Theoretical Study for Adjusting ANOVA Procedures For The One-Way Random Effects Model Under Correlated Errors Within Class And Across Classes

In this study, a method was developed to adjust the procedures of analysis of variance for the one - way balanced random effects model when error limits across classes are not independent, while the assumption of independence between the error terms among the different classes is essential in the analysis of variance. The variance-covariance structure and the correlation coefficients for the error limits were defined. The 𝜌 1 was assumed as the correlation coefficient of error terms within the class and the 𝜌 2 is the correlation coefficient for error terms in different classes. The work performed by deriving correction factor and calculating the expectation of the mean sum of squares of errors and treatments as well as correcting the f-statistic. MSC..


Introduction
The analysis of variance (ANOVA) is a popular parametric statistical technique used to analyze data in a variety of fields such as biostatistics and ecology Sahai and Ageel (2012) [1].It is used to study the effect of an independent linear variable (factor) on the dependent variable (response variable) by testing hypotheses about the equivalence of two or more populations (or treatments) through analysis of observed sample variance [2].Analysis of variance offers many advantages including being a simple statistical analysis that can be performed and interpreted, robust to some violations of certain assumptions, and produce the desired results [3][4][5].As with any parametric test, some assumptions must be satisfied in order to arrive at a reasonable and acceptable conclusion.The usual assumption of the analysis of variance approach is that error measurements in the model are independent normal random variables with zero means and homogeneity variances.However, the assumption of independence of observations and error terms is rarely confirmed, and ignoring correlations between observations can lead to unreliable statistical conclusions [6][7][8].Some studies have discussed the correlation between observations within the same class, and others have investigated the effect of the correlation between random errors within the same group.Fisher et al. (1950) [9] presented the intraclass correlation coefficient, which measures the similarity of observations of the same class.Since then, many researchers have studied intraclass correlation coefficients in the analysis of variance in several research areas, including epidemiology, genetics, psychology, and sensitivity analysis [10].Smith and Lewis (1980) [11] provided a technique for correcting the usual F-tests to construct unbiased F-tests by assuming the existence of a correlation between observations in the same class in a k-way factorial experiment.The study showed that ignoring the effect of intraclass correlation can strongly affect a type I error probability at a significance level of 0.05.Pavur and Levis (1983) [12] found a general form of the correlation matrix to produce unbiased F tests in a k-way factorial experiment.Scariano and Davenport (1984) [13] introduced a theoretical study of corrected F-tests in the general linear model to examine the role of these statistics in the hypotheses test when a known, specified form of covariance structure is provided in the presence of correlated errors.Pavur and Davenport (1985) [7] provided a technique for adjusting the usual F-test when observations are correlated.The researchers found that by increasing the number of observations per cell in a factorial experiment, small correlations can affect the F statistic and significantly amplifies the type I error.Consequently, the uncorrected F statistics may abolish the outcomes of the analysis of variance (ANOVA) because it is not a true representation of the sample.AL-KAABAWI (2007) [14] studied the effect of correlations between observations on the type I error rates for multiple comparison procedures for 3-way crossed balanced model.Pavur (1988) [15] showed that the number of repeats in one-way ANOVA model can amplify small correlations and thus ignoring these correlations can easily inflate type I error.Mallick et al. (2020) [16] studied the effect of the first-order autoregressive correlated errors in the one-way and two-way models with repeated measurements.The modified F-test statistics is the classical F statistic multiplied by a constant that is a function of a maximum likelihood estimate of the correlation coefficient.The modified procedure led to obtaining a better result of analysis of variance.
Analysis of variance problems for autocorrelated data were addressed using nonparametric and parametric approaches.The parametric methods consider more robust than non-parametric methods.The parametric methods modify the quadratic forms and their degrees of freedom in the numerator and denominator of the F ratio.Andersen et al. (1981) [17] corrected the degrees of freedom of the distribution of the F statistic based on the number of groups, time points, and autocorrelation coefficient of random errors in the two-way ANOVA model.Lund et al. (2016) [18] replaced autocorrelated data in the F ratio by estimated prediction errors where errors are stationary time series.This approach allows for retaining the classical distribution of the null hypothesis with the usual degrees of freedom.
The previous studies addressed the problem of correlations between data in the same group for various models but have not handled the issues of correlations between errors in the different groups in the one-way random effect model.The present study provides the theory side for modification of the analysis of variance for the oneway random effect model in the presence of correlated errors in the same group and also between groups.The covariance and correlation structures of errors were proposed such that random errors in the same group have the same correlation coefficient  1 and in the different groups have the same correlation coefficient  2 .The test statistic was corrected by modifying the distributions of the two sums of squares the between and within groups according to the proposed correlation structure and under null and alternative hypotheses.The expected values of sums of squares were derived under suggested correlation conditions

