Variance Component Test Research Articles

Exact Tests for Variance Components

Ofversten (1993) presented methods for obtaining exact F-tests of variance components in three unbalanced mixed linear models. He developed methods for models with one random factor, models with nested classifications, and models with interaction between two random factors. In Section 2, we review and extend Wald's test, which is a generalization of one of Ofversten's methods. Ofversten's other method is a generalization of Khuri (1987, 1990) and Khuri and Littell (1987). It does not apply to the one random factor model. In Section 3, we present a unified treatment of Ofversten's second method. This treats all of Ofversten's examples and gives a formal justification to Ofversten's claim that the method extends quite generally. Section 3 also gives a formal condition for when the method does not require resampling from the residuals. The unified treatment is conceptually simpler than Ofversten's treatment in that it uses a single orthonormal basis rather than a series of matrix decompositions that change slightly from example to example. There do not seem to be any particular computational advantages to the unified treatment. The unified treatment also appears to generalize Ofversten's method in that it applies in some cases where Wald's test also applies. Section 4 compares the two tests when both apply. All of the arguments are given in terms of vector space ideas (cf. Christensen, 1987).

A Note on Exact Tests for Variance Components in Unbalanced Random and Mixed Linear Models

Ofversten (1993) proposed a method for constructing exact tests for variance components in unbalanced mixed linear models. The method is a generalization of a technique employed by Khuri (1987), Khuri and Littell (1987), and Khuri (1990) for testing variance components in random models. This approach employs a partitioning of the residual sums of squares to derive a set of quadratic forms that are independently distributed as chi-squared random variables. These quadratic forms are used to construct an exact test statistic for testing the null hypothesis that a variance component is equal to zero. Although the actual value of the test statistic depends on the particular partitioning, the distribution of the test statistic is invariant to this choice. Ofversten refers to this technique as resampling. Resampling tests are recommended with slightly unbalanced data because the properties of these tests are similar to the corresponding uniformly most powerful tests with balanced data. An inequality that can be used to place a bound on the power of a resampling test is derived in this paper.

Estimation in mixed models via layer triangular transformation

A new alternative for estimating parameters in some mixed linear models is presented and illustrated. The approach is based on the same layer triangular transformation of the model matrix as proposed earlier by Öfversten (Biometrics 49 (1993)) for the derivation of some exact tests for variance components. Using the same preliminary transformation for both testing and estimation gives us an integrated set of procedures for the full analysis of some widely used mixed linear models. The proposed procedures are computationally stable and space saving.

Variance Components Testing in the Longitudinal Mixed Effects Model

This article discusses the asymptotic behavior of likelihood ratio tests for nonzero variance components in the longitudinal mixed effects linear model described by Laird and Ware (1982, Biometrics 38, 963-974). Our discussion of the large-sample behavior of likelihood ratio tests for nonzero variance components is based on the results for nonstandard testing situations by Self and Liang (1987, Journal of the American Statistical Association 82, 605-610).

Tests Based on Admissible Estimators in Two Variance Components Models

In the paper a new class of tests for variance components in two variance components model is presented. The tests are based on nonnegative admissible invariant quadratic estimators of variance components. A comparison of the tests with the known ones are given. Two-way classification models corresponding to block designs are considered separately. Examples of these models are given, for which the power functions of the tests are computed by using Imhof’s procedure, and compared with the attainable upper bound obtained by using the Neyman–Pearson Lemma.

Hypotheses tests for variance components in some multivariate mixed models

Abstract A multivariate mixed model involving exactly one random effect is considered and a locally best invariant (LBI) test is derived for testing the significance of the multivariate variance component corresponding to the random effect. For balanced models, the likelihood ratio test (LRT) is also derived for the same testing problem. In general, the LBI test is different from Pillai's trace test, except for balanced models. The standard multivariate tests based on Wilk's Λ, Pillai's trace and Roy's maximum root are also valid tests for this problem and simulated powers of these tests and the LBI test are reported for several bivariate unbalanced one-way random models. As expected, the LBI test has some advantage over the others for local alternatives. For several bivariate balanced one-way random models, simulation results are also given on the power of the LRT and the tests based on Wilk's Λ, Pillai's trace and Roy's maximum root in order to compare these tests.

Testing hypotheses in mixed linear models

The paper proposes a new class of tests for variance components in general mixed models. A selective literature survey on testing hypotheses in unbalanced mixed analysis of variance shows the need of a theory of approximate tests. Using a compromise on unbiasedness a new optimality criterion is introduced. It is shown that C.R. Rao's MINQUE is locally sufficient in this context, and the class of locally best locally unbiased invariant tests is constructed. Examples show a good approximation of the nominal significance level on a large region of the hypothesis and a high power of the resulting test.

Exact Tests for Variance Components in Unbalanced Mixed Linear Models

A method for deriving exact tests for variance components in some unbalanced mixed linear models is presented. The derivation is based on a new kind of preliminary orthogonal transformation and a subsequent resampling procedure. The resulting tests are based on mutually independent sums of squares which, under the null hypothesis, are distributed as scalar multiples of chi-square variates. For balanced data, the tests are identical to the traditional F tests.

