An Adjustment to the Bartlett's Test for Small Sample Size

The Bartlett's test (1937) for equality of variances is based on the χ² distribution approximation. This approximation deteriorates either when the sample size is small (particularly < 4) or when the population number is large. According to a simulation investigation, we find a similar varying trend for the mean differences between empirical distributions of Bartlett's statistics and their χ² approximations. By using the mean differences to represent the distribution departures, a simple adjustment approach on the Bartlett's statistic is proposed on the basis of equal mean principle. The performance before and after adjustment is extensively investigated under equal and unequal sample sizes, with number of populations varying from 3 to 100. Compared with the traditional Bartlett's statistic, the adjusted statistic is distributed more closely to χ² distribution, for homogeneity samples from normal populations. The type I error is well controlled and the power is a little higher after adjustment. In conclusion, the adjustment has good control on the type I error and higher power, and thus is recommended for small samples and large population number when underlying distribution is normal.     

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 equality of generalized variances of k multivariate normal populations

Generalized variance is a measure of dispersion of multivariate data. Comparison of dispersion of multivariate data is one of the favorite issues for multivariate quality control, generalized homogeneity of multidimensional scatter, etc. In this article, the problem of testing equality of generalized variances of k multivariate normal populations by using the Bartlett's modified likelihood ratio test (BMLRT) is proposed. Simulations to compare the Type I error rate and power of the BMLRT and the likelihood ratio test (LRT) methods are performed. These simulations show that the BMLRT method has a better chi-square approximation under the null hypothesis. Finally, a practical example is given.     

A Note on an Unbiased Test of Homogeneity of Variances

A necessary and sufficient condition for unbiasedness of the test of homogeneity of variances in normal samples is derived in a convenient form. In the case of two samples, it is shown that Bartlett's test is the only unbiased test of homogeneity of variances. A simple alternative proof of the unbiasedness of Bartlett's test in the general case is also provided.     

EXACT DISTRIBUTION OF CERTAIN GENERAL TEST STATISTICS IN MULTIVARIATE ANALYSIS

In this paper, exact solution of Wilks' type-B integral equation has been obtained in its most general form as a series of weighted gamma distributions. This general result then gives the distributions of many test statistics in multivariate analysis. In particular the distributions of Wilks' Λ, the sphericity test criterion, and Bartlett's test statistic, are derived in easily computable form.     

Sequential analysis methodology for a Poisson GLMM with applications to multicenter randomized clinical trials

Sequential analyses in clinical trials have ethical and economic advantages over fixed sample size methods. The sequential probability ratio test (SPRT) is a hypothesis testing procedure which evaluates data as it is collected. The original SPRT was developed by Wald for one-parameter families of distributions and later extended by Bartlett to handle the case of nuisance parameters. However, Bartlett's SPRT requires independent and identically distributed observations. In this paper we show that Bartlett's SPRT can be applied to generalized linear model (GLM) contexts. Then we propose an SPRT analysis methodology for a Poisson generalized linear mixed model (GLMM) that is suitable for our application to the design of a multicenter randomized clinical trial that compares two preventive treatments for surgical site infections. We validate the methodology with a simulation study that includes a comparison to Neyman–Pearson and Bayesian fixed sample size test designs and the Wald SPRT.     

Dependence Estimation for High‐frequency Sampled Multivariate CARMA Models

The paper considers high‐frequency sampled multivariate continuous‐time autoregressive moving average (MCARMA) models and derives the asymptotic behaviour of the sample autocovariance function to a normal random matrix. Moreover, we obtain the asymptotic behaviour of the cross‐covariances between different components of the model. We will see that the limit distribution of the sample autocovariance function has a similar structure in the continuous‐time and in the discrete‐time model. As a special case, we consider a CARMA (one‐dimensional MCARMA) process. For a CARMA process, we prove Bartlett's formula for the sample autocorrelation function. Bartlett's formula has the same form in both models; only the sums in the discrete‐time model are exchanged by integrals in the continuous‐time model. Finally, we present limit results for multivariate MA processes as well, which are not known in this generality in the multivariate setting yet.     

A test for equality of variances with censored samples

A variance homogeneity test for type II right-censored samples is proposed. The test is based on Bartlett's statistic. The asymptotic distribution of the statistic is investigated. The limiting distribution is that of a linear combination of i.i.d. chi-square variables with 1 degree of freedom. By using simulation, the critical values of the null distribution of the modified Bartlett's statistic for testing the homogeneity of variances of two normal populations are obtained when the sample sizes and censoring levels are not equal. Also, we investigate the properties of the proposed test (size, power and robustness). Results show that the distribution of the test statistic depends on the censoring level. An example of the use of the new methodology in animal science involving reproduction in ewes is provided.     

Generalizations of barlett's and hartley's tests of homogeneity using “overall variability”

The generalized variance plays on important and useful role as a measure to compare overall variability of different populations in biological sciences (Goodman, 1968; Kocherlakota and Kocherlakota, 1983; Sokai, 1965). Here we present simple and elegant multivariate analogues to Bartlett's and Hartley's tests of homogeneity. Large sample distributions of the statistics are presented and the practical usefulness of the tests are demonstrated throught several examples.     

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 new discrete distribution induced by the Luria–Delbrück mutation model

In this paper, we study a limiting distribution induced by Bartlett's formulation of the Luria–Delbrück mutation model. We establish the validity of the probability generating function and devise an algorithm for computing the probability mass function. Maximum-likelihood estimation and asymptotic behaviour of the distribution are considered.     

