The length heuristic for simultaneous hypothesis tests

We observe a sequence of test statistics T = (T 1 , T 2 , ..., T J ), each of which is approximately N(0,1) under the null hypothesis H 0 , but which are correlated with each other. Not being certain which T j is best, we use the test statistic T max = max{T 1 , ..., T J ). For a given observed value of T max , say c, what is the significance probability pr(T max > c)? Define the length of T to be Σ j L =2 arccos(p j-1,j ), where p j-1,j is the null hypothesis correlation between T j-1 and T j . Hotelling's theorem on the volume of tubes leads to a length-based bound on pr(T max > c) that usually beats the Bonferroni bound. It is easy to improve Hotelling's bound. Several examples show that in favourable circumstances the improved bounds can be good approximations to the actual value of pr(T max > c).

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

[2]  Daniel Q. Naiman,et al.  Conservative Confidence Bands in Curvilinear Regression , 1986 .

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

[4]  Ioan Tomescu,et al.  Hypertrees and Bonferroni inequalities , 1986, J. Comb. Theory, Ser. B.

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

[6]  D. Siegmund,et al.  On Hotelling's Approach to Testing for a Nonlinear Parameter in Regression , 1989 .

[7]  D. Siegmund,et al.  Maximally Selected Chi Square Statistics , 1982 .

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

[9]  B. Efron.  Logistic Regression, Survival Analysis, and the Kaplan-Meier Curve , 1988 .

[10]  D. Naiman,et al.  INCLUSION-EXCLUSION-BONFERRONI IDENTITIES AND INEQUALITIES FOR DISCRETE TUBE-LIKE PROBLEMS VIA EULER CHARACTERISTICS , 1992 .

[11]  Robert Bohrer,et al.  Sharp one-sided confidence bounds for linear regression over intervals , 1972 .

[12]  B. Efron,et al.  On the correlation of angular position with time of occurrence of gamma-ray bursts , 1995 .

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

[14]  B. Efron.  Student's t-Test under Symmetry Conditions , 1969 .

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

[16]  Melinda H. McCann,et al.  A Path Length Inequality for the Multivariate-t Distribution, with Applications to Multiple Comparisons , 1996 .

[17]  R. Madsen,et al.  P values for tests using a repeated significance test design , 1982 .

[18]  Rupert G. Miller.  Simultaneous Statistical Inference , 1966 .

[19]  K. Worsley.  An improved Bonferroni inequality and applications , 1982 .

[20]  S J Pocock,et al.  Allocation of patients to treatment in clinical trials. , 1979, Biometrics.

[21]  Fred M. Hoppe,et al.  Iterating bonferroni bounds , 1985 .

[22]  David J. Hunter.  An upper bound for the probability of a union , 1976, Journal of Applied Probability.

[23]  George K. Francis,et al.  Sharp One-Sided Confidence Bounds Over Positive Regions , 1972 .