Open Access
September 2016 Using Scheffé projections for multiple outcomes in an observational study of smoking and periodontal disease
Paul R. Rosenbaum
Ann. Appl. Stat. 10(3): 1447-1471 (September 2016). DOI: 10.1214/16-AOAS942


In an observational study of the effects caused by treatments, a sensitivity analysis asks about the magnitude of bias from unmeasured covariates that would need to be present to alter the conclusions of a naive analysis that presumes adjustments for measured covariates remove all biases. When there are two or more outcomes in an observational study, these outcomes may be unequally sensitive to unmeasured biases, and the least sensitive finding may concern a combination of several outcomes. A method of sensitivity analysis is proposed using Scheffé projections that permits the investigator to consider all linear contrasts in two or more scored outcomes while controlling the family-wise error rate. In sufficiently large samples, the method will exhibit insensitivity to bias that is greater than or equal to methods, such as the Bonferroni–Holm procedure, that focus on individual outcomes; that is, Scheffé projections have larger design sensitivities. More precisely, if the least sensitive linear combination is a single one of the several outcomes, then the design sensitivity using Scheffé projections equals that using a Bonferroni correction, but if the least sensitive combination is a nontrivial combination of two or more outcomes, then Scheffé projections have larger design sensitivities. This asymptotic property is examined in terms of finite sample power of sensitivity analyses using simulation. The method is applied to a replication with recent data of a well-known study of the effects of smoking on periodontal disease. In the example, the comparison that is least sensitive to bias from unmeasured covariates combines results for lower and upper teeth, but emphasizes lower teeth. This pattern would be difficult to anticipate prior to examining the data, but Scheffé’s method permits use of this unanticipated pattern without fear of capitalizing on chance.


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Paul R. Rosenbaum. "Using Scheffé projections for multiple outcomes in an observational study of smoking and periodontal disease." Ann. Appl. Stat. 10 (3) 1447 - 1471, September 2016.


Received: 1 August 2015; Revised: 1 January 2016; Published: September 2016
First available in Project Euclid: 28 September 2016

zbMATH: 06775273
MathSciNet: MR3553231
Digital Object Identifier: 10.1214/16-AOAS942

Keywords: Causal inference , design sensitivity , observational study , Scheffe projection , sensitivity analysis

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.10 • No. 3 • September 2016
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