sen {fugue}

Sensitivity Analysis for a Matched Comparison 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, for instance, a mean. 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 confidence intervals, use function senCI(). The method is described in Li and Rosenbaum (2019); see also Rosenbaum (2007,2013).

Usage

sen(y, z, mset, gamma = 1, inner = NULL, trim = NULL, lambda = 1/2,
     tau = 0, 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.

tau

The null hypothesis asserts that the treatment has an additive effect, tau. By default, tau=0, so by default the null hypothesis is Fisher's sharp null hypothesis of no treatment effect.

alternative

If alternative="greater", the null hypothesis of a treatment effect of tau is tested against the alternative of a treatment effect larger than tau. If alternative="less", the null hypothesis of a treatment effect of tau is tested against the alternative of a treatment effect smaller than tau. In particular, alternative="less" is equivalent to: (i) alternative="greater", (ii) y replaced by -y, and (iii) tau replaced by -tau. See the note for discussion of two-sided sensitivity analyses.

Details

The novel element in the fugue package is the automatic use of different \psi-functions for matched sets of different sizes. These \psi-functions have been selected to approximately equate the design sensitivities in sets of unequal sizes when the errors are Normal and the additive effect is half the standard deviation of a matched pair difference; see Li and Rosenbaum (2019). If you disable this automatic feature by manually setting a single value for inner and trim, then the results will agree with senm() in the R package sensitivitymult. For instance, using both sen() in the fugue package and senm() in the sensitivitymult package will yield the same deviate and P-value in: data(nh1and3) attach(nh1and3) sen(homocysteine,z,mset,inner=0,gamma=1.9) senm(homocysteine,z,mset,inner=0,trim=3,gamma=1.9) Note that the sensitivitymult package is intended to implement methods from Rosenbaum (2016,2019) that are not implemented in the fugue package.

For the given \Gamma, sen() computes the upper bound on the 1-sided P-value testing the null hypothesis of an additive treatment effect tau against the alternative hypothesis of a treatment effect larger than tau. By default, sen() tests the null hypothesis of no treatment effect against the alternative of a positive treatment effect. The P-value is an approximate P-value based on a Normal approximation to the null distribution; see Rosenbaum (2007).

Matched sets of unequal size are weighted using weights that would be efficient in a randomization test under a simple model with additive set and treatment effects and errors with constant variance; see Rosenbaum (2007).

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

pval

The upper bound on the 1-sided P-value.

deviate

The deviate that was compared to the Normal distribution to produce pval.

statistic

The value of the M-statistic.

expectation

The maximum expectation of the M-statistic for the given \Gamma.

variance

The maximum variance of the M-statistic among treatment assignments that achieve the maximum expectation. Part of the separable approximation.

Note

The function sen() performs 1-sided tests. One approach to a 2-sided, \alpha-level test does both 1-sided tests at level \alpha/2, and rejects the null hypothesis if either 1-sided test rejects. Equivalently, a bound on the two sided P-value is the smaller of 1 and twice the smaller of the two 1-sided P-values. This approach views a 2-sided test as two 1-sided tests with a Bonferroni correction; see Cox (1977, Section 4.2). In all cases, this approach is a valid large sample test: a true null hypothesis is falsely rejected with probability at most \alpha if the bias in treatment assignment is at most \Gamma; so, this procedure is entirely safe to use. For a randomization test, \Gamma=1, this Bonferroni procedure is not typically conservative. For large \Gamma, this Bonferroni procedure tends to be somewhat conservative.

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, X. 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. (2014). Weighted M-statistics with superior design sensitivity in matched observational studies with multiple controls. J. Am. Statist. Assoc. 109 1145-1158. (R package sensitivitymw) <doi:10.1080/01621459.2013.879261>

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>

Rosenbaum, P. R. (2019). Combining planned and discovered comparisons in observational studies. Biostatistics, to appear. <doi.org/10.1093/biostatistics/kxy055>

Examples

# Reproduces results from Table 3 of Li and Rosenbaum (2019)
data(nh1and3)
attach(nh1and3)
sen(homocysteine,z,mset,gamma=1)
sen(homocysteine,z,mset,gamma=1.9)
sen(homocysteine,z,mset,inner=0,gamma=1.9)
amplify(1.9,c(3,3.5,4))
detach(nh1and3)