Testing for random individual effects using recursive residuals

This paper derives a simple ANOVA-F-statistic which tests for random individual effects in a one-way error component model, using recursive residuals. Power comparisons are performed for this F-test when it is computed using true disturbances and recursive residuals from a panel data regression. Under the null, both statistics have an exact F distribution. The standardized version of the Breusch and Pagan (1980) Lagrange Multiplier test (SLM) as well as a fixed effects F-statistic (FE) recommended by Moulton and Randolph (1989), are also included in this comparison. The exact power function can be computed in all cases using Imhof's (1961) procedure. Our results suggest that the F-test based on recursive residuals is inferior to the popular SLM and FE tests based on computational simplicity, power comparisons and its sensitivity to the K observations starting the recursion.     

An empirical study of five multivariate tests for the single-factor repeated measures model

A Monte Carlo study was used to examine the Type I error and power values of five multivariate tests for the single-factor repeated measures model The performance of Hotelling's T2 and four nonparametric tests, including a chi-square and an F-test version of a rank-transform procedure, were investigated for different distributions, sample sizes, and numbers of repeated measures. The results indicated that both Hotellings T* and the F-test version of the rank-transform performed well, producing Type I error rates which were close to the nominal value. The chi-square version of the rank-transform test, on the other hand, produced inflated Type I error rates for every condition studied. The Hotelling and F-test version of the rank-transform procedure showed similar power for moderately-skewed distributions, but for strongly skewed distributions the F-test showed much better power. The performance of the other nonparametric tests depended heavily on sample size. Based on these results, the F-test version of the rank-transform procedure is recommended for the single-factor repeated measures model.     

Power comparisons between the F-test,the studentised range test,and an optimal test of the equality of several normal means

The power assessment of tests of the equality of k normal means such as the k treatment means in a one-way fixed effects analysis of variance model is addressed. Power assessment is considered in terms of a constraint on the range of the treatment means. The power properties of the standard F-test and Studentised range test are compared with those of an optimal (minimax) test procedure, which is known to maximise power levels under this constraint. It is shown that the standard test procedures compare well with the optimal test procedure, and in particular, the Studentised range test is shown to be practically as good as optimal in this setting.     

An extension of the median test to analysis of variance

In a k-way analysis of variance model, the major concern is testing for main effects and for the presence of interaction between the factors. When the assumptions of normality and equal variances are satisfied, the appropriate test to use is the usual F-test for ANOVA. However, when the normality assumption is not satisfied then a robust or nonparametric test is needed to conduct the analysis. In this paper a nonparametric method based on cell counts is proposed. Each cell is divided into L subcells based on predetermined outpoints and the resulting frequencies are laid out in a contingency table. Then the Pearson x² and tne likelihood ratio tests are performed. A comparison with the classical ANOVA F-test indicates that the proposed method is preferable when the data comes from a thick-tailed highly skewed distribution.     

Distribution of variance ratio in unbalanced random model under heterogeneity of error variances

The effects of heteroscedasticity have been studied on the mean and variance of F ratio and on the power of F-test in unbalanced one-way random model, numerically. The computed results reveal that the heteroscedasticity and unbalanoedness have combined effects. The mean and variance of F as well as the power of F-test increase with inequality of error variances under balanced and those unbalanced situations where more variable groups have larger size. The effects are of serious nature when more variable groups have smaller size.     

A closer examination on some parametric alternatives to the ANOVA F-test

In experiments, the classical (ANOVA) F-test is often used to test the omnibus null-hypothesis μ₁ = μ₂ ... = μ _j = ... = μ _n (all n population means are equal) in a one-way ANOVA design, even when one or more basic assumptions are being violated. In the first part of this article, we will briefly discuss the consequences of the different types of violations of the basic assumptions (dependent measurements, non-normality, heteroscedasticity) on the validity of the F-test. Secondly, we will present a simulation experiment, designed to compare the type I-error and power properties of both the F-test and some of its parametric adaptations: the Brown & Forsythe F^*-test and Welch’s V_w-test. It is concluded that the Welch V_w-test offers acceptable control over the type I-error rate in combination with (very) high power in most of the experimental conditions. Therefore, its use is highly recommended when one or more basic assumptions are being violated. In general, the use of the Brown & Forsythe F^*-test cannot be recommended on power considerations unless the design is balanced and the homoscedasticity assumption holds.     

