Proportional subclass numbers in two-factor ANOVA

The between-within split of total sum of squares in one-way analysis of variance (ANOVA) is intuitively appealing and computationally simple, whether balanced or not. In the balanced two-factor setting, the same heuristic and computations apply to analyse treatment sum of squares into main effects and interaction effects sums of squares. Accomplishing the same in unbalanced settings is more difficult, requiring development of tests of general linear hypotheses. However, textbooks treat unbalanced settings with proportional subclasss numbers (psn) as essentially equivalent to balanced settings. It is shown here that, while psn permit an ANOVA-like partition of sums of squares, test statistics for main effects of the two factors generally test the wrong hypotheses when the model includes interaction effects.     

Interpretation of the four types of analysis of variance tables in sas

Various computational methods exist for generating sums of squares in an analysis of variance table. When the ANOVA design is balanced, most of these computational methods will produce equivalent sums of squares for testing the significance of the ANOVA model parameters. However, when the design is unbalanced, as is frequently the case in practice, these sums of squares depend on the computational method used.- The basic reason for the difference in these sums of squares is that different hypotheses are being tested. The purpose of this paper is to describe these hypotheses in terms of population or cell means. A numerical example is given for the two factor model with interaction. The hypotheses that are tested by the four computational methods of the SAS general linear model procedure are specified.

Although the ultimate choice of hypotheses should be made by the researcher before conducting the experiment, this paper

PENDLETON,VON TRESS,AND BREMER

presents the following guidelines in selecting these hypotheses:

When the design is balanced, all of the SAS procedures will agree.

In unbalanced ANOVA designs when there are no missing cells. SAS Type III should be used. SAS Type III tests an unweighted hypothesis about cell means. SAS Types I and II test hypotheses that are functions of the ceil frequencies. These frequencies are often merely arti¬facts of the experimental process and not reflective of any underlying frequencies in the population.

When there are missing cells, i.e. no observations for some factor level combinations. Type IV should be used with caution. SAS Type IV tests hypotheses which depend     

ANOVA for unbalanced data: Use Type II instead of Type III sums of squares

Methods for analyzing unbalanced factorial designs can be traced back to Yates (1934). Today, most major statistical programs perform, by default, unbalanced ANOVA based on Type III sums of squares (Yates's weighted squares of means). As criticized by Nelder and Lane (1995), this analysis is founded on unrealistic models—models with interactions, but without all corresponding main effects. The Type II analysis (Yates's method of fitting constants) is usually not preferred because of the underlying assumption of no interactions. This argument is, however, also founded on unrealistic models. Furthermore, by considering the power of the two methods, it is clear that Type II is preferable.     

Resolving Current Controversies in Analysis of Variance

Definition of effects and calculation of sums of squares for various tests of hypotheses in unbalanced analysis of variance has been a topic of considerable interest for at least 10 years. Conceptually, these concerns apply to balanced cases as well. It is suggested that proceeding logically from highest-order effects to lowest-order effects in a careful fashion helps to resolve the difficulties pointed out by various writers, including those concerned about completely missing cells.     

Applying skovgaard's modified directed likelihood statistic to mixed linear models

The introduction of software to calculate maximum likelihood estimates for mixed linear models has made likelihood estimation a practical alternative to methods based on sums of squares. Likelihood based tests and confidence intervals, however, may be misleading in problems with small sample sizes. This paper discusses an adjusted version of the directed log-likelihood statistic for mixed models that is highly accurate for testing one parameter hypotheses. Indroduced by Skovgaard (1996, Journal of the Bernoulli Society,2,145-165), we show in mixed models that the statistic has a simple conpact from that may be obtained from standard software. Simulation studies indicate that this statistic is more accurate than many of the specialized procedure that have been advocated.     

On locally optimal invariant unbiased tests for the variance components ratio in mixed linear models

In the paper the problem of testing of two-sided hypotheses for variance components in mixed linear models is considered. When the uniformly most powerful invariant test does not exist (see e.g. Das and Sinha, in Proceedings of the second international Tampere conference in statistics, 1987; Gnot and Michalski, in Statistics 25:213–223, 1994; Michalski and Zmyślony, in Statistics 27:297–310, 1996) then to conduct the optimal statistical inference on model parameters a construction of a test with locally best properties is desirable, cf. Michalski (in Tatra Mountains Mathematical Publications 26:1–21, 2003). The main goal of this article is the construction of the locally best invariant unbiased test for a single variance component (or for a ratio of variance components). The result has been obtained utilizing Andersson’s and Wijsman’s approach connected with a representation of density function of maximal invariant (Andersson, in Ann Stat 10:955–961, 1982; Wijsman, in Proceedings of fifth Berk Symp Math Statist Prob 1:389–400, 1967; Wijsman, in Sankhyā A 48:1–42, 1986; Khuri et al., in Statistical tests for mixed linear models, 1998) and from generalized Neyman–Pearson Lemma (Dantzig and Wald, in Ann Math Stat 22:87–93, 1951; Rao, in Linear statistical inference and its applications, 1973). One selected real example of an unbalanced mixed linear model is given, for which the power functions of the LBIU test and Wald’s test (the F-test in ANOVA model) are computed, and compared with the attainable upper bound of power obtained by using Neyman–Pearson Lemma.     

