Performance of confidence interval tests for the ratio of two lognormal means applied to Weibull and gamma distribution data

Five estimation approaches have been developed to compute the confidence interval (CI) for the ratio of two lognormal means: (1) T, the CI based on the t-test procedure; (2) ML, a traditional maximum likelihood-based approach; (3) BT, a bootstrap approach; (4) R, the signed log-likelihood ratio statistic; and (5) R*, the modified signed log-likelihood ratio statistic. The purpose of this study was to assess the performance of these five approaches when applied to distributions other than lognormal distribution, for which they were derived. Performance was assessed in terms of average length and coverage probability of the CIs for each estimation approaches (i.e., T, ML, BT, R, and R*) when data followed a Weibull or gamma distribution. Four models were discussed in this study. In Model 1, the sample sizes and variances were equal within the two groups. In Model 2, the sample sizes were equal but variances were different within the two groups. In Model 3, the variances were different within the two groups and the larger variance was paired with the larger sample size. In Model 4, the variances were different within the two groups and the larger variance was paired with the smaller sample size. The results showed that when the variances of the two groups were equal, the t-test performed well, no matter what the underlying distribution was and how large the variances of the two groups were. The BT approach performed better than the others when the underlying distribution was not lognormal distribution, although it was inaccurate when the variances were large. The R* test did not perform well when the underlying distribution was Weibull or gamma distributed data, but it performed best when the data followed a lognormal distribution.     

To be or not to be valid in testing the significance of the slope in simple quantitative linear models with autocorrelated errors

In this article, the validity of procedures for testing the significance of the slope in quantitative linear models with one explanatory variable and first-order autoregressive [AR(1)] errors is analyzed in a Monte Carlo study conducted in the time domain. Two cases are considered for the regressor: fixed and trended versus random and AR(1). In addition to the classical t -test using the Ordinary Least Squares (OLS) estimator of the slope and its standard error, we consider seven t -tests with n-2\,\hbox{df} built on the Generalized Least Squares (GLS) estimator or an estimated GLS estimator, three variants of the classical t -test with different variances of the OLS estimator, two asymptotic tests built on the Maximum Likelihood (ML) estimator, the F -test for fixed effects based on the Restricted Maximum Likelihood (REML) estimator in the mixed-model approach, two t -tests with n - 2 df based on first differences (FD) and first-difference ratios (FDR), and four modified t -tests using various corrections of the number of degrees of freedom. The FDR t -test, the REML F -test and the modified t -test using Dutilleul's effective sample size are the most valid among the testing procedures that do not assume the complete knowledge of the covariance matrix of the errors. However, modified t -tests are not applicable and the FDR t -test suffers from a lack of power when the regressor is fixed and trended ( i.e. , FDR is the same as FD in this case when observations are equally spaced), whereas the REML algorithm fails to converge at small sample sizes. The classical t -test is valid when the regressor is fixed and trended and autocorrelation among errors is predominantly negative, and when the regressor is random and AR(1), like the errors, and autocorrelation is moderately negative or positive. We discuss the results graphically, in terms of the circularity condition defined in repeated measures ANOVA and of the effective sample size used in correlation analysis with autocorrelated sample data. An example with environmental data is presented.     

Interval Estimation for the Difference of Two Independent Variances

Analytical methods for interval estimation of differences between variances have not been described. A simple analytical method is given for interval estimation of the difference between variances of two independent samples. It is shown, using simulations, that confidence intervals generated with this method have close to nominal coverage even when sample sizes are small and unequal and observations are highly skewed and leptokurtic, provided the difference in variances is not very large. The method is also adapted for testing the hypothesis of no difference between variances. The test is robust but slightly less powerful than Bonett's test with small samples.     

Comparing the overlapping of two independent confidence intervals with a single confidence interval for two normal population parameters

Two overlapping confidence intervals have been used in the past to conduct statistical inferences about two population means and proportions. Several authors have examined the shortcomings of Overlap procedure and have determined that such a method distorts the significance level of testing the null hypothesis of two population means and reduces the statistical power of the test. Nearly all results for small samples in Overlap literature have been obtained either by simulation or by formulas that may need refinement for small sample sizes, but accurate large sample information exists. Nevertheless, there are aspects of Overlap that have not been presented and compared against the standard statistical procedure. This article will present exact formulas for the maximum % overlap of two independent confidence intervals below which the null hypothesis of equality of two normal population means or variances must still be rejected for any sample sizes. Further, the impact of Overlap on the power of testing the null hypothesis of equality of two normal variances will be assessed. Finally, the noncentral t-distribution is used to assess the Overlap impact on type II error probability when testing equality of means for sample sizes larger than 1.     

