Model choice in separate families: A comparison between the FBST and the Cox test

ABSTRACT The important problem of discriminating between separate families of distributions is the theme of this work. The Bayesian significance test, FBST, is compared with the celebrated Cox test. The three families most used in survival analysis, lognormal, gamma and Weibull, are considered for the discrimination. A convex combination—with unknown weights—of the three densities is used for this discrimination. After these weights have been estimated, the one with the highest value indicates the best statistical model among the three. Another important feature considered is the parameterization used. All the three densities are written as a function of the common population mean and variance. Including the weights, the number of parameters is reduced from eight (two of each density and two of the convex combination) to four (two from the common mean and variance plus two of the weights). Some numerical results from simulations are given. In these simulations, the results of FBST are compared with those obtained with the Cox test. Two real examples properly illustrate the procedures.

[1]  Gordon Johnston,et al.  Statistical Models and Methods for Lifetime Data , 2003, Technometrics.

[2]  Julio Michael Stern,et al.  A straightforward multiallelic significance test for the Hardy-Weinberg equilibrium law , 2009, Genetics and molecular biology.

[3]  Carlos Alberto de Bragança Pereira,et al.  Can a Significance Test Be Genuinely Bayesian , 2008 .

[4]  C. Robert,et al.  Testing hypotheses via a mixture estimation model , 2014, 1412.2044.

[5]  F. Proschan,et al.  Piecewise exponential estimator of the survivor function , 1991 .

[6]  B. Pereira.  Empirical comparisons of some tests of separate families of hypotheses , 1978 .

[7]  Julio Michael Stern,et al.  Bayesian evidence test for precise hypotheses , 2003 .

[8]  C. M. Reeves,et al.  Function minimization by conjugate gradients , 1964, Comput. J..

[9]  Gerhard Hommel,et al.  Multiple Hypotheses Testing , 1993 .

[10]  Julio Michael Stern,et al.  The full Bayesian significance test for mixture models: results in gene expression clustering. , 2008, Genetics and molecular research : GMR.

[11]  Basilio De Braganca Pereira.  Separate Families of Hypotheses , 2014 .

[12]  H. Haario,et al.  An adaptive Metropolis algorithm , 2001 .

[13]  Julio Michael Stern,et al.  Comparing Parameters of Two Bivariate Normal Distributions using the FBST : Full Bayesian Significance Test , 2002 .

[14]  David R. Cox,et al.  Further Results on Tests of Separate Families of Hypotheses , 1962 .

[15]  Marcelo de Souza Lauretto,et al.  FBST for Mixture Model Selection , 2005 .

[16]  David R. Wolf.  A Bayesian Reflection on Surfaces , 1999, Entropy.

[17]  Oscar Kempthore,et al.  Of what use are tests of significance and tests of hypothesis , 1976 .

[18]  Carlos Alberto de Bragança Pereira,et al.  RELATIONSHIP BETWEEN BAYESIAN AND FREQUENTIST SIGNIFICANCE INDICES , 2012 .

[19]  Julio Michael Stern,et al.  Bayesian epistemic values: focus on surprise, measure probability! , 2014, Log. J. IGPL.

[20]  Basilio de Bragança Pereira,et al.  A Comparison of Bayes Factors for Separated Models: Some Simulation Results , 2003, Commun. Stat. Simul. Comput..

[21]  J. Berger,et al.  Testing Precise Hypotheses , 1987 .

[22]  Julio Michael Stern,et al.  The Problem of Separate Hypotheses via Mixture Models , 2007 .

[23]  D. Cox.  Tests of Separate Families of Hypotheses , 1961 .

[24]  B. B. Pereira.  Choice of a survival model for patients with a brain tumour , 1981 .

[25]  Debasis Kundu,et al.  Discriminating between the Weibull and log‐normal distributions , 2004 .

[26]  V. De Gruttola,et al.  Nonparametric analysis of truncated survival data, with application to AIDS , 1988 .

[27]  C. Quesenberry,et al.  Selecting Among Probability Distributions Used in Reliability , 1982 .

[28]  D. Chakrabarty.  A New Bayesian Test to Test for the Intractability-Countering Hypothesis , 2013, 1304.5982.

[29]  Jerald F. Lawless,et al.  Statistical Models and Methods for Lifetime Data: Lawless/Statistical , 2002 .

[30]  Julio Michael Stern,et al.  SPECIAL CHARACTERIZATIONS OF STANDARD DISCRETE MODELS , 2008 .

[31]  Telba Z. Irony,et al.  A WEIBULL WEAROUT TEST: FULL BAYESIAN APPROACH , 2001 .

[32]  Shrikant I Bangdiwala.  Multiple hypotheses testing , 2017, International journal of injury control and safety promotion.

[33]  J. K. Lindsey,et al.  Comparison of Probability Distributions , 1974 .

[34]  Basílio de Bragança Pereira,et al.  Model Choice in Nonnested Families , 2016 .

[35]  B. Pereira,et al.  Bayesian significance test for discriminating between survival distributions , 2017, 1705.11073.

[36]  Julio Michael Stern,et al.  Evidence and Credibility: Full Bayesian Significance Test for Precise Hypotheses , 1999, Entropy.