Testing a Regression Model When We Have Smooth Alternatives in Mind

Goodness-of-fit tests based on residual sums of squares are standard procedures used when fitting regression models. Often we have a smooth alternative in mind, a qualitative feature that the χ²-test does not take into account. We show that the power of detecting a smooth alternative increases when we smooth the current model as well. The proposed test is shown to be able to detect any continuous local alternative tending to zero slower than n ^−1/2. Theoretical results also address minimax non-parametric hypothesis testing in Sobolev spaces. A simulation study is presented, and the procedure is applied to expenditure curve estimation.     

Powerful Parametric Tests Based on Sum‐Functions of Spacings

Assume that we have a sequence of n independent and identically distributed random variables with a continuous distribution function F, which is specified up to a few unknown parameters. In this paper, tests based on sum‐functions of sample spacings are proposed, and large sample theory of the tests are presented under simple null hypotheses as well as under close alternatives. Tests, which are optimal within this class, are constructed, and it is noted that these tests have properties that closely parallel those of the likelihood ratio test in regular parametric models. Some examples are given, which show that the proposed tests work also in situations where the likelihood ratio test breaks down. Extensions to more general hypotheses are discussed.     

A robust I-sample analysis of means type randomization test for variances for unbalanced designs

Two analysis of means type randomization tests for testing the equality of I variances for unbalanced designs are presented. Randomization techniques for testing statistical hypotheses can be used when parametric tests are inappropriate. Suppose that I independent samples have been collected. Randomization tests are based on shuffles or rearrangements of the (combined) sample. Putting each of the I samples ‘in a bowl’ forms the combined sample. Drawing samples ‘from the bowl’ forms a shuffle. Shuffles can be made with replacement (bootstrap shuffling) or without replacement (permutation shuffling). The tests that are presented offer two advantages. They are robust to non-normality and they allow the user to graphically present the results via a decision chart similar to a Shewhart control chart. A Monte Carlo study is used to verify that the permutation version of the tests exhibit excellent power when compared to other robust tests. The Monte Carlo study also identifies circumstances under which the popular Levene's test fails.     

New Section on Social Statistics

The analysis of a general k-factor factorial experiment having unequal numbers of observations per cell is complex. For the special case of a 2 ^k experiment with unequal numbers of observations per cell, the method of unweighted means provides a simple vehicle for analysis that requires no matrix inversion and can be used with existing software programs for the analysis of balanced data. All numerator sums of squares for testing main effects and interactions are χ² with one degree of freedom. In addition, for tests having one degree of freedom in any factorial experiment, the method of unweighted means may be modified to yield exact tests.     

Superiority comparisons of heterogeneous linear estimators

In this paper conditions for strong and weak superiority of a heterogeneous linear estimator over another are derived. The general results are applied to some special cases: in particular, two restricted least squares estimators are compared using the superiority conditions obtained. The weak superiority criterion is used as a basis in forming a generalization of an optimal se-quence of tests (Anderson, 1962) for searching for the best estimator when the alternative linear restrictions form a nested se-quence of hypotheses. An application of this is the determination of the correct length of lag and appropriate degree of polynomial in the estimation of polynomial distributed lag models.     

On invariant quadratic unbiased estimation of variance components

The present study deals with three different invarint quadratic unbiased estimators (IQUE) for variance components namely quadratic least squares estimators (QLSE), weighted quadratic least squares estimators (WQLSE) and Mitra type estimators (MTE). The variance and covariances of these three different estimators are presented for unbalanced one-way random model. The relative performances of these estimators are assessed based on different optimality criteria like, D-optimality, T-optimality and M-optimality together with variances of these estimators. As a result, it has been shown that MTE has optimal properties.     

STEPWISE TESTS WHEN THE TEST STATISTICS ARE INDEPENDENT

Consider the problem of simultaneously testing a nonhierarchical finite family of hypotheses based on independent test statistics. A general stepwise test is defined, of which the well known step-down and step-up tests are special cases. The step-up test is shown to dominate the other stepwise tests, including the step-down test, for situations of practical importance. When testing against two-sided alternatives, it is pointed out that if the step-up test is augmented to include directional decisions then the augmented step-up test controls the type I and III familywise error jointly at the original level q. The definition of the adjusted p values for the step-up test is justified. The results are illustrated by a numerical example.     

Large-scale simultaneous inference under dependence

Simultaneous inference allows for the exploration of data while deciding on criteria for proclaiming discoveries. It was recently proved that all admissible post hoc inference methods for the true discoveries must employ closed testing. In this paper, we investigate efficient closed testing with local tests of a special form: thresholding a function of sums of test scores for the individual hypotheses. Under this special design, we propose a new statistic that quantifies the cost of multiplicity adjustments, and we develop fast (mostly linear-time) algorithms for post hoc inference. Paired with recent advances in global null tests based on generalized means, our work instantiates a series of simultaneous inference methods that can handle many dependence structures and signal compositions. We provide guidance on the method choices via theoretical investigation of the conservativeness and sensitivity for different local tests, as well as simulations that find analogous behavior for local tests and full closed testing.     

