Two Statistics for Unconditional Exact Tests in Matched-Pairs Design

Abstract

In a recent article Hsueh et al. (Hsueh, H.-M., Liu, J.-P., Chen, J. J. (2001 Hsueh, H.-M., Liu, J.-P. and Chen, J. J. 2001. Unconditional exact tests for equivalence or noninferiority for paired binary endpoints. Biometrics, 57: 478–483. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Unconditional exact tests for equivalence or noninferiority for paired binary endpoints. Biometrics 57:478–483.) considered unconditional exact tests for paired binary endpoints. They suggested two statistics one of which is based on the restricted maximum-likelihood estimator. Properties of these statistics and the related tests are treated in this article.     

MODIFICATION OF KELLY AND RICE METHOD FOR SYNERGISM TESTS

ABSTRACT

In this paper, we present a modified Kelly and Rice method for testing synergism. This approach is consistent with Berenbaum's1-3 Berenbaum, M.C. 1977. Synergism, Additivism, and Antagonism in Immuno-Suppression: A Critical Review. Clinical and Experimental Immunology, 28: 1–18. Berenbaum, M.C. 1985. The Expected Effect of a Combination of Agents: The General Solution. Journal of Theoretical Biology, 114: 413–431. Berenbaum, M.C. 1989. What is Synergism? Pharmacological Reviews. 41: 93–141.   framework for additivity. The delta method[4] Bishop, Y.M., Fienberg, S.E. and Holland, P.W. 1975. Discrete Multivariate Analysis: Theory and Practice Massachusetts: MIT Press.  [Google Scholar] is applied to obtain the estimated variance for the predicted additivity proportion. A Monte Carlo simulation study for the evaluation of the method's performance, i.e., global overall tests for synergism, is also discussed. Kelly and Rice[5] Kelly, C. and Rice, J. 1990. Monotone Smoothing with Application to Dose-Response Curves and the Assessment of Synergism. Biometrics, 46: 1071–1085. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] do not provide a correct test statistic because the variance is underestimated. Hence, the performance of the Kelly–Rice[5] Kelly, C. and Rice, J. 1990. Monotone Smoothing with Application to Dose-Response Curves and the Assessment of Synergism. Biometrics, 46: 1071–1085. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] method is generally anti-conservative, based on the simulation findings. In addition, the overall test of synergism with χ²(r) from the modified Kelly and Rice method for larger sample sizes is better than that with χ²(1) from the modified Kelly and Rice method.     

Maximum Value of Hotelling's T 2 Statistics Based on the Successive Differences Covariance Matrix Estimator

In statistical process control applications, the multivariate T ² control chart based on Hotelling's T ² statistic is useful for detecting the presence of special causes of variation. In particular, use of the T ² statistic based on the successive differences covariance matrix estimator has been shown to be very effective in detecting the presence of a sustained step or ramp shift in the mean vector. However, the exact distribution of this statistic is unknown. In this article, we derive the maximum value of the T ² statistic based on the successive differences covariance matrix estimator. This distributional property is crucial for calculating an approximate upper control limit of a T ² control chart based on successive differences, as described in Williams et al. (2006 Williams , J. D. , Woodall , W. H. , Birch , J. B. , Sullivan , J. H. ( 2006 ). On the distribution of T ² statistics based on successive differences . J. Qual. Technol. 38 : 217 – 229 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

A Class of Improved Heteroskedasticity-Consistent Covariance Matrix Estimators

Abstract

The heteroskedasticity-consistent covariance matrix estimator proposed by White [White, H. A. (1980 White, H. A. 1980. Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48: 817–838. [Crossref], [Web of Science ®] , [Google Scholar]). Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48:817–838], also known as HC0, is commonly used in practical applications and is implemented into a number of statistical software. Cribari–Neto et al. [Cribari–Neto, F., Ferrari, S. L. P., Cordeiro, G. M. (2000 Cribari–Neto, F., Ferrari, S. L. P. and Cordeiro, G. M. 2000. Improved heteroscedasticity–consistent covariance matrix estimators. Biometrika, 87: 907–918. [Crossref], [Web of Science ®] , [Google Scholar]). Improved heteroscedasticity–consistent covariance matrix estimators. Biometrika 87:907–918] have developed a bias-adjustment scheme that delivers bias-corrected White estimators. There are several variants of the original White estimator that are also commonly used by practitioners. These include the HC1, HC2, and HC3 estimators, which have proven to have superior small-sample behavior relative to White's estimator. This paper defines a general bias-correction mechamism that can be applied not only to White's estimator, but to variants of this estimator as well, such as HC1, HC2, and HC3. Numerical evidence on the usefulness of the proposed corrections is also presented. Overall, the results favor the sequence of improved HC2 estimators.     