A Monte Carlo study of rank tests for block designs

Tests based on ranks and the F-test are compared for block designs with n observations per block-treatment combination. Com-parisons are made on level of significance and on power. Rank tests examined include the Friedman as well as those using aligned ranks, weighted ranks, and the rank transformation. It is seen that the performance of these tests in relationship to each other depends on sample size, distribution of the random error term, and the severity of the block effects.     

On testing random effects in the nested-error regression model

We derive an exact F-test for random effects in the nested-error regression model. The derivation utilizes a matrix decomposition that offers a transformation of the response vector into two independent subvectors. When the random effects are absent, the test statistic reduces to a ratio of two independent residual sums of squares that are computed by fitting a regression model using each subvector. A small simulation study compares the power of the F-test with various recent tests and shows that the proposed test has a competitive performance under small as well as large number of clusters.     

Size performance of some tests in one-way anova

Tsui and Weerahandi (1989) introduced the notion of generalized p-values and since then this idea is used to solve many statistical testing problems. Heteroskedasticity is one of the major practical problems encountered in ANOVA problems. To compare the means of several groups under heteroskedasticity approximate tests are used in the literature. Weerahandi (1995a) introduced a test using the notion of generalized p-values for comparing the means of several populations when the variances are not equal. This test is referred to as a generalized F-test.

In this paper we compare the size performance of the Generalized F-test and four other widely used procedures: the Classical F-test for ANOVA, the F-test obtained by the weighted least-squares to adjust for heteroskedasticity, the Brown-Forsythe-test, and the Welch-test. The comparison is based on a simulation study of size performance of tests applied to the balanced one-way model. The intended level of the tests is set at 0.05. While the Generalized F-test was found to have size not exceeding the intended level, as heteroskedasticity becomes severe the other tests were found to have poor size performance. With mild heteroskedasticity the Welch-test and the classical ANOVA F-test have the intended levels, and the Welch-test was found to perform better than the latter. Widely used (due to computational convenience) weighted F-test was found to have very serious size problems. The size advantage of the generalized F-test was also found to be robust even under severe deviations from the assumption of normality.     

The Effects of Misconceptions on the Properties of Friedman's Test

Friedman's test is a widely used rank-based alternative to the analysis of variance (ANOVA) F-test for identifying treatment differences in a randomized complete block design. Many texts provide incomplete or misleading information about when Friedman's test may be appropriately applied. We discuss the assumptions needed for the test and common misconceptions. We show via simulation that when the variance or skew of the treatment distributions differ, application of Friedman's test to detect differences in treatment location can result in Type I error probabilities larger than the nominal α, and even when α is unaffected, the power of the test can be less than expected.     

A Hausman test for Brownian motion

New tests are proposed for the specification of the intraday price process of a risky asset, based on open, high, low, and close prices. Under the null of a Brownian process we derive two stochastically independent, unbiased volatility estimators. For a Hausman specification test we prove its equivalence with an F-test, consider its robustness against variation in drift and volatility, and analyze the power against an Ornstein–Uhlenbeck process, as well as a random walk with alternative distributions.     

Optimum Mixture Design for the Detection of Synergistic Effects

In mixture experiments, optimal designs for the estimation of parameters, both linear and non-linear, have been discussed by several authors. Optimal designs for the estimation of a subset of parameters have also been investigated. However, designs for testing the effects of certain factors and interactions have been studied only in the context of response surface models. In this article, we attempt to find the optimum design for testing the presence of synergistic effects in a mixture model. The classical F-test has been considered and the optimum design has been obtained so as to maximize the power of the test. It is observed that the barycenters are necessarily the support points of the trace-optimal design.     

An approximation to the distribution of the ratio of two general quadratic forms with application

Based on mixed cumulants up to order six, this paper provides a four moment approximation to the distribution of a ratio of two general quadratic forms in normal variables. The approximation is applied to calculate the percentile points of modified F-test statistics for testing treatment effects when standard F-ratio test is misleading because of dependence among observations. For the special case, when data is generated by an AR(1) process, the approximation is evaluated by a simulation study. For the general SARMA (p,q)(P,Q)_s process, a modified F-test statistic Is given, and its distribution for the (0,1)(0,l)₁₂ process, is approximated by the moment approximation technique.     