Exact tests in unbalanced mixed analysis of variance

The paper proposes a new class of unbiased tests for variance components and fixed effects in unbalanced mixed analysis of variance. The class of Bartlett-Scheffè tests is discussed in this context. A new method is given for the construction of such tests for general hypotheses in general classifications. The resulting tests are shown to be minimax in the class considered.

A note on the application of diallel crosses for the analysis of genetic variation in natural populations

The complete diallel cross among homozygous lines can be a useful tool to analyze the genetic architecture of natural populations. However, it represents the natural population only approximately, in particular if the number of lines is small and the analyzed traits exhibit inbreeding depression or other forms of directional dominance. Some incorrect expected mean squares that can be found in the literature suggest tests for genetic variance components that can be misleading under such circumstances. Expected mean squares for a factorial analysis and for a modified Hayman analysis are presented and the effect of the number of lines and directional dominance is discussed.

Linear and quadratic Estimation from Inter- and Intra-Elock Sources of Information

In This Paper the Problem of Inter- and intra-block estimation of treatment effects and variance components in mixed linear model corresponding to block designs is considered. It is assumed that block effects are random and treatment effects are fixed, The considerations lead to a new characterization of balancedness and orthogonality of designs. Obtained results are applied to construction of variance component tests based on admissible estimators

Proceedings of the Second International Tampere Conference in Statistics

Keynote Addresses ANDERSON, T.W.: Multivariate linear relations. Box, G.E.P. (with R.D. MEYER): Some aspects of statistical design in quality improvement. HANNAN, E.J.: The statistical theory of linear systems. RAO, C.R.: Estimation in linear models with mixed effects: a unified theory. Invited Papers ATKINSON, A.C.: Robust regression and unmasking transformations. BAKSALARY, J. K.: Algebraic characterizations and statistical implications ofthe commutativity of orthogonal projectors. CONRADSEN, K. (with B.K. NIELSEN): Textural features useful in classification of digital images. KARIYA, T.: MTV model and its application to prediction of stock prices. KHATRI, C.G.: Quadratic forms and null robustness for elliptical distributions. MUSTONEN, S.: Editorial approach in statistical computing. PERLMAN, M.D. (with T.W. ANDERSON): Consistency of invariant tests for the multivariate analysis of variance. PRATT, J.W.: Dividing the indivisible: using simple symmetry to partition variance explained. PUKELSHEIM, F.: Majorization orderings for linear regression designs. PUKKILA, T. (with A. KALIJNEN): On the order determination of time series models. RISSANEN, J.: Complexity and information in contingency tables. SoCTT, A.J.: Generalized linear models with survey data. SINHA, B.K. (with R. DAs): Robust optimum invariant unbiased tests for variance components. SPEED, T.P.: Generalized variance component models. STYAN, G.P.H. (with R.E. HARTWIG): Partially ordered idempotent matrices. Contributed Papers ANDERSON, A.J.B.: Fortran8x as a language forstatisticalcomputation. BALDESSARI,B.: On regression analysisundertheintrinsicinference model for stratified dependent normal random variables. BRANNAS, K.: Control in the duration model framework. DANIAL, E.J. and KATTi, S.K.: The asymptotic variances of the necessary and sufficient conditions for the infinite divisibility of discrete distributions through multivariate analysis. DANIAL, E.J. and KATTI, S.K.: Computing the maximum likelihood estimates of the hyper-Poisson parameters interactively through APL computer language. DIEBOLD, F.X. and NERLOVE, M.: Factor structure in a multivariate ARCH model of exchange ratefluctuations. FAREBROTHER, R.W.: Simultaneous confidence intervals when the regressor matrix is unbalanced. FEUERVERGER, A.: Some Fourier procedures for time domain analysis of parametric and semi-parametric time series models. GELFAND, A.E.: Estimation of a restricted variance ratio. de GOOIJER, J.G. and HEUTS, R.M.J.: Higher order moments of bilinear time series processes with symmetrically distributed errors. IGNATOV, Z.G. and KAISHEV, V.K.: On the computation of distributions of serial correlation coefficients through B splines. JAJUGA, K.: Elliptically symmetric distributions and their application to classification and regression. KOLLO, T.: Asymptotic distributions of functions of means and covariances. KONAKOV, V.D.: Nonparametric density estimation: L approach. KORHONEN, P. and NARULA, S.C.: On subjectivity in statistics. KOUKOUVINOS, C. and KoUNIAS, S.: Optimal designs in linear models and Hadamard matrices. LATOUR, D., PUNTANEN, S. and STYAN, G.P.H.: Equalities and inequalities for the canonical correlations associated with some partitioned generalized inverses of a covariance matrix. LISKI, E.P.: Identifying influential data in a growth curves model. MANLY, B.F.J. and McALEVEY, L.: A randomization alternative to the Bonferroni inequality with multiple F tests. MEJZA, S.: Experiments in incomplete split-plot designs. MERIKOSKI, J.K.: Underestimating the largest eigenvalue of a covariance matrix. MILITKt, J., KvSMR, K. and CAP, J.: Comparison of some influence measures in nonlinear regression. NORDSTROM, K. and von ROSEN, D.: Algebra of subspaces with applications to problems in statistics. OSOSKOV, G.A.: Robust regression for the heavy contaminated sample. PYNNONEN, S.: Selection of variables in nonlinear discriminant analysis by information criteria. SALEM, A.M. and WAHAB, S.A.: On the determination of the amplitude of quasi-stationary phenomena. SCHAFFRIN, B.: Less sensitive tests by introducing stochastic linear hypotheses. SCHALL, R. and DUNNE, T.T.: On outliers and influence in the general multivariate normal linear model. TRENKLER, G., STAHLECKER, P., SCHIPP, B. and HERING, F.: Iterative improvements of a partial minimax estimator in regression analysis. WERNECKE, K.-D., KALB, G. and STdRZEBECKER, E.: On classification strategies in medical functional diagnostics.

