mhLS: Large Sample Test for One 2x2xk Table
In sensitivity2x2xk: Sensitivity Analysis for 2x2xk Tables in Observational Studies

Description Usage Arguments Value Note References Examples

Large sample approximation to a one-sided sensitivity analysis for a 2x2 or 2x2xk table.

1	mhLS(tab,Gamma=1,correction=FALSE)

`tab`	tab is a 2x2 or a 2x2xk contingency table, typically treatment x outcome x covariate.
`Gamma`	Gamma>=1 is the sensitivity parameter, where Gamma=1 yields a randomization test.
`correction`	If correction=TRUE, a continuity correction is used. The usual Mantel-Haenszel test for a 2x2xk table uses a continuity correction.

`pval`	Upper bound on the one-sided P-value.
`A`	A is the total count in the [1,1,j] cells of the 2x2xk table tab, summing over j=1,...,k. It is the test statistic in Birch's (1964) version of the Cochran-Mantel-Haenszel test. Birch's test is the uniformly most powerful unbiased test against a constant odds ratio alternative.
`Expectation`	Null expectation of A for the upper bounding distribution of A.
`Variance`	Null variance of A for the upper bounding distribution of A. The approximate P-value is 1-pnorm((A-Expectation)/sqrt(Variance)), unless correction=TRUE, in which case A is reduced by 1/2.

The mhLS function uses the exact null expectation and variance of A in a Normal approximation. Use mh for an exact test.

Birch, M. W. (1964). The detection of partial association, I: The 2 x 2 case. Journal of the Royal Statistical Society. Series B (Methodological), 313-324.

Cochran, W. G. (1954). Some methods for strengthening the common chi-squared tests. Biometrics, 10, 417-451.

Fu, Zhenming, Martha J. Shrubsole, Walter E. Smalley, Huiyun Wu, Zhi Chen, Yu Shyr, Reid M. Ness, and Wei Zheng (2012). Lifestyle factors and their combined impact on the risk of colorectal polyps. American Journal of Epidemiology 176, 766-776.

Mantel, N. and Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies. Journal of the National Cancer Institute 22, 719-748.

Rosenbaum, P. R. (1995). Quantiles in nonrandom samples and observational studies. Journal of the American Statistical Association, 90, 1424-1431.

Rosenbaum, P. R. (2002) Observational Studies (2nd edition). New York: Springer.

Satagopan, J. M., Offit, K., Foulkes, W., Robson, M. E. Wacholder, S., Eng, C. M., Karp, S. E. and Begg, C. B. (2001). The lifetime risks of breast cancer in Ashkenazi Jewish carriers of brca1 and brca2 mutations. Cancer Epidemology, Biomarkers and Prevention, 10, 467-473.

# The first example is from Satagopan, et al. (2001), Table 2.
# It is a case-control study of breast cancer and
# BRCA1+ mutations for women aged <40 and women aged >=40.
ageLT40<-matrix(c(18,51,11,673),2,2)
ageGE40<-matrix(c(39,652,21,2699),2,2)
rownames(ageLT40)<-c("BRCA1+","Negative")
rownames(ageGE40)<-c("BRCA1+","Negative")
colnames(ageLT40)<-c("Case","Control")
colnames(ageGE40)<-c("Case","Control")
names(dimnames(ageLT40))<-c("Mutation","Breast Cancer")
names(dimnames(ageGE40))<-c("Mutation","Breast Cancer")
both<-array(c(18,51,11,673,39,652,21,2699),c(2,2,2))
dimnames(both)<-list(c("BRCA1+","Negative"),
      c("Case","Control"),c("LT40","GE40"))
names(dimnames(both))<-c("Mutation","Breast Cancer","Age")

mhLS(ageLT40,Gamma=9.3)
mhLS(ageLT40,Gamma=9.3,correction=TRUE)
mhLS(ageGE40,Gamma=9.3)
mhLS(ageGE40,Gamma=9.3,correction=TRUE)

#Compare with the exact test:
mh(ageLT40,Gamma=9.3)
mh(ageGE40,Gamma=9.3)

mhLS(both,Gamma=9.3)
mhLS(both,Gamma=9.3,correction=TRUE)

#Compare with the exact test
mh(both,Gamma=9.3)

#Compare with the adaptive test
adaptmhLS(ageLT40,ageGE40,Gamma=9.3)