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Table 3 Lower and upper bounds for 10-year counterfactual risks and per-protocol effects among individuals 55–64 years old (units = cases/100 persons for risks and risk differences)

From: Bounding the per-protocol effect in randomized trials: an application to colorectal cancer screening

  CRC incidence CRC mortality All-cause mortality
  Lower bound Upper bound Lower bound Upper bound Lower bound Upper bound
No Assumptions
 Risk under no screening 1.2 17.4 0.4 16.6 9.1 25.3
 Risk under screening 0.2 84.0 0.03 83.9 1.1 84.9
 Risk difference −17.2 82.8 −16.5 83.5 −24.2 75.8
 Risk ratio 0.01 68.95 0.00 214.54 0.04 9.32
Instrumental conditions
 Overall
  Risk under no screeninga 1.4 0.4 10.2
  Risk under screening 0.7 35.9 0.1 35.3 4.3 39.5
  Risk difference −0.7 34.5 −0.3 34.9 −5.9 29.3
  Risk ratio 0.49 24.88 0.28 80.64 0.42 3.86
 Among the “never-takers” (35 %)b
  Risk under no screeninga 1.5 0.7 16.2
  Risk under screening 0.0 100.0 0.0 100.0 0.0 100.0
  Risk difference −1.5 98.5 −0.7 99.3 −16.2 83.8
  Risk ratio 0.00 64.95 0.00 141.35 0.00 6.17
 Among the “compliers” (65 %)a,b
  Risk under no screening 1.4 0.3 7.0
  Risk under screening 1.1 0.2 6.7
  Risk difference −0.3 −0.1 −0.3
  Risk ratio 0.79 0.66 0.96
Instrumental conditions and additive effect homogeneitya
 Risk under no screening 1.4 0.4 10.2
 Risk under screening 1.1 0.3 9.9
 Risk difference −0.3 −0.1 −0.3
 Risk ratio 0.80 0.77 0.97
Instrumental conditions and multiplicative effect homogeneitya
 Risk under no screening 1.4 0.4 10.2
 Risk under screening 1.1 0.3 9.8
 Risk difference −0.3 −0.1 −0.5
 Risk ratio 0.79 0.66 0.96
  1. aPoint identification is achieved under these conditions in the NORCCAP trial
  2. bIn this particular study the distribution of compliance types is known given instrumental conditions. In other study designs, identifying the counterfactual risks and treatment effects within compliance types requires an additional assumption of an assumed feasible distribution of compliance types