A Variance Estimator for Systematic Sampling from a Deliberately Ordered Population

ABSTRACT

The systematic sampling (SYS) design (Madow and Madow, 1944 Madow , L. H. , Madow , W. G. ( 1944 ). On the theory of systematic sampling . Ann. Math. Statist. 15 : 1 – 24 .[Crossref] , [Google Scholar]) is widely used by statistical offices due to its simplicity and efficiency (e.g., Iachan, 1982 Iachan , R. ( 1982 ). Systematic sampling a critical review . Int. Statist. Rev. 50 : 293 – 303 .[Crossref], [Web of Science ®] , [Google Scholar]). But it suffers from a serious defect, namely, that it is impossible to unbiasedly estimate the sampling variance (Iachan, 1982 Iachan , R. ( 1982 ). Systematic sampling a critical review . Int. Statist. Rev. 50 : 293 – 303 .[Crossref], [Web of Science ®] , [Google Scholar]) and usual variance estimators (Yates and Grundy, 1953 Yates , F. , Grundy , P. M. ( 1953 ). Selection without replacement from within strata with probability proportional to size . J. Roy. Statist. Soc. Series B 1 : 253 – 261 . [Google Scholar]) are inadequate and can overestimate the variance significantly (Särndal et al., 1992 Särndal , C. E. , Swenson , B. , Wretman , J. H. ( 1992 ). Model Assisted Survey Sampling . New York : Springer-Verlag , Ch. 3 .[Crossref] , [Google Scholar]). We propose a novel variance estimator which is less biased and that can be implemented with any given population order. We will justify this estimator theoretically and with a Monte Carlo simulation study.     

SIMULTANEOUS CONFIDENCE REGIONS IN AN EXTENDED GROWTH CURVE MODEL WITH k HIERARCHICAL WITHIN-INDIVIDUALS DESIGN MATRICES

ABSTRACT

We consider an extended growth curve model with k hierarchical within-individuals design matrices. The model includes the one whose mean structure consists of polynomial growth curves with k different degrees. First we propose certain simple estimators of the mean and covariance parameters which are closely related to the MLE's. Using these estimators we construct simultaneous confidence regions for each or all of k growth curves which is an extension of Fujikoshi.[2] Fujikoshi, Y. 1999. Simultaneous Confidence Intervals in an Extended Growth Curves. Commun. Statist.–-Theor. Meth., 28: 671–682. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] A numerical example with k = 3 is also given.     

A Nonparametric Shewhart-Type Synthetic Control Chart

In this article, we provide a nonparametric Shewhart-type synthetic control chart based on the signed-rank statistic to monitor shifts in the known in-control process median. The synthetic control chart is a combination of a signed-rank chart due to Bakir (2004 Bakir , S. T. ( 2004 ). A distribution-free Shewhart quality control chart based on signed-ranks . Quality Engineering 16 : 613 – 623 .[Taylor & Francis Online] , [Google Scholar]) and a conforming run length chart due to Bourke (1991 Bourke , P. D. ( 1991 ). Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection . Journal of Quality Technology 23 : 225 – 238 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The operation and design of the chart are discussed and the performance of the chart has been studied. The chart has an attractive average run length behavior as compared to the parametric control chart for a class of symmetric continuous process distributions. The proposed chart performs better than the nonparametric signed-rank chart given by Bakir (2004 Bakir , S. T. ( 2004 ). A distribution-free Shewhart quality control chart based on signed-ranks . Quality Engineering 16 : 613 – 623 .[Taylor & Francis Online] , [Google Scholar]) and Chakraborti and Eryilmaz (2007 Chakraborti , S. , Eryilmaz , S. (2007). A nonparametric Shewhart-type signed-rank control chart based on runs. Communications in Statistics—Simulation and Computation 36:335–356.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

