Testing equality of two intercepts for the parallel regression model with non sample prior information

This paper proposes tests for equality of intercepts of two simple regression models when non sample prior information is available on the equality of two slopes. For three different scenarios on the values of the slope, namely (i) unknown (unspecified), (ii) known (specified), and (iii) suspected, we derive the unrestricted test (UT), restricted test (RT), and pre-test test (PTT) for testing the equality of intercepts. The test statistics, their sampling distributions, and power functions of the tests are obtained. Comparison of power function and size of the tests reveals that the PTT has a reasonable dominance over the UT and RT.     

Bootstrap test for a structural break under possible heteroscedasticity

In this article, we consider the Wald test statistic for testing equality between the sets of regression coefficients in two linear regression models when the disturbance variances may possibly be unequal. This test can be also used as a test for a structural break. However, it is well known that the test based on the Wald test statistic suffers from severe size distortion in small sample when the disturbance variances of the two regression models are unequal. Our simulation results show that substantial improvements are made when the bootstrap methods are applied.     

Shrinkage Estimators of Intercept Parameters of Two Simple Regression Models with Suspected Equal Slopes

Estimators of the intercept parameter of a simple linear regression model involves the slope estimator. In this article, we consider the estimation of the intercept parameters of two linear regression models with normal errors, when it is a priori suspected that the two regression lines are parallel, but in doubt. We also introduce a coefficient of distrust as a measure of degree of lack of trust on the uncertain prior information regarding the equality of two slopes. Three different estimators of the intercept parameters are defined by using the sample data, the non sample uncertain prior information, an appropriate test statistic, and the coefficient of distrust. The relative performances of the unrestricted, shrinkage restricted and shrinkage preliminary test estimators are investigated based on the analyses of the bias and risk functions under quadratic loss. If the prior information is precise and the coefficient of distrust is small, the shrinkage preliminary test estimator overperforms the other estimators. An example based on a medical study is used to illustrate the method.     

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.     

Effects of non-normality and mild heteroscedasticity on estimators in regression

Several estimators are examined for the simple linear regression model under a controlled, experimental situation with multiple observations at each design point. The model is examined under normal and non-normal error distributions and mild heterogeneity of variances across the chosen design points. We consider the ordinary, generalized, and estimated generalized least squares estimators and several examples of M estimators. The asymptotic properties of the M estimator using the Huber ψ are presented under these conditions for the multiple regression model. A simulation study is also presented which indicates that the M estimator possesses strong robustness properties under the presence of both non-normality and mild heteroscedasticity o£ errors. Finally, the M estimates are compared to the least squares estimates in two examples.     

Permutation Tests for Linear Models

Several approximate permutation tests have been proposed for tests of partial regression coefficients in a linear model based on sample partial correlations. This paper begins with an explanation and notation for an exact test. It then compares the distributions of the test statistics under the various permutation methods proposed, and shows that the partial correlations under permutation are asymptotically jointly normal with means 0 and variances 1. The method of Freedman & Lane (1983) is found to have asymptotic correlation 1 with the exact test, and the other methods are found to have smaller correlations with this test. Under local alternatives the critical values of all the approximate permutation tests converge to the same constant, so they all have the same asymptotic power. Simulations demonstrate these theoretical results.     

The application of the durbin-watson test to the dynamic regression model under normal and non-normal errors

Until recently, a difficulty with applying the Durbin-Watson (DW) test to the dynamic linear regression model has been the lack of appropriate critical values. Inder (1986) used a modified small-disturbance distribution (SDD) to find approximate critical values. King and Wu (1991) showed that the exact SDD of the DW statistic is equivalent to the distribution of the DW statistic from the regression with the lagged dependent variables replaced by their means. Unfortunately, these means are unknown although they could be estimated by the actual variable values. This provides a justification for using the exact critical values of the DW statistic from the regression with the lagged dependent variables treated as non-stochastic regressors. Extensive Monte Carlo experiments are reported in this paper. They show that this approach leads to reasonably accurate critical values, particularly when two lags of the dependent variable are present. Robustness to non-normality is also investigated.     

The application of the durbin-watson test to the dynamic regression model under normal and non-normal errors

Until recently, a difficulty with applying the Durbin-Watson (DW) test to the dynamic linear regression model has been the lack of appropriate critical values. Inder (1986) used a modified small-disturbance distribution (SDD) to find approximate critical values. King and Wu (1991) showed that the exact SDD of the DW statistic is equivalent to the distribution of the DW statistic from the regression with the lagged dependent variables replaced by their means. Unfortunately, these means are unknown although they could be estimated by the actual variable values. This provides a justification for using the exact critical values of the DW statistic from the regression with the lagged dependent variables treated as non-stochastic regressors. Extensive Monte Carlo experiments are reported in this paper. They show that this approach leads to reasonably accurate critical values, particularly when two lags of the dependent variable are present. Robustness to non-normality is also investigated.     

