senfmCI {sensitivityfull}

Sensitivity Analysis for a Confidence Interval in a Full Match.

Description

In a full match, each matched set contains either one treated individual and one or more controls or one control and one or more treated individuals. Uses Huber's M-statistic as the basis for a confidence interval for an additive constant treatment effect, \tau. Performs either a randomization inference or an analysis of sensitivity to departures from random assignment. The confidence interval inverts the test in the function senfm() in the sensitivityfull package.

Usage

senfmCI(y,treated1,gamma=1,inner=0,trim=3,lambda=1/2,
                  alpha=0.05,twosided=TRUE,upper=TRUE)

Arguments

y

If there are I matched sets and the largest matched set contains J individuals, then y is an I by J matrix with one row for each matched set. If matched set i contains one treated individual and k controls, where k is at least 1 and at most J-1, then y[i,1] is the treated individual's response, y[i,2],...,y[i,k+1] are the responses of the k controls, and y[i,k+2],...,y[i,J] are equal to NA. If matched set i contains one control and k>1 treated individuals, then y[i,1] is the control's response, y[i,2],...,y[i,k+1] are the responses of the k treated individuals, and y[i,k+2],...,y[i,J] are equal to NA. Although y may, and typically does, contain NA's, y[i,1] and y[i,2] must not be NA for all i.

If you have matched pairs, not matched sets, use the senmwCI function in the sensitivitymw package.

treated1

The vector treated1 is a logical vector of length I, where treated1[i]=TRUE if there is one treated subject in matched set i and treated1[i]=FALSE if there is more than one treated subject in matched set i.

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=0 and inner=Inf. 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.

Details

For the given \Gamma, senfmCI() inverts the test in the function senfm() 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).

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 senfm().

Author(s)

Paul R. Rosenbaum.

References

Hansen, B. B. (2007). Optmatch. R News 7 18-24. (R package optmatch) (Optmatch can create an optimal full match.)

Hansen, B. B. and Klopfer, S. O. (2006). Optimal full matching and related designs via network flows. J. Comput. Graph. Statist. 15 609-627. (R package optmatch)

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. (1991). A characterization of optimal designs for observational studies. J. Roy. Statist. Soc. B 53 597-610. (Introduces full matching.)

Rosenbaum, P. R. (2007). Sensitivity analysis for m-estimates, tests and confidence intervals in matched observational studies. Biometrics 63 456-64. (R package sensitivitymv)

Rosenbaum, P. R. (2013). Impact of multiple matched controls on design sensitivity in observational studies. Biometrics 69 118-127. (Introduces inner trimming.)

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)

Examples

# The artificial example that follows has I=9
# matched sets.  The first 3 sets have one treated
# individual and two controls with treated subjects
# in column 1.  The next 3 sets are
# matched pairs, with treated subjects in column 1.
# The next 3 sets have one control and two treated
# subjects, with the control in column 1.  Simulated
# from a Normal distribution with an additive effect
# of tau=1.

y<-c(2.2, 1.4, 1.6, 2.4, 0.7, 1.3, 1.2, 0.6, 0.3,
0.5, -0.1, -1.3, -0.3, 0.1, 0.4, 3.0, 1.1, 1.4, -0.8,
0.1, 0.8, NA, NA, NA, 1.1, 0.5, 1.8)
y<-matrix(y,9,3)
treated1<-c(rep(TRUE,6),rep(FALSE,3))

# Randomization interval and point estimate, Huber scores:
senfmCI(y,treated1)

# Uses senfm() to show how senfmCI() inverts the test.
senfm(y,treated1,tau=0.6172307) #P-value is 0.025
senfm(y,treated1,tau=2.0612746,alternative = "less") #P-value is 0.025
senfm(y,treated1,tau=1.345436) #Statistic is 0

senfmCI(y,treated1,gamma=1.5) #Sensitivity of two-sided CI
# The next two calculations relate one and two-sided intervals
senfmCI(y,treated1,gamma=1.5,twosided=FALSE,upper=TRUE,alpha=0.025)
senfmCI(y,treated1,gamma=1.5,twosided=FALSE,upper=FALSE,alpha=0.025)

# If an estimator is approximately Normal, then +/- a standard
# error is approximately a 2/3 confidence interval.  Going the
# other way, people sometimes suggest looking at a 2/3
# confidence interval as analogous to +/- a standard error.
senfmCI(y,treated1,gamma=1.5,alpha=1/3)

# For an additional example, install and load package sensitivitymw
# library(sensitivitymw)
# The mercury data is 397 triples, 1 treated, 2 controls.
# It is the example in Rosenbaum (2014).
# data(mercury)
# help(mercury)
# In this balanced design, senmwCI() and senfmCI() give the same CI.
# senmwCI(mercury,gamma=3)
# senfmCI(mercury,rep(TRUE,397),gamma=3,twosided=FALSE)