Bayesian One- Way Repeated Measurements Model Based on Bayes Quadratic Unbiased Estimator

In this paper, bayesian approach based on Bayes quadratic unbiased estimator is employed to the linear oneway 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. The prior information obtained by using variance analysis technique to represent prior estimates of the parameters of the model. Then, the prior distribution is considered as a uniform distribution.


1.INTRODUCTION
The repeated measures model had its origins in analysis of variance , so the term repeated measures ANOVA is often used to refer to the generic design . In fact , the independent variables in repeated measures can be any combination of categorical and continuous variables . Repeated measurements occur frequently in observational studied which are longitudinal in nature , and in experimental studies incorporating repeated measures design . 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 a linear or nonlinear function of time . Repeated measures designs , on the other hand, entail one or more response variables being measured repeatedly on each individuals over arrange of conditions. Here the metameter may be time or it may be set of experimental conditions (e.g., dose levels of a drug) . Repeated measurements model has been investigated by many researchers as Vonesh and Chinchilli in (1997) discussed the univariate repeated measurements model , analysis of variance model (RM ANOVA), [10]. Al-Mouel in (2004) studied the multivariate repeated measures models and comparison of estimators, [1]. Al-Mouel and Wang in (2004) presented the sphericity test for the one-way multivariate repeated measurements analysis of variance model . They studied the asymptotic expansion of the sphericity test for the one -way multivariate repeated measurements analysis of variance model, [2].Mohaisen and Swadi in (2014) studied Bayesian approach based on Markov Chain Monte Carlo, which is employed to making inferences on the one-way repeated measurements model, bayesian approach is employed to making inferences on the one-way repeated measurements model as mixed model and they 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, they identified the analytic form of the Bayes factor and showed that the Bayes factor is consistent, [4], [5], [6], [7] .
Variance covariance components play an important role in many fields of sciences. There are several procedures to estimate variance components.Rao in (1971) has introduced the minimum norm quadratic estimation (MINQUE) method. In this approach a quadratic estimator is sought that satisfies the minimum norm criterion, [8]. Stuchly in (1989) presentedthe necessary and sufficient conditions forthe existence and an explicit expression for the Bayes invariant quadratic unbiased estimate of the linear function of the variance components for the mixed linear model with three unknown variance components in the normal case, [9]. Witkovsky in (1996) has considered (MINQUE) approach with first order autoregressive disturbances, [11]. Fathy in (2006) presented Bayes invariant quadratic unbiased estimator (BAIQUE) . he used Bayesian approach to estimate the covariance functions of the regionalized variables which appear in the spatial covariance structure in mixed linear model, [3]. This paper deals with Bayesian estimation of one-way repeated measurements model. Bayesian approach based on Bayes quadratic unbiased estimator of 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. The prior information obtained by using variance analysis technique to represent prior estimates of the parameters of the model. Then, the prior distribution is considered as a uniform distribution. yijk is the response measurement at time k for unit i within group j, µ is the overall mean , τj is the added effect for treatment group j ,

2.One-Way Repeated Measurements Model
is the random effect for due to experimental unit i within treatment group j , is the added effect for time k , (τγ)jk is the added effect for the group j × 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 for each k=1,…,p for each j=1,…,q And we assumed that the e ijk and δ i(j) are indepndent with Sum of squares due to groups, subjects(group), time, group*time and residuals are then defined respectively as follows: Where is the overall mean.
is the mean for group j.
p is the mean for the i th subject in group j.
is the mean for time k.

3.Bayes Quadratic Unbiased Estimator
From (3) we have ∼ , 2 + σ δ 2 I nq , (5) we can write the model (3) as where M are known nqp × nqp projection matrix and According to property projection matrix M we have Since the matrix M symmetric and idempotent i.e where ( ) is an ( × ) symmetric matrix. The parametersθ 1 and θ 2 are estimate by linear function estimation which proposed by Rao (1974) as follows: where ℓ = (ℓ 1 , ℓ 2 )́ and = ( 1 , 2 ). The linear function estimated by the following quadratic form: where A is an × symmetric matrix which must be found from the data. If we have the prior distribution function g( ) of the parameter , then the quadratic loss function is: And the risk function is: (20) Where is the second moment for θ i , and the matrix of the second moment is The matrix is To solve (22) use Lagrange method according to unbiasedness condition to get minimum value assume that Where is Lagrange Multipliers.
Derive (23) for the A, and by equality to zero get: Then derive (23) for , and by equality to zero get: We can write (26) as Then we can write the system (27) and (28) as O is an ( 2 + 2) × 1 unknown values vector in the system, B is an ( 2 + 2) × 1 constants vector in the system,

4.Prior distribution
Assuming that little is known about variance components for one-way repeated measurement model, it makes sense to put an improper uniform prior on this components, then we will assume that the prior distribution on 1 and 2 are

5.Example (The storage experiment)
In this section, we illustrate the effectiveness of the our methodology. We have choosing the data set which an experiment was conducted during winter season 2008-2009 in one of unhted plastic house which belong to tomato development project in Basrah / Agriculture directorate of Basrah (Khor Alzubiar) in order to investigate the effect of calcium on growth and yield of cultivars of cucumber (Sayff) and the storage temperature on storage capability and quality. The field experiment included 84 variable treatments which were the interaction of three factors of three storage temperatures, which were room temperature, 5°and 12 ° and two concentration of calcium chloride 0, 1, 2, 3 % and three storage period 0 , 10 , 15 , 20 , 25 , 30 day. After harvest, the fruit was treated with CaCl2 . 2H2O for five minute, then it was stored in three temperatures. Then the chemical changes characteristics of the fruit was reviewed during the storage at 5 days periods. the design of the experiment was done according to the model (1).   We next applied our methodology Bayes quadratic unbiased estimator to the storage experiment data. The values of variance components for the one-way repeated measurement model based on this method are 1 = 1 2 = 1.601 and 2 = 2 2 = 0.2312. Then we can see that the values of variance components for the one-way repeated measurement model obtained in both ANOVA and Bayes quadratic unbiased estimator are nearly alike and encouraging.

6.CONCLUSIONS
The conclusions obtained throughout this paper are as follows: (1) The linear function estimated by the quadratic form =́ where satisfies the unbiasedness condition i.e. = and α satisfies theminimizing property of the Bayes risk function.
(2) There is significant effectfor calcium chloride on storage capability for cucumber fruits under different temperatures.