In the factor-analytic model, let some of the factors be known (i.e., the factor loadings are given in advance; they may e.g. be obtained from some previous analyses). However, their covariance matrix may, or may not, be known. The remaining factors (if any) are assumed to be uncorrelated among themselves and to the first set. For this model, the maximum likelihood equations are obtained and an iterative method for the solution is proposed.

In a multiple (or multivariate) regression model where the predictors are subject to errors of measurement with a known variance-covariance structure, two-sample hypotheses are formulated for (i) equality of regressions on true scores and (ii) equality of residual variances (or covariance matrices) after regression on true scores. The hypotheses are tested using a large-sample procedure based on maximum likelihood estimators. Formulas for the test statistic are presented; these may be avoided in practice by using a general purpose computer program. The procedure has been applied to a comparison of learning in high schools using achievement test data.

Maximum likelihood estimates of the free parameters, and an asymptotic likelihood-ratio test, are given for the hypothesis that one or more elements of a covariance matrix are zero, and/or that two or more of its elements are equal. The theory applies immediately to a transformation of the covariance matrix by a known nonsingular matrix. Estimation is by Newton's method, starting conveniently from a closed-form least-squares solution.

Numerical illustrations include a test for equality of diagonal blocks of a covariance matrix, and estimation of quasi-simplex structures.

In the method of multiple paired comparisons the dominance of object j over object k is observed upon p attributes. The present paper develops a covariance analysis for these paired comparisons in terms of a linear model which includes scale, bias, and interaction effects, along with s covariants upon which the comparisons are presumably dependent. The covariance model gives rise to adjusted parameter estimates and hypothesis tests for the residual pairwise layout from which the effects of the s covariants have been removed. These estimation and testing procedures are illustrated with an analysis of political judgment data, and their relevance to the general problem of residual scaling is discussed.

This paper is concerned with the study of similarities and differences in factor structures between different groups. A common situation occurs when a battery of tests has been administered to samples of examinees from several populations.

A very general model is presented, in which any parameter in the factor analysis models (factor loadings, factor variances, factor covariances, and unique variances) for the different groups may be assigned an arbitrary value or constrained to be equal to some other parameter. Given such a specification, the model is estimated by the maximum likelihood method yielding a large sample x2 of goodness of fit. By computing several solutions under different specifications one can test various hypotheses.

The method is capable of dealing with any degree of invariance, from the one extreme, where nothing is invariant, to the other extreme, where everything is invariant. Neither the number of tests nor the number of common factors need to be the same for all groups, but to be at all interesting, it is assumed that there is a common core of tests in each battery that is the same or at least content-wise comparable.

Following the work of Tucker, Cooper, and Meredith, image and anti-image covariance matrices from a correlation matrix that may be singular are derived.

The present study was undertaken to estimate the (co)variance components and genetic parameters of body weights recorded in Landlly piglets from birth to weaning at weekly intervals (w0 to w6). The data pertained to body weights of 2462 piglets, born to 91 sires and 159 dams across different generations during a 7-year period from 2014 to 2020. Five animal models (I–V), differentiated by inclusion or exclusion of maternal effects with or without covariance between maternal and direct genetic effects, were fitted on the data using the Bayesian algorithm. The analyses were implemented by Gibbs sampling in the BLUPF90 program and Markov chain Monte Carlo (MCMC) methodology was used to draw samples of posterior distribution pertaining to (co)variance components. Based on deviance information criteria (DIC), model V with inclusion of direct additive genetic, direct maternal genetic and permanent environmental effect of dam as random factors along with covariance between direct additive and maternal effects best fitted the data on pre-weaning traits (w0 to w5). Whereas, model I incorporating only the direct additive genetic effect best fitted the weaning weight (w6) data in Landlly piglets. The posterior mean estimates of direct heritability under the best models for W0 to W6 were 0.13, 0.19, 0.29, 0.13, 0.26, 0.32 and 0.46, respectively. Inclusion of the maternal component helped in better partitioning of variance for different body weights in Landlly piglets. The maternal heritability ranged from 0.06 to 0.14, while the litter heritability ranged from 0.11 to 0.15 for pre-weaning weights (W0 to W5) under the best-fit models. The influence of maternal environment was greater than maternal genetic effect from birth to 4th week of age. The results implied that variations in body weight of Landlly pigs were genetically controlled to moderate levels (especially w2 and w4) with contributions from direct additive and maternal genotype that can be exploited by designing efficient breeding programmes.

We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries, that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose components have correlation either zero or one or periodically oscillating. Moreover, the same type of behavior is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.

The Marcinkiewicz strong law, limn→∞(1 / n1/p)∑k=1n(Dk - D) = 0 almost surely with p ∈ (1, 2), is studied for outer products Dk = {XkX̅kT}, where {Xk} and {X̅k} are both two-sided (multivariate) linear processes (with coefficient matrices (Cl), (C̅l) and independent and identically distributed zero-mean innovations {Ξ} and {Ξ̅}). Matrix sequences Cl and C ̅l can decay slowly enough (as |l| → ∞) that {Xk,X ̅k} have long-range dependence, while {Dk} can have heavy tails. In particular, the heavy-tail and long-range-dependence phenomena for {Dk} are handled simultaneously and a new decoupling property is proved that shows the convergence rate is determined by the worst of the heavy tails or the long-range dependence, but not the combination. The main result is applied to obtain a Marcinkiewicz strong law of large numbers for stochastic approximation, nonlinear function forms, and autocovariances.

For a time series, a plot of sample covariances is a popular way to assess its dependence properties. In this paper we give a systematic characterization of the asymptotic behavior of sample covariances of long-memory linear processes. Central and noncentral limit theorems are obtained for sample covariances with bounded as well as unbounded lags. It is shown that the limiting distribution depends in a very interesting way on the strength of dependence, the heavy-tailedness of the innovations, and the magnitude of the lags.

The effects of low-level aircraft over-flights on behaviour of harlequin ducks (Histrionicus histrionicus) breeding in central Labrador were quantified during 2000–2002. The Canadian Department of National Defence supports a low-level training programme in the 130 000 km2 Military Training Area of Labrador involving military jets. The Institute for Environmental Monitoring and Research (IEMR) undertakes scientific research into environmental impacts of low-level military jet over-flights. A suite of 17 behavioural categories of paired male and female harlequin ducks was modelled, and a canonical variable representing alert behaviour, inactivity on the water and decreased inactivity out of water in response to over-flights represented 73.1% of the variance in the data cluster and provided marked separation of disturbed and undisturbed groups. Behavioural responses of harlequin ducks to military jets were 23 times stronger than their responses to floatplanes, helicopters and military cargo planes, and the significant interaction of aircraft type and noise indicated that noise may be the primary stressor affecting behaviour. A quadratic response of the canonical variable to noise generated from aircraft during standardized 30-minute observation periods was defined. The multivariate analyses were more robust because they indicated covariance in behavioural categories associated with disturbance that was not originally detected in univariate analyses, suggesting the importance of integrating behaviours other than overt responses. The significant effects of military jet over-flights on harlequin duck behaviour emphasize the need to evaluate potential population consequences of aircraft disturbance.