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In this paper a mixed model with nested random effects was used to model the cross-sectional area increment of lodgepole pine based on relevant explanatory variables. This model was used to show that minimum monthly temperature, monthly precipitation, and foliar biomass are positively related to the cross-sectional area increment, while an ordinal variable approximating lower trunk position and maximum monthly temperature are negatively related. It was shown that annual growth is better explained by monthly maximum and minimum temperatures than by average annual temperature and that the use of climate variables provided more reliable estimates for growth prediction than the use of the growth and density measurements from previous years. The Canadian Journal of Statistics 39: 185–189; 2011 © 2011 Statistical Society of Canada Resume: Dans cet article, un modele mixte avec des effets aleatoires emboites est utilise pour representer les accroissements transversaux de l’aire du pin de Murray a l’aide de variables explicatives pertinentes. Ce modele a permis de montrer que la temperature mensuelle minimale, la quantite de precipitation mensuelle et la biomasse foliaire sont positivement reliees a l’accroisement transversal d’aire tandis qu’une variable ordinale approximant la position de la partie inferieure du tronc et la temperature mensuelle maximale sont negativement reliees. Il est montre que la croissance annuelle est mieux expliquee par les temperatures mensuelles maximale et minimale que par la temperature annuelle moyenne et que l’utilisation des variables de climats procure des estimes plus fiables de la prediction de croissance que l’utilisation des observations sur la croissance et les mesures de densite des annees precedentes. La revue canadienne de statistique 39: 185–189; 2011 © 2011 Societe statistique du Canada

Reliability measures have been well studied over many years. Such measures have been thoroughly studied for two-factor models. Motivated by a medical research problem, point and confidence interval estimates of the intraclass correlation coefficient are extended to models containing three crossed random factors-subjects, raters, and occasions. The estimation is conducted using both analysis of variance and Monte Carlo Markov chain methods.

In studies of disease inheritance, it is more convenient to collect family data by first locating an affected individual and then enquiring about the status of his or her relatives. Although the different categories of children classified by disease, sex, and other covariates may have a particular multinomial distribution among families of a given size, the numbers as ascertained do not have the same distribution because of unequal probabilities of selection of families. The introduction of weighted distributions to correct for ascertainment bias in the estimation of parameters in the classical segregation model can be traced to Fisher in 1934. This theory was presented in a general formulation by C. R. Rao at the First International Symposium on Classical and Contagious Distributions in 1963. Further expansion on the topic was given by C. R. Rao in the ISI Centenary Volume published in 1985. The effects of different two-phase sampling designs on the estimation of parameters in the classical segregation m...

Summary Beginning with Galton, Various authors have proposed non-iterative estimators of the correlation coefficient that measure the degree of similarity among siblings. However, these extimators lose efficiency appreciable for certain ranges of values of the correlation coefficient. An estimator formed by an algebraic combination of two non-iterative extimators based on Smith (1957) is shown by simulation to be nearly full efficent for teh full range of values of the coefficent. A new estimator given by the best linear combination of these extimators offers a practical improvement for large values of the coefficent. The discussion conceludes with an example.

SUMMARY In genetics, the term interclass correlation has been used to refer to the parent-offspring and son-daughter correlations within families. Under the assumption of variable family sizes and normality, we derive the asymptotic variance of the maximum likelihood estimator of the sondaughter interclass correlation.

Methods to unravel the genetic determinants of non-Mendelian diseases lie at the next frontier of statistical approaches for human genetics. It is generally agreed that, before proceeding with segregation or linkage analysis, the trait under study ought to be shown to exhibit familial correlation. By coding dichotomous traits as binary variables, a single robust approach in the estimation of pedigree correlations, rather than two distinct approaches, can be used to assess the potential heritability of a trait, and, latterly, to examine the mode of inheritance. The asymptotic theory to conduct hypothesis tests and confidence intervals for correlations among different members of nuclear families is well established but is applicable only if the nuclear families are independent. As a further contribution to the literature, we derive the asymptotic sampling distribution of correlations between random variables among arbitrary pairs of members in extended families for the Pearson product-moment estimator with generalized weights. This derivation is done without assuming normality of the traits. The sampling distribution is shown to be asymptotically normal to first order, and hence large-sample hypothesis tests and confidence intervals with estimates of the variances and correlation coefficients are proposed. Discussion concludes with an example and a suggestion for future research.

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.

SUMMARY A noniterative method for estimating the interclass correlation coefficient is derived from the technique of weighted sums of squares. The asymptotic variance of this estimator is derived under the assumption of normality. Through extensive Monte Carlo simulations, the sample variance of this estimator is found not to differ greatly from that of the maximum likelihood estimator. An important problem in genetics is the estimation of the degree of resemblance between a mother and her children as measured by the interclass correlation coefficient. Because parents produce differing numbers of offspring during their reproductive period, the maximum likelihood estimation of the interclass correlation presents computational difficulties requiring an iterative solution. Therefore, a number of noniterative estimators have been proposed in the literature. These estimators will be shown to be associated with special cases of the generalized estimator derived from the weighted sums of squares of measurements on parents and offspring. Because our purpose is to provide a comprehensive approach to the noniterative point and interval estimation problems for the interclass correlation coefficient, the asymptotic variance of the proposed estimator is derived under the assumption of normality. Moreover, the sample variances of two special cases of the proposed estimator are compared to that of the maximum likelihood estimator through Monte Carlo simulations. By a theoretical argument, it is shown that the estimator proposed by Srivastava (1984) has a uniformly smaller asymptotic variance than the ensemble estimator proposed by Rosner, Donner & Hennekens (1977). Smith (1957) used the weighted sums-of-squares approach for the estimation problem concerning the intraclass correlation coefficient which measures the degree to which siblings resemble one another. Our results together with those of Smith (1957) provide a unified approach to estimation problems concerning familial correlations among a parent and offspring.

Asymptotically normal estimators of interclass and intraclass correlations are derived for more than two quantitative characteristics of parent and siblings in a simple random sampling of families that have different numbers of offspring. These estimators are proposed as an alternative to the maximum likelihood estimators, which can be found only by iterative methods requiring prohibitively large amounts of computation. The asymptotic variances of the proposed estimators are also given. In an illustrative example, these easily computable estimators are seen to be comparable to the corresponding maximum likelihood estimators.