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Clarifying and Contextualizing Sensitivity to Unmeasured Confounders Tipping Point Analyses

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Vanderbilt Biostatistics Departmental Seminar

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Clarifying and Contextualizing Sensitivity to Unmeasured Confounders Tipping Point Analyses

  1. 1. Clarifying and Contextualizing Sensitivity to Unmeasured Confounding Tipping Point Analyses Lucy D’Agostino McGowan Robert Greevy, Jr
  2. 2. https://ryan-j-apps.shinyapps.io/ seminarSurvival/
  3. 3. motivation
  4. 4. review of 90 observational studies 2015
  5. 5. review of 90 observational studies 2015 45.6% mentioned unmeasured confounding as a limitation
  6. 6. review of 90 observational studies 2015 45.6% mentioned unmeasured confounding as a limitation 4.4% included a quantitative sensitivity analysis 😑
  7. 7. "In a study with binary outcomes and binary exposures the relative risk may be off by a factor of 2, but unlikely to be off more than that." van Belle, G. (2011). Statistical Rules of Thumb. Wiley.
  8. 8. Stampfer, M. J., & Colditz, G. A. (1991). Estrogen replacement therapy and coronary heart disease: A quantitative assessment of the epidemiologic evidence. Preventive Medicine, 20(1), 47–63. http://doi.org/ 10.1016/0091-7435(91)90006-P
  9. 9. Stampfer, M. J., & Colditz, G. A. (1991). Estrogen replacement therapy and coronary heart disease: A quantitative assessment of the epidemiologic evidence. Preventive Medicine, 20(1), 47–63. http://doi.org/ 10.1016/0091-7435(91)90006-P RR 0.5 (0.43,0.56)
  10. 10. Stampfer, M. J., & Colditz, G. A. (1991). Estrogen replacement therapy and coronary heart disease: A quantitative assessment of the epidemiologic evidence. Preventive Medicine, 20(1), 47–63. http://doi.org/ 10.1016/0091-7435(91)90006-P
  11. 11. Roumie, C. L., Min, J. Y., D’Agostino McGowan, L., Presley, C., Grijalva, C. G., Hackstadt, A. J., et al. (2017). Comparative Safety of Sulfonylurea and Metformin Monotherapy on the Risk of Heart Failure: A Cohort Study. Journal of the American Heart Association, 6(4), e005379. http://doi.org/10.1161/JAHA.116.005379
  12. 12. Roumie, C. L., Min, J. Y., D’Agostino McGowan, L., Presley, C., Grijalva, C. G., Hackstadt, A. J., et al. (2017). Comparative Safety of Sulfonylurea and Metformin Monotherapy on the Risk of Heart Failure: A Cohort Study. Journal of the American Heart Association, 6(4), e005379. http://doi.org/10.1161/JAHA.116.005379 HR 1.32 (1.21,1.43)
  13. 13. background
  14. 14. Cornfield 1959
  15. 15. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  16. 16. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  17. 17. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  18. 18. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  19. 19. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  20. 20. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  21. 21. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  22. 22. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  23. 23. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  24. 24. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  25. 25. Cornfield 1959 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  26. 26. Cornfield 1959 9 R1 R0 = p1RU + (1 p1)R¯U p0RU + (1 p0)R¯U
  27. 27. Cornfield 1959 p1 p0 = R1 R0 + 1 p0 R¯U RU  (1 p0) R1 R0 (1 p1)
  28. 28. Cornfield 1959 p1 p0 = R1 R0 + 1 p0 R¯U RU  (1 p0) R1 R0 (1 p1) +
  29. 29. Cornfield 1959 p1 p0 > R1 R0
  30. 30. Cornfield 1959 p1 p0 > 9
  31. 31. Cornfield 1959 the proportion of hormone-X- producers among cigarette smokers must be at least 9 times greater than that of nonsmokers...
