mscorev {submax}

Computes M-scores for Permuational M-tests.

Description

Of limited interest to most users, function mscorev() computes M-scores. A similar function function is in the package sensitivitymv.

Usage

mscorev(ymat, inner = 0, trim = 3, lambda = 0.5)

Arguments

ymat

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.

Although y can contain NA's, y[i,1] and y[i,2] must not be NA for every i. That is, every matched set must have at least one treated subject and one control.

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.

If uncertain about inner, trim and lambda, then use the defaults.

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.

Value

Generally, a matrix with the same dimensions as ymat containing the M-scores.

Note

The example reproduces Table 3 in 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). Specifically, the total score in set (row) i is divided by the number ni of individuals in row i, as in expression (8) in Rosenbaum (2007).

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. (2007) Sensitivity analysis for m-estimates, tests and confidence intervals in matched observational studies. Biometrics, 2007, 63, 456-464. <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.)

Examples

# The example reproduces Table 3 in Rosenbaum (2007).
data(tbmetaphase)
mscorev(tbmetaphase,trim=1)