submax: Effect Modification Using the Submax Method in Observational...
In submax: Effect Modification in Observational Studies Using the Submax Method

Description Usage Arguments Details Value Note Author(s) References Examples

Effect modification means that the magnitude or stability of a treatment effect varies with observed covariates. When there is effect modification, causal conclusions may be less sensitive to unmeasured biases in a subgroup in which the treatment effect is larger or more stable. The submax or subgroup maximum method looks at an overall test and subgroup tests, correcting for multiple testing using the joint distribution of the tests. The submax method was proposed by Lee et al. (2017).

1	submax(y, cmat, gamma = 1, alternative = "greater", alpha = 0.05, rnd = 2, fast=FALSE)

`y`	In general, y is either a matrix with I rows for I matched sets or a vector of length I for I matched pairs. The y values are outcomes, perhaps after scoring (e.g., Huber-Maritz M-scores) or ranking (e.g., Wilcoxon) to produce a robust test. If y contains the outcomes themselves, then the outcomes are permuted in the manner of a permutational t-test. More precisely, y contains the scores q[gij] discussed in section 2.1 of Lee et al. (2017). If every matched set gi is a pair, then y may be a vector of treated-minus-control pair differences, q[gi1]-q[gi2]. If there are I matched pairs, then y can be either a vector of length I giving the I treated-minus-control pair differences, or y can be a 2-column matrix with treated responses in the first column and control responses in the second column. 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.
`cmat`	A matrix with one row for each matched set in y. For each column of cmat, a statistical test is done. Typically, the first column is (1,...,1) and refers to a test that uses all of the matched sets. If the second column is 1 for matched sets of women and 0 for matched sets of men, then the second test is restricted to matched sets of women.
`gamma`	gamma is the sensitivity parameter Γ, where Γ ≥ 1. Setting Γ = 1 is equivalent to assuming ignorable treatment assignment given the matched sets, and it performs a within-set randomization test.
`alternative`	If alternative="greater", the null hypothesis of no treatment effect is tested against the alternative of a treatment effect larger than zero. If alternative="less", the null hypothesis of no treatment effect is tested against the alternative of a treatment effect smaller than 0. In particular, alternative="less" is equivalent to: (i) alternative="greater", (ii) y replaced by -y. See the note for discussion of two-sided sensitivity analyses.
`alpha`	The global null hypothesis of no effect is tested at simultaneous level α in the presence of a bias of at most Γ.
`rnd`	The correlation matrix of the dim(cmat)[2] test statistics is returned, rounded to rnd digits. The critical.constant is also rounded to rnd digits.
`fast`	Determines the speed and accuracy of the determination of the critical.constant used in testing. fast=TRUE is faster but less precise, less stable. fast=FALSE is slower but more precise, more stable. See details.

The submax procedure is developed by Lee et al. (2017), and the example reproduces analyses from that paper.

The submax() function rejects the null hypothesis at level α in the presence of a bias in treatment assignment of at most Γ if maxdeviate is greater than or equal to critical.constant.

The global null hypothesis of no effect is tested at simultaneous level α in the presence of a bias of at most Γ. If the global null is true, and the bias in treatment assignment is at most Γ, then the probability of falsely rejecting the global null hypothesis is at most α. The test looks at the largest of dim(cmat)[2] standardized test statistics and corrects for multiple testing using their joint distribution. The joint distribution is approximated by a multivariate Normal distribution, so the entire procedure is a large sample approximation.

The function score() in this package may be helpful in creating the matrics y and cmat that are arguments of the submax function.

The sensitivity bound is based on the separable approximation described in Gastwirth, Krieger and Rosenbaum (2000); see also Rosenbaum (2007).

The critical.constant is determined rather precisely by a call to the qmvnorm() function in the mvtnorm package. The qmvnorm() function uses random numbers, but this should produce negligible variability in the critical.constant. However, if you call submax() twice, there will be a tiny change in the critical.constant. There is a trade-off between precision and speed, and you can alter that trade-off – make submax faster or more precise – by editing the call to qmvnorm() in the rcode for submax(). For almost all users, there will be no need to alter the code.

maxdeviate

The submax or subgroup maximum statistic. It is the maximum standardized deviate for the several columns of cmat. In Lee et al. (2017), this is D[Γ max]

.

`critical.constant`	If maxdeviate >= critical.constant, the global null hypothesis of no treatment effect is rejected at level α in the presence of a bias of at most Γ
`deviates`	There is one standardized deviate for each column of cmat, and the maximum of these is maxdeviate. If deviate[j] > critical.constant, then deviate[j] would lead to rejection of the global null hypothesis.
`correlation`	The correlation matrix of the deviates. By default, it is printed to two digits for easy viewing. By changing rnd=2, it can be produced with additional digits, perhaps for use in further computations.
`detail`	Reminds the user of the value of α and Γ.

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.

A 2-sided, α-level test may be obtained by performing two one-sided tests, each at level α/2. This is a safe approach, but for Γ > 1 it is slightly conservative. Cox (1977) suggests that we view a two-sided test as two one-sided tests with a Bonferroni correction for testing two hypotheses.

Paul R. Rosenbaum

Cox, D. R. (1977). The role of signficance tests (with Discussion). Scand. J. Statist. 4, 49-70.

Gastwirth, J. L., Krieger, A. M. and Rosenbaum, P. R. (2000). Asymptotic separability in sensitivity analysis. J. Roy. Statist. Soc. B. 62 545-555. <doi:10.1111/1467-9868.00249>

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

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

Rosenbaum, P. R. (2007). Sensitivity analysis for m-estimates, tests and confidence intervals in matched observational studies. Biometrics 63 456-64. (See section 4.) <doi:10.1111/j.1541-0420.2006.00717.x>

# Reproduces parts of Table 2 of Lee et al. (2017).
data(Active)
submax(Active$delta,Active[,1:7],gamma=1.77,alternative="less",fast=TRUE)
amplify(1.77,c(2,3,4))

# Reproduces the closed-testing analysis in
#Section 4 of Lee et al. (2017)
submax(Active$delta,Active[,c(3,5)],gamma=1.4,alternative="less",fast=TRUE)

# See also the examples for the score() function.