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.     

Computing Wald criteria for nested hypotheses

We present a numerically convenient procedure for computing Wald criteria for nested hypotheses. Similar to Szroeter’s (1983) generalized Wald test, the suggested procedure does not require explicit derivation of the restrictions implied by the null hypothesis and hence its use might eliminate an intricate step in testing linear and nonlinear hypotheses. We show that the traditional Wald test, Szroeter’s (1983) generalized Wald test and our procedure are asymptotically equivalent under H₀. A class of nonlinear transformations of the restrictions for which the Wald statistic is asymptotically invariant is discussed. Finally, we illustrate the use of our procedure for testing the common factor restrictions in a dynamic regression model.     

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.     

Power and Sample Size for Fixed-Effects Inference in Reversible Linear Mixed Models

ABSTRACT

Despite the popularity of the general linear mixed model for data analysis, power and sample size methods and software are not generally available for commonly used test statistics and reference distributions. Statisticians resort to simulations with homegrown and uncertified programs or rough approximations which are misaligned with the data analysis. For a wide range of designs with longitudinal and clustering features, we provide accurate power and sample size approximations for inference about fixed effects in the linear models we call reversible. We show that under widely applicable conditions, the general linear mixed-model Wald test has noncentral distributions equivalent to well-studied multivariate tests. In turn, exact and approximate power and sample size results for the multivariate Hotelling–Lawley test provide exact and approximate power and sample size results for the mixed-model Wald test. The calculations are easily computed with a free, open-source product that requires only a web browser to use. Commercial software can be used for a smaller range of reversible models. Simple approximations allow accounting for modest amounts of missing data. A real-world example illustrates the methods. Sample size results are presented for a multicenter study on pregnancy. The proposed study, an extension of a funded project, has clustering within clinic. Exchangeability among the participants allows averaging across them to remove the clustering structure. The resulting simplified design is a single-level longitudinal study. Multivariate methods for power provide an approximate sample size. All proofs and inputs for the example are in the supplementary materials (available online).     

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 modification of the partial sequential probability ratio test

For the problem of discriminating between two simple hypoth¬eses concerning a Koopman - Darmois parameter, a modification of the partial sequential probability ratio test is proposed where instead of drawing only one fixed sample, two fixed samples are drawn and then Wald's SPRT is started. The OC and the ASN func¬tions are derived. Numerical comparisons are made with Wald's and Read's procedures for testing the normal mean with known variance. For some parameter values, the test procedure has a lower ASN than that of Read's procedure.     

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.     

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.     

A Note on Bartlett's M Test for Homogeneity of Variances

After pointing out a drawback in Bartlett's chi-square approximation, we suggest a simple modification and a Gamma approximation to improve Bartlett's M test for homogeneity of variances.     

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.     

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.     

The effect of spillover on the Granger causality test

In this paper, we investigate the properties of the Granger causality test in stationary and stable vector autoregressive models under the presence of spillover effects, that is, causality in variance. The Wald test and the WW test (the Wald test with White's proposed heteroskedasticity-consistent covariance matrix estimator imposed) are analyzed. The investigation is undertaken by using Monte Carlo simulation in which two different sample sizes and six different kinds of data-generating processes are used. The results show that the Wald test over-rejects the null hypothesis both with and without the spillover effect, and that the over-rejection in the latter case is more severe in larger samples. The size properties of the WW test are satisfactory when there is spillover between the variables. Only when there is feedback in the variance is the size of the WW test slightly affected. The Wald test is shown to have higher power than the WW test when the errors follow a GARCH(1,1) process without a spillover effect. When there is a spillover, the power of both tests deteriorates, which implies that the spillover has a negative effect on the causality tests.     

Approximate is Better than “Exact” for Interval Estimation of Binomial Proportions

For interval estimation of a proportion, coverage probabilities tend to be too large for “exact” confidence intervals based on inverting the binomial test and too small for the interval based on inverting the Wald large-sample normal test (i.e., sample proportion ± z-score × estimated standard error). Wilson's suggestion of inverting the related score test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes. The 95% score interval has similar behavior as the adjusted Wald interval obtained after adding two “successes” and two “failures” to the sample. In elementary courses, with the score and adjusted Wald methods it is unnecessary to provide students with awkward sample size guidelines.     

