Unbalanced One-way Random Effects Research Articles

Limiting the Effects of Single-Member Families in the Estimation of the Intraclass Correlation

SUMMARY Because the maximum likelihood estimator for the intraclass correlation p does not exist in closed form for the unbalanced one-way random-effects model, several noniterative estimators have been proposed for p in normal populations. Estimators that exclude all families of only one member have been recommended in the literature of familial correlations; however, these estimators lose efficiency appreciably for certain ranges of values of p. Indeed, no single estimator can have the highest efficiency along the full range of the intraclass correlation coefficient. Three proposed estimators, each formed by combining a pair of noniterative estimators of which one constituent estimator excludes families having only one member, are shown by simulation to be nearly fully efficient for the full range of values of p. A mathematical expression is given that can be used to calculate the large-sample standard errors of all known noniterative estimators of the intraclass correlation.

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.

Estimation of variance components in an unbalanced one-way classification

In the unbalanced one-way random effects model the weighted least squares approach with estimated weights is used to develop a relatively simple estimator of variance components. As the number of classes increases, the proposed estimator is seen not only to be best asymptotically normal but also to be asymptotically equivalent to the maximum likelihood estimator.

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.