Comparison of Wald , Score , and Likelihood Ratio Tests for Response Adaptive Designs

Data collected from response adaptive designs are dependent. Traditional statistical methods need to be justified for the use in response adaptive designs. This paper generalizes the Rao’s score test to response adaptive designs and introduces a generalized score statistic. Simulation is conducted to compare the statistical powers of the Wald, the score, the generalized score and the likelihood ratio statistics. The overall statistical power of the Wald statistic is better than the score, the generalized score and the likelihood ratio statistics for small to medium sample sizes. The score statistic does not show good sample properties for adaptive designs and the generalized score statistic is better than the score statistic under the adaptive designs considered. When the sample size becomes large, the statistical power is similar for the Wald, the sore, the generalized score and the likelihood ratio test statistics. MSC: 62L05, 62F03

[1]  Calyampudi R. Rao.  Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  Xikui Wang,et al.  Asymptotically efficient estimation in response adaptive trials , 2008 .

[3]  Connie Page,et al.  An adaptive randomized design with application to estimation , 2001 .

[4]  G. Molenberghs,et al.  Likelihood Ratio, Score, and Wald Tests in a Constrained Parameter Space , 2007 .

[5]  Connie Page,et al.  Estimation after adaptive allocation , 2000 .

[6]  R. Mukerjee.  Comparison of tests in the multiparameter case. part I. second-order power , 1990 .

[7]  Comparison of tests in the multiparameter case. part II. a third-order optimality property of Rao's test , 1990 .

[8]  Pranab Kumar Sen,et al.  Large Sample Methods in Statistics: An Introduction with Applications , 1993 .

[9]  R. Gelber,et al.  Reduction of maternal-infant transmission of human immunodeficiency virus type 1 with zidovudine treatment. Pediatric AIDS Clinical Trials Group Protocol 076 Study Group. , 1994, The New England journal of medicine.

[10]  M. Woodroofe,et al.  Approximate bias calculations for sequentially designed experiments , 1998 .

[11]  Xikui Wang,et al.  Goodness-of-fit test for response adaptive clinical trials , 2007 .

[12]  Anastasia Ivanova,et al.  A play-the-winner-type urn design with reduced variability , 2003 .

[13]  John C. W. Rayner,et al.  The asymptotically optimal tests , 1997 .

[14]  J. Ghosh.  Higher order asymptotics for the likelihood ratio, Rao's and Wald's tests , 1991 .

[15]  A. Wald.  Tests of statistical hypotheses concerning several parameters when the number of observations is large , 1943 .

[16]  J. Ghosh,et al.  Test statistics arising from quasi likelihood: Bartlett adjustment and higher-order power , 2001 .

[17]  Hung T. Nguyen,et al.  Asymptotic Normality of Maximum Likelihood Estimators , 1989 .

[18]  William F. Rosenberger,et al.  Asymptotically best response-adaptive randomization procedures , 2006 .

[19]  C. Heyde.  Quasi-likelihood and its application : a general approach to optimal parameter estimation , 1998 .

[20]  S. D. Silvey,et al.  The Lagrangian Multiplier Test , 1959 .

[21]  Anastasia Ivanova,et al.  BIAS CALCULATIONS FOR ADAPTIVE URN DESIGNS , 2001 .

[22]  Anil K. Bera,et al.  Rao's score, Neyman's C(α) and Silvey's LM tests: an essay on historical developments and some new results , 2001 .

[23]  Calyampudi R. Rao,et al.  Score Test: Historical Review and Recent Developments , 2005 .

[24]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[25]  Bing Li.  Sensitivity of Rao's score test, the Wald test and the likelihood ratio test to nuisance parameters , 2001 .

[26]  Masanobu Taniguchi,et al.  Third-order asymptotic properties of a class of test statistics under a local alternative , 1991 .

[27]  J. Aitchison,et al.  Maximum-Likelihood Estimation of Parameters Subject to Restraints , 1958 .

[28]  F. R.,et al.  WALD . LIKELIHOOD RATIO , AND LAGRANGE MULTiPLIER TESTS , 1999 .

[29]  N Stallard,et al.  Optimal Adaptive Designs for Binary Response Trials , 2001, Biometrics.

[30]  Calyampudi R. Rao,et al.  Comparison of LR, Score, and Wald Tests in a Non-IID Setting , 1997 .

[31]  Lee-Jen Wei,et al.  Play the winner for phase II/III clinical trials. , 1996, Statistics in medicine.

[32]  William F. Rosenberger,et al.  Optimality, Variability, Power , 2003 .

[33]  A. Buse.  The Likelihood Ratio, Wald, and Lagrange Multiplier Tests: An Expository Note , 1982 .

[34]  Yanqing Yi,et al.  Response Adaptive Designs with a Variance‐penalized Criterion , 2009, Biometrical journal. Biometrische Zeitschrift.

[35]  B. Sutradhar,et al.  Monte carlo comparison of wald's, likelihood ratio and rao's tests , 1993 .

[36]  T. Ferguson.  A Course in Large Sample Theory , 1996 .