Methods to test for equality of two normal distributions

Statistical tests for two independent samples under the assumption of normality are applied routinely by most practitioners of statistics. Likewise, presumably each introductory course in statistics treats some statistical procedures for two independent normal samples. Often, the classical two-sample model with equal variances is introduced, emphasizing that a test for equality of the expected values is a test for equality of both distributions as well, which is the actual goal. In a second step, usually the assumption of equal variances is discarded. The two-sample t test with Welch correction and the F test for equality of variances are introduced. The first test is solely treated as a test for the equality of central location, as well as the second as a test for the equality of scatter. Typically, there is no discussion if and to which extent testing for equality of the underlying normal distributions is possible, which is quite unsatisfactorily regarding the motivation and treatment of the situation with equal variances. It is the aim of this article to investigate the problem of testing for equality of two normal distributions, and to do so using knowledge and methods adequate to statistical practitioners as well as to students in an introductory statistics course. The power of the different tests discussed in the article is examined empirically. Finally, we apply the tests to several real data sets to illustrate their performance. In particular, we consider several data sets arising from intelligence tests since there is a large body of research supporting the existence of sex differences in mean scores or in variability in specific cognitive abilities.     

The two-sample Winsorized t

This paper extends the single-sample Winsorized t to the case of two independent samples. Statistics are defined for both the cases of equal and unequal population variances. Monte Carlo techniques were used to find the approximate distributions of these statistics as well as to evaluate their performances in comparison to the trimmed t . It was found that as far as the power is concerned, the differences between the Winsorized t and the trimmed t are so small that the choice is immaterial for practical purposes.     

Johnson's transformation two-sample trimmed t and its bootstrap method for heterogeneity and non-normality

The present study investigates the performance of Johnson's transformation trimmed t statistic, Welch's t test, Yuen's trimmed t , Johnson's transformation untrimmed t test, and the corresponding bootstrap methods for the two-sample case with small/unequal sample sizes when the distribution is non-normal and variances are heterogeneous. The Monte Carlo simulation is conducted in two-sided as well as one-sided tests. When the variance is proportional to the sample size, Yuen's trimmed t is as good as Johnson's transformation trimmed t . However, when the variance is disproportional to the sample size, the bootstrap Yuen's trimmed t and the bootstrap Johnson's transformation trimmed t are recommended in one-sided tests. For two-sided tests, Johnson's transformation trimmed t is not only valid but also powerful in comparison to the bootstrap methods.     

Comparing J independent groups with a method based on trimmed means

The ANOVA-F test is the most popular and commonly used procedure for comparing J independent groups. However, it is well known that this method is very sensitive to non-normality, which has led to the derivation of alternative techniques based on robust estimators. In this work, ANOVA-F-test, trimmed mean Welch test, bootstrap-t trimmed mean Welch test, Schrader and Hettmansperger method with trimmed means, a percentile bootstrap method with trimmed means and a newly proposed method were compared in terms of both the Type I error probability and power. The proposed method compares well with ANOVA-F and other alternatives under various situations.     

Homogeneity of variances in normal linear regression with a change point

Two methods for testing the equality of variances in straight lines regression with a change point are considered. One is likelihood ratio test and the other is Bayesian confidence interval, based on the highest posterior density for the ratio of variances, using non-informative priors. Methods are applied to the renal transplant data analyzed by Smith and Cook(1980) and Stephens(1994).     

An approximate conditional analysis of the linear functional relationship

The linear functional relationship is exhibited as an approximate conditional normal model. The estimation of the slope of the linear functional relationship is approximately equivalent to the estimation of the common mean of a set of normal distributions having the same mean and different variances in known ratios. These known ratios are approximate ancillary statistics that specify the observed precisions of the samples from the different normal distributions. The analysis conditions on these observed precisions. This conditional method is compared with several standard unconditional methods. Several numerical examples are given.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

Testing parallelism of regression lines against ordered alternatives

For the problem of testing the equality of slopes of several regression lines against the alternative that the slopes are in increasing (decreasing) order of magnitude, two types of tests are proposed. These are the likelihood ratio test and a test that depends on suitable linear combination of one group statistics. Rank analogues of the two tests are also examined.     

A note on homogeneity tests of covariance matrices

In this note we propose two procedures for testing homogeneity of co-variance matrices that are both extensions of Hartley's (1940) test for equality of variances. The first is a two-stage procedure where the first step is a simple test for equality of the largest eigenvalues, and corresponding eigenvectors, of the covariance matrices. The second is based on projection pursuit and seems harder to apply in practice.     

Hypothesis Testing in Multivariate Linear Models with Randomly Missing Data

Tests for mean equality proposed by Weerahandi (1995) and Chen and Chen (1998), tests that do not require equality of population variances, were examined when data were not only heterogeneous but, as well, nonnormal in unbalanced completely randomized designs. Furthermore, these tests were compared to a test examined by Lix and Keselman (1998), a test that uses a heteroscedastic statistic (i.e., Welch, 1951) with robust estimators (20% trimmed means and Winsorized variances). Our findings confirmed previously published data that the tests are indeed robust to variance heterogeneity when the data are obtained from normal populations. However, the Weerahandi (1995) and Chen and Chen (1998) tests were not found to be robust when data were obtained from nonnormal populations. Indeed, rates of Type I error were typically in excess of 10% and, at times, exceeded 50%. On the other hand, the statistic presented by Lix and Keselman (1998) was generally robust to variance heterogeneity and nonnormality.     

Robust procedures for multi-sample location problems with unequal group variances

Robust procedures are proposed for testing the equality of several group means without assuming the equality of group variances. These statistics are obtained by modifying Welch's W and Brown-Forsythe's F^* using a trimmed mean and a sine-wave M estimator.Approximate distributions of these new statistics are obtained under normality. Their performances are evaluated by Monte Carlo sampling experiments under various long-tailed symmetric distributions     

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.