Two Random Covariates in Repeated Measurement ...

Basrah journal of science (A)

Vol. 32(1), 130-140, 2014

Two Random Covariates in Repeated Measurement Model Huda Zeki Naji Dept.of Mathematics/ College of Science / University of Basrah/ Basrah /Iraq [email protected]

This research is devoted to study of one-way Multivariate repeated measurements analysis of covariance model (MRM ANCOVA), which contains one between-units factor (Group with q levels) ,one within-units factor(Time with p levels) and two random covariates ( Z 1 , Z 2 ) . For this model the two covariate is time-independent, that is measured only once. the test statistics of various hypotheses on between-unites factors, within-units factors and the interaction between them are given.

: (One-Way MRMM): One-Way Multivariate Repeated Measures Model, ,

U * is

: Wilks distribution

P  P orthogonal matrix , (Wishart) the multivariate- Wishart distribution ,

(ANCOVA) analysis of variance and covariance contain on the covariates , ( Z 1 , Z 2 ) Concomitant Variate or Covariates.

021

Two Random Covariates…………

Naji, H.Z.

:Repeated measurements analysis is widely

(1.1) Covariate in One-Way Multivariate

used in many fields , for example, the health

Repeated Measurements Design :

and life science, epidemiology , biomedical,

There is a variety of possibilities for the

agricultural,

between units factors in a one-way design.

psychological,

educational research and so on (see, Huynh

In a randomized one-way MRM experiment,

and Feldt (1970)[9] , and Vonesh and

the experimental units are randomized to one between-units factor

industrial,

Chinchilli

(Group with q

(1997)[12]).

Repeated

measurements analysis of variance, often

levels) ,one within-units factor (Time with p

referred to as randomized block and split-

levels) and two random covariates ( Z 1 , Z 2 )

plot designs (see Bennett and Franklin

. For this model the two covariate is time-

(1954)[7],

independent, that is measured only once.

Sendeecor

and

Cochran

(1967)[11]). Repeated measurements is a

For convenience ,we define the following

term used to describe data in which the

linear model and parameterization for the

response variable for each experimental

one-way repeated measurements design with

unite is observed on multiple occasions and

one between units factor incorporation two

possibly

covariates:-

conditions

under

different

(see,Vonesh

experimental

and

(1997)[12]). Y        ( )  ( Z  Z )   ( Z  Z )  e ijk j k jk 1ij 1.. 1 2ij 2.. 2 ijk

Where (i  1,..., n ) is an index for experimental unit of level ( j ) j ( j  1,..., q)

is an index for levels of the between-units factor (Group).

(k  1,..., p) is an index for levels of the within-units factor (Time).

020

…(1.1)

Chinchilli

Basrah journal of science (A)

Vol. 32(1), 130-140, 2014

Yijk  (Yijk1 ,..., Yijkr ) is the response measurement of within-units factors (Time) for unit i within

treatment factors (Group).   (1 ,...,  r ) is the over all mean.  j  ( j1 ,..., jr ) is the added effect of the j th level of the treatment factor (Group).

 k  ( k1 ,...,  kr ) is the added effect of the k th level of Time.  ( ) jk  ( ) jk1 ,..., ( ) jkr  is the added effect of the interaction between the units factor (Group) at

level of (Time). Z1ij  (Z1ij1 ,..., Z1ijr ) is the value of covariate Z1 at time k for unit i within group j . Z1..  (Z1..1 ,..., Z1..r ) is the mean of covariate Z1 over all experimental units.

1  (11,..., 1r ) is the slope corresponding to covariate Z1 . Z 2ij  (Z 2ij1 ,..., Z 2ijr ) is the value of covariate Z 2 at time k for unit i within group j .

Z 2..  (Z 2..1 ,..., Z 2..r ) is the mean of covariate Z 2 over all experimental units.

 2  ( 21,...,  2r ) is the slope corresponding to covariate Z 2 . eijk  (eijk1 ,..., eijkr ) is the random error at time k for unit i within group j.

