# Sensitivity Analysis for a Matched Comparison in an Observational Study. 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. Matched sets of different sizes use different $\psi$-functions, creating what is called a fugue statistic. Performs either a randomization test (Gamma=1) or an analysis of sensitivity to departures from random assignment (Gamma>1). For confidence intervals, use function senCI(). The method is described in Li and Rosenbaum (2019); see also Rosenbaum (2007,2013). ```r sen(y, z, mset, gamma = 1, inner = NULL, trim = NULL, lambda = 1/2, tau = 0, alternative = "greater") ``` ## 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. If the largest matched set has k controls, then inner is either a scalar or a vector with k=length(inner). If inner is a scalar, then the same value of inner is used, regardless of the number of controls. Otherwise, inner[1] is used with one control, inner[2] is used with two controls, etc. If inner is NULL, default values of inner=c(.8,.8,.6,.4,0,0,0,...,0) are used. - `trim`: inner and trim together define the $\psi$-function for the M-statistic. If the largest matched set has k controls, then trim is either a scalar or a vector with k=length(trim). If trim is a scalar, then the same value of trim is used, regardless of the number of controls. Otherwise, trim[1] is used with one control, trim[2] is used with two controls, etc. If trim is NULL, default values of trim=c(3,3,...,3) are used. For each i, 0 <= inner[i] < trim[i] < Inf. - `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. ## Details The novel element in the fugue package is the automatic use of different $\psi$-functions for matched sets of different sizes. These $\psi$-functions have been selected to approximately equate the design sensitivities in sets of unequal sizes when the errors are Normal and the additive effect is half the standard deviation of a matched pair difference; see Li and Rosenbaum (2019). If you disable this automatic feature by manually setting a single value for inner and trim, then the results will agree with senm() in the R package sensitivitymult. For instance, using both sen() in the fugue package and senm() in the sensitivitymult package will yield the same deviate and P-value in: data(nh1and3) attach(nh1and3) sen(homocysteine,z,mset,inner=0,gamma=1.9) senm(homocysteine,z,mset,inner=0,trim=3,gamma=1.9) Note that the sensitivitymult package is intended to implement methods from Rosenbaum (2016,2019) that are not implemented in the fugue package. For the given $\Gamma$, sen() 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, sen() 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, 2018). ## 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. Huber, P. (1981) Robust Statistics. New York: John Wiley. (M-estimates based on M-statistics.) Li, X. and Rosenbaum, P. R. (2019) Maintaining high constant design sensitivity in observational studies with matched sets of varying sizes. Manuscript. 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) 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. (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. Rosenbaum, P. R. (2018). Sensitivity analysis for stratified comparisons in an observational study of the effect of smoking on homocysteine levels. The Annals of Applied Statistics, 12(4), 2312-2334. Rosenbaum, P. R. (2019). Combining planned and discovered comparisons in observational studies. Biostatistics, to appear. ## Author(s) Xinran Li and Paul R. Rosenbaum. ## Note The function sen() 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. The examples reproduce some results from Li and Rosenbaum (2019). ## Examples ```r # Reproduces results from Table 3 of Li and Rosenbaum (2019) data(nh1and3) attach(nh1and3) sen(homocysteine,z,mset,gamma=1) sen(homocysteine,z,mset,gamma=1.9) sen(homocysteine,z,mset,inner=0,gamma=1.9) amplify(1.9,c(3,3.5,4)) detach(nh1and3) ```

sen function