A side sensitive modified group runs control chart to detect shifts in the process mean

Gadre and Rattihalli [5 Gadre, M. P. and Rattihalli, R. N. 2006. Modified group runs control charts to detect increases in fraction non-conforming and shifts in the process mean. Commun. Stat. Simul. Comput., 35: 225–240. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]] have introduced the Modified Group Runs (MGR) control chart to identify the increases in fraction non-conforming and to detect shifts in the process mean. The MGR chart reduces the out-of-control average time-to-signal (ATS), as compared with most of the well-known control charts. In this article, we develop the Side Sensitive Modified Group Runs (SSMGR) chart to detect shifts in the process mean. With the help of numerical examples, it is illustrated that the SSMGR chart performs better than the Shewhart's X¯ chart, the synthetic chart [12 Wu, Z. and Spedding, T. A. 2000. A synthetic control chart for detecting small shifts in the process mean. J. Qual. Technol., 32: 32–38. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]], the Group Runs chart [4 Gadre, M. P. and Rattihalli, R. N. 2004. A group runs control chart for detecting shifts in the process mean. Econ. Qual. Control, 19: 29–43. [Crossref] [Google Scholar]], the Side Sensitive Group Runs chart [6 Gadre, M. P. and Rattihalli, R. N. 2007. A side sensitive group runs control chart for detecting shifts in the process mean. Stat. Methods Appl., 16: 27–37. [Crossref], [Web of Science ®] [Google Scholar]], as well as the MGR chart [5 Gadre, M. P. and Rattihalli, R. N. 2006. Modified group runs control charts to detect increases in fraction non-conforming and shifts in the process mean. Commun. Stat. Simul. Comput., 35: 225–240. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]]. In some situations it is also superior to the Cumulative Sum chart p9 Page, E. S. 1954. Continuous inspection schemes. Biometrika, 41: 100–114. [Crossref], [Web of Science ®] [Google Scholar]] and the exponentially weighed moving average chart [10 Roberts, S. W. 1959. Control chart tests based on geometric moving averages. Technometrics, 1: 239–250. [Taylor & Francis Online] [Google Scholar]]. In the steady state also, its performance is better than the above charts.     

EVALUATION OF POWER OF TESTS INVOLVING COVARIATES UNDER NONCONSTANT VARIANCE

In many experiments where pre-treatment and post-treatment measurements are taken, investigators wish to determine if there is a difference between two treatment groups. For this type of data, the post-treatment variable is used as the primary comparison variable and the pre-treatment variable is used as a covariate. Although most of the discussion in this paper is written with the pre-treatment variable as the covariate the results are applicable to other choices of a covariate. Tests based on residuals have been proposed as alternatives to the usual covariance methods. Our objective is to investigate how the powers of these tests are affected when the conditional variance of the post-treatment variable depends on the magnitude of the pre-treatment variable. In particular, we investigate two cases. [1] Crager, Michael R. 1987. Analysis of Covariance in Parallel-Group Clinical Trials With Pretreatment Baselines. Biometrics, 43: 895–901. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] The conditional variance of the post-treatment variable gradually increases as the magnitude of the pre-treatment variable increases. (In many biological models this is the case.) [2] Knoke, James D. 1991. Nonparametric Analysis of Covariance for Comparing Change in Randomized Studies with Baseline Values Subject to Error. Biometrics, 47: 523–533. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] The conditional variance of the post-treatment variable is dependent upon natural or imposed subgroups contained within the pre-treatment variable. Power comparisons are made using Monte Carlo techniques.     

