐๐๐๐๐ฟ2)
๐๐๐โ(1+๐+๐+๐๐)๐=1 + ๐ธ(๐๐๐๐)
๐๐๐๐๐ ๐ธ(๐๐๐๐)
= ๐๐๐๐,1 + ๐ค๐๐๐ , and
๐ฃ๐๐(๐๐๐๐๐ ๐๐๐๐๐๐๐๐ ) = 2๐ก๐ (๐๐๐๐
2) + 4๐ธ(๐๐๐๐)๐๐๐๐๐
2 ๐ธ(๐๐๐๐).
Note
๐ค๐๐๐ = ๐ธ(๐๐๐๐)๐๐๐๐๐ ๐ธ(๐๐๐๐)
= (๐, ๐๐ , ๐พ๐ , (๐๐พ)๐ , ๐ฟ๐๐๐โ(1+๐+๐+๐๐)๐ )๐๐๐๐
๐1
๐๐๐2๐๐๐๐
[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐ 0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)
2๐๐ฟ2
(๐๐2+๐๐๐๐
๐ฟ
2)๐๐2 )๐ผ๐๐๐โ(1+๐+๐+๐๐)
]
๐๐๐๐๐ ๐๐๐๐ (๐, ๐๐, ๐พ๐, (๐๐พ)
๐, ๐ฟ๐๐๐โ(1+๐+๐+๐๐)๐
)๐
= (๐, ๐๐ , ๐พ๐ , (๐๐พ)๐ , ๐ฟ๐๐๐โ(1+๐+๐+๐๐)๐ )
[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)2๐๐ฟ
2
(๐๐2+๐๐๐๐๐ฟ
2)๐๐2 ) ๐ผ๐๐๐โ(1+๐+๐+๐๐)]
(๐, ๐๐ , ๐พ๐ , (๐๐พ)๐ , ๐ฟ๐๐๐โ(1+๐+๐+๐๐)๐ )
๐
= ๐ฟ๐๐๐โ(1+๐+๐+๐๐)๐ (
(๐๐๐)2๐๐ฟ2
(๐๐2+๐๐๐๐๐ฟ
2)๐๐2 ) ๐ผ๐๐๐โ(1+๐+๐+๐๐)๐ฟ๐๐๐โ(1+๐+๐+๐๐) = โ
(๐๐๐)2๐๐ฟ2๐ฟ๐2
1+๐๐๐๐๐ฟ2
๐๐๐โ(1+๐+๐+๐๐)๐=1 ,
since ๐ ๐ข๐๐ ๐ฟ๐ < โ , we have
๐ค๐๐๐ โค ๐ ๐ข๐๐ ๐ฟ๐2 ๐๐๐โ
๐๐๐๐๐ฟ2
1+๐๐๐๐๐ฟ2
๐๐๐๐=1 = ๐(๐๐๐ ๐๐๐๐,1), then ๐ค๐๐๐ = ๐(๐๐๐
2).
Now we can consider ๐ฟ๐2 > 0 , where ๐ โฅ 1 is a fixed positive integer and a sufficiently large ๐๐๐ with
๐๐๐๐๐ฟ2
1+๐๐๐๐๐ฟ2 โฅ
1
2 . Then for such ๐๐๐ and ๐.
๐ค๐๐๐ = โ๐๐๐2๐๐ฟ
2๐ฟ๐2
1+๐๐๐๐๐ฟ2
๐๐๐โ(1+๐+๐+๐๐)๐=1 โฅ
๐๐๐2๐๐ฟ2
1+๐๐๐๐๐ฟ2 ๐ฟ๐
2
= ๐๐๐ ๐๐๐๐๐ฟ
2
1+๐๐๐๐๐ฟ2 ๐ฟ๐
2 โฅ ๐๐๐๐ฟ๐2
2.
