senmCI {sensitivitymult}

Sensitivity Analysis for a Confidence Interval.

Description

Each matched set contains one treated individual and one or more controls. Uses Huber's M-statistic as the basis for a confidence interval for an additive constant treatment effect, \tau. Produces either a randomization based confidence interval or an analysis of sensitivity to departures from random assignment. Also produces a point estimate for randomization inference or an interval of point estimates for a sensitivity analysis. For tests, use function senm(). The method is described in Rosenbaum (2007,2013).

Usage

senmCI(y, z, mset, gamma=1, inner=0, trim=3, lambda=1/2,
                  alpha=0.05, twosided=TRUE, upper=TRUE, TonT=FALSE)

Arguments

y

A vector of responses with no missing data.

z

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

mset

Matched set indicators, 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 returns a randomization-based confidence interval.

inner

inner and trim together define the \psi-function for the M-statistic. The default values yield a version of Huber's \psi-function, while setting inner = 0 and trim = Inf uses the mean within each matched set. The \psi-function is an odd function, so \psi(w) = -\psi(-w). For w \ge 0, the \psi-function is \psi(w)=0 for 0 \le w \le inner, is \psi(w)= trim for w \ge trim, and rises linearly from 0 to trim for inner < w < trim.

An error will result unless 0 \le inner \le trim.

Taking trim < Inf limits the influence of outliers; see Huber (1981). Taking inner > 0 often increases design sensitivity; see Rosenbaum (2013).

trim

inner and trim together define the \psi-function for the M-statistic. See inner.

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 will cover the true \tau with probability at least 1-\alpha.

twosided

If twosided==TRUE, then a two-sided 1-\alpha confidence interval is constructed. If twosided==FALSE, then a one-sided 1-\alpha confidence interval is constructed. The two sided interval is the intersection of two one-sided 1-\alpha/2 intervals.

upper

If twosided==TRUE, then upper is ignored. If twosided==FALSE and upper=TRUE, then an upper 1-\alpha confidence interval is constructed. If twosided==FALSE and upper=FALSE, then a lower 1-\alpha confidence interval is constructed.

TonT

TonT refers to the effect of the treatment on the treated; see Rosenbaum and Rubin (1985, equation 1.1.1) The default is TonT=FALSE. If TonT=FALSE, then the total score in matched set i is divided by the number ni of individuals in set i, as in expression (8) in Rosenbaum (2007). This division by ni has few consequences when every matched set has the same number of individuals, but when set sizes vary, dividing by ni is intended to increase efficiency by weighting inversely as the variance; see the discussion in section 4.2 of Rosenbaum (2007). If TonT=TRUE, then the division is by ni-1, not by ni, and there is a further division by the total number of matched sets to make it a type of mean. If TonT=TRUE and trim=Inf, then the statistic is the mean over matched sets of the treated minus mean-control response, so it is weighted to estimate the average effect of the treatment on the treated.

Details

For the given \Gamma, senmCI() inverts the test in the function senm() to produce the confidence interval. That is, it tests every \tau and retains the values not rejected at level \alpha.

The test is a large sample approximation based on a Normal approximation to the null distribution; see Rosenbaum (2007).

If TonT=FALSE, 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).

Value

PointEstimates

The interval of point estimates of \tau. If gamma=1, then the interval is a single point estimate.

ConfidenceInterval

The confidence interval for \tau.

description

Reminder of the coverage rate, gamma, and type of interval.

Note

In a sensitivity analysis, a one-sided confidence interval is not conservative; however, two-sided intervals formed as the intersection of two one-sided 1-\alpha/2 intervals are somewhat conservative. See the discussion of two-sided tests in the documentation for senm().

Author(s)

Paul R. Rosenbaum.

References

Huber, P. (1981) Robust Statistics. New York: John Wiley. (M-estimates based on M-statistics.)

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. (1993). Hodges-Lehmann point estimates of treatment effect in observational studies. Journal of the American Statistical Association, 88, 1250-1253. (Introduces sensitivity analysis for point estimates.)

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., & Rubin, D. B. (1985). The bias due to incomplete matching. Biometrics, 41, 103-116.

Examples

data(teeth)
attach(teeth)
#
# Note: Computing confidence intervals takes a few moments
# The calls to senmCI() are commented to meet time requirements
# for cran examples.  Remove the comment symbol to run them.
#
# The calculations that follow reproduce the intervals from
# section 5.1 of Rosenbaum (2016, p. 1466)
# senmCI(either4low,smoker,mset,trim=2.5,gamma=1.5)
# senmCI(either4up,smoker,mset,trim=2.5,gamma=1.5)
# Confidence interval using the mean by inverting the
# permuational t-test.
# senmCI(either4low,smoker,mset,trim=Inf,TonT=TRUE)
dif<-either4low[smoker==1]-either4low[smoker==0] # Matched pair differences
mean(dif) # Equals the point estimate above
t.test(dif) # But permutational t-interval and t-interval differ
# Sensitivity analysis using the mean difference
# senmCI(either4low,smoker,mset,gamma=1.5,trim=Inf,TonT=TRUE)
detach(teeth)