One-way Random Effects Model Research Articles

Nonparametric estimation in one-way random effects models

We present a rank based method for obtaining point and interval estimates of a scale version of the intraclass correlation coefficient in a one-way random effects model. When compared to the method of Arvesen and Schmitz (1970), the new method is not only applicable to a broader class of situations, but also much easier to implement. Results of a simulation study indicate that the new procedure compares favorably with the Arvesen-Schmitz procedure and the classical normal theory procedure especially If the random components have heavy tailed distributions.

Robustness and power of tests for a null variance ratio

SUMMARY A locally optimal test for a null variance ratio is considered in the context of the one-way random effects model, when normality is assumed. Using an asymptotic design sequence with an increasing number of groups with bounded but unequal sizes, this test has the correct asymptotic level of significance for nonnormal data, a property which is not shared by the competing Wald test. Asymptotic power calculations demonstrate that the locally optimal test may be more powerful than the Wald test even in situations where the actual significance level of the Wald test overstates the nominal level.

Prediction-interval procedures and (fixed-effects) confidence-interval procedures for mixed linear models

A general approach is presented for devising an approximate 100(1-α)% prediction interval for an unobservable random variable w based on the value of an observable random vector y. It is assumed theat E(w) and E(y) are ;linear combinations of unknown parameters β1,β2,…,βpand that the joint distribution of w-E(w) and yE(y) is symmetric and known up to the value of a vector θ of unknown parameters, as would be the case if y followed a mixed linear model (with normally distributed random effects and errors) and w were a linear combination of the model's fixed and random effects. Various implementations of the proposed approach are illustrated (and comparisons among them made) in the context of the Behrens-Fisher problem and the problem of making inferences about a group mean under a balanced one-way random-effects model.

Improved Estimators for Ratios of Variance Components

Abstract The problem of estimating a ratio of variance components in the balanced one-way random effects model is considered. It is shown that in terms of mean squared error, the ML, REML (or truncated ANOVA), and Bayes modal estimators (using the noninformative prior) are inadmissible. An estimator that dominates all three is derived. Two other estimators that are adaptive in nature are also introduced. The new estimators are shown to possess much-improved mean squared error properties. The results easily extend to balanced higher-way random or mixed effects models.

Improved Estimators for Ratios of Variance Components

Abstract The problem of estimating a ratio of variance components in the balanced one-way random effects model is considered. It is shown that in terms of mean squared error, the ML, REML (or truncated ANOVA), and Bayes modal estimators (using the noninformative prior) are inadmissible. An estimator that dominates all three is derived. Two other estimators that are adaptive in nature are also introduced. The new estimators are shown to possess much-improved mean squared error properties. The results easily extend to balanced higher-way random or mixed effects models.

A Comparison of Confidence Interval Methods for the Intraclass Correlation Coefficient

Different methods of obtaining confidence intervals for the intraclass correlation coefficient rho in the unbalanced one-way random-effects model are investigated, focusing on applications to family studies. Methods based on simple modifications of formulas for the case of equal group sizes are found to provide adequate coverage at small to moderate values of rho. A method based on the large-sample standard error of the sample intraclass correlation, as derived by Smith (1956, Annals of Human Genetics 21, 363-373), is shown to provide consistently good coverage at all values of rho. A method proposed by Thomas and Hultquist (1978, Annals of Statistics 6, 582-587) also provides consistently good coverage, but generates mean interval widths substantially greater than those generated by Smith's method at values of rho likely to arise in practice.

A Review of Inference Procedures for the Intraclass Correlation Coefficient in the One-Way Random Effects Model

Summary Recent theory and methodology for inferences concerning the intraclass correlation coefficient are reviewed, under the assumption of an underlying random effects model. Topics discussed include point and interval estimation, significance-testing for nonzero values of the intraclass correlation, and inference procedures in multiple samples.

Test of hypothesis in one-way random effects model with unequal error variances

An approximate test for testing a hypothesis in the one-way random effects model with unequal error variances is proposed. The validity of the procedure is evaluated by a simulation study.

