Levene type tests for the ratio of two scales

Tests for the equality of variances are of interest in many areas such as quality control, agricultural production systems, experimental education, pharmacology, biology, as well as a preliminary to the analysis of variance, dose–response modelling or discriminant analysis. The literature is vast. Traditional non-parametric tests are due to Mood, Miller and Ansari–Bradley. A test which usually stands out in terms of power and robustness against non-normality is the W50 Brown and Forsythe [Robust tests for the equality of variances, J. Am. Stat. Assoc. 69 (1974), pp. 364–367] modification of the Levene test [Robust tests for equality of variances, in Contributions to Probability and Statistics, I. Olkin, ed., Stanford University Press, Stanford, 1960, pp. 278–292]. This paper deals with the two-sample scale problem and in particular with Levene type tests. We consider 10 Levene type tests: the W50, the M50 and L50 tests [G. Pan, On a Levene type test for equality of two variances, J. Stat. Comput. Simul. 63 (1999), pp. 59–71], the R-test [R.G. O'Brien, A general ANOVA method for robust tests of additive models for variances, J. Am. Stat. Assoc. 74 (1979), pp. 877–880], as well as the bootstrap and permutation versions of the W50, L50 and R tests. We consider also the F-test, the modified Fligner and Killeen [Distribution-free two-sample tests for scale, J. Am. Stat. Assoc. 71 (1976), pp. 210–213] test, an adaptive test due to Hall and Padmanabhan [Adaptive inference for the two-sample scale problem, Technometrics 23 (1997), pp. 351–361] and the two tests due to Shoemaker [Tests for differences in dispersion based on quantiles, Am. Stat. 49(2) (1995), pp. 179–182; Interquantile tests for dispersion in skewed distributions, Commun. Stat. Simul. Comput. 28 (1999), pp. 189–205]. The aim is to identify the effective methods for detecting scale differences. Our study is different with respect to the other ones since it is focused on resampling versions of the Levene type tests, and many tests considered here have not ever been proposed and/or compared. The computationally simplest test found robust is W50. Higher power, while preserving robustness, is achieved by considering the resampling version of Levene type tests like the permutation R-test (recommended for normal- and light-tailed distributions) and the bootstrap L50 test (recommended for heavy-tailed and skewed distributions). Among non-Levene type tests, the best one is the adaptive test due to Hall and Padmanabhan.     

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.     

A dual problem of calibration of design weights

In this note, a dual problem to the calibration of design weights of the Deville and Särndal [Calibration estimators in survey sampling, J. Amer. Statist. Assoc. 87 (1992), pp. 376–382] method has been considered. We conclude that the chi-squared distance between the design weights and the calibrated weights equals the square of the standardized Z-score obtained by the difference between the known population total of the auxiliary variable and its corresponding Horvitz and Thompson [A generalization of sampling without replacement from a finite universe, J. Amer. Statist. Assoc. 47 (1952), pp. 663–685] estimator divided by the sample standard deviation of the auxiliary variable to obtain the linear regression estimator in survey sampling.     

A new unrelated question randomized response model

In this paper, we extend the work of Gjestvang and Singh [A new randomized response model, J. R. Statist. Soc. Ser. B (Methodological) 68 (2006), pp. 523–530] to propose a new unrelated question randomized response model that can be used for any sampling scheme. The interesting thing is that the estimator based on one sample is free from the use of known proportion of an unrelated character, unlike Horvitz et al. [The unrelated question randomized response model, Social Statistics Section, Proceedings of the American Statistical Association, 1967, pp. 65–72], Greenberg et al. [The unrelated question randomized response model: Theoretical framework, J. Amer. Statist. Assoc. 64 (1969), pp. 520–539] and Mangat et al. [An improved unrelated question randomized response strategy, Calcutta Statist. Assoc. Bull. 42 (1992), pp. 167–168] models. The relative efficiency of the proposed model with respect to the existing competitors has been studied.     

