25. Linear Models for Unbalanced Data. By S. R. Searle. ISBN 0 471 84096 3. Wiley, Chichester, 1987. 536 pp. £45.85.

Linear Models for Unbalanced Data

SUMMARY Tests for fixed-effect factors in unbalanced mixed models have previously used t-tests on a contrastby-contrast basis or Wald statistics without a universally accepted method of calculating the denominator degrees of freedom. This situation has arisen because the variances of different contrasts are differently weighted sums of the variance components with associated degrees of freedom that are not necessarily equal. A simultaneous F-test for differences between all levels of a fixed-effect factor can be derived by forming new contrasts, by rotation of the original contrasts, with variances that are close to being the same weighted sum of variance components. The associated degrees of freedom for these new contrasts are nearly equal. A small simulation study shows the appropriateness of a X2 approximation to the distribution of the weighted sums of variance components. Three simple examples are used to demonstrate the effects of rotation. The last of these examples is also used to compare the proposed simultaneous F-test with the distribution of the Wald statistic obtained by numerical simulation. The method of rotations is then applied to data on the range size of mountain hares (Lepus timidus) to assess the evidence for a two-way interaction between season and habitat. Analysis of data from balanced experiments is traditionally performed by an analysis of variance in which the total sum of squared deviations of observations about their mean is partitioned into sums of squares attributable to either the different treatment effects or the different random effects. Under standard distributional assumptions, each random effect sum of squares follows a multiple of a X2-distribution, this multiple being the product of the degrees of freedom (the number of independent error contrasts whose square has the required expectation) and a linear combination of the different variance components, the coefficients in this linear combination being fixed by the design. Under any null hypothesis concerning the absence of treatment effects, the corresponding treatment sum of squares also follows such a multiple of a X2-distribution. Furthermore, the linear combination of variance components in the multiple for the treatment sum of squares is identical to the linear combination for one of the random-effect sum of squares; hence, hypotheses about the treatment effects can be tested by dividing treatment mean squares by the appropriate error mean square to form a variance ratio with an F-distribution. The benefits of experimental design accrue, first, through the variance of all contrasts for any given treatment factor or interaction having the same expectation under the null hypothesis; second, through this variance being as small as possible; and, third, through there being as many error contrasts as possible whose variance is the same as that for the treatment factor or interaction. Analysis of data that lacks these properties of balance is much less straightforward, yet there are many areas in which, due to the nature of experimental material, the treatment effects cannot be applied in a balanced fashion, and so such unbalanced data are the norm rather than the

http://www.jstor.org/stable/pdfplus/2533859.pdf

Estimation of Denominator Degrees of Freedom of F-Distributions for Assessing Wald Statistics for Fixed-Effect Factors in Unbalanced Mixed Models

The variability among data may have quite complex structure and the splitting of variation into its individual components is important in many areas of applications. In particular, the study of variation through a class of linear models known as random and mixed models is a central topic in statistics with wide ramifications in both theory and applications. Sometimes these models are called components of variance models since the interest in the model lies in investigating the variances of the effects of factor levels which are assumed to be randomly selected from a large population of levels. These variances are also known as variance components since the total variance can be expressed as the sum of these variances and the error variance. Knowledge of the variance components, i.e., the information on the relative importance and magnitude of the sources of variability in the data is of critical importance in many problems since the isolation of the total variance into the individual components of variance would result in the physical elimination of the sources of variability in question.

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A bibliography on variance components an introduction and an update: 1984-2002

The 25-page Bibliography in Applied Regression Analysis, 2nd edition, by N.R Draper and H. Smith, published by Wiley in 1981, was previously extended by a list of selected references available during 1988-89,1990-91 and 1992-93, plus a few older references. See Communications in Statistics-Theory and Methods,19(4), 1205-1229 (1990), 21(9), 2415-2437 (1992) and 23(9), 2701-2731 (1994). Here is a further update covering the years 1994-97. As before, items were chosen on the basis of their perceived relevance to practical applications (sometimes rather widely interpreted). The classification system used here is that of the 3rd edition of the book, published in 1998

Applied regression analysis bibliography update 1994-97

. 1. Survival of newly emerged jack pine budworm Choristoneura pinus pinus is related to the density of available pollen cones (microsporangiate strobili) produced by its host tree, jack pine Pinus banksiana.



2. A 7-year time series of observations from a plot network in Ontario, Canada, compared the propensity of jack pine to produce pollen cones, τ, on trees that were either defoliated or undisturbed by the jack pine budworm.



3. Non-defoliated jack pine trees have a high propensity to produce pollen cones. More than one-third of these trees produced pollen cones in every year of the series. Propensity varied significantly among plots and trees. Temporal patterns in propensity were also highly variable but within a plot propensity was often autocorrelated in time.



4. Defoliation by the jack pine budworm was associated with forest plots composed of the oldest and the largest trees and with the fewest trees per hectare. Within a plot, outbreaks lasted 3 or 4 years although individual trees were only defoliated in 1 or 2 years.



5. The propensity to produce pollen cones in jack pine was reduced in the years after defoliation. The most pronounced reductions in propensity occurred where defoliation was most severe.



6. The reduction in propensity to produce pollen cones resulting from previous defoliation, coupled with the dependence of jack pine budworm survival on the availability of pollen cones, induces a lagged, negative feedback between the density of the consumer and that of its resource.



7. The lagged, density-dependent relationship between jack pine budworm and its jack pine host contributes to oscillatory dynamics of the jack pine budworm. Comparison of the outbreak behaviour of jack pine budworm with that of the closely related eastern spruce budworm C. fumiferana suggests that differences in the strength of the host-plant interaction may account for differences in the relative frequency of outbreaks in the respective systems.

A lagged, density‐dependent relationship between jack pine budworm Choristoneura pinus pinus and its host tree Pinus banksiana

The testing of both variance components and fixed effects in an unbalanced mixed model has relied on approximate techniques, particularly, Satterthwaite's approximation of the test statistics. The derived tests have unknown distributions, both under the null and alternative hypotheses, due to the lack of independence and chi-squaredness of the mean squares involved. Hence, the appeal for exact testing techniques is understandable. This article presents exact tests concerning the variance components of the random effects and estimable linear functions of the fixed effects in an unbalanced mixed two-way cross-classification with interaction model. The derivations are based on techniques similar to those applied by Khuri and Littell (1987, Biometrics 43, 545-560) to the same model, but with all random effects. The proposed methodology requires that the data under consideration contain no empty cells.

Jose Gallo

Papers

Exact tests for the random and fixed effects in an unbalanced mixed two-way cross-classification model