Small sample size comparisons of tests for homogeneity of variances by Monte-Carlo

A number of tests are available for testing the equality of several population variances. Some are claimed to be robust. We compared six of those claimed robust procedures by Monte Carlo simulated experiments, particularly for cases of small and unequal sample sizes. Our results show that the jack-knife test compares favorably with the other tests.     

A step-down heteroscedastic multiple comparison procedure

All-pairs power in a one-way ANOVA is the probability of detecting all true differences between pairs of means. Ramsey (1978) found that for normal distributions having equal variances, step-down multiple comparison procedures can have substantially more all-pairs power than single-step procedures, such as Tukey’s HSD, when equal sample sizes are randomly sampled from each group. This paper suggests a step-down procedure for the case of unequal variances and compares it to Dunnett's T3 technique. The new procedure is similar in spirit to one of the heteroscedastic procedures described by Hochberg and Tamhane (1987), but it has certain advantages that are discussed in the paper. Included are results on unequal sample sizes.     

Two separate effects of variance heterogeneity on the validity and power of significance tests of location

Heterogeneity of variances of treatment groups influences the validity and power of significance tests of location in two distinct ways. First, if sample sizes are unequal, the Type I error rate and power are depressed if a larger variance is associated with a larger sample size, and elevated if a larger variance is associated with a smaller sample size. This well-established effect, which occurs in t and F tests, and to a lesser degree in nonparametric rank tests, results from unequal contributions of pooled estimates of error variance in the computation of test statistics. It is observed in samples from normal distributions, as well as non-normal distributions of various shapes. Second, transformation of scores from skewed distributions with unequal variances to ranks produces differences in the means of the ranks assigned to the respective groups, even if the means of the initial groups are equal, and a subsequent inflation of Type I error rates and power. This effect occurs for all sample sizes, equal and unequal. For the t test, the discrepancy diminishes, and for the Wilcoxon–Mann–Whitney test, it becomes larger, as sample size increases. The Welch separate-variance t test overcomes the first effect but not the second. Because of interaction of these separate effects, the validity and power of both parametric and nonparametric tests performed on samples of any size from unknown distributions with possibly unequal variances can be distorted in unpredictable ways.     

The split sample permutation t-tests

Without the exchangeability assumption, permutation tests for comparing two population means do not provide exact control of the probability of making a Type I error. Another drawback of permutation tests is that it cannot be used to test hypothesis about one population. In this paper, we propose a new type of permutation tests for testing the difference between two population means: the split sample permutation t-tests. We show that the split sample permutation t-tests do not require the exchangeability assumption, are asymptotically exact and can be easily extended to testing hypothesis about one population. Extensive simulations were carried out to evaluate the performance of two specific split sample permutation t-tests: the split in the middle permutation t-test and the split in the end permutation t-test. The simulation results show that the split in the middle permutation t-test has comparable performance to the permutation test if the population distributions are symmetric and satisfy the exchangeability assumption. Otherwise, the split in the end permutation t-test has significantly more accurate control of level of significance than the split in the middle permutation t-test and other existing permutation tests.     

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.     

Robust test for means when population variances are unequal

We consider the problem of testing the equality of two population means when the population variances are not necessarily equal. We propose a Welch-type statistic, say T^* _c, based on Tiku!s ‘1967, 1980’ modified maximum likelihood estimators, and show that this statistic is robust to symmetric and moderately skew distributions. We investigate the power properties of the statistic T^* _c; T^* _c clearly seems to be more powerful than Yuen's ‘1974’ Welch-type robust statistic based on the trimmed sample means and the matching sample variances. We show that the analogous statistics based on the ‘adaptive’ robust estimators give misleading Type I errors. We generalize the results to testing linear contrasts among k population means     

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.     

Levels of significance of some two-sample tests wkem observations are from compound normal distributions

This paper presents an investigation of the behavior of the levels of significance of the two-sample t and its related tests and the Mann-Whitney test when the samples are randomly drawn from mixtures of two normal populations (compound normals) and when the sample sizes are small (combined sample sizes ? 15). The use of the compound normal allows for investigation when the underlying populations are unequal, nonnormal, heterogeneous in variances, unimodal or bimodal, possessing smaller than normal kurtosis or containing contamination. The exact distribution of the t and its related tests are given. However, they are not readily amenable to calculations. Most of the numerical results presented were obtained by simulations     

Least significant spacing for ‘one versus the rest’ normal populations

Consider sample means from k(≥2) normal populations where the variances and sample sizes are equal. The problem is to find the ‘least significant difference’ or ‘spacing’ (LSS) between the two largest means, so that if an observed spacing is larger we have confidence 1 - α that the population with largest sample mean also has the largest population mean.

