Testing Equality Of Means Research Articles

Testing equality of means in one-way ANOVA using three and four moment approximations

In this study, we focus on two test statistics for testing the equality of treatment means in one-way analysis of variance (ANOVA). The first one is the well known Cochran ($C_{LS}$) test statistic based on least squares (LS) estimators and the second one is robust version of it ($RC_{MML}$) based on modified maximum likelihood (MML) estimators. These two test statistics are asymptotically distributed as chi-square. However, distributions of them are unknown for small samples. Therefore, three-moment chi-square and four moment $F$ approximations to the null distributions of $C_{LS}$ and $RC_{MML}$ are derived inspired by Tiku and Wong [19]. To investigate the small and moderate sample properties of these tests based on the mentioned approximations, an extensive Monte-Carlo simulation study is performed when the underlying distribution is long-tailed symmetric (LTS). Simulation results show that four-moment $F$ approximation provides better approximation than the three-moment chi-square approximation for both $C_{LS}$ and $RC_{MML}$ tests. Therefore, the simulated Type I error rates and powers of the $C_{LS}$ and $RC_{MML}$ test statistics are calculated using four-moment $F$ approximation. According to simulation results, $RC_{MML}$ test is more powerful than the corresponding $C_{LS}$ test.

Testing for equality of means with equal and unequal variances

In this paper, we are interested in comparing the conventional t –test with the proposed t – test for testing equality of means with unequal and equal variances. Here, we proposed harmonic mean of variances as an alternative to the pooled sample variance when there is heterogeneity of variances. Two sets of secondary data were obtained from Agricultural Development Project (KWADP) and the Ministry of Agriculture in Ilorin, Kwara State to demonstrate the two test statistics used and the results show that the proposed t – test statistic is found to be appropriate than the conventional t – test statistic when we have unequal variances but the conventional t – test perform better when we have equal variances.

Analyzing partially paired data: when can the unpaired portion(s) be safely ignored?

Partially paired data, either with incompleteness in one or both arms, are common in practice. For testing equality of means of two arms, practitioners often use only the portion of data with complete pairs and perform paired tests. Although such tests (referred as ‘naive paired tests’) are legitimate, their powers might be low as only partial data are utilized. The recently proposed ‘P-value pooling methods’, based on combining P-values from two tests, use all data, have reasonable type-I error control and good power property. While it is generally believed that ‘P-value pooling methods’ are superior to ‘naive paired tests’ in terms of power as the former use more data than the latter, no detailed power comparison has been done. This paper aims to compare powers of ‘naive paired tests’ and ‘P-value pooling methods’ analytically and our findings are counterintuitive, i.e. the ‘P-value pooling methods’ do not always outperform the naive paired tests in terms of power. Based on these results, we present guidance on how to select the best test for testing equality of means with partially paired data.

Testing equality of means in partially paired data with incompleteness in single response.

In testing differentially expressed genes between tumor and healthy tissues, data are usually collected in paired form. However, incomplete paired data often occur. While extensive statistical researches exist for paired data with incompleteness in both arms, hardly any recent work can be found on paired data with incompleteness in single arm. This paper aims to fill this gap by proposing some new methods, namely, P-value pooling methods and a nonparametric combination test. Simulation studies are conducted to investigate the performance of the proposed methods in terms of type I error and power at small to moderate sample sizes. A real data set from The Cancer Genome Atlas (TCGA) breast cancer study is analyzed using the proposed methods.

Testing the equality of quantiles for several normal populations

ABSTRACTMany procedures exist for testing equality of means or medians to compare several independent distributions. However, the mean or median do not determine the entire distribution. In this article, we propose a new small-sample modification of the likelihood ratio test for testing the equality of the quantiles of several normal distributions. The merits of the proposed test are numerically compared with the existing tests—a generalized p-value method and likelihood ratio test—with respect to their sizes and powers. The simulation results demonstrate that proposed method is satisfactory; its actual size is very close to the nominal level. We illustrate these approaches using two real examples.

Two-stage test of means of unordered pairs.

The problem of testing equality of means of a bivariate normal distribution on the basis of a sample of size n has been considered when the labels of the observations are either missing or not known. The problem may arise in many applied settings, especially in genetics. Classical likelihood ratio test fails here because of identifiability problems. We propose a two-stage testing procedure using a recently developed test in the context of penalized splines. The proposed testing procedure is found to outperform the tests proposed in the literature. Copyright © 2017 John Wiley & Sons, Ltd.