Bartlett's Test Applied to Variance Component Models

Exact methods for testing equality between variance components obtained from several cases of the same type of balanced orthogonal design are discussed. In particular, methods for successively testing equality of a number of components using Bartlett's tests are outlined for univariate and multivariate responses. Two clinical trial examples of repeated‐measures data are presented.     

A revisit to test the equality of variances of several populations

We revisit the problem of testing homoscedasticity (or, equality of variances) of several normal populations which has applications in many statistical analyses, including design of experiments. The standard text books and widely used statistical packages propose a few popular tests including Bartlett's test, Levene's test and a few adjustments of the latter. Apparently, the popularity of these tests have been based on limited simulation study carried out a few decades ago. The traditional tests, including the classical likelihood ratio test (LRT), are asymptotic in nature, and hence do not perform well for small sample sizes. In this paper we propose a simple parametric bootstrap (PB) modification of the LRT, and compare it against the other popular tests as well as their PB versions in terms of size and power. Our comprehensive simulation study bursts some popularly held myths about the commonly used tests and sheds some new light on this important problem. Though most popular statistical software/packages suggest using Bartlette's test, Levene's test, or modified Levene's test among a few others, our extensive simulation study, carried out under both the normal model as well as several non-normal models clearly shows that a PB version of the modified Levene's test (which does not use the F-distribution cut-off point as its critical value), and Loh's exact test are the “best” performers in terms of overall size as well as power.     

Likelihood-based confidence limits for the common odds ratio

In this paper we derive two likelihood-based procedures for the construction of confidence limits for the common odds ratio in K 2 × 2 contingency tables. We then conduct a simulation study to compare these procedures with a recently proposed procedure by Sato (Biometrics 46 (1990) 71–79), based on the asymptotic distribution of the Mantel-Haenszel estimate of the common odds ratio. We consider the situation in which the number of strata remains fixed (finite), but the sample sizes within each stratum are large. Bartlett's score procedure based on the conditional likelihood is found to be almost identical, in terms of coverage probabilities and average coverage lengths, to the procedure recommended by Sato, although the score procedure has some edge, in some instances, in terms of average coverage lengths. So, for ‘fixed strata and large sample’ situation Bartlett's score procedure can be considered as an alternative to the procedure proposed by Sato, based on the asymptotic distribution of the Mantel-Haenszel estimator of the common odds ratio.     

Sampled autocovariance and autocorrelation results for linear time processes

We derive an exact formula for the covariance between the sampled autocovariances at any two lags for a finite time series realisation from a general stationary autoregressive moving average process. We indicate, through one particular example, how this result can be used to deduce analogous formulae for any nonstationary model of the ARUMA class, a generalisation of the ARIMA models. Such formulae then allow us to obtain approximate expressions for the convariances between all pairs of serial correlations for finite realisations from the ARUMA model. We also note that, in the limit as the series length n → ∞, our results for the ARMA class retrieve those of Bartlett (1946). Finally, we investigate an improvement to the approximation that is obtained by applying Bartlett's general asymptotic formula to finite series realisations. That such an improvement should exist can immediately be seen by consideration of out results for the simplest case of a white noise process. However, we deduce the final improved approapproximation, for general models, in two ways - from (corrected) results due to Davies and Newbold (1980), and by an alternative approach to theirs.     

The Exact Null Distribution of Bartlett's Criterion

In this paper the exact null distribution of Bartlett's criterion for testing the homogeneity of variances in normal samples with unequal sizes is derived. The most general form of the density function is obtained by using contour integration. The expression for the cumulative distribution, being a series in simple algebraic functions, seems quite tractable for computation of the exact critical values. In the special case of equal sample sizes, some indication of the relation of the work of others to our series expansions has also been given.     

Estimation du comportement asymptotique des autocovariances et autocorrelations empiriques de processus multivariéeas

Statistics based on the sample autocovariances are widely used in time-series analysis. Estimators of the asymptotic covariance between the sample autocovariances are commonly derived from the so-called Bartlett's formula. However, this formula essentially holds for linear processes. This entails that for a wide range of nonlinear time series the above-mentioned estimators are not suitable. In this paper the behaviour of an alternative estimator is studied within the framework of centered or uncentered multivariate strongly mixing processes. Applications to differential functions of sample autocovariances, such as the sample autocorrelations, are considered.     

An evaluation of change-point estimators for a sequence of normal observations with unknown parameters

Performance of maximum likelihood estimators (MLE) of the change-point in normal series is evaluated considering three scenarios where process parameters are assumed to be unknown. Different shifts, sample sizes, and locations of a change-point were tested. A comparison is made with estimators based on cumulative sums and Bartlett's test. Performance analysis done with extensive simulations for normally distributed series showed that the MLEs perform better (or equal) in almost every scenario, with smaller bias and standard error. In addition, robustness of MLE to non-normality is also studied.     

Equivalence of Certain Chi-Squared Test Statistics

In likelihood analysis of categorized data, it is well known that within a restricted class of log-linear models the likelihood kernels for multinomial and product multinomial sampling distributions are identical. In practical terms the estimation procedure for one is appropriate for the other. There does not appear to be a widespread realization that a similar result holds for a wide class of the Grizzle, Starmer, and Koch (1969) weighted least squares techniques. In this report such a fundamental relationship is explicitly presented and illustrated through two analyses of Bartlett's (1935) data.     

Non Uniform Bounds on Geometric Approximation Via Stein's Method and w-Functions

In this article, we use Stein's method and w-functions to give uniform and non uniform bounds in the geometric approximation of a non negative integer-valued random variable. We give some applications of the results of this approximation concerning the beta-geometric, Pólya, and Poisson distributions.