Mathematical Model
The mathematical model of the balanced one-way random effects model can be written as follows: where   is the  ℎ observation corresponding to the  ℎ treatment group, μ is the general mean,   is the random effect corresponding to the  ℎ treatment group of the factor,   is the random error due to the  ℎ observation, at the  ℎ treatment group.For model (1), the   and   are independent and identical distributed normal random variables with means 0 and variances  1 2 and  2 2 , respectively.We adopted an error covariance structure based on the assumption that the errors are correlated.Errors occurring within the same group have a certain correlation, denoted as  1 , and errors occurring between groups have correlation, called  2 .The correlation coefficients  1 and  2 are called the intra-class and between-class error correlation coefficients, respectively, and the covariance structure of the errors is given by ( ) Main effects are combined to find the observed value, as we can see from the mathematical model for each ANOVA design.Therefore, the covariance structure between observations due to the correlation between errors in the model ( 1) is given by ( ) ( ) The sums of squares of the total, treatments (between groups), and errors (within groups) are given by ( ) ( ) We begin to prove (1) and ( 2), respectively.First, from (1), we have ( ) .. .

= + +
where From covariance structure in equation ( 2) and ( 3), then it easily follows that  ̅ .. is a normal random variable with ( ) Proof.Both the total sum of squares and the degrees of freedom have the addition property, so according to Cochrane's theorem [18], and under the null hypothesis, - ( ) ( )

 
From equation (3), it is easy to show that ( ) ( )

 
From theorem 1 we can obtain that ( ) Substituting equations ( 11), ( 14) and (15) into equation ( 13) we get that When the null hypothesis is true, Let each observation in a set of n random samples is normally distributed with the same variance, 22 12 +  , where  1 2 is the variance of random effects and  2 2 is the variance of random errors.

H : 0
proof.We know that  and  are independent.By theorem 2, the ( ) ( )

The proof of point (2), under H
A is true, the numerator ( ) ( ) , where C is correction factor.This implies that under   , the corrected statistic,  * has a noncentral Fdistribution with degrees of freedom ( ) , and non-centrality parameter  .

Correction Factor
This section discusses the values of the correction factor C to make sure that F tests are correctly performed under certain correlation patterns for random errors.It has been demonstrated that the modified test statistic  * is the usual F statistic multiplied by the correction factor, ( It is important to note that in the case  1 =  2 , the corrected statistic is identical to the classical F statistic, consequently, no correction is required.In the case the factor of correction is not equivalent to one, then there is a need for correction [14] [20-21].The values of the correction factor rely on the values of the correlation coefficients that include three cases.In case one, the values of both  1 and  2 are positive ; In case two are both negative, and in case three  1 is positive and  2 negative.In all three cases, values  1 >  2 produce a positive correlation coefficient.Tables 1, 2 and 3 show that the values of C increase as  2 increase and decrease as  1 increase for the three studied cases of correlation coefficients.As result, the values of the correction factor are restricted between 0 and 1.In addition, tables 1, 2 and 3 show that the increase in the size of the replications minimizes the value of the correlation factor where 2, 8, 40, and 80 were specified to be the replication size.The * mentions that this factor could not be calculated according to condition 12 >  , which guarantees a positive value of the correction factor .

□ Theorem 3 .
Under the null hypothesis the expected sums squares are given by 1.
the expected sum squares of treatments is given by from the definition of the F distribution, the corrected  * − statistic has a central F-distribution with under  0 is true.
Chi-square distribution regardless of whether  0 or H A is true .Then we can shown that we prove (2) in the same way.Again, from model (1), we can get

Table 1 :
Correction factor for = 2 and 8 k and for negative values of   and   .

and for negative values of
and   .

Table 3 : correction factor for
= 2 and 8 k and for positive values of   and   .

The * mentions that this factor could not be calculated according to condition 12 >  , which guarantees a positive value of the correction factor .Table 4 : correction factor for
and for positive values of   and   .

Table 5 :
correction factor for = 2 and 8 k and for positive values of   and negative values of   .