Testing for random effects and serial correlation in spatial autoregressive models

This paper constructs and evaluates tests for random effects and serial correlation in spatial autoregressive panel data models. In these models, ignoring the presence of random effects not only produces misleading inference but inconsistent estimation of the regression coefficients. Two different estimation methods are considered: maximum likelihood and instrumental variables. For each estimator, optimal tests are constructed: Lagrange multiplier in the first case; Neyman's C(α)$C(\alpha )$ in the second. In addition, locally size-robust tests, for individual hypotheses under local misspecification of the unconsidered parameter, are constructed. Extensive Monte Carlo evidence is presented.     

Testing the equality of the variances of two linear models

Testing the equality of variances of two linear models with common β-parameter is considered. A test based on least squares residuals (ASR test) is proposed, and it is shown that this test is invariant under the group of scale and translation changes. For some special cases, it is also proved that this test has a monotone power function. Finding the exact critical values of this test is not easy; an approximation is given to facilitate the computation of these. The powers of the BLUS test, the F-test and the new test are computed for various alternatives and compared in a particular case. The proposed test seems to be locally more powerful than the alternative tests.     

A Test for Slope Heterogeneity in Fixed Effects Models

Typical panel data models make use of the assumption that the regression parameters are the same for each individual cross-sectional unit. We propose tests for slope heterogeneity in panel data models. Our tests are based on the conditional Gaussian likelihood function in order to avoid the incidental parameters problem induced by the inclusion of individual fixed effects for each cross-sectional unit. We derive the Conditional Lagrange Multiplier test that is valid in cases where N → ∞ and T is fixed. The test applies to both balanced and unbalanced panels. We expand the test to account for general heteroskedasticity where each cross-sectional unit has its own form of heteroskedasticity. The modification is possible if T is large enough to estimate regression coefficients for each cross-sectional unit by using the MINQUE unbiased estimator for regression variances under heteroskedasticity. All versions of the test have a standard Normal distribution under general assumptions on the error distribution as N → ∞. A Monte Carlo experiment shows that the test has very good size properties under all specifications considered, including heteroskedastic errors. In addition, power of our test is very good relative to existing tests, particularly when T is not large.     

On the relation between Hodges-Lehmann efficiency and Pitman efficiency

The relation between Hodges–Lehmann efficiency and Pitman efficiency is studied in the context of testing one-sided hypotheses about a real-valued parameter. It is first shown that for tests based on sums of independently and identically distributed observations local Hodges-Lehmann efficiency is equivalent to Pitman efficiency. Then, it is proved that this equivalence also carries over to tests based on two broad classes of M -estimators for the location problem. In all cases considered explicit formulas of the Hodges-Lehmann efficacies are obtained.     

General lack of fit tests based on families of groupings

Lack of fit tests based on groupings of the observations are developed. These tests are first applied to models with replication. In this case, the classic Fisher test assumes that the true model is contained in the one-way ANOVA model. However, Christensen [(2003). Significantly insignificant F tests. Amer. Statist. 57, 27–32] has noted that small values of the F-statistic may indicate lack of fit due to features which are not part of the proposed model. Such model inadequacy is called within-cluster lack of fit, whereas the standard Fisher lack of fit is called between-cluster lack of fit. Typically, lack of fit exists as a combination of these two pure types, and can be extremely difficult to detect depending on the nature of the mixture. In this paper, the one-way ANOVA model is embedded in larger models using groupings of the observations, which provides tests with good power for detecting all of the above types of model inadequacies, including mixtures. In particular, several such tests are considered, each based on a different grouping of the observations, and the multiple testing approach of Baraud et al. [(2003). Adaptive tests of linear hypotheses by model selection. Ann. Statist. 31, 225–251] is followed. More generally, the preceding testing procedure based on families of groupings is extended to the case of nonreplication. For this case, it is proposed that such families be determined by linear orders on the predictors based on disjoint parallel tubes in predictor space. Test statistics follow the cluster-based regression lack of fit tests presented by Christensen [(1989). Lack of fit based on near or exact replicates. Ann. Statist. 17, 673–683; (1991). Small sample characterizations of near replicate lack of fit tests. J. Amer. Statist. Assoc. 86, 752–756], by considering the groupings as determining special types of clusterings. In order to detect general lack of fit, several such tests are again considered, each based on a different grouping of the observations, and the multiple testing approach given by Baraud et al. [(2003). Adaptive tests of linear hypotheses by model selection. Ann. Statist. 31, 225–251] is followed. Simulation results illustrating the power of the proposed testing procedure are given.     

On the distribution of quadratic forms and applications in the analysis of repeated measurements

In this paper a unified approach is given to the distribution of scalar quadratic forms for dependent variables. Necessary and sufficient conditions are found for the sums of squares of the various hierarchical layers in ANOVA to be distributed as multiples of chi-square variables. Results concerning the usual univariate F-tests in ANOVA of repeated measurements are derived as a special case.