For the parameterization to be of full rank, we impose the following set of conditions : q

 j 1

q

j

0

 ( ) jk  0 j 1

p

,  k

0

k 1

p

,  ( ) jk

0

for each

j, k  1,..., p

k 1

,s

We assume that eijk is independent with eijk  (eijk1 ,..., eijkr ) ~

i.i.d

N r ( 0,  e ) 021

Two Random Covariates…………

Naji, H.Z.

Z1ij  (Z1ij1 ,..., Z1ijr ) ~

i.i.d

Z 21ij  (Z 2ij1 ,..., Z 2ijr ) ~

Where

Nr

i.i.d

…(1.2)

N r ( 0,  Z1 ) N r ( 0,  Z 2 )

is denoted to the multivariate ‫ـــ‬normal distribution , and e ,  Z ,  Z is r  r positive 1

2

definite matrix.

Where the variance matrix and covariance ∑ of the model (1.1) satisfy the assumption of compound symmetry. i.e.

= I p   *1  J p   Z  J p   Z 1

Where :

2

 *1   e   Z1   Z 2

 *1          *1              *1  

,

  Z  Z 1

…(1.3)

2

I p denote the P  P identity matrix. J p denote the P  P matrix of one's. and  be the Kroneker product operation of two matrices. (1.2) Transforming the one-way Repeated measurements Analysis of Covariance (ANCOVA) model : In this section we use an orthogonal matrix to transform the observations Yijk for i  1, .n j , * Let U be any P  P

j  1, , q , k  1, , p

orthogonal matrix is partitioned as follows:

022

Basrah journal of science (A)

Vol. 32(1), 130-140, 2014 

1 2

U  ( p jp *

…(1.4)

U)

denote the P 1 vector of one's , U  is ( p  1)  p matrix .

Where

 e  p( Z1   Z 2 ) 0  0    0  e  0   Co(Yij* )           0 0  e  

...(1.5)

(1.3 ) Analysis of Covariance (ANCOVA) for the One-Way Repeated Measurements Model : In this section, we study the ANCOVA for the effects of between-units factors and within-units factors for the One-way RM model (1.1). Also we give the null hypotheses which is concerned with these effects and the interaction between them, and the test statistics for them. Now Y  Yij p * ij1



1 2

 1 jp    p

p

Y k 1

ijk 1

 Yijk 2  Yijkr    p k 1 p k 1 

1

p

1

p



From (1.1), we obtain: Yij*1   *   *j  (Z1*ij  Z1*.. )1*  (Z 2*ij  Z 2*.. ) 2*  eij*1

Then the set of vectors * * (Y111 , , Yn*111 ), (Y121 , , Yn*2 21 ), , (Y1*q1 , , Yn*q q1 )

Have mean vectors X 1  P   P 1 X 2  P   P 2  X q  P   P q

023

Two Random Covariates…………

Naji, H.Z.

Respectively, and each of them has covariance matrix e  p(Z  Z ) . 1

2

So, the null hypothesis of the same treatment effects are: H 01   1     q  0 * The ANCOVA based on the set of transformed observations above the Yij1,s provides the

ANCOVA for between-units effects. This leads to the following form for the sum square terms SS G ، SS u (Group) : q

SSG   n j (Y j*1  Y1* )(Y j*1  Y1* ) j 1

SS u (Group)   C (C ) q

nj

j 1 i 1

Where C = Yij*1  Y j*1  2Yij*  2Y j*  1*Z1*ij   2*Z 2*ij  q

nj

Y j*1 

Y i 1

nj

* ij1

, Y1* 

nj

 Y j 1 i 1

q

* ij1

n

, Yij*1 

nj

q

 Yij*1 j 1 i 1

njq

,

Yk* 

Thus SS G ~ Wr (q  1,  e  p( Z1   Z 2 )) SS u (Group) ~ Wr (n  q,  e  p( Z1   Z 2 ))

Where

denote the multivariate-Wishart distribution.