Pseudo-Bayes and pseudo-empirical Bayes estimators in randomized response sampling

In this article, new pseudo-Bayes and pseudo-empirical Bayes estimators for estimating the proportion of a potentially sensitive attribute in a survey sampling have been introduced. The proposed estimators are compared with the recent estimator proposed by Odumade and Singh [Efficient use of two decks of cards in randomized response sampling, Comm. Statist. Theory Methods 38 (2009), pp. 439–446] and Warner [Randomized response: A survey technique for eliminating evasive answer bias, J. Amer. Statist. Assoc. 60 (1965), pp. 63–69].     

Finite sample performance of frequency- and time-domain tests for seasonal fractional integration

Testing the order of integration of economic and financial time series has become a conventional procedure prior to any modelling exercise. In this paper, we investigate and compare the finite sample properties of the frequency-domain tests proposed by Robinson [Efficient tests of nonstationary hypotheses, J. Amer. Statist. Assoc. 89(428) (1994), pp. 1420–1437] and the time-domain procedure proposed by Hassler, Rodrigues, and Rubia [Testing for general fractional integration in the time domain, Econometric Theory 25 (2009), pp. 1793–1828] when applied to seasonal data. The results presented are of empirical relevance as they provide some guidance regarding the finite sample properties of these tests.     

Mutiple three-decision procedure for comparing several exponential treatments with the best control

In this paper, the three-decision procedures to classify p treatments as better than or worse than one control, proposed for normal/symmetric probability models [Bohrer, Multiple three-decision rules for parametric signs. J. Amer. Statist. Assoc. 74 (1979), pp. 432–437; Bohrer et al., Multiple three-decision rules for factorial simple effects: Bonferroni wins again!, J. Amer. Statist. Assoc. 76 (1981), pp. 119–124; Liu, A multiple three-decision procedure for comparing several treatments with a control, Austral. J. Statist. 39 (1997), pp. 79–92 and Singh and Mishra, Classifying logistic populations using sample medians, J. Statist. Plann. Inference 137 (2007), pp. 1647–1657]; in the literature, have been extended to asymmetric two-parameter exponential probability models to classify p(p≥1) treatments as better than or worse than the best of q(q≥1) control treatments in terms of location parameters. Critical constants required for the implementation of the proposed procedure are tabulated for some pre-specified values of probability of no misclassification. Power function of the proposed procedure is also defined and a common sample size necessary to guarantee various pre-specified power levels are tabulated. Optimal allocation scheme is also discussed. Finally, the implementation of the proposed methodology is demonstrated through numerical example.     

Exact confidence intervals for the shape parameter of the gamma distribution

In this paper exact confidence intervals (CIs) for the shape parameter of the gamma distribution are constructed using the method of Bølviken and Skovlund [Confidence intervals from Monte Carlo tests. J Amer Statist Assoc. 1996;91:1071–1078]. The CIs which are based on the maximum likelihood estimator or the moment estimator are compared to bootstrap CIs via a simulation study.     

A block bootstrap comparison for sparse chains

In this paper we apply the sequential bootstrap method proposed by Collet et al. [Bootstrap Central Limit theorem for chains of infinite order via Markov approximations, Markov Processes and Related Fields 11(3) (2005), pp. 443–464] to estimate the variance of the empirical mean of a special class of chains of infinite order called sparse chains. For this process, we show that we are able to compute numerically the true value of the standard error with any fixed error.

Our main goal is to present a comparison, for sparse chains, among sequential bootstrap, the block bootstrap method proposed by Künsch [The jackknife and the Bootstrap for general stationary observations, Ann. Statist. 17 (1989), pp. 1217–1241] and improved by Liu and Singh [Moving blocks jackknife and Bootstrap capture week dependence, in Exploring the limits of the Bootstrap, R. Lepage and L. Billard, eds., Wiley, New York, 1992, pp. 225–248] and the bootstrap method proposed by Bühlmann [Blockwise bootstrapped empirical process for stationary sequences, Ann. Statist. 22 (1994), pp. 995–1012].     

