Zaykin et al. (2002) proposed combining L=length(p) independent P-values p by taking the product of the P-values that are no larger than a truncation point trunc, namely w = prod(p^(p<=trunc)). Computes the one P-value for the combination w.
truncatedPbg(p, trunc = 0.2)Returns the one P-value for the truncated product. For trunc < 1, the distribution is not continuous, but rather attaches positive probability to a P-value of 1.
a vector of P-values that are either independent or stochastically larger than the uniform distribution on the cube (see Rosenbaum (2011, Definition 1))
trucation point. Computes the product of all P-values in p that are no larger than trunc. For trunc = 1, performs Fisher's method of combining independent P-values.
Paul R. Rosenbaum
Zaykin et al. (2002) proposed combining L=length(p) independent P-values p by taking the product of the P-values that are no larger than a truncation point trunc, namely w = prod(p^(p<=trunc)). For trunc = 1, this is Fisher's method for combining independent P-values. The method also works for certain kinds of fairly inconsequential dependence; see Rosenbaum (2011, section 2).
trucatedP computes the one P-value for the combination w using the formula in Zaykin et al. (2002). The equivalent function truncatedPbg computes the exact same P-value for w using a binomial mixture of gamma distributions, as discussed by Hsu et al. (2013, section 3.1).
The truncated product or Fisher's method (trunc = 1) may be used for sensitivity analyses with evidence factors; see Rosenbaum (2011) and the mtm example below. The truncated product with trunc < 1 is useful in combining P-value upper bounds produced by sensitivity analyses, for instance those produced by senmv. These upper bounds eventually approach 1 for larger values of the sensitivity parameter, and trunc < 1 eliminates these, often increasing power. See Hsu et al. (2013) for comparisons.
Hsu, J. Y., Small, D. S. and Rosenbaum, P. R. (2013) <doi:10.1080/01621459.2012.742018> Effect modification and design sensitivity in observational studies. Journal of the American Statistical Association, 108, 135-148.
Meibian, Z., Zhijian, C., Qing, C. et al. (2008) Investigating DNA damage in tannery workers occupationally exposed to tivalent chromium using the comet assay. Mutation Research 654, 45-51.
Rosenbaum, P. R. (2010) <doi:10.1093/biomet/asq019> Evidence factors in observational studies. Biometrika, 97, 333-345.
Rosenbaum, P. R. (2011) <doi:10.1198/jasa.2011.tm10422> Some approximate evidence factors in observational studies. Journal of the American Statistical Association, 106, 285-295. <doi:10.1198/jasa.2011.tm10422>
Rosenbaum, P. R. (2015). Two R packages for sensitivity analysis in observational studies. Observational Studies, 1(1), 1-17. Free on-line at obsstudies.org
Rosenbaum, P. R. (2021) <doi:10.1201/9781003039648> Replication and Evidence Factors in Observational Studies. Chapman and Hall/CRC.
Zaykin, D. V., Zhivotovsky, L. A., Westfall, P. H. and Weir, B. S. (2002) Truncated product method of combining P-values. Genetic Epidemiology, 22, 170-185. <doi:10.1002/gepi.0042>
Zhang, K., Small, D. S., Lorch, S., Srinivas, S. and Rosenbaum, P. R. (2011) Using split samples and evidence factors in an observational study of neonatal outcomes. Journal of the American Statistical Association, 106, 511-524.
# Evidence factor example: see note above.
data(mtm)
senmv(-mtm,gamma=11.7,trim=1)
senmv(-mtm[,2:3],gamma=2.1,trim=1)
senmv(-mtm,gamma=12,trim=1)
senmv(-mtm[,2:3],gamma=3,trim=1)
truncatedPbg(c(0.05167572,0.1527849),trunc=1)
truncatedPbg(c(0.05167572,0.1527849),trunc=.2)
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