score {submax}

Creates M-scores for Use by submax().

Description

The score() function is an optional aid in using the submax() function. The submax() function may be used on its own, but the user must then create the y matrix and the cmat matrix. The score() function can be used to create the y matrix and cmat matrix for use in the submax() function. It creates Huber-Maritz M-scores.

Usage

score(y, z, mset, x, expandx = FALSE, scale = "closed", inner = 0, trim = 3,
lambda = 1/2, xnames=NULL)

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.

x

A matrix of binary covariates. These are the potential effect modifiers. If x is a binary vector for one covariate, it will be reshaped into a matrix. An error will result if x does not have 1 or 0 coordinates. The matrix x should length(y) rows.

expandx

If expandx=FALSE, then x is used as is. If expandx=TRUE, then a column of 1's is added to x, as well as 1-x[,j] for each column j. For example, if x is a single column, 1 for female, 0 for male, then with expandx=TRUE, x will be replaced by 3 columns, all 1's, female, and 1-female=male.

scale

scale determines how the observations are scaled in computing M-scores. scale must equal "closed"" or "global" or "interaction". If you use the mean (or equivalently the total) as your test statistic (by setting inner=0 and trim=Inf), then scaling is not needed and the scale parameter is ignored. If scale=global, then all observations are used in computing a single scale factor. This is appropriate when testing Fisher's global hypothesis of no treatment effect at all. If scale=interaction, then the scale is determined separately in each of the unique groups formed from the interaction of all of the columns of x. This can be reasonable if the interaction groups are not too small. If x had a column for men, a column for women, a column for people over 50, a column for people under 50, then there would be 4 interaction groups, such as men under 50. If scale=closed, then a single scale factor is computed using every y[i] such that x[i,j]=1 for at least one j. With scale=closed, the score() function will return matrices y and cmat that only contain the rows i such that x[i,j]=1 for at least one j. For instance, if your hypotheses are confined to women, then the men will be excluded. score=closed is useful in closed testing, as described in Lee et al. (2017). If x contains a column of 1's, then scale=closed and scale=global are equivalent. Of course, x will contain a column of 1's if you set expandx=TRUE. You can use scale=interaction only if sets are exactly matched for all effect modifiers in x, but this is not required for scale=global or scale=closed. If you set scale=interaction when sets are not exactly matched, you will receive a warning and scale will be reset to the default of closed.

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 or total 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.

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.

xnames

If xnames=NULL and x is a matrix or data.frame with column names, then those names are used to label output. If xnames is not null, and x is a matrix, then xnames must be a vector of dim(x)[2] names, and these names are used to label output. If x is a vector, not a matrix, then the output will be easier to read if you give it a name, xnames="a.name". This is particularly true if x is a vector and expandx=TRUE.

Details

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 and inner = 0 does no trimming and is similar to a mean or a weighted mean.

Taking inner > 0 often increases design sensitivity; see Rosenbaum (2013). This is most evident with matched pairs, where inner=0.5 may be a good choice.

An M-statistic similar to a trimmed mean is obtained by setting inner=0, trim=1, and 1-lambda to the total amount to be trimmed from both tails. For example, inner=0, trim=1, lambda=0.9 trims 10 percent, perhaps 5 percent from each tail. Arguably, inner=0, trim=1, and lambda=0.99 is very much like a mean, but also safer than a mean.

In general, each call to submax() tests a global null hypothesis of no treatment effect, but does this by looking in several subgroups defined by pretreatment covariates, that is, by potential effect modifiers. We often want to say something about the subgroups themselves, not the global null hypothesis. This is possible by using closed testing (Marcus et al. 1976), which may entail several calls to submax(). Before using closed testing, it is suggested that you read the section in Lee et al. (2017) discussing closed testing.

Matched sets of unequal size are weighted with a view to efficiency. See the documentation for mscorev() and the references mentioned there.

Value

y

A matrix of M-scores suitable for use as the y matrix in the submax() function.

cmat

A matrix of comparisons suitable for use as the cmat matrix in the submax funtion.

detail

A data.frame reminding you of the settings that produced the M-scores. It contains inner, trim, lambda, scale, permutational.t, and anyinexact, where permutational.t=TRUE if you set inner=0 and trim=Inf, and anyinexact=TRUE if some of the potential effect modifiers are not exactly matched.

Note

Several other packages use M-scores, so their examples and documentation may be helpful. These include sensitivitymv, sensitivitymw, sensitivitymult, sensitivityfull, and senstrat. In particular, sensitivitymw and sensitivitymult compute sensitivity analyses for confidence intervals.

If you have inexactly matched sets, but you want to use scale=interaction, then you must discard the inexactly matched sets before using score().

Author(s)

Paul R. Rosenbaum.

References

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

Lee, K., Small, D. S., & Rosenbaum, P. R. (2017). A new, powerful approach to the study of effect modification in observational studies. arXiv:1702.00525.

Marcus, R., Eric, P., & Gabriel, K. R. (1976). On closed testing procedures with special reference to ordered analysis of variance. Biometrika, 63, 655-660.

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.)

Examples

data(mercury)
attach(mercury)
# The mercury data has two binary covariates, black and female,
# that will be considered as potential effect modifiers.
# Both black and female are not exactly matched.  Of 397
# matched sets, 72 contain three blacks, 319 contain no
# blacks, 3 contain one black, and 3 contain 2 blacks.
table(table(mset,black)[,2])
# A similar situation arises with females.
table(table(mset,female)[,2])
# When considering females as an effect modifier, only
# sets exactly matched for female are used, etc.  A
# set that is inexact for black may be used when looking
# at females, providing that set is exactly matched for female.


male<-1-female
nonblack<-1-black
everyone<-rep(1,dim(mercury)[1])
x<-cbind(everyone,female,male,black,nonblack)
sc<-score(methylmercury,fish,mset,x)

# At gamma=4, the global null of no effect is
# rejected at alpha=0.05 by every subgroup test
submax(sc$y,sc$cmat,gamma=4,fast=TRUE)

# What does expandx do?
sc<-score(methylmercury,fish,mset,cbind(female,black))
head(sc$cmat)
sc<-score(methylmercury,fish,mset,cbind(female,black),expandx=TRUE)
head(sc$cmat)

# Using exandx with a vector: remember to give it a name.
sc<-score(methylmercury,fish,mset,female,xnames="Female",expandx=TRUE)
head(sc$cmat)

 ## Not run: 
# For closed testing, the process is repeated with fewer columns.
# In general, if cmat has L columns, closed testing may require
# up to (2^L)-1 tests.  Here are two of those tests.
  sc<-score(methylmercury,fish,mset,cbind(female,black))
  submax(sc$y,sc$cmat,gamma=4)
# Note that the critical.constant has become smaller, making it
# easier to reject a component hypothesis when fewer hypotheses
# are tested.
  sc<-score(methylmercury,fish,mset,female,xnames="Female")
  submax(sc$y,sc$cmat,gamma=4,fast=TRUE)
# Use of closed testing is discussed in Lee et al. (2017).

# For a two-sided test, change alpha and do 2 tests.
  submax(sc$y,sc$cmat,gamma=4,alpha=0.025,alternative = "greater")
  submax(sc$y,sc$cmat,gamma=4,alpha=0.025,alternative = "less")
# So we reject in the positive direction in all 5 component tests.
  
## End(Not run)
detach(mercury)