Most powerful test sequences with early stopping options

Sequential likelihood ratio testing is found to be most powerful in sequential studies with early stopping rules when grouped data come from the one-parameter exponential family. First, to obtain this elusive result, the probability measure of a group sequential design is constructed with support for all possible outcome events, as is useful for designing an experiment prior to having data. This construction identifies impossible events that are not part of the support. The overall probability distribution is dissected into stage specific components. These components are sub-densities of interim test statistics first described by Armitage, McPherson and Rowe (1969) that are commonly used to create stopping boundaries given an α-spending function and a set of interim analysis times. Likelihood expressions conditional on reaching a stage are given to connect pieces of the probability anatomy together. The reduction of the support caused by the adoption of an early stopping rule induces sequential truncation (not nesting) in the probability distributions of possible events. Multiple testing induces mixtures on the adapted support. Even asymptotic distributions of inferential statistics are mixtures of truncated distributions. In contrast to the classical result on local asymptotic normality (Le Cam 1960), statistics that are asymptotically normal without stopping options have asymptotic distributions that are mixtures of truncated normal distributions under local alternatives with stopping options; under fixed alternatives, asymptotic distributions of test statistics are degenerate.

[1]  Tze Leung Lai,et al.  Power, sample size and adaptation considerations in the design of group sequential clinical trials , 2003 .

[2]  J. Haybittle,et al.  Repeated assessment of results in clinical trials of cancer treatment. , 1971, The British journal of radiology.

[3]  H. Luschgy,et al.  Stable Convergence and Stable Limit Theorems , 2015 .

[4]  N. Flournoy,et al.  Asymptotic properties of maximum likelihood estimators with sample size recalculation , 2019, Statistical Papers.

[5]  Heinz Schmidli,et al.  Sample size re‐estimation for clinical trials with longitudinal negative binomial counts including time trends , 2018, Statistics in medicine.

[6]  S. Pocock.  Group sequential methods in the design and analysis of clinical trials , 1977 .

[7]  V. Bening,et al.  On rate of convergence in distribution of asymptotically normal statistics based on samples of random size , 2012 .

[8]  B. Efron.  Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency) , 1975 .

[9]  S. Karlin,et al.  The Theory of Decision Procedures for Distributions with Monotone Likelihood Ratio , 1956 .

[10]  Michael R. Chernick,et al.  Statistical Monitoring of Clinical Trials: a Unified Approach , 2007, Technometrics.

[11]  N. Flournoy,et al.  Two-Stage Adaptive Optimal Design with Fixed First-Stage Sample Size , 2012 .

[12]  D. Blackwell.  Conditional Expectation and Unbiased Sequential Estimation , 1947 .

[13]  Y. Nikitin,et al.  Asymptotic Efficiency of Nonparametric Tests , 1995 .

[14]  Katharina Burger,et al.  Counting Processes And Survival Analysis , 2016 .

[15]  C. Mehta,et al.  Exact inference for adaptive group sequential designs , 2013, Statistics in medicine.

[16]  R. Simes,et al.  An improved Bonferroni procedure for multiple tests of significance , 1986 .

[17]  Chengqing Wu,et al.  ESTIMATION FOLLOWING A GROUP SEQUENTIAL TEST FOR DISTRIBUTIONS IN THE ONE-PARAMETER EXPONENTIAL FAMILY , 2006 .

[18]  Gerald S. Rogers,et al.  Mathematical Statistics: A Decision Theoretic Approach , 1967 .

[19]  H. F. Dodge,et al.  A method of sampling inspection , 1929 .

[20]  Walter Bartky,et al.  Multiple Sampling with Constant Probability , 1943 .

[21]  W. J. Hall,et al.  Unbiased estimation following a group sequential test , 1999 .

[22]  Andreas N. Philippou,et al.  Asymptotic Distribution of the Likelihood Function in the Independent not Identically Distributed Case , 1973 .

[23]  N. Flournoy,et al.  The Effects of Adaptation on Inference for Non-Linear Regression Models with Normal Errors , 2018, 1812.03970.

[24]  P. Armitage,et al.  Design and analysis of randomized clinical trials requiring prolonged observation of each patient. I. Introduction and design. , 1976, British Journal of Cancer.

[25]  D L DeMets,et al.  Interim analysis: the alpha spending function approach. , 1994, Statistics in medicine.

[26]  B. Turnbull,et al.  Group Sequential Methods with Applications to Clinical Trials , 1999 .