The multivariate Wilks test (Wilks 1932) : Tw1 

SS u (Group) SS u (Group)  SS G

, When

is true 024

n Y j 1

j

n

* jk

Basrah journal of science (A)

Vol. 32(1), 130-140, 2014

Tw1 ~  r (n  q, q  1) * The ANCOVA based on the set of transformed observations the Yijk ,s for each k  2, , p has

the model which is partitioned as follows: * Yijk*   k*  ( )*jk  (Z1*ij  Z1*.. )1*  (Z 2*ij  Z 2*.. ) 2*  eijk

Then from above analysis we test the following hypotheses : H 02 :  2*     *p  0

H 03 : ( ) *j 2    ( )*jp  0

The ANCOVA based on the set of transformed observations above , the

Yij*2 , , Yijp* provides

the ANCOVA for within-units effects. This leads to the following forms for the sum square terms : p

SS Time  n (Yk* (Yk* )) k 2

nj

p

q





SS TimeGroup   n j (Y jk*  Yk* )(Y jk*  Yk* ) , Y jk* 

Y i 1

k  2 j 1

p

q

nj

SSTimeUmit (Group)   G(G) k 2 j 1 i 1

Where G= (Yijk*  Y jk*  2Yij*  2Y j*  1* Z1*ij   2* Z 2*ij )

Then from above sum square terms , we have : SSTime ~ Wr ( p  1), e  025

nj

* ijk

, k  2,..., p

Two Random Covariates…………

Naji, H.Z.

SSTimeGroup ~ Wr ( p  1)(q  1),  e  SSTimeUnit(Group) ~ Wr ( p  1)(n  q), e 

The multivariate Wilks test : Tw4 

SS TimeUnit(Group) SS TimeUnit(Group)  SS Time

, when

is true

Tw4 ~  r ( p  1)(n  q), ( p  1)

Tw5 

SS TimeUnit(Group) SSTimeUnit(Group)  SS TimeGroup

, when

is true

Tw5 ~  r ( p  1)(n  q), ( p  1)(q  1)

The one-way MRM ANCOVA with one between-unit factor (Group) and two covariates ( Z 1 , Z 2 ) that are time-independent

026

Basrah journal of science (A)

Source

Vol. 32(1), 130-140, 2014

D.F

SS

Wilks Criterion TW1 

Group

q 1

SS G

Unit(Group)

nq

SSUnit(Group)

SSUnit(Group) SSUnit(Group)  SS G

Between

TW4 

Time

p 1

SSTime

TW5 

With in

Time Group

( p  1)  (q  1)

SSTimeGroup

Time Unit (Group)

( p  1)  (n  q)

SSTimeUnit(Group)

SS TimeUnit(Group) SS TimeUnit(Group)  SS Time

SS TimeUnit(Group) SS TimeUnit(Group)  SS TimeGroup

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Spherisity

test

for

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Two Random Covariates…………

Naji, H.Z.

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028

‫)‪Basrah journal of science (A‬‬

‫‪Vol. 32(1), 130-140, 2014‬‬

‫العاملٌن المرافقٌن العشوائٌٌن فً نموذج قٌاسات متكرر متعدد المتغٌرات‬ ‫هدى زكً ناجً‬ ‫قسم الرٌاضٌات‪ /‬كلٌة العلوم‪/‬جامعة البصرة‬ ‫‪[email protected]‬‬

‫ٌتضمن هذا البحث دراسة احد نماذج تحلٌل التباٌن المشترك متعددالمتغٌرات للقٌاسات المتكره )‪ (MRMANCOVA‬حٌث‬ ‫ٌحتوي هذا النموذج على عامل واحد بٌن الوحدات ) ‪ (Group with q levels‬وعامل واحد ضمن الوحدات ‪(Group with‬‬ ‫) ‪p levels‬كما ٌحتوي على عاملٌن مستقلٌن متغٌٌرٌن عشوائٌا ً هما‪ z1‬و‪ ، z2‬الهدف من هذا البحث هو دراسة كل من‬ ‫المتغٌرٌن المستقلٌن على انهما مستقالن عن الزمن اي انهما ٌقاسا مره واحده فً كل مستوى من مستوٌات التجربه وتتضمن‬ ‫الدراسه كل الفرضٌات واﻻختبارات اﻻحصائٌه للعوامل بٌن الوحدات والعوامل ضمن الوحدات والتفاعل بٌنهم‬

‫‪031‬‬