Analysis of Type I Error Rates of Univariate and Multivariate Procedures in Repeated Measures Designs

We compared the robustness of univariate and multivariate statistical procedures to control Type I error rates when the normality and homocedasticity assumptions were not fulfilled. The procedures we evaluated are the mixed model adjusted by means of the SAS Proc Mixed module, and Bootstrap-F approach, Brown–Forsythe multivariate approach, Welch–James multivariate approach, and Welch–James multivariate approach with robust estimators. The results suggest that the Kenward Roger, Brown–Forsythe, Welch–James, and Improved Generalized Aprroximate procedures satisfactorily kept Type I error rates within the nominal levels for both the main and interaction effects under most of the conditions assessed.     

Testing the equality of several means when the population variances are unequal

A comparative study is made of three tests, developed by James (1951), Welch (1951) and Brown & Forsythe (1974). James presented two methods of which only one is considered in this paper. It is shown that this method gives better control over the size than the other two tests. None of these methods is uniformly more powerful than the other two. In some cases the tests of James and Welch reject a false null hypothesis more often than the test of Brown & Forsythe, but there are also situations in which it is the other way around.

We conclude that for implementation in a statistical software package the very complicated test of James is the most attractive. A practical disadvantage of this method can be overcome by a minor modification.     

A Parametric Bootstrap Solution to Two-way MANOVA without Interaction under Heteroscedasticity: Fixed and Mixed Models

In this article, we propose a parametric bootstrap (PB) test for heteroscedastic two-way multivariate analysis of variance without Interaction. For the problem of testing equal main effects of factors, we obtain a PB approach and compare it with existing modified Brown–Forsythe (MBF) test and approximate Hotelling T² (AHT) test by an extensive simulation study. The PB test is a symmetric function in samples, and does not depend on the chosen weights used to define the parameters uniquely. Simulation results indicate that the PB test performs satisfactorily for various cell sizes and parameter configurations when the homogeneity assumption is seriously violated, and tends to outperform the AHT test for moderate or larger samples in terms of power and controlling size. The MBF test, the AHT test, and the PB test have similar robustness to violations of underlying assumptions. It is also noted that the same PB test can be used to test the significance of random effect vector in a two-way multivariate mixed effects model with unequal cell covariance matrices.     

Robust analysis of variance: a simulation study

Traditional analysis-of-variance (ANOVA) is based on 'normality' and 'homogeneity' assumptions. If either or both of these assumptions are violated, then the one-way ANOVA may not be as powerful as robust analysis-of-variance (RANOVA) alternatives. We report the results of a simulation study of alternatives to ANOVA: Welch (W*), the first and second methods of James (J1*, 3J11*), Brown- Forsythe (BF*), a Box (B*) procedure, and the Kruskal-Wallis (KW*) procedure. Random samples from 14 distributions—uniform (0, 1), normal (0, 1), contaminated normal, SLATE, SLACU, SLASH, double exponential, Cauchy, half-normal, chi-squared (two degrees of freedom), chi-squared (four degrees of freedom) log normal, gamma (1, 2) and beta (2, 5)—were generated using a composite linear congruential generator. Corresponding test satis tics were computed and the empirical size for each test is given for three nominal a values (0.10, 0.05, 0.01). For k, we choose 3, 4 and 6. The sample sizes and combinations of sample sizes were chosen at 4, 6, 8, 10, 15 and 20. We then propose an adaptive algorithm based on an ancillary statistic that selects an ANOVA/RANOVA procedure for either symmetric or asymmetric data distributions, and for equal or unequal sample sizes.     

Some comparative studies on testing parallelism of several straight lines under heteroscedastic variances

This paper examines the robustness of the Welch test, the James test as well as Tan's ANOVA test (to be referred as F_β test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.     

On a Berry-Esséen theorem for a Studentized jackknife L-estimate

Consider a linear function of order statistics (“L-estimate”) which can be expressed as a statistical function T(F_n) based on the sample cumulative distribution function F_n. Let T*(F_n) be the corresponding jackknifed version of T(F_n), and let V²_n be the jackknife estimate of the asymptotic variance of n ^1/2T(F_n) or n ^1/2T*(F_n). In this paper, we provide a Berry-Esséen rate of the normal approximation for a Studentized jackknife L-estimate n^1/2[T*(F_n) - T(F)]/V_n, where T(F) is the basic functional associated with the L-estimate.     