When the variance is known it is shown that the maximum LSS occurs when k = 2, provided a < .2723. In other words, for any value of k we may use the usual (one-tailed) least significant difference to demonstrate that one population has a population mean greater than (or equal to) the rest.

When the variance is estimated bounds are obtained for the confidence which indicate that this last result is approximately correct.     

A comparison of selection procedures for selecting the best normal population under heterogeneity of variance

This study examines the comparative probabilities of making a correct selection when using the means procedure (M), the medians procedure (D) and the rank-sum procedure (S) to correctly select the normal population with the largest mean under heterogeneity of variance. The comparison is conducted by using Monte-Carlo simulation techniques for 3, 4, and 5 normal populations under the condition that equal sample sizes are taken from each population. The population means and standard deviations are assumed to be equally-spaced. Two types of heterogeneity of variance are considered: (1) associating larger means with larger variances, and (2) associating larger means with smaller variances.     

Comparison of Three Sampling Designs

Three sampling designs are considered for estimating the sum of k population means by the sum of the corresponding sample means. These are (a) the optimal design; (b) equal sample sizes from all populations; and (c) sample sizes that render equal variances to all sample means. Designs (b) and (c) are equally inefficient, and may yield a variance up to k times as large as that of (a). Similar results are true when the cost of sampling is introduced, and they depend on the population sampled.     

A test for bivariate two sample location problem

A consistent test for difference in locations between two bivariate populations is proposed, The test is similar as the Mann-Whitney test and depends on the exceedances of slopes of the two samples where slope for each sample observation is computed by taking the ratios of the observed values. In terms of the slopes, it reduces to a univariate problem, The power of the test has been compared with those of various existing tests by simulation. The proposed test statistic is compared with Mardia's(1967) test statistics, Peters-Randies(1991) test statistic, Wilcoxon's rank sum test. statistic and Hotelling' T² test statistic using Monte Carlo technique. It performs better than other statistics compared for small differences in locations between two populations when underlying population is population 7(light tailed population) and sample size 15 and 18 respectively. When underlying population is population 6(heavy tailed population) and sample sizes are 15 and 18 it performas better than other statistic compared except Wilcoxon's rank sum test statistics for small differences in location between two populations. It performs better than Mardia's(1967) test statistic for large differences in location between two population when underlying population is bivariate normal mixture with probability p=0.5, population 6, Pearson type II population and Pearson type VII population for sample size 15 and 18 .Under bivariate normal population it performs as good as Mardia' (1967) test statistic for small differences in locations between two populations and sample sizes 15 and 18. For sample sizes 25 and 28 respectively it performs better than Mardia's (1967) test statistic when underlying population is population 6, Pearson type II population and Pearson type VII population     

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.     

New monte carlo results on the robustness of the anova f,w and f statistics

Because the usual F test for equal means is not robust to unequal variances, Brown and Forsythe (1974a) suggest replacing F with the statistics F or W which are based on the Satterthwaite and Welch adjusted degrees of freedom procedures. This paper reports practical situations where both F and W give * unsatisfactory results. In particular, both F and W may not provide adequate control over Type I errors. Moreover, for equal variances, but unequal sample sizes, W should be avoided in favor of F (or F ), but for equal sample sizes, and possibly unequal variances, W was the only satisfactory statistic. New results on power are included as well. The paper also considers the effect of using F or W only after a significant test for equal variances has been obtained, and new results on the robustness of the F test are described. It is found that even for equal sample sizes as large as 50 per treatment group, there are practical situations where the F test does not provide adequately control over the probability of a Type I error.     

The two-sample -test with a known ratio of variances

We consider the two-sample t-test where error variances are unknown but with known relationships between them. This situation arises, for example, when two measuring instruments average different number of replicates to report the response. In particular we compare our procedure with the usual Satterthwaite approximation in the two sample t-test with variances unequal. Our procedure uses the knowledge of a known ratio of variances while the Satterthwaite approximation assumes only that the two variances are unequal. Simulations show that our procedure has both better size and better power than the Satterthwaite approximation. Finally, we consider an extension of our results to the General Linear Model.     

The power functions of the likelihood ratio tests for a simple tree ordering in normal means: unequal weights

Likelihood ratio tests are considered for two testing situations; testing for the homogeneity of k normal means against the alternative restricted by a simple tree ordering trend and testing the null hypothesis that the means satisfy the trend against all alternatives. Exact expressions are given for the power functions for k = 3 and 4 and unequal sample sizes, both for the case of known and unknown population variances, and approximations are discussed for larger k. Also, Bartholomew’s conjectures concerning minimal and maximal powers are investigated for the case of equal and unequal sample sizes. The power formulas are used to compute powers for a numerical example.