Exact Tests for the Main Effects Variance Components in an Unbalanced Random Two-Way Model

Abstract : Exact tests are derived for testing hypotheses concerning the variance components of the main effects in an unbalanced random two-way crossed classification with interaction model. The tests are based on four sums of squares which are distributed independently as scalar multiples of chi-squared variates. These sums of squares can also be used to find an exact test concerning the interaction variance component, and to obtain simultaneous confidence intervals on all continuous functions of the model's variance components. A study is made concerning the power of the proposed tests, including a comparison with other approximate tests. (Author)

Applying Wald's Variance Component Test

In this note a generalization of a variance component test that was first suggested by Wald is examined. Necessary and sufficient conditions are given for the test to be applicable in a mixed linear model. A uniqueness property of the test in terms of degrees of freedom is also obtained.

On a theorem of graybill

Graybill (1976) gives a theorem for testing variance components in balanced random models. Unfortunately, the theorem does not hold true. We pin down the reason that causes the theorem to fail and give a correct version of it.

The power function and an approximation for testing variance components in the presence of interaction in two‐way random effects models

AbstractIn this paper we consider a two‐way layout random‐effects model with interaction proportional to the product of two random components and derive the distributions of the sums of squares and the F‐ratio under the usual normality assumption for the random effects. Approximations to these distributions by finite series Laguerre polynomial expansions are then developed. Some numerical results are given to illustrate the applications of the theory.

Uanova: a fortran IV program for unbalanced nested analysis of variance

Abstract A FORTRAN IV program (UANOVA) to carry out unbalanced analyses of variance, associated variance component testing using synthesized error mean squares and degrees of freedom where appropriate, and the generation of summary statistics is presented. Data transforms are available to assist in meeting the requirements of the ANOVA method, and the data structure is defined as part of the input stream. The summary statistics include estimates of the population mean and variance, and confidence limits for the mean and variance components. The latter are approximations for levels higher than the first, however their calculation affords on indication of the imprecision of the point estimates. The program has been verified, where possible, with published data sets, results of these tests are presented.

Variance Component Testing in Unbalanced Nested Designs

Abstract The commonly used approximate F test for variance component testing in unbalanced nested designs was adapted from procedures developed for balanced cross classifications. However, in the unbalanced case, the analysis of variance mean squares generally are not independent or chi-square type variables, as assumed by the test procedure. Estimates of the test size disturbance introduced by employing the approximate procedure for selected unbalanced three-level nested designs indicate that the disturbances are small for a wide range of variance component ratios for extremely unbalanced design situations. Extensions to higher level unbalanced nested designs are discussed.

Variance Component Testing in Unbalanced Nested Designs

Variance Component Testing in Unbalanced Nested Designs

Type-I Error and Power of a Test Involving a Satterthwaite's Approximate F-Statistic

In certain types of statistical analysis, it is often desirable to test the general hypothesis that the relation CliO + ' * + CmOm = Cm+l0m+i + * * * + c,A, holds among the variances or variance components Oi , where the c, are known real numbers. These components of variances have independent mean square estimates v, based on ni degrees of freedom, such that nivi/O, has a chi-square, x2, distribution. Specific examples of such tests are the F-test for equality of two variances, Bartlett's test of homogeneity of variances, the two-tailed Behrens-Fisher problem, and testing variance components in some ANOVA with random effects. Also Welch (1936) shows the usefulness of being able to test such relations, and Comstock and Robinson (1948) showed that such relations occur in certain types of genetic experiments. The exact distributions of linear combinations of independent x2 distributed random variables have been derived by several authors, e.g. see Hsu (1938), Robbins and Pittman (1949), Box (1954), Ruben (1963) and Wang (1967). However, such distributions are too complicated for practical usage. When considering the problem of testing the relation