A Bayesian Approach for Nonlinear Regression Models with Continuous Errors

Abstract

In this paper we develop a Bayesian analysis for the nonlinear regression model with errors that follow a continuous autoregressive process. In this way, unequally spaced observations do not present a problem in the analysis. We employ the Gibbs sampler, (see Gelfand, A., Smith, A. (1990 Gelfand, A. and Smith, A. 1990. Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc., 85: 398–409. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Sampling based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85:398–409.), as the foundation for making Bayesian inferences. We illustrate these Bayesian inferences with an analysis of a real data-set. Using these same data, we contrast the Bayesian approach with a generalized least squares technique.     

An Estimator for the Survival Function on the Basis of Observable Censoring Times

Abstract

Chiu [Chiu, S. N. (1999 Chiu, S. N. 1999. An unbiased estimator for the survival function of censored data. Commun. Statist. - Theory Meth., 28(9): 2249–2260. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). An unbiased estimator for the survival function of censored data. Commun. Statist. - Theory Meth. 28(9):2249–2260.] proposed a nonparametric estimator for the survival function which is based on observable censoring times in the general censoring model. His estimator is less efficient than the Product-Limit estimator. Considering an informative censoring model this drawback can partially be overcome. This is shown by a nonparametric, uniformly consistent estimator based on observable censoring times within the simple Koziol–Green model. Some asymptotic properties of the new estimator are investigated and it is compared with the well-known ACL-estimator.     

The Product-Limit Estimate as an Inverse-Probability-Weighted Average

Abstract

For randomly censored data, (Satten, G. A., Datta S. (2001 Satten, G. A. and Datta, S. 2001. The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. Ass., 55: 207–210. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. Ass. 55:207–210) showed that the Kaplan–Meier estimator (product-limit estimator (PLE)) can be expressed as an inverse-probability-weighted average. In this article, we consider the other two PLEs: the truncation PLE and the censoring-truncation PLE. For the data subject to left-truncation or both left-truncation and right-censoring, it is shown that these two PLEs can be expressed as inverse-probability-weighted averages.     

A Two-Stage Sequential Sampling Scheme for Warner's Randomized Response Model

Abstract

In this article we revisit Warner's (Warner, S. L. (1965 Warner, S. L. 1965. Randomized Response: a survey technique for elimination evasive answer bias. Journal of the American Statistical Association, 60: 63–69. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Randomized response: a survey technique for elimination evasive answer bias. Journal of the American Statistical Association 60:63–69) randomize response model for estimating the proportion of “sensitive” attributed in a population and propose a two-stage sequential sampling procedure for it. We show that the new procedure can potentially reduce, on the average, the number of interviewees surveyed in the study while allowing instances with smaller error of estimation. The properties and some of the attractive features of the randomized response two-stage sequential sampling estimation procedure is discussed as illustrated.     

Skewing and Generalized Jackknifing in Kernel Density Estimation

Abstract

Kernel methods are very popular in nonparametric density estimation. In this article we suggest a simple estimator which reduces the bias to the fourth power of the bandwidth, while the variance of the estimator increases only by at most a moderate constant factor. Our proposal turns out to be a fourth order kernel estimator and may be regarded as a new version of the generalized jackknifing approach (Schucany W. R., Sommers, J. P. (1977 Schucany, W. R. and Sommers, J. P. 1977. Improvement of kernel type estimators. Journal of the American Statistical Association, 72: 420–423. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Improvement of Kernal type estimators. Journal of the American Statistical Association 72:420–423.) applied to kernel density estimation.     

Uniqueness of the Maximum Likelihood Estimate of the Weibull Distribution Tampered Failure Rate Model

Abstract

Bhattacharyya and Soejoeti (Bhattacharyya, G. K., Soejoeti, Z. A. (1989 Bhattacharyya, G. K. and Soejoeti, Z. A. 1989. Tampered failure rate model for step-stress accelerated life test. Commun. Statist.—Theory Meth., 18(5): 1627–1643. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Tampered failure rate model for step-stress accelerated life test. Commun. Statist.—Theory Meth. 18(5):1627–1643.) pro- posed the TFR model for step-stress accelerated life tests. Under the TFR model, this article proves that the maximum likelihood estimate of the shape parameters is unique for the Weibull distribution in a multiple step-stress accelerated life test, and investigates the accuracy of the maximum likelihood estimate using the Monte-Carlo simulation.     

