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Table 3 Estimates of ORSCL, ORVF, 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