Unbalanced Model Research Articles

Nonnegative unbiased estimability of linear combinations of two variance components

Abstract For a general mixed model with two variance components θ1 and θ2, a criterion for a function q1θ1+q2θ2 to admit an unbiased nonnegative definite quadratic estimator is established in a form that allows answering the question of existence of such an estimator more explicitly than with the use of the criteria known hitherto. An application of this result to the case of a random one-way model shows that for many unbalanced models the estimability criterion is expressible directly by the largest of the numbers of observations within levels, thus extending the criterion established by LaMotte (1973) for balanced models.

Confidence Intervals on Linear Combinations of Variance Components in the Unbalanced One-Way Classification

A problem that can occur in a variance component analysis is the estimation and construction of confidence intervals for linear combinations of the variance components. This article considers the unbalanced one-way classification model and develops a procedure that can be used to construct a confidence interval on a linear combination of the among- and within-group variances. A simulation study suggests that the procedure provides an interval that in most cases has an achieved confidence coefficient at least as great as the stated level.

Quadratic estimation in mixed linear models with two variance components

Abstract Explicit expressions for Bayes invariant quadratic estimates, biased and unbiased, are presented and proved to cover the entire class of admissible estimates in the considered classes. An unbalanced genetic model is studied for demonstration.

Textbook Analysis of Covariance -- Is It Correct?

Abstract : For unbalanced Model I ANOVA data there is a controversy in the literature -- should the partial SS's be computed hierarchically or should they be computed after fitting all other model terms including higher-order interactions? A similar problem exists in the Model I analysis of covariance (ANCOVA), yet the usual prescription given in textbooks uses only the non-hierarchical approach. Using two classic examples of ANCOVA, this paper examines the two approaches to ANCOVA calculations. (Author)

Cumulants of bilinear forms and quadratic forms and their applications

In this paper, mixed cumulants of general bilinear forms, quadratic forms and linear forms are given up to order 4. These results are useful in deriving approximations to the distributions of sums of squares and F-ratios from unbalanced analysis of variance models from non-normal universes. As an illustration, the results are applied to derive mixed cumulants of sums of squares in a three stages nested unbalanced random effects models from non-normal universes.

On Approximating the Null and Nonnull Distributions of the F Ration i Unbalanced Random-Effect Models from Nonnormal Universes

On Approximating the Null and Nonnull Distributions of the F Ration i Unbalanced Random-Effect Models from Nonnormal Universes

On Approximating the Null and Nonnull Distributions of theFRatio in Unbalanced Random-Effect Models from Nonnormal Universes

Abstract In this article we consider the unbalanced random-effects model as given in (1.1) with random variables from nonormal universes. Assuming that the αi 's are independently distributed of the eij 's, an approximation to the joint distribution of the mean squares (and hence the F ratio) by using Laguerre polynomials is derived. These results are then used to examine the effects of departure from normality on the traditional F-ratio distribution.

Computing Expected Mean Squares

SUMMARY The evaluation of the expected mean squares arising from the analysis of unbalanced mixed ANOVA models has long been an analytically intractable problem. Irl this paper we present a theoretically and computationally simple technique for computation of the coefficients in the expected mean squares.

The State and Economic Development in India 1890—1947: The Case of Oil

The role of the State in promoting Indian economic development in the nineteenth century is one of several aspects of modern Indian economic history which have been ‘re-interpreted’ in recent years. The conventional wisdom once portrayed the policy of the British government in India as one essentially geared to serving British economic interests. By means of ‘discriminatory interventionism’ in economic affairs, it was argued, the Government encouraged the development of a primary commodity export economy, with all its attendant defects, in India. However, over the last two decades the reputation of the Government of India has undergone a rather noticeable transformation. Economic imperialists became, first, benevolent nightwatchmen, and then ‘development-orientated’ officials formulating an embryonic unbalanced growth model for Indian development. Parallel with this improvement in the Government of India's reputation has been a deterioration in the economic reputations of certain other governments in nineteenth-century developing economies, governments whose performances used to be favourably compared with that of the British in India. In the cases of Japan and Tsarist Russia, for instance, both the extent and effectiveness of State intervention in the economy has been questioned, and there has been an increasing recognition of the primacy of non-governmental factors in the economic growth of those countries. Given the ideological and organizational parameters limiting the range of possible activity by any nineteenth century government in its economy, the performance of governments in other developing countries of the period, and the political constraints imposed by being a subordinate section of a world-wide Empire, it is no longer possible regard the actions of British officials in India as wicked, and many would now regard them as almost respectable.