On Application of Development of Test Statistic for Testing Unequal Group Variances

In this paper, an already proposed test-statistic for testing equality of means under unequal population variances is applied. When the group variances differ, using pooled sample variance will give an inappropriate result as a single value for the variances. This kind of problem in statistics is commonly referred to as Brehen-Fisher problem in the k-sample location problems. A proposed unbiased sample harmonic mean of variances 2 HS was examined and found useful for unequal variances which have received a considerable attention in the area of medical and biological sciences. Little or nothing has been achieved in social sciences that form a major part of this work. Data from the six geopolitical zones on road crashes in Nigeria from the year 2004 to 2013 was used to ascertain the consistency of the result with the literature which was found useful and relevant for the proposed developed test statistic. It was observed that using this proposed test statistic, the number of road crashes was significant in some geopolitical zones in Nigeria which was ordinarily latent to pool sample variance.

A study of resampling methods for testing equality of means in paired population

The goal of this study is to compare the resampling methods including bootstrap and permutation tests against classical methods for paired samples. A simulation study was conducted to see the performance of both parametric and nonparametric methods under various assumptions such as non-normal populations and small or large sample sizes. The results of the simulation study are with respect to type I error and power of the test.

Paired Sample Tests with Unordered Observations

The problem of testing equality of means of a bivariate normal distribution on the basis of a sample of size n has been considered when the labels of the observations are not known. Two tests based on the sample moments of absolute values of the differences of the observations in each pair are suggested. Their power properties are investigated. It is interesting to observe that for small values of the parameter one is more powerful than the other while for larger values it is the other way around. AMS (2000) Subject Classification : 62F03, 62P10.

Testing Equality of Means Under Order Restrictions Using Multiple Contrasts

The likelihood ratio test for equality of ordered means is known to have power characteristics that are generally superior to those of competing procedures. Difficulties in implementing this test have led to the development of alternative approaches, most of which are based on contrasts. While orthogonal contrasts can be chosen to simplify the distribution theory, we propose a class of tests that is easy to implement even if the contrasts used are not orthogonal. An overall measure of significance may be obtained by using Fisher's combination statistic to combine the dependent p-values arising from these contrasts. This method can be easily implemented for testing problems involving unequal sample sizes and any partial order, and has power properties that compare well with those of the likelihood ratio test and other contrast-based tests.

임상시험에서 중도탈락을 고려한 표본크기의 결정

임상시험에서 피험자수는 검정가설, 변수값의 분산과 유효차이, 검정력과 유의수준 등에 의해 결정되어진다. 일반적으로 수학적으로 계산된 피험자수에 중도탈락 예상치를 고려한 피험자수를 추가하여 최종적인 실험참가자수를 결정하는데 본 논문에서는 이론적인 계산식에서부터 중도탈락을 고려하여 피험자수를 결정하는 방법을 제안한다. 임상시험에서 많은 자료는 경시적(longitudinal) 형태를 갖고, ITT(intention to treat) 실험의 경우 중도탈락이 생기면 결측값으로 처리하지 않고 탈락직전에 관측된 값을 최종값으로 대체하는 LOCF(last observation carried forward) 방법을 주로 사용한다. 이러한 LOCF 방법은 피험자수 계산에 사용했던 분산과 유효차이 값의 가정에 왜곡을 가져오기 때문에 우리가 원하는 검정력을 보장 받지 못할 수 있다. 본 연구에서는 중도탈락률에 관한 정보를 포함하는 피험자수의 결정식을 제안하고 평균의 동일성 검정 경우에 검정력을 비교하여 이러한 산출방식이 합리적임을 실증하였다. The sample size in a clinical trial is determined by the hypothesis, the variance of observations, the effect size, the power and the significance level. Dropouts in clinical trials are inevitable, so we need to consider dropouts on the determination of sample size. It is common that some proportion corresponding to the expected dropout rate would be added to the sample size calculated from a mathematical equation. This paper proposes new equations for calculating sample size dealing with dropouts. Since we observe data longitudinally in most clinical trials, we can use a last observation to impute for missing one in the intention to treat (ITT) trials, and this technique is called last observation carried forward(LOCF). But LOCF might make deviations on the assumed variance and effect size, so that we could not guarantee the power of test with the sample size obtained from the existing equation. This study suggests the formulas for sample size involving information about dropouts and shows the properties of the proposed method in testing equality of means.