Similarly to the previous calculation, we have
๐ธ(๐๐๐๐)๐๐๐๐๐2 ๐ธ(๐๐๐๐) = (๐, ๐
๐ , ๐พ๐ , (๐๐พ)๐ , ๐ฟ๐๐๐โ(1+๐+๐+๐๐)๐ )๐๐๐๐
๐1
(๐๐๐)3๐๐๐๐
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[ 0 01ร๐ 01ร๐ 01ร๐๐ 01ร๐๐
0๐ร1 0๐ร๐ 0๐ร๐ 0๐ร๐๐ 0๐ร๐๐
0๐ร1
0๐๐ร1
0๐๐ร1
0๐ร๐
0๐๐ร๐
0๐๐ร๐
0๐ร๐ 0๐ร๐๐ 0๐ร๐๐ 0๐๐ร๐ 0๐๐ร๐๐ 0๐๐ร๐๐
0๐๐ร๐ 0๐๐ร๐๐ ( (๐๐๐)
2๐๐ฟ2
(๐๐2+๐๐๐๐
๐ฟ
2)๐๐2 )2
๐ผ๐๐๐โ(1+๐+๐+๐๐)]
๐๐๐๐๐ ๐๐๐๐ (๐, ๐๐, ๐พ๐, (๐๐พ)
๐, ๐ฟ๐๐๐โ(1+๐+๐+๐๐)๐
)๐
=1
๐๐๐๐ฟ๐๐๐โ((1+๐+๐+๐๐)๐ (
(๐๐๐)2๐๐ฟ2
(๐๐2+๐๐๐๐๐ฟ
2)๐๐2 )
2๐ผ๐๐๐โ(1+๐+๐+๐๐)๐ฟ๐๐๐โ(1+๐+๐+๐๐)
=1
๐๐๐โ
(๐๐๐)4๐๐ฟ4๐ฟ๐2
(1+๐๐๐๐๐ฟ2)2
๐๐๐โ(1+๐+๐+๐๐)๐=1 .
Since ๐ธ(๐๐๐๐)๐๐๐๐๐2 ๐ธ(๐๐๐๐) โค โ
(๐๐๐)3๐๐ฟ4๐ฟ๐2
(1+๐๐๐๐๐ฟ2)2
๐๐๐๐=1 โค ๐๐๐ ๐ ๐ข๐๐ ๐ฟ๐
2โ(๐๐๐)2๐๐ฟ
4
(1+๐๐๐๐๐ฟ2)2
๐๐๐๐=1 = ๐( ๐๐๐๐๐๐๐,2) (as in case
of ๐ธ(๐๐๐๐)๐๐๐๐๐ ๐ธ(๐๐๐๐)).
Now for previous ๐๐๐ and ๐ we can find
๐ธ(๐๐๐๐)๐๐๐๐๐2 ๐ธ(๐๐๐๐) = โ
(๐๐๐)3๐๐ฟ4๐ฟ๐2
(1+๐๐๐๐๐ฟ2)2
๐๐๐โ(1+๐+๐+๐๐)๐=1 โฅ ๐๐๐ ๐ฟ๐
2 (๐๐๐๐๐ฟ
2
1+๐๐๐๐๐ฟ2)2
โฅ ๐ฟ๐2
4 ๐๐๐.
By combining value of ๐๐๐๐,1 and the previous result, get
๐ฃ๐๐(๐๐๐๐๐ ๐๐๐๐๐๐๐๐ ) โค ๐(๐๐๐ ๐๐๐๐,1).
Furthermore, note that ๐ธ(๐๐๐๐)๐๐๐๐๐ ๐ธ(๐๐๐๐) โฅ ๐ธ(๐๐๐๐)
๐๐๐๐๐
2 ๐ธ(๐๐๐๐).
By the Chebshev inequality[13] get
๐๐ {|1
๐ค๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ โ
๐ธ(๐๐๐๐๐ ๐๐๐๐๐๐๐๐)
๐ค๐๐๐| > ํ } โค
๐ฃ๐๐(๐๐๐๐๐ ๐๐๐๐๐๐๐๐)
๐ค๐๐๐2 2 = ๐(๐๐๐โ2)
๐๐๐โโโ 0, where ํ > 0 .
Since lim๐๐๐โโ๐ธ(๐๐๐๐
๐ ๐๐๐๐๐๐๐๐)
๐ค๐๐๐= lim๐๐๐โโ (
๐๐๐๐,1
๐ค๐๐๐+๐ค๐๐๐
๐ค๐๐๐) = 1
โด lim๐๐๐โโ1
๐ค๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ = 1 , in probability โ
Lemma(5):
Suppose we have the mixed one- way repeated measurements model, then
lim๐๐๐โโ1
๐ค๐๐๐log
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)= 0
Proof:
Since det(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐) โค โ (๐๐
2 + ๐๐2๐ฅ๐2 + โ ๐๐
2๐ฅ๐๐2 + โ ๐๐พ
2๐ฅ๐๐2 + โ ๐(๐๐พ)
2๐ฅ๐๐2๐๐
๐=1
๐
๐=1
๐
๐=1 +๐๐๐๐=1
โ ๐๐ฟ2๐๐
๐=1 ๐ฅ๐๐2 ) (from the property of the determinant of the positive definite matrix)
= (๐๐๐)๐๐๐โ (๐๐2
๐๐๐+(๐๐2๐ฅ๐2+โ ๐๐
2๐ฅ๐๐2 +โ ๐๐พ
2๐ฅ๐๐2 +โ ๐(๐๐พ)
2 ๐ฅ๐๐2๐๐
๐=1๐๐=1
๐๐=1 +โ ๐๐ฟ
2๐๐๐=1 ๐ฅ๐๐
2
๐๐๐ )
๐๐๐๐=1 .