Modern trends in the theory of robustness2

The first part of this paper contains an introduction into robust statistics with special emphasis on some difficult conceptual points and on recent results concerning the intuitive basis of robustness theory. A brief review of some classical robustness concepts and results follows, and the last part discusses a number of research areas in robustness which are of current interest, including treatment of arbitrary unstructured data and of unbalanced linear models.

On tests in variance components models2

A procedure for finding exact tests for some hypotheses on variance components in unbalanced models is proposed. It is based on F-distributed statistics got by an orthogonal decomposition of the sample space.

On the kinetic energy of the divergent and nondivergent flow in the atmosphere

The kinetic energy of horizontal flow in a hydrostatic atmosphere is divided into the kinetic energies of its divergent and nondivergent components. The law of conversion between these two energies for large-scale flows in the atmosphere is derived and discussed using balanced and unbalanced models of circulations in the atmosphere. It is shown that the total potential energy is converted into the kinetic energy of the divergent flow which, in turn, is converted into the kinetic energy of the nondivergent flow. These energy conversions are equal in a so-called balanced model, but may differ in a model based on the primitive equations. The analysis shows that the kinetic energy reservoir of the divergent part of the flow may play a quasi-catalytic role in the energy conversion between the total potential energy and the kinetic energy of the nondivergent flow. Two sets of data, the history data generated by the NCAR GCM of a January simulation, and the NMC wind data and FNWC height field of August 1 to 15 of 1970, were used to evaluate the energies and energy conversions. The computations confirm the direction of energy conversions in the theoretical argument and the quasi-catalytic nature of the kinetic energy of the divergent part of the flow. The energy conversions between the total potential energy and the kinetic energy of the divergent flow occurs mainly in the lower layers of the atmosphere at middle latitudes where the baroclinic eddy activities are dominant. The energy conversion between the kinetic energies of the divergent and the nondivergent flows is mostly carried out by the Coriolis effect. In other words, the quasi-geostrophic model explains most of this energy conversion. The introduction of a more sophisticated model gives small improvement as far as the magnitude of energy is concerned. DOI: 10.1111/j.2153-3490.1976.tb00697.x

On the kinetic energy of the divergent and nondivergent flow in the atmosphere

The kinetic energy of horizontal flow in a hydrostatic atmosphere is divided into the kinetic energies of its divergent and nondivergent components. The law of conversion between these two energies for large-scale flows in the atmosphere is derived and discussed using balanced and unbalanced models of circulations in the atmosphere. It is shown that the total potential energy is converted into the kinetic energy of the divergent flow which, in turn, is converted into the kinetic energy of the nondivergent flow. These energy conversions are equal in a so-called balanced model, but may differ in a model based on the primitive equations. The analysis shows that the kinetic energy reservoir of the divergent part of the flow may play a quasi-catalytic role in the energy conversion between the total potential energy and the kinetic energy of the nondivergent flow. Two sets of data, the history data generated by the NCAR GCM of a January simulation, and the NMC wind data and FNWC height field of August 1 to 15 of 1970, were used to evaluate the energies and energy conversions. The computations confirm the direction of energy conversions in the theoretical argument and the quasi-catalytic nature of the kinetic energy of the divergent part of the flow. The energy conversions between the total potential energy and the kinetic energy of the divergent flow occurs mainly in the lower layers of the atmosphere at middle latitudes where the baroclinic eddy activities are dominant. The energy conversion between the kinetic energies of the divergent and the nondivergent flows is mostly carried out by the Coriolis effect. In other words, the quasi-geostrophic model explains most of this energy conversion. The introduction of a more sophisticated model gives small improvement as far as the magnitude of energy is concerned.