[27]  V. Ulyanov,et al.  Second-Order Chebyshev–Edgeworth and Cornish–Fisher Expansions for Distributions of Statistics Constructed from Samples with Random Sizes , 2020 .

[28]  Geert Molenberghs,et al.  Estimation After a Group Sequential Trial , 2014, Statistics in Biosciences.

[29]  P. O'Brien,et al.  A multiple testing procedure for clinical trials. , 1979, Biometrics.

[30]  Nancy Flournoy,et al.  A Birth and Death Urn for Ternary Outcomes: Stochastic Processes Applied to Urn Models , 2000 .

[31]  W. Rosenberger,et al.  Random norming aids analysis of non-linear regression models with sequential informative dose selection , 2019, Journal of Statistical Planning and Inference.

[32]  Tze Leung Lai,et al.  Asymptotic approximations for error probabilities of sequential or fixed sample size tests in exponential families , 1999 .

[33]  KyungMann Kim.  Group Sequential Methods with Applications to Clinical Trials , 2001 .

[34]  Eric V. Slud,et al.  Sequential Linear Rank Tests for Two-Sample Censored Survival Data , 1984 .

[35]  Haiying Wang,et al.  A new bounded log-linear regression model , 2014 .

[36]  W. Shih,et al.  A sample size adjustment procedure for clinical trials based on conditional power. , 2002, Biostatistics.

[37]  B. K. Ghosh,et al.  CURTAILED AND UNIFORMLY MOST POWERFUL SEQUENTIAL TESTS , 1980 .

[38]  Anastasios A. Tsiatis,et al.  Repeated Significance Testing for a General Class of Statistics Used in Censored Survival Analysis , 1982 .

[39]  N. Flournoy,et al.  Distribution theory following blinded and unblinded sample size re-estimation under parametric models , 2019, Commun. Stat. Simul. Comput..

[40]  Brent R Logan,et al.  A group sequential test for treatment effect based on the Fine–Gray model , 2018, Biometrics.

[41]  Christopher Jennison,et al.  Adaptive and nonadaptive group sequential tests , 2006 .

[42]  Tze Leung Lai,et al.  Weak Convergence of Time-Sequential Censored Rank Statistics with Applications to Sequential Testing in Clinical Trials , 1991 .

[43]  E. S. Pearson,et al.  On the Problem of the Most Efficient Tests of Statistical Hypotheses , 1933 .

[44]  William F. Rosenberger,et al.  A BIRTH AND DEATH URN FOR RANDOMIZED CLINICAL TRIALS: ASYMPTOTIC METHODS , 2016 .

[45]  Margaret Sullivan Pepe,et al.  Conditional estimation after a two-stage diagnostic biomarker study that allows early termination for futility. , 2012, Statistics in medicine.

[46]  Geert Molenberghs,et al.  On random sample size, ignorability, ancillarity, completeness, separability, and degeneracy: Sequential trials, random sample sizes, and missing data , 2014, Statistical methods in medical research.

[47]  Frank Bretz,et al.  Clinical Trials Impacted by the COVID-19 Pandemic: Adaptive Designs to the Rescue? , 2020, Statistics in biopharmaceutical research.

[48]  B. Gnedenko,et al.  Random Summation: Limit Theorems and Applications , 1996 .

[49]  James M. Robins,et al.  Semiparametric Efficiency and its Implication on the Design and Analysis of Group-Sequential Studies , 1997 .

[50]  N. Flournoy,et al.  Asymptotics in response-adaptive designs generated by a two-color, randomly reinforced urn , 2009, 0904.0350.

[51]  V. Korolev,et al.  From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes , 2019, Probability, Combinatorics and Control.

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

[53]  Le Cam,et al.  Locally asymptotically normal families of distributions : certain approximations to families of distributions & thier use in the theory of estimation & testing hypotheses , 1960 .

[54]  M. Crowder.  Maximum Likelihood Estimation for Dependent Observations , 1976 .

[55]  J. Andel.  Sequential Analysis , 2022, The SAGE Encyclopedia of Research Design.

[56]  P. Armitage,et al.  Repeated Significance Tests on Accumulating Data , 1969 .

[57]  A. Tamhane,et al.  Some Drawbacks of the Simes Test in the Group Sequential Setting , 2020 .

[58]  I. Marschner,et al.  Meta‐analysis of clinical trials with early stopping: an investigation of potential bias , 2013, Statistics in medicine.