And we have
det(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐) = (๐๐๐)๐๐๐(
๐๐2
๐๐๐+ ๐๐
2)(๐๐2
๐๐๐+ ๐๐
2)๐(๐๐2
๐๐๐+ ๐๐พ
2)๐(๐๐2
๐๐๐+ ๐(๐๐พ)
2 )๐๐(๐๐2
๐๐๐+
๐๐ฟ2)๐๐๐โ(1+๐+๐+๐๐) (from lemma 1).
โ logdet(๐๐
2๐ผ๐๐๐+๐โ0๐๐+๐โ1๐
๐)
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)= logdet(๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐โ1๐
๐) โ log det(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ +
๐๐โ1,๐๐๐๐)
โค {(๐๐๐)log(๐๐๐) + โ log (๐๐2
๐๐๐+(๐๐2๐ฅ๐2+โ ๐๐
2๐ฅ๐๐2 +โ ๐๐พ
2๐ฅ๐๐2 +โ ๐(๐๐พ)
2 ๐ฅ๐๐2๐๐
๐=1๐๐=1
๐๐=1 +โ ๐๐ฟ
2๐๐๐=1 ๐ฅ๐๐
2
๐๐๐ )
๐๐๐๐=1 } โ {๐๐๐log(๐๐๐) +
log (๐๐2
๐๐๐+ ๐๐
2) + ๐log (๐๐2
๐๐๐+ ๐๐
2) + ๐log (๐๐2
๐๐๐+ ๐๐พ
2) + ๐๐log (๐๐2
๐๐๐+ ๐(๐๐พ)
2 ) + ๐๐๐ โ (1 + ๐ + ๐ +
๐๐)log (๐๐2
๐๐๐+ ๐๐ฟ
2)} = ๐(1) โ ๐(1) = ๐(1).
Then
1
๐ค๐๐๐log
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)
๐๐๐โโโ 0
Theorem(2):
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Suppose that the true model is the mixed one- way repeated measurements model (the model under ๐ป1in 5),
and let ๐1๐๐๐
denote the true distribution of the data, with probability density function
๐1(yijk|ฮผฯj, ฮดi(j), ฮณk, (ฯฮณ)jk) = ๐ท{(yijk โ (ฮผ + ฯj + ฮดi(j) + ฮณk+ (ฯฮณ)jk))/๐๐}, where ๐ท(โ) is the standard normal
density. Then the Bayes factor is consistent under the alternative hypothesis ๐ป1 i.e
lim๐๐๐โโ ๐ต01 = 0 in probability ๐1๐๐๐
.
Proof:
log ๐ต01 = 1
2log
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐)โ1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐) โ1โ(๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐โ1๐
๐)โ1)๐๐๐๐
=
1
2log
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)+
1
2log
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐)โ1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐) โ1โ(๐๐
2๐ผ๐๐๐ +
๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1)๐๐๐๐ โ
1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1
โ(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1)๐๐๐๐
= log๏ฟฝฬ๏ฟฝ01 โ1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐) โ1โ(๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1)๐๐๐๐ โ
1
2๐๐๐๐๐
((๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1โ(๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐โ1๐
๐)โ1)๐๐๐๐ , where
log๏ฟฝฬ๏ฟฝ01 =
1
2log
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐)โ1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐) โ1โ(๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1)๐๐๐๐ .
From lemma (1), we have
det(๐๐
2๐ผ๐๐๐+๐โ0๐๐+๐๐โ1,๐๐๐
๐)
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐)=
(๐๐๐)๐๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2
๐๐๐+๐๐ฟ
2)๐๐๐โ(1+๐+๐+๐๐)
๐๐๐1+๐+๐+๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
=๐๐๐๐๐๐โ(1+๐+๐+๐๐)(
๐๐2
๐๐๐+๐๐ฟ
2)
๐๐๐โ(1+๐+๐+๐๐)
๐๐2๐๐๐โ(1+๐+๐+๐๐)
= (๐๐2+๐๐๐๐๐ฟ
2
๐๐2 )
๐๐๐โ(1+๐+๐+๐๐)
=(1 +๐๐๐๐๐ฟ
2
๐๐2 )
๐๐๐โ(1+๐+๐+๐๐)
= ๐(๐๐๐)
โน log (1 +๐๐๐๐๐ฟ
2
๐๐2 )
๐๐๐โ(1+๐+๐+๐๐)
= ๐(๐๐๐).