On the robustness of the linear hypothesis test procedure in the singular linear model with implied restrictions

The linear hypothesis test procedure is considered in the restricted linear modelsM _r = {y, Xβ |Rβ = 0, σ ²V} andM _r ^* = {y, Xβ |ARβ = 0, σ ²V}. Necessary and sufficient conditions are derived under which the statistic providing anF-test for the linear hypothesisH ₀:Kβ=0 in the modelM _r ^* (M_r) continues to be valid in the modelM _r (M _r ^* ); the results obtained cover the case whereM _r ^* is replaced by the general Gauss-Markov modelM = {y, Xβ, σ ²V}.     

On testing more IFRA ordering-II

ABSTRACT

Suppose F and G are two life distribution functions. It is said that F is more IFRA (increasing failure rate average) than G (written by F ? _*G) if G^{? 1}F(x) is star-shaped on (0, ∞). In this paper, the problem of testing H₀: F = _*G against H₁: F ? _*G and F ≠ _*G is considered in both cases when G is known and when G is unknown. We propose a new test based on U-statistics and obtain the asymptotic distribution of the test statistics. The new test is compared with some well-known tests in the literature. In addition, we apply our test to a real data set in the context of reliability.     

On the construction of orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set

Complete sets of orthogonal F-squares of order n = s^p, where g is a prime or prime power and p is a positive integer have been constructed by Hedayat, Raghavarao, and Seiden (1975). Federer (1977) has constructed complete sets of orthogonal F-squares of order n = 4t, where t is a positive integer. We give a general procedure for constructing orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set, where n is not necessarily a prime or prime power. In particular, we show how to construct sets of orthogonal F-squares of order n = 2s^p, where s is a prime or prime power and p is a positive integer. These sets are shown to be near complete and approach complete sets as s and/or p become large. We have also shown how to construct orthogonal arrays by these methods. In addition, the best upper bound on the number t of orthogonal F(n, λ₁), F(n, λ₂), …, F(n, λ₁) squares is given.     

A non standard χ2-test of fit for testing uniformity with unknown limits

A χ²-test of fit for testingH ₀ “X～U(a,b), a,b unknown” is suggested. It is nonstandard because the usual regularity assumptions are not satisfied. The asymptotic distribution of the test statistic underH ₀ is derived. The error probabilities of the first kind are investigated by Monte Carlo simulation for samples of small and medium size.     

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.     

Comparison of k independent,zero‐heavy lognormal distributions

Lachenbruch ( 1976 , 2001 ) introduced two‐part tests for comparison of two means in zero‐inflated continuous data. We are extending this approach and compare k independent distributions (by comparing their means, either overall or the departure from equal proportion of zeros and equal means of nonzero values) by introducing two tests: a two‐part Wald test and a two‐part likelihood ratio test. If the continuous part of the distributions is lognormal then the proposed two test statistics have asymptotically chi‐square distribution with $2(k-1)$ degrees of freedom. A simulation study was conducted to compare the performance of the proposed tests with several well‐known tests such as ANOVA, Welch ( 1951 ), Brown & Forsythe ( 1974 ), Kruskal–Wallis, and one‐part Wald test proposed by Tu & Zhou ( 1999 ). Results indicate that the proposed tests keep the nominal type I error and have consistently best power among all tests being compared. An application to rainfall data is provided as an example. The Canadian Journal of Statistics 39: 690–702; 2011. © 2011 Statistical Society of Canada     

A bayesian approach to prediction in analysis of variance settings

This paper takes the results of Lindley and smith ( 1972 ) one step further, by finding the predictive distribution of an observation y^* whose distribution is normal, and centred at A^* ₁θ₁ We then apply this distribution to the case of prediction based on data obtained in one and two wau ANOVA situations. For Example, it turns out that for two way ANOVA with interaction, the predictive mean, (which we would use as the predictor) is a weighted combination of sample main effects and interaction effects     

