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

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

	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

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).