And from lemma (4) ๐ค๐๐๐ = ๐(๐๐๐2). Then
1
๐ค๐๐๐log
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐)= ๐(๐๐๐โ1)
๐๐๐โโโ 0.
Then,
1
๐ค๐๐๐log๏ฟฝฬ๏ฟฝ01 =
1
2(
1
๐ค๐๐๐ log
det(๐1,๐๐๐)
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐)โ
1
๐ค๐๐๐๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐) โ1โ(๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ +
๐๐โ1,๐๐๐๐)โ1)๐๐๐๐).
Since (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐) โ1 โ (๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐๐โ1,๐๐๐๐)โ1= ๐๐๐๐, then
1
๐ค๐๐๐๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐) โ1โ(๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1)๐๐๐๐ =
1
๐ค๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐. from lemma
(4), lim๐๐๐โโ1
๐ค๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ = 1 , then
1
๐ค๐๐๐log๏ฟฝฬ๏ฟฝ01
๐๐๐โโโ =
1
2(
1
๐ค๐๐๐ log
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐)โ
1
๐ค๐๐๐๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐) โ1โ(๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ +
๐๐โ1,๐๐๐๐)โ1)๐๐๐๐)
๐๐๐โโโ =
1
2(0 โ 1) =
โ1
2
โด1
๐ค๐๐๐log๏ฟฝฬ๏ฟฝ01
๐๐๐โโโ
โ1
2 , in probability ๐1
๐๐๐.
Now from lemma (5),
1
๐ค๐๐๐ 1
2log
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det (๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐) ๐๐๐โโโ 0, and โ
1
2๐๐๐๐๐ ((๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1โ(๐๐
2๐ผ๐๐๐ +
๐โ0๐๐ + ๐โ1๐
๐)โ1)๐๐๐๐ is nonpositive random variable since (๐๐
2๐ผ๐๐๐ + ๐โ0๐๐ + ๐๐โ1,๐๐๐
๐)โ1โ
(๐๐2๐ผ๐๐๐ + ๐โ0๐
๐ + ๐โ1๐๐)โ1
is a nonnegative definite matrix.
Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
156
Therefore, by combining the above results, there exist ๐ถ2 > 0 such that
lim๐๐๐โโ sup1
๐ค๐๐๐log ๐ต01 โค
โ๐ถ2
2, with probability tending to 1.
Then , ๐ต01๐๐๐โโโ 0, in probability ๐1
๐๐๐. โ
4.Conclusions
1- The posterior distribution ๐1(๐๐๐๐|๐๐๐๐) is the multivariate normal with
๐ธ(๐๐๐๐|๐๐๐๐ , ๐๐2) = โ2(โ2 + ๐๐
2๐ผ๐)โ1๐๐๐๐ , where N=1+q+p+qp+nq and
๐๐๐(๐๐๐๐|๐๐๐๐, ๐๐2) = (โ2
โ1 + (๐๐2๐ผ๐)
โ1)โ1 = โ2(โ2 + ๐๐2๐ผ๐ )
โ1๐๐2๐ผ๐, where
๐๐๐๐ = ฮผ+ ฯj + ฮดi(j) + ฮณk+ (ฯฮณ)jk + eijk,for, ๐ = 1,2, โฆ , ๐, ๐ = 1,โฆ , ๐ ๐๐๐ ๐ = 1,โฆ , ๐, and
โ2 = ๐โ0๐๐ + ๐โ1๐
๐ = ๐
[
๐๐2 01ร๐ 01ร๐
0๐ร1 ๐๐2๐ผ๐ร๐ 0๐ร๐
0๐ร1 0๐ร๐ ๐๐พ2๐ผ๐ร๐
01ร๐๐ 0๐ร๐๐ 0๐ร๐๐
0๐๐ร1 0๐๐ร๐ 0๐๐ร๐ ๐(๐๐พ)2 ๐ผ๐๐ร๐๐]
๐๐ + ๐
[ ๐๐ฟ112 0 โฏ 0
0 ๐๐ฟ122 โฏ 0
โฎ0
โฎ0
โฑ โฎโฏ ๐๐ฟ๐๐
2]
๐๐
2-The Bayes factor for testing problem ๐ป0: yijk = ฮผ + ฯj + ฮณk + (ฯฮณ)jk + eijk versus
๐ป1: yijk= ฮผ + ฯj + ฮดi(j) + ฮณk + (ฯฮณ)jk + eijk is given by:
๐ต01 =
โ(๐๐๐)๐๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐
(๐๐2
๐๐๐+๐๐พ
2)๐
(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐
(๐๐2
๐๐๐+๐๐ฟ
2)๐๐
โ๐๐๐1+๐+๐+๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
exp{ โ1
2๐๐๐2๐๐๐๐๐
(๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐๐ โ๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐]
โ1
๐๐๐๐๐ )๐๐๐๐}.