  32. 32. Cornfield 1959 Bross 1966
  33. 33. Cornfield 1959 Bross 1966 Schlesselman 1978
  34. 34. Cornfield 1959 Bross 1966 Schlesselman 1978 RRobs = RRadj p1 + (1 p1) p0 + (1 p0)
  35. 35. Cornfield 1959 Bross 1966 Schlesselman 1978 RRobs = RRadj p1 + (1 p1) p0 + (1 p0) R1 R0
  36. 36. Cornfield 1959 Bross 1966 Schlesselman 1978 RRobs = RRadj p1 + (1 p1) p0 + (1 p0)
  37. 37. Cornfield 1959 Bross 1966 Schlesselman 1978 RRobs = RRadj p1 + (1 p1) p0 + (1 p0) RU R¯U
  38. 38. Cornfield 1959 Bross 1966 Schlesselman 1978 RRobs = RRadj p1 + (1 p1) p0 + (1 p0)
  39. 39. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983
  40. 40. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998
  41. 41. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998RRadj = RRobs p0 + (1 p0) p1 + (1 p1)
  42. 42. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998RRadj = RRobs p0 + (1 p0) p1 + (1 p1)
  43. 43. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998RRadj = RRobs p0 + (1 p0) p1 + (1 p1)
  44. 44. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998RRadj = RRobs p0 + (1 p0) p1 + (1 p1) HRadj HRobs
  45. 45. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998RRadj = RRobs p0 + (1 p0) p1 + (1 p1) HRadj HRobs
  46. 46. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998RRadj = RRobs p0 + (1 p0) p1 + (1 p1) HRadj HRobs
  47. 47. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998 what we fit P(Y = 1|X, Z) = exp{↵⇤ + ⇤ X + ✓⇤0 Z}
  48. 48. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998 what we wish we fit exp{↵ + X + ✓0 Z}(exp{ X }pX,Z + (1 pX,Z))
  49. 49. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998 what we wish we fit exp{↵ + X + ✓0 Z}(exp{ X }pX,Z + (1 pX,Z))
  50. 50. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998 what we wish we fit exp{↵ + X + ✓0 Z}(exp{ X }pX,Z + (1 pX,Z))PX PX
  51. 51. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998 what we wish we fit exp n ↵ + log{e 0 p0 + (1 p0)} + ✓ + log e 1 p1 + (1 p1) e 0 p0 + (1 p0) ◆ X + ✓0 Z o
  52. 52. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998 what we wish we fit exp n ↵ + log{e 0 p0 + (1 p0)} + ✓ + log e 1 p1 + (1 p1) e 0 p0 + (1 p0) ◆ X + ✓0 Z o
  53. 53. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998 solving for what we wish we fit = ⇤ log e 1 p1 + (1 p1) e 0 p0 + (1 p0)
  54. 54. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998RRadj = RRobs p0 + (1 p0) p1 + (1 p1)
  55. 55. Cornfield 1959 Bross 1966 Schlesselman 1978 Rosenbaum & Rubin 1983 Lin 1998 Ding & VanderWeele 2016
  56. 56. tipping point
  57. 57. what will tip our confidence bound to cross 1
  58. 58. what will tip our confidence bound to cross 1 assuming the sensitivity parameters are fixed and known, we can extend these methods to confidence bounds
  59. 59. RRadj = RRobs p0 + (1 p0) p1 + (1 p1)
  60. 60. LBadj = LBobs p0 + (1 p0) p1 + (1 p1)
  61. 61. 1 LBadj = LBobs p0 + (1 p0) p1 + (1 p1)
  62. 62. 1 LBadj = LBobs p0 + (1 p0) p1 + (1 p1)
  63. 63. (LBobs, p0, p1) = LBobs(1 p0) (1 p1) p1 LBobsp0
  64. 64. (LBobs, p0, p1) = LBobs(1 p0) (1 p1) p1 LBobsp0
  65. 65. "in order for our association to no longer be significant, there would need to exist an unmeasured confounder of size that is prevalent in p1 of the exposed population and p0 of the unexposed population”