An Extension of the T-Method to General Unbalanced Models of Fixed Effects

1. Introduction and Summary Among the various procedures for simultaneous interval estimation, Tukey's T-method gives the shortest confidence intervals for pairwise contrasts when it is applicable. (Cf. Scheffé, 1959, Ch. 3; Miller, 1966, Ch. 2.) However, the T-method in its exact formulation (excluding one way anova, see Spjøtvoll and Stoline, 1973) is restricted to well-balanced designs. In this paper we discuss a generalized T-type procedure for simultaneous interval estimation of all estimable parametric functions in general fixed effects, univariate linear models. The new method, which we subsequently abbreviate as the GT method, is an extension of a generalized T-procedure for the unbalanced one-way anova model given in Spjøtvoll and Stoline (1973). In Section 2 we derive the new procedure in full-rank and in less than full-rank models. Some important features of the procedure are discussed in Section 3. In order to gain insight into the types of models where the new procedure is expected to produce satisfactory confidence intervals for pairwise contrasts, we illustrate, in Section 4, the application of the GT method in some unbalanced designs. In Section 5 we discuss a modified GT procedure which constitutes a combination of the GT1 in Hochberg (1974) and the GT methods. The proofs of some of the theorems are given in an Appendix for purposes of a convenient reading.

Nonparametric results for two-way ANOVA with fixed effects, no interactions, and unequal subclasses

Considered are unbalanced cross-classification models for two-way analysis of variance with fixed effects and no interaction terms. At least one observation occurs for each combination of factor levels, each factor has at least four levels, and the data are continuous. Special definitions with a median basis are used for contrasts in each of the two types of effects and for the constant term. However, the parameter values resulting from the special definitions are the same as occur for normality models and other important kinds of models. Subject only to independence among the error terms, approximate tests, confidence intervals, and median estimates are developed for stated contrasts in each of the kinds of effects. In general, an exact median estimate occurs for the constant term (the only procedure for this parameter). Exact tests and intervals are developed for some cases of a moderately general nature. Sign test types of procedures provide the basis for some of the exact results, but others can be based on the numerous procedures developed for investigating symmetry about zero. The models and investigation methods presented are based on the linearized nonlinear regression approach.

Sequential analysis of variance for irregular designs: some Monte Carlo results

Sequential analysis of variance was first dealt with by Johnson (1953) for equal numbers in groups and the results were extended to unequal numbers by Jones (1968) and Jones and Hinkelmann (1969). The average sample number (A.S.N.) obtained for an irregular design is usually lower than that for equal subclass numbers and this advantage is enhanced if sampling starts with the design. In this article Monte Carlo results are given for two classifications models with fixed effects, namely, one-way classification and twoway classification without interaction. Different methods are adopted to obtain the minimum starting design for unbalanced models and the resulting A.S.N. of these are compared. In all cases the probabilities of errors of the two kinds are taken to be 0.05.

On Expectations, Variances, and Covariances of ANOVA Mean Squares by 'Synthesis'

Hartley [1967] has given a general procedure for finding directly the numerical values in the formulas for expectations, variances and covariances of ANOVA mean squares. His procedure is applicable for any analysis of variance and any unbalanced random model with design matrices having exactly one 1 in each row and the remaining elements zero. This paper extends these results to general design matrices and also to any unbalanced mixed model. Generalization to correlated errors and/or correlated random components is also considered. Simple and direct proofs are provided with the help of the 'trace' of a matrix.

Note on Testing Hypotheses in an Unbalanced Random Effects Model

Note on Testing Hypotheses in an Unbalanced Random Effects Model

Note on testing hypotheses in an unbalanced random effects model

Note on testing hypotheses in an unbalanced random effects model