Statistical Science

When Is a Sensitivity Parameter Exactly That?

Paul Gustafson and Lawrence C. McCandless

Full-text: Open access


Sensitivity analysis is used widely in statistical work. Yet the notion and properties of sensitivity parameters are often left quite vague and intuitive. Working in the Bayesian paradigm, we present a definition of when a sensitivity parameter is “pure,” and we discuss the implications of a parameter meeting or not meeting this definition. We also present a diagnostic with which the extent of violations of purity can be visualized.

Article information

Statist. Sci., Volume 33, Number 1 (2018), 86-95.

First available in Project Euclid: 2 February 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian inference misclassification missing data selection bias sensitivity analysis


Gustafson, Paul; McCandless, Lawrence C. When Is a Sensitivity Parameter Exactly That?. Statist. Sci. 33 (2018), no. 1, 86--95. doi:10.1214/17-STS632.

Export citation


  • Daniels, M. J. and Hogan, J. W. (2008). Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis. Monographs on Statistics and Applied Probability 109. Chapman & Hall/CRC, Boca Raton, FL.
  • Geneletti, S., Best, N., Toledano, M., Elliott, P. and Richardson, S. (2013). Uncovering selection bias in case–control studies using Bayesian post-stratification. Stat. Med. 32 2555–2570.
  • Greenland, S. (1996). Basic methods for sensitivity analysis of biases. Int. J. Epidemiol. 25 1107–1116.
  • Greenland, S. (2003). The impact of prior distributions for uncontrolled confounding and response bias: A case study of the relation of wire codes and magnetic fields to childhood leukemia. J. Amer. Statist. Assoc. 98 47–54.
  • Greenland, S. (2005). Multiple-bias modelling for analysis of observational data. J. Roy. Statist. Soc. Ser. A 168 267–306.
  • Gustafson, P. (2005). On model expansion, model contraction, identifiability, and prior information: Two illustrative scenarios involving mismeasured variables (with discussion). Statist. Sci. 20 111–140.
  • Gustafson, P. (2015). Bayesian Inference for Partially Identified Models: Exploring the Limits of Limited Data. Chapman & Hall/CRC Press, Boca Raton, FL.
  • Gustafson, P., Le, N. D. and Saskin, R. (2001). Case-control analysis with partial knowledge of exposure misclassification probabilities. Biometrics 57 598–609.
  • Gustafson, P., McCandless, L. C., Levy, A. R. and Richardson, S. (2010). Simplified Bayesian sensitivity analysis for mismeasured and unobserved confounders. Biometrics 66 1129–1137.
  • Kahneman, D. (2011). Thinking, Fast and Slow. Macmillan, New York.
  • Lash, T. L., Fox, M. P. and Fink, A. K. (2009). Applying Quantitative Bias Analysis to Epidemiologic Data. Springer, New York.
  • Lin, D. Y., Psaty, B. M. and Kronmal, R. A. (1998). Assessing the sensitivity of regression results to unmeasured confounders in observational studies. Biometrics 54 948–963.
  • MacLehose, R. F. and Gustafson, P. (2012). Is probabilistic bias analysis approximately Bayesian? Epidemiology 23 151–158.
  • McCandless, L. C., Gustafson, P. and Levy, A. R. (2007). Bayesian sensitivity analysis for unmeasured confounding in observational studies. Stat. Med. 26 2331–2347.
  • Phillips, C. V. (2003). Quantifying uncertainty in epidemiologic studies. Epidemiology 14 459–466.
  • Rosenbaum, P. R. (2002). Observational Studies, 2nd ed. Springer, New York.
  • Rosenbaum, P. R. (2010). Design of Observational Studies. Springer, New York.
  • Rosenbaum, P. R. and Rubin, D. B. (1983). Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. J. Roy. Statist. Soc. Ser. B 45 212–218.
  • Scharfstein, D. O., Daniels, M. J. and Robins, J. M. (2003). Incorporating prior beliefs about selection bias into the analysis of randomized trials with missing outcomes. Biostatistics 4 495–512.
  • Xia, M. and Gustafson, P. (2012). A Bayesian method for estimating prevalence in the presence of a hidden sub-population. Stat. Med. 31 2386–2398.
  • Xia, M. and Gustafson, P. (2014). Bayesian sensitivity analyses for hidden sub-populations in weighted sampling. Canad. J. Statist. 42 436–450.