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Sensitivity analysis for stratified comparisons in an observational study of the effect of smoking on homocysteine levels. (English) Zbl 1412.62176
Summary: Sensitivity bounds for randomization inferences exist in several important cases, such as matched pairs with any type of outcome or binary outcomes with any type of stratification, but computationally feasible bounds for any outcome in any stratification are not currently available. For instance, with 20 strata, some large, others small, there is no currently available, computationally feasible sensitivity bound testing the null hypothesis of no treatment effect in the presence of a bias from nonrandom treatment assignment of a specific magnitude. The current paper solves the general problem; it uses an inequality formed by taking a one-step Taylor approximation from a near extreme solution, known as the separable approximation, where the concavity of the underlying function ensures that the Taylor approximation is, at worst, conservative. In practice, the separable approximation and the one-step movement away from it provide computationally feasible lower and upper bounds, thereby providing both a usable, perhaps slightly conservative statement, together with a check that the conservative statement is not unduly conservative. In every example that I have tried, the upper and lower bounds barely differ, although with some effort one can construct examples in which the separable approximation gives a \(P\)-value of 0.0499 and the Taylor approximation gives 0.0501. The new inequality holds in finite samples, so it strengthens certain existing asymptotic results, additionally simplifying the proof of those results. The method is discussed in the context of an observational study of the effects of smoking on homocysteine levels, a possible risk factor for several diseases including cardiovascular disease, thrombosis and Alzheimer’s disease. This study contains two evidence factors, the comparison of smokers and nonsmokers and the comparison of smokers to one another in terms of recent nicotine exposure. A new \(\mathtt{R}\) package, \(\mathtt{senstrat}\), implements the procedure and illustrates it with the example from the current paper.

MSC:
62P10 Applications of statistics to biology and medical sciences; meta analysis
62F03 Parametric hypothesis testing
62J10 Analysis of variance and covariance (ANOVA)
Software:
senstrat; R
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References:
[1] Bazzano, L. A., He, J., Muntner, P., Vupputuri, S. and Whelton, P. K. (2003). Relationship between cigarette smoking and novel risk factors for cardiovascular disease in the United States. Ann. Intern. Med.138 891–897.
[2] Bertsekas, D. P. (2009). Convex Optimization Theory. Athena Scientific, Nashua, NH. · Zbl 1242.90001
[3] Bickel, P. J. and van Zwet, W. R. (1978). Asymptotic expansions for the power of distribution free tests in the two-sample problem. Ann. Statist.6 937–1004. · Zbl 0378.62047
[4] Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge Univ. Press, Cambridge. · Zbl 1058.90049
[5] Centers for Disease Control (2016). Biomonitoring summary: Cotinine. CAS No. 486-56-6. Available at https://www.cdc.gov/biomonitoring/Cotinine_BiomonitoringSummary.html, dated December 27, 2016.
[6] Cornfield, J., Haenszel, W., Hammond, E. C., Lilienfeld, A. M., Shimkin, M. B. and Wynder, E. L. (1959). Smoking and lung cancer: Recent evidence and a discussion of some questions. J. Natl. Cancer Inst.22 173–203.
[7] Egleston, B. L., Scharfstein, D. O. and MacKenzie, E. (2009). On estimation of the survivor average causal effect in observational studies when important confounders are missing due to death. Biometrics65 497–504. · Zbl 1167.62083
[8] Fisher, R. A. (1935). The Design of Experiments. Oliver & Boyd, Edinburgh.
[9] Fogarty, C. B. and Small, D. S. (2016). Sensitivity analysis for multiple comparisons in matched observational studies through quadratically constrained linear programming. J. Amer. Statist. Assoc.111 1820–1830.
[10] Gastwirth, J. L., Krieger, A. M. and Rosenbaum, P. R. (2000). Asymptotic separability in sensitivity analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol.62 545–555. · Zbl 0953.62041
[11] Gilbert, P. B., Bosch, R. J. and Hudgens, M. G. (2003). Sensitivity analysis for the assessment of causal vaccine effects on viral load in HIV vaccine trials. Biometrics59 531–541. · Zbl 1210.62161
[12] Hankey, G. J. and Eikelboom, J. W. (1999). Homocysteine and vascular disease. Lancet354 407–413.
[13] Hansen, B. B. (2004). Full matching in an observational study of coaching for the SAT. J. Amer. Statist. Assoc.99 609–618. · Zbl 1117.62349
[14] Hodges, J. L. Jr. and Lehmann, E. L. (1962). Rank methods for combination of independent experiments in analysis of variance. Ann. Math. Stat.33 482–497. · Zbl 0112.10303
[15] Hosman, C. A., Hansen, B. B. and Holland, P. W. (2010). The sensitivity of linear regression coefficients’ confidence limits to the omission of a confounder. Ann. Appl. Stat.4 849–870. · Zbl 1194.62089
[16] Huber, P. J. (1981). Robust Statistics. Wiley, New York. · Zbl 0536.62025
[17] Lehmann, E. L. (1975). Nonparametrics: Statistical Methods Based on Ranks. Holden-Day, San Francisco, CA. · Zbl 0354.62038
[18] Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer Texts in Statistics. Springer, New York. · Zbl 1076.62018
[19] Liu, W., Kuramoto, J. and Stuart, E. (2013). Sensitivity analysis for unobserved confounding in nonexperimental prevention research. Prev. Sci.14 570–580.