Estimation of Fixed and Random Effects in the Functional Nonlinear Errors‐in‐Variable Model

AbstractConsider an experiment where a nonlinear continuous functional relationship exists between y and X. Assume that this relationship has been measured at n replicated points of X from each of t treatments or populations. Assume further that the X are fixed unknown vectors and that the location parameter v is either a fixed unknown vector or a vector of random variables. In the first case various linear hypotheses are to be tested about v, such as tests for main effects and interaction; in the second case, the mean and variance of the random variable v are to be estimated. A two‐step procedure based on asymptotic theory is presented to test hypotheses or develop estimates for the fixed effects or random effects functional errors‐in‐variable model. An example of a one‐way random effects model is given.

The analysis of binomial data by a generalized linear mixed model

SUMMARY Methods for generalized linear models are extended to provide estimates of location and variance parameters for mixed models fitted to binomial data formed by classifying samples from an underlying normal distribution. The method estimates the parameters directly on the underlying scale. For a balanced one-way random effects model, the variance estimator simplifies to the usual analysis of variance one. The estimation of variances and the prediction of random effects for binomial traits is required by animal breeders. The predictors given are analogous to best linear unbiased predictors (Henderson, 1973) but differ from those presented by Harville & Mee (1984).

A multivariate one-way classification model with random effects

In this paper a multivariate generalization of the one-way random effects model is investigated, maximum likelihood estimators are obtained, and the likelihood ratio test is derived for an hypothesis on the rank of the covariance matrix of the random effect vectors. Properties of the likelihood ratio test are investigated. A sequential procedure for determining the rank of the covariance matrix of the random effect vectors is presented.

Anova and minque type of estimators for the one-way random effects model

The one-way random effects model with unequal variances and unequal sample sizes is considered. Estimation of the variances, variance of a single observation (total variance), and the standard error of the unweighted mean are considered. Precision of the Analysis of Variance and Unweighted Sums of Squares type of estimators and the Minimum Norm Quadratic Unbiased Estimators with a priori weights are examined.

Bayesian shrinkage estimates for regression coefficients in m populations

For a linear regression model over m populations with separate regression coefficients but a common error variance, a Bayesian model is employed to obtain regression coefficient estimates which are shrunk toward an overall value. The formulation uses Normal priors on the coefficients and diffuse priors on the grand mean vectors, the error variance, and the between-to-error variance ratios. The posterior density of the parameters which were given diffuse priors is obtained. From this the posterior means and variances of regression coefficients and the predictive mean and variance of a future observation are obtained directly by numerical integration in the balanced case, and with the aid of series expansions in the approximately balanced case. An example is presented and worked out for the case of one predictor variable. The method is an extension of Box & Tiao's Bayesian estimation of means in the balanced one-way random effects model.

Use of prior information for estimating the variance components

The MINQUE and its modifications are considered for estimating the variances of the balanced one-way random effects model. The effects of the a priori values on the estimators of the variances are examined in detail. The Mean Square Errors of the estimators are compared for variations in the prior values of the unknown variances.

Interval Estimation for the Unbalanced Case of the One-Way Random Effects Model

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.

Selecting a test level for random effects

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

Effect of Non-normality on Inferences About Variance Components

Bayesian methods are utilized to analyse the one-way random effect model yij = μ + ai + eij in which eij are assumed normal, N(0, σ2 e ), and ai are assumed to have a mixture of two normals, .95N(−.05o(k – 1)σ, σ2) + .05N(.95o(k – l)σ, k 2o2). It is shown by a moderately sized sample that inferences regarding σ2 e = var (eij ) are insensitive but those of σ2 e = var (ai ) are very sensitive to changes of the two non-normality parameters (k, o). The data are also used to illustrate what inferences can be made about (k, o). Further, for σ = 0 it is found in general that extreme observations have less importance in comparison to others in the posterior expectation of σ2 e 2. Finally, the ways of assessing the contributions from ai and eii to the variation of yij are considered.

Bayesian Estimation of Means for the Random Effect Model

The problem of estimating the means in the one-way random effect model yjk = θj+ejk is considered from a Bayesian viewpoint. Posterior distributions of the θj are obtained under the assumption that the θj are independently drawn from a Normal population N(θ, σ22) and that the ejk are independent random errors having a N(0, σ21) distribution. It is shown that the posterior distributions of the θj are clustered more closely together than are the corresponding distributions for a fixed effect model. A numerical example is given.