Local polynomial quasi-likelihood regression with truncated and dependent data

The local polynomial quasi-likelihood estimation has several good statistical properties such as high minimax efficiency and adaptation of edge effects. In this paper, we construct a local quasi-likelihood regression estimator for a left truncated model, and establish the asymptotic normality of the proposed estimator when the observations form a stationary and α-mixing sequence, such that the corresponding result of Fan et al. [Local polynomial kernel regression for generalized linear models and quasilikelihood functions, J. Amer. Statist. Assoc. 90 (1995), pp. 141–150] is extended from the independent and complete data to the dependent and truncated one. Finite sample behaviour of the estimator is investigated via simulations too.     

Doubly robust empirical likelihood inference in covariate-missing data problems

Missing covariate data occurs often in regression analysis. We study methods for estimating the regression coefficients in an assumed conditional mean function when some covariates are completely observed but other covariates are missing for some subjects. We adopt the semiparametric perspective of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866] on regression analyses with missing covariates, in which they pioneered the use of two working models, the working propensity score model and the working conditional score model. A recent approach to missing covariate data analysis is the empirical likelihood method of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503], which effectively combines unbiased estimating equations. In this paper, we consider an alternative likelihood approach based on the full likelihood of the observed data. This full likelihood-based method enables us to generate estimators for the vector of the regression coefficients that are (a) asymptotically equivalent to those of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the working propensity score model is correctly specified, and (b) doubly robust, like the augmented inverse probability weighting (AIPW) estimators of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Am Statist Assoc. 1994;89:846–866]. Thus, the proposed full likelihood-based estimators improve on the efficiency of the AIPW estimators when the working propensity score model is correct but the working conditional score model is possibly incorrect, and also improve on the empirical likelihood estimators of Qin, Zhang and Leung [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the reverse is true, that is, the working conditional score model is correct but the working propensity score model is possibly incorrect. In addition, we consider a regression method for estimation of the regression coefficients when the working conditional score model is correctly specified; the asymptotic variance of the resulting estimator is no greater than the semiparametric variance bound characterized by the theory of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866]. Finally, we compare the finite-sample performance of various estimators in a simulation study.     

A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.     

Bayesian meta-analysis using skewed elliptical distributions

Meta-analysis refers to a quantitative method for combining results from independent studies in order to draw overall conclusions. We consider hierarchical models including selection models under a skewed heavy tailed error distribution proposed originally by Chen, Dey, and Shao [M. H. Chen, D. K. Dey, Q. M. Shao, A new skewed link model for dichotomous quantal response data, J. Amer. Statist. Assoc. 94 (1983), pp. 1172–1186.] and Branco and Dey [D. Branco and D.K. Dey, A general class of multivariate skew-elliptical distributions, J. Multivariate Anal. 79, pp. 99–113.]. These rich classes of models combine the information of independent studies, allowing investigation of variability both between and within studies and incorporating weight functions. We constructed a detailed computational scheme under skewed normal and skewed Student's t distribution using the MCMC method. Bayesian model selection was conducted by Bayes factor under a different skewed error. Finally, we illustrated our methodology using a real data example taken from Johnson [M.F. Johnson, Comparative efficacy of Naf and SMFP dentifrices in caries prevention: a meta-analysis overview, J Eur. Organ. Caries Res. 27 (1993), pp. 328–336.].     

A new estimator of entropy and its application in testing normality

In this paper, we introduce a new estimator of entropy of a continuous random variable. We compare the proposed estimator with the existing estimators, namely, Vasicek [A test for normality based on sample entropy, J. Roy. Statist. Soc. Ser. B 38 (1976), pp. 54–59], van Es [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Correa [A new estimator of entropy, Commun. Statist. Theory and Methods 24 (1995), pp. 2439–2449] and Wieczorkowski-Grzegorewski [Entropy estimators improvements and comparisons, Commun. Statist. Simulation and Computation 28 (1999), pp. 541–567]. We next introduce a new test for normality. By simulation, the powers of the proposed test under various alternatives are compared with normality tests proposed by Vasicek (1976) and Esteban et al. [Monte Carlo comparison of four normality tests using different entropy estimates, Commun. Statist.–Simulation and Computation 30(4) (2001), pp. 761–785].     