The Research on Two Kinds of Restricted Biased Estimators Based on Mean Squared Error Matrix

Sakall?oglu et al. (2001 Sakall?oglu , Kaç?ranlar , Akdeniz ( 2001 ). Mean squared error comparisons of some biased estimators . Commun. Statist. Theor. Meth. 30 : 347 – 361 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) dealt with the comparisons among the ridge estimator, Liu estimator, and iteration estimator. Akdeniz and Erol (2003 Akdeniz , F. , Erol , H. ( 2003 ). Mean squared error matrix comparisons of some biased estimators in linear regression . Commun. Statist. Theor. Meth. 32 : 2389 – 2413 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) have compared the (almost unbiased) generalized ridge regression estimator with the (almost unbiased) generalized Liu estimator in the matrix mean squared error sense. In this article, we study the ridge estimator and Liu estimator with respect to linear equality restriction, and establish some sufficient conditions for the superiority of the restricted ridge estimator over the restricted Liu estimator and the superiority of the restricted Liu estimator over the restricted ridge estimator under mean squared error matrix, respectively. Furthermore, we give a numerical example.     

Inference Using Latent Variables for Mixtures of Distributions for Censored Data with Partial Identification

Abstract

In this article two methods are proposed to make inferences about the parameters of a finite mixture of distributions in the context of partially identifiable censored data. The first method focuses on a mixture of location and scale models and relies on an asymptotic approximation to a suitably constructed augmented likelihood; the second method provides a full Bayesian analysis of the mixture based on a Gibbs sampler. Both methods make explicit use of latent variables and provide computationally efficient procedures compared to other methods which deal directly with the likelihood of the mixture. This may be crucial if the number of components in the mixture is not small. Our proposals are illustrated on a classical example on failure times for communication devices first studied by Mendenhall and Hader (Mendenhall, W., Hader, R. J. (1958 Mendenhall, W. and Hader, R. J. 1958. Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika, 45: 504–520. [Crossref], [Web of Science ®] , [Google Scholar]). Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45:504–520.). In addition, we study the coverage of the confidence intervals obtained from each of the methods by means of a small simulation exercise.     

Nonparametric Bivariate Estimation with Left-Truncated and Right-Censored Data

ABSTRACT

In this article we consider estimating the bivariate survival function observations where one of the components is subject to left truncation and right censoring and the other is subject to right censoring only. Two types of nonparametric estimators are proposed. One is in the form of inverse-probability-weighted average (Satten and Datta, 2001 Satten , G. A. , Datta , S. ( 2001 ). The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average . Amer. Statist. 55 : 207 – 210 . [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and the other is a generalization of Dabrowska's 1988 Dabrowska , D. M. ( 1988 ). Kaplan–Meier estimate on the plane . Ann. Statist. 18 : 308 – 325 . [Google Scholar] estimator. The two are then compared based on their empirical performances.     

RANDOM PERMUTATION TESTING IN MULTIPLE LINEAR REGRESSION

We developed an alternative random permutation testing method for multiple linear regression, which is an improvement over the existing one proposed by [1] Kennedy, P. E. 1995. Randomization tests in econometrics. Journal of Business and Economic Statistics, 13: 85–94. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] or [2] Freedman, D. and Lane, D. 1983. A nonstochastic interpretation of reported significance levels. Journal of Business and Economic Statistics, 1: 292–298. [Taylor & Francis Online] , [Google Scholar].     

Asymptotic Skewness in Exponential Family Nonlinear Models

In this article, we give an asymptotic formula of order n ^?1/2, where n is the sample size, for the skewness of the distributions of the maximum likelihood estimates of the pa-ra-meters in exponencial family nonlinear models. We generalize the result by Cordeiro and Cordeiro (2001 Cordeiro , H. H. , Cordeiro , G. M. ( 2001 ). Skewness for parameters in generalized linear models . Commun. Statist. Theor. Meth. 30 : 1317 – 1334 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The formula is given in matrix notation and is very suitable for computer implementation and to obtain closed form expressions for a great variety of models. Some special cases and two applications are discussed.