The use of covariate information in the sequential analysis of dichotomous response experiments

A method is presented for the sequential analysis of experiments involving two treatments to which response is dichotomous. Composite hypotheses about the difference in success probabilities are tested, and covariate information is utilized in the analysis. The method is based upon a generalization of Bartlett’s (1946) procedure for using the maximum likelihood estimate of a nuisance parameter in a Sequential Probability Ratio Test (SPRT). Treatment assignment rules studied include pure randomization, randomized blocks, and an adaptive rule which tends to assign the superior treatment to the majority of subjects. It is shown that the use of covariate information can result in important reductions in the expected sample size for specified error probabilities, and that the use of covariate information is essential for the elimination of bias when adaptive assignment rules are employed. Designs of the type presented are easily generated, as the termination criterion is the same as for a Wald SPRT of simple hypotheses.     

GMM inference in spatial autoregressive models

In this study, we investigate the finite sample properties of the optimal generalized method of moments estimator (OGMME) for a spatial econometric model with a first-order spatial autoregressive process in the dependent variable and the disturbance term (for short SARAR(1, 1)). We show that the estimated asymptotic standard errors for spatial autoregressive parameters can be substantially smaller than their empirical counterparts. Hence, we extend the finite sample variance correction methodology of Windmeijer (2005 Windmeijer, F. (2005). A finite sample correction for the variance of linear efficient two-step GMM estimators. Journal of Econometrics 126(1):25–51.[Crossref], [Web of Science ®] , [Google Scholar]) to the OGMME for the SARAR(1, 1) model. Results from simulation studies indicate that the correction method improves the variance estimates in small samples and leads to more accurate inference for the spatial autoregressive parameters. For the same model, we compare the finite sample properties of various test statistics for linear restrictions on autoregressive parameters. These tests include the standard asymptotic Wald test based on various GMMEs, a bootstrapped version of the Wald test, two versions of the C(α) test, the standard Lagrange multiplier (LM) test, the minimum chi-square test (MC), and two versions of the generalized method of moments (GMM) criterion test. Finally, we study the finite sample properties of effects estimators that show how changes in explanatory variables impact the dependent variable.     

A Comparison of the Cost-efficiencies of the Sequential, Group-sequential, and Variable-sample-size-sequential Probability Ratio Tests

Wald and Wolfowitz (1948) have shown that the Sequential Probability Ratio Test (SPRT) for deciding between two simple hypotheses is, under very restrictive conditions, optimal in three attractive senses. First, it can be a Bayes-optimal rule. Second, of all level α tests having the same power, the test with the smallest joint-expected number of observations is the SPRT, where this expectation is taken jointly with respect to both data and prior over the two hypotheses. Third, the level α test needing the fewest conditional-expected number of observat ions is the SPRT, where this expectation is now taken with respect to the data conditional on either hypothesis being true. Principal among the strong restrictions is that sampling can proceed only in a one-at-a-time manner. In this paper, we relax some of the conditions and show that there are sequential procedures that strictly dominate the SPRT in all three senses. We conclude that the third type of optimality occurs rarely and that decision-makers are better served by looking for sequential procedures that possess the first two types of optimality. By relaxing the one-at-a-time sampling restriction, we obtain optimal (in the first two senses) variable-s ample-size- sequential probability ratio tests.     

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.     

Testing linear hypotheses in high-dimensional regressions

For a multivariate linear model, Wilk's likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative hypothesis requires complex analytic approximations, and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say p≤20. On the other hand, assuming that the data dimension p as well as the number q of regression variables are fixed while the sample size n grows, several asymptotic approximations are proposed in the literature for Wilk's Λ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilk's test in a high-dimensional context, specifically assuming a high data dimension p and a large sample size n. Based on recent random matrix theory, the correction we propose to Wilk's test is asymptotically Gaussian under the null hypothesis and simulations demonstrate that the corrected LRT has very satisfactory size and power, surely in the large p and large n context, but also for moderately large data dimensions such as p=30 or p=50. As a byproduct, we give a reason explaining why the standard chi-square approximation fails for high-dimensional data. We also introduce a new procedure for the classical multiple sample significance test in multivariate analysis of variance which is valid for high-dimensional data.