The Effects of Correlated Data and Correction Procedures for F-Test in Unbalanced Two Way Model

لاقتـسا نأ ورفلا دحأ ربتعت تادهاشملا ةيل ض نيابتلا ليلحت لودج يف ةيـساـسلأا . نمضتت ثـيح يـعيبطلا عيزوــتلا عـبتتو لـثامتم اهعـيزوت نوكيو ةلقتـسم تاريغتم جذومنلا يف ءاطـخلأا دودح نوك ةـسناجتم تانيابتو ةيرفـص تلادعمب . تانايبلا ةيللاقتـسا مدع ريثأت حيـضوت مت ةـساردلا هذه يف ليلحت يف هذـهل نياـبتلا ليدـعتل ةـقيرط ريوطت مت امك بعـشتم نزاوـتم ريغ نيـهاجتا يذ تـباث جذومنل نيابتلا ةطبترملا تادهاشملا . طرـش ةـلازإ ريثأـت ىلع ءوضلا طلسنسو ةطبارتم نوكت ءاطخلأا دودح نأ ينعي اذه مجم لدعم عقوت باـسح ةطـساوب كلذو تايضرفلا رابتخا ىلع ةيللاقتـسلاا كلذكو ،ءاطـخلأل تاعبرملا عو ا حيحـصتو جذومنلا اذه يف تاجلاعملل لإ ءاصح F لماعلا تاريثأت رابتـخلا . عـيمج اـجذومن ضرـتفن اذهل نيابتلا سفن كلتمت هيف تاـسايقلا 2 σ يلي امك ةفرـعم ةينـبب عتـمتت كرتـشملا نيابتلا ةفوفـصمو : جوز لك أت يتلا تاـسايقلا نم نم يت : ) 1 ( ةدحولا سـفن نم نكلو ةفلـتخم تاراركت ) . 2 ( ةدـحولا سـفنب ةفلـتخم ةيئزج تادحو ) . 3 ( تادـحو نيابت اهل ةفلـتخم 1 2 2 2 3 2 , , ρ σ ρ σ ρ σ يلاوتلا ىلع .


Introduction
The assumption of independence of observation in ANOVA table may seem like a reasonable assumption in examining data using experimental designs .Rarely is the independent assumption verified in an ANOVA consequently, the analysis of data from experimental design is often hampered by lack of technique to correct the usual F-test for the effect of correlation. Some researches have been devoted to show that slight dependence in the form of serial correlation and interclass correlation can quickly inflate the Type I error rate. See [4], [8], [1] and [3] have shown that in an experimental design, certain forms of dependence can quickly invalidate the result of ANOVA. How small experiment is shown see [6] and see [5] for 2-way corssedand balanced nested classifications respectively.
The aim of this work explains a method for adjusting ANOVA table when observation are correlated, that is, when the error terms are correlated and focus on the effects of departures from independence assumption on hypothesis by determining the expected mean squares for errors as well as treatments for un balanced two way nested fixed effects model (balanced model is special case of un balanced model)and correcting the F statistics for testing the factor effect.

1-Defining the model
The model of study occurs in ANOVA, when we have a number of independent experimental units and each experimental unit has the same number of experimental observation and each experimental observation receives different number of treatment. See [2], [10], [11].
In this study, we assume that is the overall mean, i  is the effect of the ith experimental unit and ij  is the effect of the ith nested experimental observation in the jth experimental unit; such that  (1) and (3) we have The model we consider in this part, which is defined by equations (2), (5), is unbalanced 2-way nested effects model. Then from equation (1) the ijk e are distributed normally with mean zero and ) , ( ) , ( First we discuss ANOVA, concerns with the F statistic for the equality of factor level means, and we will discuss correction for correlation.

2-Analysis of Variance
In this section see [2], [9] and [10] we will partition the total sum of square and degrees of freedom from the model given in equation (5), in the same way that we partition the total sum of square and degrees of freedom in the ANOVA for unbalanced 2-way nested effects model when the observation are independent .Where we partition the total sum of square in the ANOVA table for unbalanced 2-way nested effects model to sum of square of factor A, sum of square of factor B and sum of square of error when the observation are dependent as follows  . Thus, we will determine the expected mean squares E(MS). When the observation are not independent by using equation (2) each of the MSSA, MSSB and MSSe.

Determining expected mean squares sum of square of error E(MSe)
Here, we use equation (8d) and the corresponding degrees of freedom for SSe Now using equation (5) we get Now by taking mathematical expectation for (12) we get Now using the two equations (1), (2) and (6), we can determine h A for h=1,2,3 as follows By substituting equation (15a), (15b) and (15c) in equation (13) we get

Determining expected mean squares sum of square of factor B E(MSB)
Using equation (8c) and the corresponding degrees of freedom for SSB Now using equation (5) By using equation (8c), (17) Now by taking the mathematical expectation for equation (20) we get Using equation (2) and (6) we can determine Substituting equation (23a), (23b), (23c) And (23d) in equation (21)

4.Analysis of variance table
Analysis of variance table for the model of study, which is given in equation (5), is given in table 1.  and SST are given in equation (8)

The F test for equality of Factor Level Means
In the last section, we have prepared the ANOVA table, So in this section we will discuss two case for two null hypothesis to know whether the factor level means are equal for the first factor and for the second factor, which is the nested factor. Hence, the alternative conclusions, which we want to, consider for the model of study, are the following .for the first factor is

Correcting for correlation's
The correcting factor 2 1 C and C which is given in equation (37), (39) could be in the following three cases: Case1: the correcting factor =1 Case2: the correcting factor >1 Case3: the correcting factor <1 When we are in the first case, then no correction is needed to correct F test . otherwise the correction is essential and we cannot ignore it.
To find the effect of correlation's for testing the factors effects in model (5), the correcting factors         We can note from table 4,5, that the true  level in flate /deflate when the correction factor greater/less than 1,and this leads to have a bigger /smaller rejection region for the complete null hypothesis on testing factors. Hence the uncorrected F test can be liberal for some tests and conservative for others in ANOVA when the correlation structure is ignored.

6.The Relationship between  and true 
The relationship between values of  when data are uncorrelated and values of the true  when data are correlated for different values of the correction factor 2