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Showing papers on "Random effects model published in 1978"


Journal ArticleDOI
TL;DR: In this paper, the interval estimation of variance components is studied for the unbalanced one-way random effects model and an easily calculated function, $W$, of the harmonic mean of the class sizes and of the sample variance of class means is found and shown to be excellently approximated by a chi-square distribution.
Abstract: Interval estimation of variance components is studied for the unbalanced one-way random effects model. An easily calculated function, $W$, of the harmonic mean of the class sizes and of the sample variance of the class means is found to be important. The exact distribution of $W$ is found and is shown to be excellently approximated by a chi-square distribution. The random variable $W$ is used to construct interval estimates for (i) the between classes variance component and (ii) the ratio of the variance components and thus for the intraclass correlation and heritability. For most one-way unbalanced designs use of these approximate interval estimators will work very well.

139 citations


Journal ArticleDOI
TL;DR: In this article, the mixed model yijk, where xijkkm are known constants and βm's are unknown fixed effects, is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the β m's.
Abstract: Consider the model Yijk=μ + ai + bij + eijk (i=1, 2,…, t; j=1,2,…, Bi; k=1,2…,nij), where μ is a constant and a1,bij and eijk are distributed independently and normally with zero means and variances σ2adij and σ2, respectively, where it is assumed that the di's and dij's are known In this paper procedures for estimating the variance components (σ2, σ2a and σ2b) and for testing the hypothesis σ2b = 0 and σ2a = 0 are presented In the last section the mixed model yijk, where xijkkm are known constants and βm's are unknown fixed effects (m = 1, 2,…,p), is transformed to a fixed effect model with equal variances so that least squares theory can be used to draw inferences about the βm's

4 citations



Journal ArticleDOI
TL;DR: In this paper, a method for selecting an a-level to use when testing for group difference in a one-way classification random effects model is presented, where the a level is chosen to make the power of the test equal to.5 when the parameters are such that between group mean square and total mean square are equally good minimum expected squared error estimators.
Abstract: A method is presented for selecting an a-level to use when testing for group difference in a one-way classification random effects model. The a-level is chosen to make the power of the test equal to .5 when the parameters are such that between group mean square and total mean square are equally good minimum expected squared error estimators of the variance of y the estimator of the mean

1 citations


01 Jan 1978
TL;DR: In this paper, it is shown that the random effect approach and the fixed effect approach yield the same estimate for the slopes, the "within" estimate, when the "between" estimate is subject to a relatively large error, as is generally done in regression analysis.
Abstract: In empirical analysis of data consisting of repeated observations on economic units (time series on a cross section) it is often assumed that the coefficients of the quantitiative variables (slopes) are the same, whereas the coefficients of the qualitative variables (intercepts or effects) vary over units or periods. This is the constant-slope variableintercept framework. In such an analysis an explicit account should be taken of the statistical dependence that exists between the quantitative variables and the effects. It is shown that when this is done, the random effect approach and the fixed effect approach yield the same estimate for the slopes, the "within" estimate. Any matrix combination of the "within" and "between" estimates is generally biased. When the "within" estimate is subject to a relatively large error a minimum mean square error can be applied, as is generally done in regression analysis. Such an estimator is developed here from a somewhat different point of departure.

Dissertation
01 Jan 1978
TL;DR: In this paper, the robustness of the power in analysis of variance in relation to the departures from the in-built assumptions (i) equality of variance of the errors, (ii) statistical independence, and (iii) normality of the error in fixed and random effects models is studied.
Abstract: The object of the present work is to study the robustness of the power in Analysis of Variance in relation to the departures from the in-built assumptions (i) equality of variance of the errors, (ii) statistical independence of the errors, and (iii) normality of the errors in fixed and random effects models. It is difficult if not impossible, to conduct an exhaustive study of the problem, because the above assumptions can be violated in many ways. However, a general model and some important particular models have been used to obtain fairly conclusive evidence regarding the robustness of the power in Analysis of Variance. In order to obtain the power value in relation to the departure from the usual test assumptions, the general linear hypothesis model is considered. The power values when the assumptions of equality of variances and independence of errors are violated, are obtained and presented in Table IA and IB. The result suggests that in the above model, for tests regarding the inference about means, the power value is greatly affected by the inequality of error variances but only slightly affected by the serially correlated error variables. By using the permutation theory an approximate method is developed to study the effect of non-normality of the errors on the probability of type two errors in the above situation. Having studied the most general case in Analysis of Variance some particular models are discussed to investigate certain important aspects of the problem that are generated by these models. First of all fixed model one-way classification is considered to investigate whether it could show a different picture for unequal replication. The results so obtained are presented in Table IIA and IIB. They indicate that the power value is greatly affected by the inequality of error variances and unequal group sizes. This procedure is easily modified to handle the random model. Another particular case of the general linear model, that is fixed effect model two-way classification, is discussed. The results so obtained are presented in Table IIIA and IIIB. They indicate that in two-way classification for the between Column test, the power value is greatly affected by the inequality of column variances but only slightly affected by the serially correlated within rows error variables. Again this procedure is easily modified to handle the random model. The use of simulation methods for calculating the power values in the case of non-normal errors is discussed. One and two-way classifications are considered for the fixed effect model. The Erlangian and contaminated normal distribution are taken as examples of a non-normal error distribution. The results obtained by these methods are given in Table IVA and IVB which indicate that for the inference concerning means, the power calculated under normal theory is only slightly affected by the non-normality of the errors. Finally, the effect of non-normality on the power in analysis of variance for a random effect model is also discussed by a simulation method. One and two-way classification are considered for this model and the Erlangian and contaminated normal distributions are taken as examples of non-normality. The results obtained by these methods are given in Tables VA and VB which indicate that non-normality has little effect on the power of the test.