Comparison op some two sample tests for equality of variances

The F-test, F max-test and Bartlett's test are compared on the basis of power for the purpose of testing the equality of variances in two normal populations. The power of each test is expressed as a linear combination of F-probabilities. Bartlett's test is noted to be unbiased, UMPU, consistent against all alterna¬tives and the test which yields minimum length confidence intervals on the ratio of the variancesλ=σ₁ ² /σ² ₂ The two samples Bartlett critical values, although not recognized as such, are found in the works of other authors. Tables of the powers of each test are given for various values of λ, levels of significance a and the respective sample sizes, n₁ and n₂.     

The equivalence between likelihood ratio test and F-test for testing variance component in a balanced one-way random effects model

In mixed linear models, it is frequently of interest to test hypotheses on the variance components. F-test and likelihood ratio test (LRT) are commonly used for such purposes. Current LRTs available in literature are based on limiting distribution theory. With the development of finite sample distribution theory, it becomes possible to derive the exact test for likelihood ratio statistic. In this paper, we consider the problem of testing null hypotheses on the variance component in a one-way balanced random effects model. We use the exact test for the likelihood ratio statistic and compare the performance of F-test and LRT. Simulations provide strong support of the equivalence between these two tests. Furthermore, we prove the equivalence between these two tests mathematically.     

Analysis of binary data for the balanced one-way classification

A test is proposed for testing the equality of proportions based on the data available from a one-way classification having t treatment conditions and n binary observations per treatment. The test statistic B is a constant multiple of the F-statistic which results when the analysis of variance procedure for the one-way classification is applied to the data and, hence, is computationally simple. The statistic B from this binary analysis of variance (BIANOVA) is distributed asymptotically as a chi-square random variable. The proposed test is uniformly more powerful than either the F-test indicated above or the Pearson chi-square test; however, the attained empirical level of significance is frequently higher than for either of these competitors and usually higher than the stated level of significance for smaller values of n (say n ≤ 20).     

Nearly exact tests in factorial experiments using the aligned rank transform

A procedure is studied that uses rank-transformed data to perform exact and estimated exact tests, which is an alternative to the commonly used F-ratio test procedure. First, a common parametric test statistic is computed using rank-transformed data, where two methods of ranking-ranks taken for the original observations and ranks taken after aligning the observations-are studied. Significance is then determined using either the exact permutation distribution of the statistic or an estimate of this distribution based on a random sample of all possible permutations. Simulation studies compare the performance of this method with the normal theory parametric F-test and the traditional rank transform procedure. Power and nominal type I error rates are compared under conditions when normal theory assumptions are satisfied, as well as when these assumptions are violated. The method is studied for a two-factor factorial arrangement of treatments in a completely randomized design and for a split-unit experiment. The power of the tests rivals the parametric F-test when normal theory assumptions are satisfied, and is usually superior when normal theory assumptions are not satisfied. Based on the evidence of this study, the exact aligned rank procedure appears to be the overall best choice for performing tests in a general factorial experiment.     

Reconsideration of the F-test of the analysis of variance: the semi-bayesian significance test

The usual F-test of the analysis of variance is reconsidered within the Bayesian framework, In terms of predictive distributions, This leads to the notion of semi-Bayesian significance test, so called because it consists in only probabilizing the space of nuisance parameters, thus bringing a general principle for “eliminating” nuisance parameters, or more exactly incorporating information about these parameters. The approach is shown to extend the F-tests, by allowing the testing of hypotheses of non-zero effects.     

Joint impact of multiple observations on a subset of variables in multiple linear regression

In multiple linear regression analysis, each observation affects the fitted regression equation differently and has varying influences on the regression coefficients of the different variables. Chatterjee & Hadi (1988) have proposed some measures such as DSSE_ij (Impact on Residual Sum of Squares of simultaneously omitting the ith observation and the jth variable), F_j (Partial F-test for the jth variable) and F_j(i) (Partial F-test for the jth variable omitting the ith observation) to show the joint impact and the interrelationship that exists among a variable and an observation. In this paper we have proposed more extended form of those measures DSSE_IJ, F_J and F_J(I) to deal with the interrelationships that exist among the multiple observations and a subset of variables by monitoring the effects of the simultaneous omission of multiple variables and multiple observations.     

线性回归模型的统计检验关系辨析

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