Maximin clusters for nonreplicated multiresponse lack of fit tests

Khuri (Technometrics 27 (1985) 213) and Levy and Neill (Comm. Statist. A 19 (1990) 1987) presented regression lack of fit tests for multiresponse data with replicated observations available at points in the experimental region, thereby extending the classical univariate lack of fit test given by Fisher (J. Roy. Statist. Soc. 85 (1922) 597). In this paper, multivariate tests for lack of fit in a linear multiresponse model are derived for the common circumstance in which replicated observations are not obtained. The tests are based on the union–intersection principle, and provide multiresponse extensions of the univariate tests for between- and within-cluster lack of fit introduced by Christensen (Ann. of Statist. 17 (1989) 673; J. Amer. Statist. Assoc. 86 (1991) 752). Since the properties of these tests depend on the choice of multivariate clusters of the observations, a multiresponse generalization of the maximin power clustering criterion given by Miller, Neill and Sherfey (Ann. of Statist. 26 (1998) 1411; J. Amer. Statist. Assoc. 94 (1999) 610) is also developed.     

A comparative study of doubly robust estimators of the mean with missing data

Doubly robust (DR) estimators of the mean with missing data are compared. An estimator is DR if either the regression of the missing variable on the observed variables or the missing data mechanism is correctly specified. One method is to include the inverse of the propensity score as a linear term in the imputation model [D. Firth and K.E. Bennett, Robust models in probability sampling, J. R. Statist. Soc. Ser. B. 60 (1998), pp. 3–21; D.O. Scharfstein, A. Rotnitzky, and J.M. Robins, Adjusting for nonignorable drop-out using semiparametric nonresponse models (with discussion), J. Am. Statist. Assoc. 94 (1999), pp. 1096–1146; H. Bang and J.M. Robins, Doubly robust estimation in missing data and causal inference models, Biometrics 61 (2005), pp. 962–972]. Another method is to calibrate the predictions from a parametric model by adding a mean of the weighted residuals [J.M Robins, A. Rotnitzky, and L.P. Zhao, Estimation of regression coefficients when some regressors are not always observed, J. Am. Statist. Assoc. 89 (1994), pp. 846–866; D.O. Scharfstein, A. Rotnitzky, and J.M. Robins, Adjusting for nonignorable drop-out using semiparametric nonresponse models (with discussion), J. Am. Statist. Assoc. 94 (1999), pp. 1096–1146]. The penalized spline propensity prediction (PSPP) model includes the propensity score into the model non-parametrically [R.J.A. Little and H. An, Robust likelihood-based analysis of multivariate data with missing values, Statist. Sin. 14 (2004), pp. 949–968; G. Zhang and R.J. Little, Extensions of the penalized spline propensity prediction method of imputation, Biometrics, 65(3) (2008), pp. 911–918]. All these methods have consistency properties under misspecification of regression models, but their comparative efficiency and confidence coverage in finite samples have received little attention. In this paper, we compare the root mean square error (RMSE), width of confidence interval and non-coverage rate of these methods under various mean and response propensity functions. We study the effects of sample size and robustness to model misspecification. The PSPP method yields estimates with smaller RMSE and width of confidence interval compared with other methods under most situations. It also yields estimates with confidence coverage close to the 95% nominal level, provided the sample size is not too small.     

Point and confidence interval estimates for a global maximum via extreme value theory

The aim of this paper is to provide some practical aspects of point and interval estimates of the global maximum of a function using extreme value theory. Consider a real-valued function f:D→? defined on a bounded interval D such that f is either not known analytically or is known analytically but has rather a complicated analytic form. We assume that f possesses a global maximum attained, say, at u*∈D with maximal value x*=max _u  f(u)?f(u*). The problem of seeking the optimum of a function which is more or less unknown to the observer has resulted in the development of a large variety of search techniques. In this paper we use the extreme-value approach as appears in Dekkers et al. [A moment estimator for the index of an extreme-value distribution, Ann. Statist. 17 (1989), pp. 1833–1855] and de Haan [Estimation of the minimum of a function using order statistics, J. Amer. Statist. Assoc. 76 (1981), pp. 467–469]. We impose some Lipschitz conditions on the functions being investigated and through repeated simulation-based samplings, we provide various practical interpretations of the parameters involved as well as point and interval estimates for x*.     