Certain Multi Sample Tests for Inverse Gaussian Populations

This paper considers problems of testing equality of means assuming coefficients of variation to be common and homogeneity of coefficients of variation in inverse Gaussian populations. The likelihood ratio, Lagrange-Multiplier, and Wald tests are developed and Monte-Carlo simulations are conducted to compare the powers of these tests for selected sample sizes and various alternatives. The simulation results report that the likelihood ratio test is consistently more powerful than the other two tests.

Robust Two-Sample Statistics for Testing Equality of Means: A Simulation Study

When testing the equality of the means from two independent normally distributed populations given that the variances of the two populations are unknown but assumed equal, the classical two-sample t-test is recommended. If the underlying population distributions are normal with unequal and unknown variances, either Welch's t-statistic or Satterthwaite's Approximate F-test is suggested. However, Welch's procedure is non-robust under most non-normal distributions. There is a variable tolerance level around the strict assumptions of data independence, homogeneity of variances and normality of the distributions. Few textbooks offer alternatives when one or more of the underlying assumptions are not defensible.

Methods for testing equality of means of health care costs in a paired design study

Methods for testing equality of means of health care costs in a paired design study

Testing Equality of Means and Simultaneous Confidence Intervals in Repeated Measures with Missing Data

In this paper, repeated measures with intraclass correlation model is considered when the observations are missing at random. An exact test for the equality of the mean components and simultaneous confidence intervals (Scheffe and Bonferroni inequality types) are given for linear contrasts of the mean components when the missing observations are of a monotone type. When the missing observations are not of the monotone type, the maximum likelihood estimates are obtained numerically by iterative methods given in Srivastava and Carter (1986). These estimators are then used to obtain asymptotic tests and confidence intervals for the equality of mean components and linear contrasts, respectively. An example is given to illustrate the method.

A comparison of power approximations for satterthwaite's test

When testing equality of means from two independent normal populations, many statisticians prefer heterogeneity tolerant tests. Moser, Stevens, and Watts described the noncentral density and a numerical integration algorithm for computing power. We present simple and accurate approximations for the power of the Satterthwaite test statistic. Two advantages accrue. First, the approximations substantially reduce the computational burden for tasks such as plotting power curves. Second, the approximations substantially simplify the programming and thereby make power calculations more widely available. Four methods of power approximation are evaluated for test sizes of .001, .01, .05, and .10, sample sizes of 6 and 51, variance ratios of 1 and 10, and noncentrality parameters from 0 to 50 by 1. A method based on a ratio of expected values is recommended due to its accuracy and simplicity.

A Note on Testing Equality of Means when Sample Sizes are Random with Application

Summary The problem of testing the equality of means of two normal populations is considered when independent random samples of random sizes are given with the total number of observations from both populations being n fixed number. An application in forestry is discussed. Key w0~d.s: Test of hypothesis; Mixture of distribution; Random sample size.

Testing Equality of Means in the Presence of Correlation and Missing Data

AbstractA statistic, derived from the combination of two dependent tests, is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one or both variates. The null distribution of the statistic is approximated by a well‐known distribution. The empirical powers of the statistic are computed and compared with some of the known statistics. The comparisons support the use of the proposed test.

Testing equality of means in a bivariate normal population under possible presence of correlation coefficient

In this paper, we have considered for testing the hypothesis H 0: μ 1 = μ 2, in a bivariate normal population N(μ 1,μ 2,σ 2,σ 2,ϱ) when it is suspected that correlation coefficient ϱ, may be equal to zero. The expressions for the power and size have been obtained and recommendations regarding use or not of the proposed procedure have also been attempted.

Coping with variance heterogeneity

An approximate solution to the problem of testing equality of means of normal populations, when the variances may differ, is developed. The proposed solution utilizes the F-test with a random critical function. A series expansion based on results of Gideon and Gurland (1976) yields the true level of significance of the test. For a small number of populations in the test the technique is viable, but for a larger number a simulation study is made to compare the behavior of this test with that of some other well-known tests. The proposed test appears to provide better control of the true significance level, especially for small sample sizes. Futhermore, since it utilizes the ordinary F-statistic, it is simpler and more easily computed than some others.