  66. 66. "in order for our association to no longer be significant, there would need to exist an unmeasured confounder of size that is prevalent in p1 of the exposed population and p0 of the unexposed population”
  67. 67. (LBobs, p0, p1) = LBobs(1 p0) (1 p1) p1 LBobsp0
  68. 68. (LBobs, p0, p1) = LBobs(1 p0) (1 p1) p1 LBobsp0
  69. 69. p1 = 0.096 p0 = 0.038
  70. 70. p1 = 0.096 p0 = 0.038 = 2.34
  71. 71. calculate it
  72. 72. LB = 1.21 p1 = 0.096 p0 = 0.038 (LBobs, p0, p1) = LBobs(1 p0) (1 p1) p1 LBobsp0
  73. 73. LB = 1.21 p1 = 0.096 p0 = 0.038 0.1 (LBobs, p0, p1) = LBobs(1 p0) (1 p1) p1 LBobsp0
  74. 74. LB = 1.21 p1 = 0.096 p0 = 0.038 0.1 0 (LBobs, p0, p1) = LBobs(1 p0) (1 p1) p1 LBobsp0
  75. 75. > devtools::install_github(“LucyMcGowan/tipr”) > library(‘tipr’) > tip(p1 = 0.1, p0 = 0, lb = 1.21, ub = 1.43) [1] 3.1
  76. 76. > tip(p1 = 0.1, p0 = 0, lb = 1.21, ub = 1.43, explanation = TRUE) [1] "An unmeasured confounder of size 3.1 with a prevalence of 0.1 in the exposed population and 0 in the unexposed population would tip your (1.21,1.43) result to nonsignificance."
  77. 77. lucy.shinyapps.io/tipr
  78. 78. anchor it
  79. 79. p1 = 0.096 p0 = 0.038
  80. 80. p1 = 0.096 p0 = 0.038 = 2.34
  81. 81. A hypothetical unobserved binary confounder with a 10% prevalence difference between the therapies would need to have an association with heart failure of HR=3.1 to tip this analysis to nonsignificance at a 5% level.
  82. 82. For a comparison from the observed confounders, baseline heart failure history had a prevalence difference of 5.8% in the pre-matching cohort and an association with CHF of HR=2.34; thus an unmeasured confounder of this magnitude would be insufficient to tip this analysis to statistical insignificance.
  83. 83. write up
  84. 84. 1. state primary analysis result
  85. 85. 1. state primary analysis result
  86. 86. 1. state primary analysis result 2. calculate the size & prevalence of a hypothetical tipping point confounder
  87. 87. 1. state primary analysis result 2. calculate the size & prevalence of a hypothetical tipping point confounder
  88. 88. 1. state primary analysis result 2. calculate the size & prevalence of a hypothetical tipping point confounder 3. anchor this in an example from your data
  89. 89. conclusion
  90. 90. we have presented a useful, easily implemented, and intuitively understood approach to allow researchers to assess the potential impact of unmeasured confounders in observational research
  91. 91. we have presented a useful, easily implemented, and intuitively understood approach to allow researchers to assess the potential impact of unmeasured confounders in observational research can be applied to both past and future research, allowing readers to understand the sensitivity of studies that do not include such an analysis, and allowing future investigators to readily include such an analysis.
  92. 92. On the other hand, it took us embarrassingly long to clue in to the lung cancer/cigarette thing, so I guess the real lesson is "figuring out which ideas are true is hard." https://xkcd.com/1592
  93. 93. tid bits
  94. 94. what about multiple unmeasured confounders?
  95. 95. bias factors
  96. 96. design sensitivity
  97. 97. next steps
  98. 98. tutorial
  99. 99. tutorial bias factors
  100. 100. tutorial bias factors design sensitivity
  101. 101. tutorial bias factors design sensitivity puppets

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