# Sensitivity Analysis for a Full Match in an Observational Study. 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 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 senfmCI(). ```r senfm(y, treated1, gamma = 1, inner = 0, trim = 3, lambda = 1/2, tau = 0, alternative="greater") ``` ## 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 can contain NA's, y[i,1] and y[i,2] must not be NA for every i. - `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 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 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. - `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. ## Details For the given $\Gamma$, senfm() 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, senfm() 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). ## Returns - **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. ## References Cox, D. R. (1977). The role of signficance tests (with Discussion). Scand. J. Statist. 4, 49-70. 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) Rosenbaum, P. R. and Small, D. S. (2016). An adaptive Mantel-Haenszel test for sensitivity analysis in observational studies. Biometrics, to appear. ## Author(s) Paul R. Rosenbaum. ## Note The function senfm() 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 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 and sensitivity2x2xk. sensitivitymv is for matched sets with one treated subject and a variable number of controls. sensitivitymw is for matched sets with one treated subject and a fixed number of controls, including matched pairs. For their special cases, sensitivitymv and sensitivitymw contain additional features not available in sensitivityfull. sensitivitymw is faster and computes confidence intervals and point estimates. sensitivitymw also implements methods from Rosenbaum (2014). sensitivity2x2xk is for 2x2xk contingency tables, treatment x outcome x covariates; see Rosenbaum and Small (2016). Rosenbaum (2007) describes the method for matching with variable numbers of controls, but only very minor adjustments are required for full matching, and senfm() implements these adjustments. ## Examples ```r # 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 test of no effect, Huber scores: senfm(y,treated1) #Sensitivity analysis, Huber scores: senfm(y,treated1,gamma=2) #Randomization test of tau=1 vs tau>1 senfm(y,treated1,tau=1) #Randomization test of tau=1 vs tau<1 senfm(y,treated1,tau=1,alternative="less") #Same randomization test of tau=1 vs tau<1 senfm(-y,treated1,tau=-1) #Sensitivity analysis testing tau=1 at gamma=2 senfm(y,treated1,tau=1,gamma=2,alternative="greater") senfm(y,treated1,tau=1,gamma=2,alternative="less") # For an additional example, install and load package sensitivitymv # The following example is a match with variable controls. # So this example has one treated subject per matched set. # Both mscorev (in sensitivitymv) and mscoref (in sensitivityfull) # reproduce parts of the example in Rosenbaum (2007, Section 4). # data(tbmetaphase) # senmv(tbmetaphase,gamma=2,trim=1) # senfm(tbmetaphase,rep(TRUE,15),gamma=2,trim=1) # senmv(tbmetaphase,gamma=2,trim=1,tau=0.94) # senfm(tbmetaphase,rep(TRUE,15),gamma=2,trim=1,tau=.94) # senmv(tbmetaphase,gamma=2,trim=1,tau=0.945) # senfm(tbmetaphase,rep(TRUE,15),gamma=2,trim=1,tau=.945) # mscoref(tbmetaphase,rep(TRUE,15),trim=1) ```

senfm function