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Table 1 Bias in estimating the association between uptake of B and prior use of A

From: Appropriate and inappropriate methods for investigating the “gateway” hypothesis, with a review of the evidence linking prior snus use to later cigarette smoking

  Neither (i) A only (ii) B only (iii) Both – A first (iv) Both – B first (v)
Time 0 10000 0 0 0 0
 Start using A 250 i → ii     
 Start using B 500 i → iii     
Time 1 10000 − 250 − 500 = 9250 250 500 0 0
 Start using A 231 i → ii   13 iii → v   
 Start using B 463 i → iii 13 ii → iv    
Time 2 9250 − 231 − 463 = 8556 250 + 231 − 13 = 469 500 + 463 − 13 = 950 13 13
 Start using A 214 i → ii   24 iii → v   
 Start using B 428 i → iii 23 ii → iv    
Time 3 8556 − 214 − 428 = 7915 469 + 214 − 23 = 659 950 + 428 − 24 = 1354 13 + 23 = 36 13 + 24 = 36
 Start using A 198 i → ii   34 iii → v   
 Start using B 396 i → iii 33 ii → iv    
Time 4 7915 − 198 − 396 = 7321 659 + 198 − 33 = 824 1354 + 396 − 34 = 1716 36 + 33 = 69 36 + 34 = 70
P1: Proportion who start B among A first users 69/(824 + 69) =0.08
P2: Proportion who start B among others (1716 + 70)/(7321 + 1716 + 70) =0.20
P3: Proportion who start B among never A users 1716/(7321 + 1716) =0.19
OR1 as defined in (using P1 and P2) (0.08/(1-0.08))/(0.20/(1-0.20)) =0.34
OR2 as defined in (using P1 and P3) (0.08/(1-0.08))/(0.19/(1-0.19)) =0.36
RR stratified by period (using columns i and ii) Period 1–2 (13/250)/(463/9250) =1
  Period 2–3 (23/469)/(428/8556) =1
  Period 3–4 (33/659)/(396/7915) =1
  1. The example involves four time periods, with starting of A and B assumed to be independent. Rows in normal type style indicate time points, and rows in italicized type style indicate the periods between the time points. In each period, 2.5% of those who have never used A by the previous time point start A, and 5% of those who have never used B by the previous time point start B. The arrows and roman numerals indicate the column to which starters are re-allocated at the next time point. It is assumed that no individual starts both products in a single period. The values are shown rounded to nearest whole numbers but are calculated to full precision (so that there are some apparent discrepancies in the additions).