Likelihood ratio tests for a dose‐response effect using multiple nonlinear regression models

We consider the problem of testing for a dose-related effect based on a candidate set of (typically nonlinear) dose-response models using likelihood-ratio tests. For the considered models this reduces to assessing whether the slope parameter in these nonlinear regression models is zero or not. A technical problem is that the null distribution (when the slope is zero) depends on non-identifiable parameters, so that standard asymptotic results on the distribution of the likelihood-ratio test no longer apply. Asymptotic solutions for this problem have been extensively discussed in the literature. The resulting approximations however are not of simple form and require simulation to calculate the asymptotic distribution. In addition, their appropriateness might be doubtful for the case of a small sample size. Direct simulation to approximate the null distribution is numerically unstable due to the non identifiability of some parameters. In this article, we derive a numerical algorithm to approximate the exact distribution of the likelihood-ratio test under multiple models for normally distributed data. The algorithm uses methods from differential geometry and can be used to evaluate the distribution under the null hypothesis, but also allows for power and sample size calculations. We compare the proposed testing approach to the MCP-Mod methodology and alternative methods for testing for a dose-related trend in a dose-finding example data set and simulations.

[1]  R. Marcus,et al.  The powers of some tests of the equality of normal means against an ordered alternative , 1976 .

[2]  A. Gorin.  ON THE VOLUME OF TUBES , 1983 .

[3]  C. Chatfield.  Model uncertainty, data mining and statistical inference , 1995 .

[4]  Holger Dette,et al.  Dose response signal detection under model uncertainty , 2015, Biometrics.

[5]  Donald W. K. Andrews,et al.  Admissibility of the Likelihood Ratio Test When a Nuisance Parameter is Present Only Under the Alternative , 1995 .

[6]  D. Andrews.  Admissibility of the lidelihood ratio test when the parameter space is restricted under the alternative , 1996 .

[7]  Andrew P Grieve,et al.  ASTIN: a Bayesian adaptive dose–response trial in acute stroke , 2005, Clinical trials.

[8]  Say Beng Tan,et al.  Dose Finding Studies , 2009 .

[9]  P Hougaard,et al.  Testing effect of a drug using multiple nested models for the dose-response. , 2015, Biometrics.

[10]  K. Fang,et al.  Generalized Multivariate Analysis , 1990 .

[11]  H. Hotelling.  Tubes and Spheres in n-Spaces, and a Class of Statistical Problems , 1939 .

[12]  Angelika Bayer,et al.  A First Course In Probability , 2016 .

[13]  P. Diaconis,et al.  Testing for independence in a two-way table , 1985 .

[14]  Nils Lid Hjort,et al.  Model Selection and Model Averaging: Contents , 2008 .

[15]  M. Krams,et al.  Innovative Approaches for Designing and Analyzing Adaptive Dose-Ranging Trials , 2007, Journal of biopharmaceutical statistics.

[16]  C. Ritz,et al.  Likelihood ratio tests in curved exponential families with nuisance parameters present only under the alternative , 2005 .

[17]  Iain M. Johnstone,et al.  Hotelling's Theorem on the Volume of Tubes: Some Illustrations in Simultaneous Inference and Data Analysis , 1990 .

[18]  D. Andrews,et al.  Optimal Tests When a Nuisance Parameter Is Present Only Under the Alternative , 1992 .

[19]  S. Ross.  A First Course in Probability , 1977 .

[20]  B. M. Pötscher,et al.  MODEL SELECTION AND INFERENCE: FACTS AND FICTION , 2005, Econometric Theory.

[21]  Williams Da,et al.  A test for differences between treatment means when several dose levels are compared with a zero dose control. , 1971 .

[22]  Y. Shao,et al.  Asymptotics for likelihood ratio tests under loss of identifiability , 2003 .

[23]  S. Li.  Concise Formulas for the Area and Volume of a Hyperspherical Cap , 2011 .

[24]  C. R. Rao,et al.  Pattern recognition based on scale invariant discriminant functions , 1988, Inf. Sci..

[25]  H. Dette,et al.  Model selection versus model averaging in dose finding studies , 2015, Statistics in medicine.

[26]  Frank Bretz,et al.  Model‐based dose finding under model uncertainty using general parametric models , 2013, Statistics in medicine.

[27]  R. Davies.  Hypothesis testing when a nuisance parameter is present only under the alternative , 1977 .

[28]  Nils Lid Hjort,et al.  Model Selection and Model Averaging , 2001 .

[29]  Arjun K. Gupta,et al.  Elliptically contoured models in statistics , 1993 .

[30]  F Bretz,et al.  Combining Multiple Comparisons and Modeling Techniques in Dose‐Response Studies , 2005, Biometrics.

[31]  Stephen E. Fienberg,et al.  Testing Statistical Hypotheses , 2005 .

[32]  George A. F. Seber,et al.  Linear regression analysis , 1977 .

[33]  Daniel Q. Naiman,et al.  Volumes of Tubular Neighborhoods of Spherical Polyhedra and Statistical Inference , 1990 .