Robust tests for the equality of variances for clustered data

Tests for the equality of variances are often needed in applications. In genetic studies the assumption of equal variances of continuous traits, measured in identical and fraternal twins, is crucial for heritability analysis. To test the equality of variances of traits, which are non-normally distributed, Levene [H. Levene, Robust tests for equality of variances, in Contributions to Probability and Statistics, I. Olkin, ed. Stanford University Press, Palo Alto, California, 1960, pp. 278–292] suggested a method that was surprisingly robust under non-normality, and the procedure was further improved by Brown and Forsythe [M.B. Brown and A.B. Forsythe, Robust tests for the equality of variances, J. Amer. Statis. Assoc. 69 (1974), pp. 364–367]. These tests assumed independence of observations. However, twin data are clustered – observations within a twin pair may be dependent due to shared genes and environmental factors. Uncritical application of the tests of Brown and Forsythe to clustered data may result in much higher than nominal Type I error probabilities. To deal with clustering we developed an extended version of Levene's test, where the ANOVA step is replaced with a regression analysis followed by a Wald-type test based on a clustered version of the robust Huber–White sandwich estimator of the covariance matrix. We studied the properties of our procedure using simulated non-normal clustered data and obtained Type I error rates close to nominal as well as reasonable powers. We also applied our method to oral glucose tolerance test data obtained from a twin study of the metabolic syndrome and related components and compared the results with those produced by the traditional approaches.     

Optimum designs in complete two-way layouts

In the usual two-way layout of ANOVA (interactions are admitted) let nij ? 1 be the number of observations for the factor-level combination(i, j). For testing the hypothesis that all main effects of the first factor vanish numbers $\text{n}{}^1\text{ij}$ are given such that the power function of the F-test is uniformly maximized (U-optimality), if one considers only designs (nij) for which the row-sums ni are prescribed. Furthermore, in the (larger) set of all designs for which the total number of observations is given, all D-optimum designs are constructed.     

Robustness of the t and U tests under combined assumption violations

SUMMARY When the assumptions of parametric statistical tests for the difference between two means are violated, it is commonly advised that non-parametric tests are a more robust substitute. The history of the investigation of this issue is summarized. The robustness of the t -test was evaluated, by repeated computer testing for differences between samples from two populations of equal means but non-normal distributions and with different variances and sample sizes. Two common alternatives to t -Welch's approximate t and the Mann-Whitney U -test-were evaluated in the same way. The t -test is sufficiently robust for use in all likely cases, except when skew is severe or when population variances and sample sizes both differ. The Welch test satisfactorily addressed the latter problem, but was itself sensitive to departures from normality. Contrary to its popular reputation, the U -test showed a dramatic 'lack of robustness' in many cases-largely because it is sensitive to population differences other than between means, so it is not properly a 'non-parametric analogue' of the t -test, as it is too often described.     

Allocating observations between two normal populations to maximize the half slope of the F-test

Consider two independent normal populations. Let R denote the ratio of the variances. The usual procedure for testing H₀: R = 1 vs. H₁: R = r, where r≠1, is the F-test. Let θ denote the proportion of observations to be allocated to the first population. Here we find the value of θ that maximizes the rate at which the observed significance level of the F-test converges to zero under H₁, as measured by the half slope.     

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.     

A comparison of the independent-samples t-test and the paired-samples t-test when the observations are nonnegatively correlated pairs

An account of the behavior of the independent-samples t-test when applied to homoschedastic bivariate normal data is presented, and a comparison is made with the paired-samples t-test. Since the significance level is not violated when applying the independent-samples t-test to data which consist of positively correlated pairs and since the estimate of the variance is based on a larger number of ‘degrees of freedom’, the results suggest that when the sample size is small, one should not worry much about the possible existence of weak positive correlation. One may do better, powerwise, to ignore such correlation and use the independent-samples t-test, as though the samples were independent.     

Testing Variability in the Two-Sample Case

A number of robust methods for testing variability have been reported in previous literature. An examination of these procedures for a wide variety of populations confirms their general robustness. Shoemaker's improvement of the F test extends that test use to a realistic variety of population shapes. However, a combination of the Brown–Forsythe and O'Brien methods based on testing kurtosis is shown to be conservative for a wide range of sample sizes and population distributions. The composite test is also shown to be more powerful in most conditions than other conservative procedures.