Each matched set contains one treated individual and one or more controls. Uses Huber's M-statistic as the basis for the test; see Maritz (1979). Matched sets of different sizes use different \(\psi\)-functions, creating what is called a fugue statistic. Performs either a randomization test (\(\Gamma=1\)) or an analysis of sensitivity to departures from random assignment (\(\Gamma>1\)). For hypothesis tests, use function sen(). The method is described in Li and Rosenbaum (2019); see also Rosenbaum (2007,2013).
senCI(y, z, mset, gamma = 1, inner = NULL, trim = NULL, lambda = 1/2,
alpha = 0.05, alternative = "greater")A vector of responses with no missing data.
Treatment indicator, z=1 for treated, z=0 for control with length(z)==length(y).
Matched set indicator, 1, 2, ..., sum(z) with length(mset)==length(y). Matched set indicators should be either integers or a factor.
gamma is the sensitivity parameter \(\Gamma\), where \(\Gamma \ge 1\). Setting \(\Gamma = 1\) is equivalent to assuming ignorable treatment assignment given the matched sets, and it performs a within-set randomization test.
inner and trim together define the \(\psi\)-function for the M-statistic. If the largest matched set has k controls, then inner is either a scalar or a vector with k=length(inner). If inner is a scalar, then the same value of inner is used, regardless of the number of controls. Otherwise, inner[1] is used with one control, inner[2] is used with two controls, etc. If inner is NULL, default values of inner=c(.8,.8,.6,.4,0,0,0,...,0) are used.
inner and trim together define the \(\psi\)-function for the M-statistic. If the largest matched set has k controls, then trim is either a scalar or a vector with k=length(trim). If trim is a scalar, then the same value of trim is used, regardless of the number of controls. Otherwise, trim[1] is used with one control, trim[2] is used with two controls, etc. If trim is NULL, default values of trim=c(3,3,...,3) are used. For each i, 0 <= inner[i] < trim[i] < Inf.
Before applying the \(\psi\)-function to treated-minus-control differences, the differences are scaled by dividing by the lambda quantile of all within set absolute differences. Typically, lambda = 1/2 for the median. The value of lambda has no effect if trim=Inf and inner=0. See Maritz (1979) for the paired case and Rosenbaum (2007) for matched sets.
An error will result unless 0 < lambda < 1.
The coverage rate of the confidence interval is 1-\(\alpha\). If the bias in treatment assignment is at most \(\Gamma\), then the confidence interval covers at rate 1-\(\alpha\).
If alternative="greater" or alternative="less", the a one-sided confidence interval is returned. If alternative="twosided", a somewhat conservative two-sided confidence interval is returned. See the discussion of two-sided tests in the documentation for sen().
For \(\Gamma>1\), an interval of point estimates is returned. \(\Gamma=1\), the interval is a point.
The confidence interval.
The confidence interval inverts the test provided by sen(). See the documentation for sen() for more information.
The upper bound on the P-value is based on the separable approximation described in Gastwirth, Krieger and Rosenbaum (2000); see also Rosenbaum (2007, 2018).
Cox, D. R. (1977). The role of signficance tests (with Discussion). Scand. J. Statist. 4, 49-70.
Huber, P. (1981) Robust Statistics. New York: John Wiley. (M-estimates based on M-statistics.)
Li, Xinran and Rosenbaum, P. R. (2019) Maintaining high constant design sensitivity in observational studies with matched sets of varying sizes. Manuscript.
Maritz, J. S. (1979). A note on exact robust confidence intervals for location. Biometrika 66 163--166. (Introduces exact permutation tests based on M-statistics by redefining the scaling parameter.)
Rosenbaum, P. R. (2007). Sensitivity analysis for m-estimates, tests and confidence intervals in matched observational studies. Biometrics 63 456-64. (R package sensitivitymv) <doi:10.1111/j.1541-0420.2006.00717.x>
Rosenbaum, P. R. (2013). Impact of multiple matched controls on design sensitivity in observational studies. Biometrics 69 118-127. (Introduces inner trimming.) <doi:10.1111/j.1541-0420.2012.01821.x>
Rosenbaum, P. R. (2015). Two R packages for sensitivity analysis in observational studies. Observational Studies, v. 1. (Free on-line.)
Rosenbaum, P. R. (2016) Using Scheffe projections for multiple outcomes in an observational study of smoking and periondontal disease. Annals of Applied Statistics, 10, 1447-1471. <doi:10.1214/16-AOAS942>
Rosenbaum, P. R. (2018). Sensitivity analysis for stratified comparisons in an observational study of the effect of smoking on homocysteine levels. The Annals of Applied Statistics, 12(4), 2312-2334. <doi:10.1214/18-AOAS1153>
# NOT RUN {
# Reproduces results from Table 3 of Li and Rosenbaum (2019)
data(nh1and3)
attach(nh1and3)
senCI(homocysteine,z,mset,gamma=1)
senCI(homocysteine,z,mset,gamma=1.9)
senCI(homocysteine,z,mset,inner=0,gamma=1.9)
amplify(1.9,c(3,3.5,4))
# Relationships between confidence intervals and P-value bounds
senCI(homocysteine,z,mset,alternative="twosided",gamma=1.75)
sen(homocysteine,z,mset,alternative="less",tau=2.21721733,gamma=1.75)
senCI(homocysteine,z,mset,alternative="less",gamma=1.75)
sen(homocysteine,z,mset,alternative="less",tau=2.159342,gamma=1.75)
detach(nh1and3)
# }
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