Abstract
One of the main computational burdens in genome-wide statistical applications is the evaluation of large scale multiple hypothesis tests. Such tests are often implemented using replication-based methods, such as the permutation test or bootstrap procedure. While such methods are widely applicable, they place a practical limit on the computational complexity of the underlying test procedure. In such cases it would seem natural to apply sequential procedures. For example, suppose we observe the first ten replications of an upper-tailed statistic under a null distribution generated by random permutations, and of those ten, five exceed the observed value. It would seem reasonable to conclude that the P-value will not be small enough to be of interest, and further replications should not be needed.
While such methods have been proposed in the literature, for example by Hall in 1983, by Besag and Clifford in 1991 and by Lock in 1991, they have not been widely applied in multiple testing applications generated by high dimensional data sets, where they would likely be of some benefit. In this article related methods will first be reviewed. It will then be shown how commonly used multiple testing procedures may be modified so as to introduce sequential procedures while preserving the validity of reported error rates. A number of examples will show how such procedures can reduce computation time by an order of magnitude with little loss in power.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Almudevar, A. (2000). Exact confidence regions for species assignment based on DNA markers. The Canadian Journal of Statistics, 28, 81–95.
Almudevar, A. (2010). A hypothesis test for equality of Bayesian network models. EURASIP Journal on Bioinformatics and Systems Biology, 2010, 10.
Benjamini, Y., & Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. The Annals of Statistics, 29, 1165–1188.
Besag, J., & Clifford, P. (1989). Generalized Monte Carlo significance tests. Biometrika, 76, 633–642.
Besag, J., & Clifford, P. (1991). Sequential Monte Carlo p-values. Biometrika, 78, 301–304.
Dudoit, S., Shaffer, J. P., & Boldrick, J. C. (2003). Multiple hypothesis testing in microarray experiments. Statistical Science, 18, 71–103.
Dudoit, S., & van der Laan, M. J. (2008). Multiple testing procedures with applications to genomics. New York: Springer.
Fay, M. P., & Follmann, D. A. (2002). Designing Monte Carlo implementations of permutation or bootstrap hypothesis tests. The American Statistician, 56, 63–70.
Hall, W. J. (1983). Some sequential tests for matched pairs: A sequential permutation test. In P. K. Sen (ed.), Contributions to statistics: essays in honour of Norman L. Johnson, (pp. 211–228). Amsterdam: North-Holland.
Ljung, L. (2007). Strong convergence of a stochastic approximation algorithm. The American Statistician, 6, 680–696.
Lock, R. H. (1991). A sequential approximation to a permutation test. Communications in Statistics: Simulation and Computation, 20, 341–363.
Medland, S., Schmitt, J., Webb, B., Kuo, P.-H., & Neale, M. (2009). Efficient calculation of empirical P-values for genome-wide linkage analysis through weighted permutation. Behavior Genetics, 39, 91–100.
Mootha, V. K., Lindgren, C. M., Eriksson, K. F., Subramanian, A., Sihag, S., Lehar, J., et al. (2003). PGC-1 α-responsive genes involved in oxidative phosphorylation are coordinately downregulated in human diabetes. Nature Genetics, 100, 605–610.
Robbins, H., & Monro, S. (1951). A stochastic approximation method. The Annals of Mathematical Statistics, 22, 400–407.
Siegmund, D. (1985). Sequential analysis: tests and confidence intervals. New York: Springer-Verlag.
Subramanian, A., Tamayo, P., Mootha, V. K., Mukherjee, S., Ebert, B. L., Gillette, M. A., et al. (2005). Gene set enrichment analysis: A knowledge-based approach for interpreting genome-wide expression profiles. Proceedings of the National Academy of Sciences of the United States of America, 102, 15545–15550.
Wald, A. (1947). Sequential analysis. New York: John Wiley and Sons.
Wald, A. (1948). Optimum character of the sequential probability ratio test. The Annals of Mathematical Statistics, 19, 326–339.
Yang, H. Y., & Speed, T. (2003). Design and analysis of comparative microarray experiments. In T. Speed (ed.) Statistical analysis of gene expression microarray data (pp. 35–92). Boca Raton, FL: Chapman and Hall.
Acknowledgement
This work was supported by NIH grant R21HG004648.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Almudevar, A. (2020). Applications of Sequential Methods in Multiple Hypothesis Testing. In: Almudevar, A., Oakes, D., Hall, J. (eds) Statistical Modeling for Biological Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-34675-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-34675-1_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-34674-4
Online ISBN: 978-3-030-34675-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)