senCI {fugue}

Sensitivity Analysis for Point Estimates and Confidence Intervals in an Observational Study.

Description

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).

Usage

senCI(y, z, mset, gamma = 1, inner = NULL, trim = NULL, lambda = 1/2,
     alpha = 0.05, alternative = "greater")

Arguments

y

A vector of responses with no missing data.

z

Treatment indicator, z=1 for treated, z=0 for control with length(z)==length(y).

mset

Matched set indicator, 1, 2, ..., sum(z) with length(mset)==length(y). Matched set indicators should be either integers or a factor.

gamma

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

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.

trim

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.

lambda

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.

alpha

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.

alternative

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().

Details

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).

Value

point.estimates

For \Gamma>1, an interval of point estimates is returned. \Gamma=1, the interval is a point.

confidence.interval

The confidence interval.

Note

The examples reproduce some results from Li and Rosenbaum (2019).

Author(s)

Xinran Li and Paul R. Rosenbaum.

References

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>

Examples

## 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)

## End(Not run)