[20] Mantel, N. (1963). Chi-square tests with one degree of freedom; extensions of the Mantel–Haenszel procedure. J. Amer. Statist. Assoc.58 690–700. · Zbl 0114.11601
[21] Maritz, J. S. (1979). A note on exact robust confidence intervals for location. Biometrika66 163–166. · Zbl 0396.62026
[22] Mehrotra, D. V., Lu, X. and Li, X. (2010). Rank-based analyses of stratified experiments: Alternatives to the van Elteren test. Amer. Statist.64 121–130. · Zbl 1319.62097
[23] Neyman, J. (1923). On the application of probability theory to agricultural experiments. Ann. Agric. Sci.10 1–51. [Translated from the Polish and edited by D. M. Da̧browska and T. P. Speed in Statist. Sci.5 (1990) 465–472. ]
[24] Pimentel, S. D., Small, D. S. and Rosenbaum, P. R. (2016). Constructed second control groups and attenuation of unmeasured biases. J. Amer. Statist. Assoc.111 1157–1167.
[25] Puri, M. L. (1965). On the combination of independent two somple tests of a general class. Rev. Inst. Int. Stat.33 229–241. · Zbl 0133.41904
[26] Rosenbaum, P. R. (1991). A characterization of optimal designs for observational studies. J. Roy. Statist. Soc. Ser. B53 597–610. · Zbl 0800.62465
[27] Rosenbaum, P. R. (1995). Quantiles in nonrandom samples and observational studies. J. Amer. Statist. Assoc.90 1424–1431. · Zbl 0899.62138
[28] Rosenbaum, P. R. (2002a). Observational Studies, 2nd ed. Springer, New York. · Zbl 0985.62091
[29] Rosenbaum, P. R. (2002b). Covariance adjustment in randomized experiments and observational studies. Statist. Sci.17 286–327. · Zbl 1013.62117
[30] Rosenbaum, P. R. (2014). Weighted \(M\)-statistics with superior design sensitivity in matched observational studies with multiple controls. J. Amer. Statist. Assoc.109 1145–1158. · Zbl 1368.62290
[31] Rosenbaum, P. R. (2017a). Observation and Experiment: An Introduction to Causal Inference. Harvard Univ. Press, Cambridge, MA. · Zbl 1372.00054
[32] Rosenbaum, P. R. (2017b). The general structure of evidence factors in observational studies. Statist. Sci.32 514–530. · Zbl 1384.62014
[33] Rosenbaum, P. R. and Krieger, A. M. (1990). Sensitivity of two-sample permutation inferences in observational studies. J. Amer. Statist. Assoc.85 493–498. · Zbl 0703.62060
[34] Rosenbaum, P. R. and Small, D. S. (2017). An adaptive Mantel–Haenszel test for sensitivity analysis in observational studies. Biometrics73 422–430. · Zbl 1371.62067
[35] Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. J. Educ. Psychol.66 688–701.
[36] Satagopan, J. M., Offit, K., Foulkes, W., Robson, Wacholder S, M. E., Eng, C. M., Karp, S. E. and Begg, C. B. (2001). The lifetime risks of breast cancer in Ashkenazi Jewish carriers of brca1 and brca2 mutations. Cancer Epidemiol. Biomark. Prev.10 467–473.
[37] Seshadri, S., Beiser, A., Selhub, J., Jacques, P. F., Rosenberg, I. H., D’agostino, R. B., Wilson, P. W. and Wolf, P. A. (2002). Plasma homocysteine as a risk factor for dementia and Alzheimer’s disease. N. Engl. J. Med.346 476–483.
[38] Wald, D. S., Law, M. and Morris, J. K. (2002). Homocysteine and cardiovascular disease: Evidence on causality from a meta-analysis. Br. Med. J.325 1202–1209.
[39] Welch, G. N. and Loscalzo, J. (1998). Homocysteine and atherothrombosis. N. Engl. J. Med.338 1042–1050.
[40] Werfel, U., Langen, V., Eickhoff, I., Schoonbrood, J., Vahrenholz, C., Brauksiepe, A., Popp, W. and Norpoth, K. (1998). Elevated DNA single-strand breakage frequencies in lymphocytes of welders exposed to chromium and nickel. Carcinogenesis19 413–418.
[41] Yu, B. B. and Gastwirth, J. L. (2005). Sensitivity analysis for trend tests: Application to the risk of radiation exposure. Biostatistics6 201–209. · Zbl 1071.62112
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