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Table 3 Estimates of ORSCLORVF, ORMH, and ORG from simulated data with a binary exposure and a binary time-varying confounder (study period = M + 1 days). Results are shown graphically in Fig. 3b

From: A novel weighting method to remove bias from within-subject exposure dependency in case-crossover studies

M variables ORSCL (95%CI) ORVF (95%CI) ORMH (95%CI) ORG (2.5–97.5pct)
1 exp(β) 4.26 (2.72–6.66) 4.27 (2.73–6.68) 4.55 (2.37–8.73) 4.26 (2.85–7.33)
  exp(γ) 1.68 (1.02–2.78) 1.86 (1.11–3.11)   1.68 (1.03–2.96)
2 exp(β) 4.04 (2.85–5.75) 4.04 (2.85–5.73) 3.82 (2.49–5.86) 4.04 (2.78–6.11)
  exp(γ) 1.84 (1.15–2.94) 2.06 (1.28–3.30)   1.84 (1.19–2.92)
3 exp(β) 3.62 (2.60–5.04) 4.11 (2.83–5.97) 4.17 (2.71–6.43) 4.10 (2.93–5.82)
  exp(γ) 2.12 (1.43–3.12) 1.93 (1.31–2.86)   2.21 (1.45–3.37)
4 exp(β) 3.91 (2.86–5.34) 4.19 (3.03–5.79) 3.91 (2.66–5.75) 4.09 (3.00–5.67)
  exp(γ) 2.14 (1.48–3.11) 2.43 (1.66–3.54)   2.20 (1.45–3.26)
5 exp(β) 4.36 (3.18–5.97) 4.07 (3.00–5.51) 4.04 (2.78–5.86) 4.02 (2.93–5.57)
  exp(γ) 1.86 (1.29–2.66) 2.35 (1.62–3.41)   1.92 (1.28–2.86)
6 exp(β) 4.78 (3.51–6.52) 3.98 (2.88–5.50) 4.15 (2.93–5.88) 4.12 (3.00–5.77)
  exp(γ) 1.92 (1.36–2.71) 2.00 (1.37–2.92)   1.87 (1.33–2.63)
7 exp(β) 4.48 (3.26–6.15) 4.42 (3.10–6.30) 4.04 (2.94–5.59)
  exp(γ) 1.89 (1.34–2.67) 2.03 (1.33–3.08)   1.86 (1.30–2.70)
8 exp(β) 4.90 (3.57–6.74) 4.07 (2.50–6.63) 3.88 (2.79–5.42) 4.07 (2.93–5.68)
  exp(γ) 1.81 (1.29–2.53) 1.76 (1.17–2.64)   1.79 (1.24–2.51)
9 exp(β) 5.24 (3.81–7.22) 4.10 (2.76–6.08) 4.47 (3.22–6.19) 4.08 (2.96–5.65)
  exp(γ) 1.77 (1.27–2.48) 2.10 (1.42–3.10)   1.75 (1.23–2.48)
10 exp(β) 5.08 (3.67–7.03) 4.13 (2.99–5.70) 4.06 (2.94–5.61)
  exp(γ) 1.77 (1.28–2.45) 1.62 (1.08–2.42)   1.76 (1.26–2.50)
11 exp(β) 5.33 (3.85–7.38) 4.07 (2.17–7.64) 4.15 (3.01–5.72) 4.07 (2.96–5.65)
  exp(γ) 1.78 (1.29–2.46) 1.86 (1.26–2.76)   1.73 (1.23–2.38)
12 exp(β) 5.61 (4.06–7.75) 4.12 (2.69–6.32) 4.51 (3.29–6.18) 4.07 (2.95–5.62)
  exp(γ) 1.92 (1.39–2.66) 1.81 (1.24–2.64)   1.86 (1.34–2.62)
13 exp(β) 5.81 (4.19–8.04) 4.06 (2.79–5.91) 4.65 (3.40–6.36) 4.02 (2.90–5.56)
  exp(γ) 1.93 (1.39–2.68) 1.87 (1.28–2.73)   1.88 (1.34–2.61)
14 exp(β) 5.60 (4.02–7.79) 4.05 (2.96–5.55) 3.99 (2.87–5.53)
  exp(γ) 1.93 (1.40–2.66) 2.10 (1.40–3.13)   1.87 (1.35–2.57)
15 exp(β) 5.83 (4.17–8.13) 3.94 (2.42–6.42) 3.85 (2.85–5.21) 3.97 (2.86–5.50)
  exp(γ) 1.91 (1.38–2.63) 2.04 (1.39–3.00)   1.86 (1.35–2.57)
16 exp(β) 6.05 (4.35–8.43) 3.98 (2.68–5.92) 3.88 (2.87–5.25) 3.98 (2.88–5.52)
  exp(γ) 2.04 (1.47–2.82) 2.11 (1.45–3.07)   1.98 (1.41–2.72)
17 exp(β) 5.94 (4.25–8.30) 4.21 (3.09–5.73) 3.97 (2.87–5.51)
  exp(γ) 2.06 (1.49–2.84) 2.19 (1.48–3.23)   2.02 (1.46–2.79)
18 exp(β) 6.14 (4.39–8.59) 4.08 (2.18–7.64) 4.13 (3.05–5.59) 3.99 (2.89–5.51)
  exp(γ) 2.02 (1.47–2.77) 2.16 (1.48–3.15)   1.95 (1.43–2.65)
19 exp(β) 6.32 (4.52–8.85) 4.00 (2.60–6.14) 3.94 (2.94–5.29) 3.98 (2.87–5.48)
  exp(γ) 2.07 (1.51–2.85) 1.98 (1.37–2.88)   2.01 (1.50–2.71)
20 exp(β) 6.49 (4.64–9.08) 3.94 (2.71–5.74) 3.99 (2.98–5.34) 3.98 (2.90–5.46)
  exp(γ) 2.16 (1.57–2.98) 2.14 (1.47–3.11)   2.05 (1.52–2.76)
21 exp(β) 6.35 (4.52–8.91) 4.16 (3.09–5.61) 3.96 (2.88–5.46)
  exp(γ) 2.20 (1.60–3.01) 2.20 (1.51–3.22)   2.10 (1.53–2.81)
  1. M is the number of control periods and study period = (M + 1) days. 1 period = 1 day for all values of M
  2. ORSCL: odds ratio by the standard conditional logistic regression; ORVF: odds ratio by the Vines and Farrington’s method; ORMH: odds ratio by the Mantel-Haenszel method; ORG: odds ratio by the Greenland’s method; 95%CI: 95% confidence interval; 2.5–97.5pct: 2.5 to 97.5 percentiles; exp(β): estimate for the exposure variable; exp(γ): estimate for the time-varying confounder