3-The approximation of ๐ต01 with design matrix of the random effects (๐) to the first ๐๐๐ terms as following
๏ฟฝฬ๏ฟฝ01 =
โ(๐๐๐)๐๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2
๐๐๐+๐๐ฟ
2)๐๐๐โ(1+๐+๐+๐๐)
โ๐๐๐1+๐+๐+๐๐(๐๐2
๐๐๐+๐๐
2)(๐๐2
๐๐๐+๐๐
2)๐(๐๐2
๐๐๐+๐๐พ
2)๐(๐๐2
๐๐๐+๐(๐๐พ)
2 )๐๐(๐๐2)๐๐๐โ(1+๐+๐+๐๐)
exp{โ1
2๐๐๐2๐๐๐๐๐
(๐๐๐๐ [๐๐2
๐๐๐๐ผ๐๐๐ + ๐ถ๐๐๐]
โ1
๐๐๐๐๐ โ๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ )๐๐๐๐}.
4- If we have the one- way repeated measurements model, then
1
๐๐๐๐,1๐ก๐ (
1
๐๐๐2๐๐๐๐ [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐ท๐๐๐]
โ1
๐๐๐๐๐ โ [๐ ๐] [
๐๐2
๐๐๐๐ผ๐๐๐ + ๐บ๐๐๐]
โ1
[๐ ๐]๐)๐๐๐โโโ 0
5-If the true model is the one-way repeated measurement model with fixed effects only then the Bayes factor is
consistent under the null hypothesis ๐ป0 . This implies that, lim๐๐๐โโ ๐ต01 = โ in probability ๐0๐๐๐
.
6- If we have the mixed one- way repeated measurements model, then
lim๐๐๐โโ1
๐ค๐๐๐log
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐โ1๐๐)
det(๐๐2๐ผ๐๐๐+๐โ0๐
๐+๐๐โ1,๐๐๐๐)= 0.
7- If the true model is the mixed one- way repeated measurements model, then the Bayes factor is consistent
under the alternative hypothesis ๐ป1.This implies that, lim๐๐๐โโ ๐ต01 = 0 in probability ๐1๐๐๐
.
5.References
[1] Aerts, M. , Claeskens, G. and Hart, J.D., (2004). Bayesian โmotivated tests of function fit and their
asymptotic frequentist properties, Ann. Statist. 32 (6), 2580-2615.
[2] Al-Mouel, A.H.S.,(2004) . Multivariate Repeated Measures Models and Comparison of Estimators, h. D.
Thesis, East China Normal University, China.
[3] Al-Mouel , A.H.S., and Wang, J.L,(2004). One โWay Multivariate Repeated Measurements analysis of
Variance Model, Applied Mathematics a Journal of Chinese Universities, 19,4,pp435- 448.
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Vol.4, No.11, 2014
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[4] Arnold, Steven F., (1981). The theory of linear models and multivariate analysis. John Wiley and
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[7] Dass, S.C. and Lee, J., (2004). A note on the consistency of Bayes factors for testing point null versus
non-parametric alternatives, J. Statist. plan. Inference 119 (1) 143-152.
[8] Ghosh, J.K. , Delampady, M. and Samanta, T., (2006). Introduction to Bayesian Analysis: Theory and
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[10] Graybill, F.A., (1976). Theory and Application of the linear Model, Duxbury press, North Scituate, Mass,
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[13] Papoulis,A.,(1997). Markoff Sequences in Probability, Random Variables, and Stochastic Processes,
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[14] Tong, Y.L., (1989). The Multivariate Normal Distribution, Springer series in statistics, Springer.
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