Some estimators and tests for accelerated hazards model using weighted cumulative hazard difference

For a censored two-sample problem, Chen and Wang [Y.Q. Chen and M.-C. Wang, Analysis of accelerated hazards models, J. Am. Statist. Assoc. 95 (2000), pp. 608–618] introduced the accelerated hazards model. The scale-change parameter in this model characterizes the association of two groups. However, its estimator involves the unknown density in the asymptotic variance. Thus, to make an inference on the parameter, numerically intensive methods are needed. The goal of this article is to propose a simple estimation method in which estimators are asymptotically normal with a density-free asymptotic variance. Some lack-of-fit tests are also obtained from this. These tests are related to Gill–Schumacher type tests [R.D. Gill and M. Schumacher, A simple test of the proportional hazards assumption, Biometrika 74 (1987), pp. 289–300] in which the estimating functions are evaluated at two different weight functions yielding two estimators that are close to each other. Numerical studies show that for some weight functions, the estimators and tests perform well. The proposed procedures are illustrated in two applications.     

Rank tests for patterned alternatives in two-way non-parametric analysis of variance

We consider the problem of testing against trend and umbrella alternatives, with known and unknown peak, in two-way layouts with fixed effects. We consider the non-parametric two-way layout ANOVA model of Akritas and Arnold (J. Amer. Statist. Assoc. 89 (1994) 336), and use the non-parametric formulation of patterned alternatives introduced by Akritas and Brunner (Research Developments in Probability and Statistics: Festschrift in honor of Madan L. Puri, VSP, Zeist, The Netherlands, 1996, pp. 277–288). The hypotheses of no main effects and of no simple effects are both considered. New rank test statistics are developed to specifically detect these types of alternatives. For main effects, we consider two types of statistics, one using weights similar to Hettmansperger and Norton (J. Amer. Statist. Assoc. 82 (1987) 292) and one with weights which maximize the asymptotic efficacy. For simple effects, we consider in detail only statistics to detect trend or umbrella patterns with known peaks, and refer to Callegari (Ph.D. Thesis, University of Padova, Italy) for a discussion about possible statistics for umbrella alternatives with unknown peaks. The null asymptotic distributions of the new statistics are derived. A number of simulation studies investigate their finite sample behaviors and compare the achieved alpha levels and power with some alternative procedures. An application to data used in a clinical study is presented to illustrate how to utilize some of the proposed tests for main effects.     

Approximations to the p-values of tests for a change-point under non-standard conditions

Three test statistics for a change-point in a linear model, variants of those considered by Andrews and Ploberger [Optimal tests when a nusiance parameter is present only under the alternative. Econometrica. 1994;62:1383–1414]: the sup-likelihood ratio (LR) statistic; a weighted average of the exponential of LR-statistics and a weighted average of LR-statistics, are studied. Critical values for the statistics with time trend regressors, obtained via simulation, are found to vary considerably, depending on conditions on the error terms. The performance of the bootstrap in approximating p-values of the distributions is assessed in a simulation study. A sample approximation to asymptotic analytical expressions extending those of Kim and Siegmund [The likelihood ratio test for a change-point in simple linear regression. Biometrika. 1989;76:409–423] in the case of the sup-LR test is also assessed. The approximations and bootstrap are applied to the Quandt data [The estimation of a parameter of a linear regression system obeying two separate regimes. J Amer Statist Assoc. 1958;53:873–880] and real data concerning a change-point in oxygen uptake during incremental exercise testing and the bootstrap gives reasonable results.