senm {sensitivitymult}

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. Performs either a randomization test or an analysis of sensitivity to departures from random assignment. For confidence intervals, use function senmCI(). The method is described in Rosenbaum (2007,2013). The senm() function is intended as a convenience for a user of the comparison() function in the sensitivitymult package. The function senm() in the sensitivitymult package is essentially the same as senmv() in the sensitivitymv package, except the format of the input to senm() resembles the format of the input to comparison(). In particular, in the sensitivitymv package, the rows of y are matched sets, not people, whereas in sensitivitymult the rows of y are people with treated and control people identified by z and matched sets identified by mset.

Usage

senm(y, z, mset, gamma = 1, inner = 0, trim = 3, lambda = 1/2,
              tau = 0, alternative="greater", TonT = FALSE)

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. 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 trim < Inf and inner = 0 uses Huber's psi function. Taking trim = Inf does no trimming and is similar to a weighted mean; see TonT. 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.

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.

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. See the examples.

Details

For the given \Gamma, senm() 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, senm() 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).

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 senm() 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.

Related packages are sensitivitymv, sensitivitymw, sensitivityfull and sensitivity2x2xk.

Author(s)

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.)

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

Examples

data(teeth)
attach(teeth)
# The following example reproduces the deviate for lower teeth
# mentioned on line 4 of Rosenbaum (2016, p. 1466).
senm(either4low,smoker,mset,trim=2.5,gamma=2.2)
# The calculation above is equivalent to using comparison()
# with weights w=c(0,1) so upper teeth are ignored.
comparison(cbind(either4up,either4low),smoker,mset,c(0,1),trim=2.5,gamma=2.2)
# The following example illustrates the permutational t-test
# which uses the mean of the pair differences as a test statistic.
senm(either4low,smoker,mset,trim=Inf,TonT=TRUE)
dif<-either4low[smoker==1]-either4low[smoker==0] # Matched pair differences
mean(dif) # Equals the test statistic above
detach(teeth)