Background
Motivating examples
Drug treatment with specific exposure pattern
Overview of real-world examples
Methods
BIAS due to conditional logistic regression for case-crossover studies
Weighting method to remove BIAS due to within-exposure dependency
Case-crossover studies with a binary exposure
Case-crossover studies with a binary exposure and a binary time-varying confounder
Simulation studies
Case period | Control periods | |
---|---|---|
Definition I | Last day | Last day |
Definition II | Last day | Half or more days |
Definition III | Last day | Any 1 day |
Definition IV | Any 1 day | Any 1 day |
Case-crossover studies of real-world data
Results
Simulation studies: comparison of methods with or without time-varing confounding
M | ORSCL (95%CI) | ORVF (95%CI) | ORMH (95%CI) | ORG (2.5–97.5pct) |
---|---|---|---|---|
1 | 4.00 (2.45–6.53) | 4.00 (2.45–6.53) | 4.00 (2.45–6.53) | 4.00 (2.57–7.33) |
2 | 4.00 (2.74–5.85) | 4.00 (2.74–5.85) | 4.00 (2.74–5.85) | 4.00 (2.80–6.00) |
3 | 3.55 (2.47–5.10) | 4.00 (2.64–6.06) | 4.00 (2.73–5.86) | 4.00 (2.80–5.72) |
4 | 3.84 (2.72–5.41) | 4.00 (2.80–5.71) | 4.00 (2.81–5.69) | 4.00 (2.82–5.64) |
5 | 4.28 (3.04–6.03) | 4.00 (2.86–5.59) | 4.00 (2.85–5.61) | 4.00 (2.84–5.73) |
6 | 4.60 (3.27–6.48) | 4.00 (2.80–5.71) | 4.00 (2.89–5.53) | 4.00 (2.84–5.70) |
7 | 4.38 (3.10–6.20) | – | 4.00 (2.87–5.57) | 4.00 (2.85–5.66) |
8 | 4.73 (3.34–6.70) | 4.00 (2.34–6.84) | 4.00 (2.89–5.54) | 4.00 (2.88–5.63) |
9 | 5.01 (3.53–7.10) | 4.00 (2.58–6.20) | 4.00 (2.91–5.50) | 4.00 (2.83–5.65) |
10 | 4.86 (3.41–6.94) | – | 4.00 (2.89–5.54) | 4.00 (2.84–5.70) |
11 | 5.09 (3.57–7.26) | 4.00 (2.00–8.00) | 4.00 (2.91–5.51) | 4.00 (2.84–5.64 |
12 | 5.33 (3.73–7.62) | 4.00 (2.49–6.42) | 4.00 (2.92–5.48) | 4.00 (2.85–5.60) |
13 | 5.54 (3.88–7.92) | 4.00 (2.64–6.06) | 4.00 (2.93–5.46) | 4.00 (2.85–5.67) |
14 | 5.39 (3.75–7.73) | – | 4.00 (2.92–5.49) | 4.00 (2.85–5.66) |
15 | 5.60 (3.90–8.05) | 4.00 (2.34–6.84) | 4.00 (2.92–5.47) | 4.00 (2.87–5.61) |
16 | 5.79 (4.02–8.33) | 4.00 (2.58–6.20) | 4.00 (2.93–5.46) | 4.00 (2.84–5.60) |
17 | 5.66 (3.92–8.18) | – | 4.00 (2.92–5.48) | 4.00 (2.84–5.65) |
18 | 5.83 (4.03–8.43) | 4.00 (2.00–8.00) | 4.00 (2.93–5.46) | 4.00 (2.84–5.64) |
19 | 6.00 (4.14–8.69) | 4.00 (2.49–6.42) | 4.00 (2.93–5.45) | 4.00 (2.84–5.61) |
20 | 6.16 (4.25–8.92) | 4.00 (2.64–6.06) | 4.00 (2.94–5.44) | 4.00 (2.85–5.65) |
21 | 6.02 (4.14–8.75) | – | 4.00 (2.93–5.46) | 4.00 (2.85–5.66) |
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) |
Days in 1 period | 1 day | 2 days | 3 days | 4 days | 7 days | 11 days |
---|---|---|---|---|---|---|
M | 21 | 10 | 6 | 4 | 2 | 1 |
One binary exposure variable only | ||||||
ORSCL exp(β) | 6.02 (4.14–8.75) | 5.30 (3.73–7.53) | 4.60 (3.27–6.48) | 5.23 (3.65–7.50) | 6.61 (3.27–13.33) | 4.00 (2.45–6.53) |
ORVF exp(β) | – | 4.00 (2.58–6.20) | 4.00 (2.80–5.71) | 4.00 (2.45–6.53) | – | 4.00 (2.45–6.53) |
ORMH exp(β) | 4.00 (2.93–5.46) | 4.00 (2.92–5.48) | 4.00 (2.89–5.53) | 4.00 (2.87–5.57) | 4.00 (2.19–7.29) | 4.00 (2.45–6.53) |
ORG exp(β) | 4.00 (2.85–5.66) | 4.00 (2.82–5.60) | 4.00 (2.84–5.70) | 4.00 (2.82–5.81) | 4.00 (2.12–9.00) | 4.00 (2.57–6.69) |
One binary exposure variable and one binary time-varying confounder | ||||||
ORSCL exp(β) | 6.35 (4.52–8.91) | 5.57 (4.04–7.68) | 4.76 (3.49–6.51) | 5.29 (3.81–7.35) | 6.79 (3.59–12.86) | 4.34 (2.76–6.82) |
exp(γ) | 2.20 (1.60–3.01) | 1.78 (1.28–2.48) | 1.80 (1.27–2.54) | 2.06 (1.39–3.06) | 1.91 (1.25–2.91) | 1.35 (0.83–2.20) |
ORVF exp(β) | – | 4.08 (2.74–6.07) | 4.02 (2.90–5.58) | 4.02 (2.57–6.29) | – | 4.31 (2.75–6.74) |
exp(γ) | 2.20 (1.51–3.22) | 1.78 (1.20–2.63) | 1.83 (1.25–2.67) | 2.02 (1.24–3.31) | 2.34 (1.43–3.85) | 1.94 (1.16–3.24) |
ORMH exp(β) | 4.16 (3.09–5.61) | 4.36 (3.15–6.03) | 4.40 (3.11–6.24) | 3.54 (2.49–5.04) | 4.11 (2.17–7.80) | 5.56 (2.73–11.30) |
ORG exp(β) | 3.96 (2.88–5.46) | 4.06 (2.93–5.61) | 4.11 (2.97–5.78) | 4.00 (2.85–5.67) | 4.04 (2.26–8.80) | 4.34 (2.86–7.40) |
exp(γ) | 2.10 (1.53–2.81) | 1.77 (1.25–2.47) | 1.76 (1.23–2.53) | 2.01 (1.39–2.83) | 1.90 (1.27–2.86) | 1.35 (0.80–2.26) |
Analyses of case-crossover studies of real-world data
Study period | 84 days | |||||
---|---|---|---|---|---|---|
Days in 1 period | 1 day | 3 days | 7 days | 14 days | 28 days | 42 days |
M | 83 | 27 | 11 | 5 | 2 | 1 |
ORSCL | 2.82 (2.14–3.71) | 2.73 (2.08–3.60) | 2.53 (1.93–3.33) | 2.28 (1.74–2.99) | 2.05 (1.54–2.74) | 1.91 (1.38–2.64) |
ORMH | 1.99 (1.58–2.51) | 1.98 (1.57–2.50) | 1.97 (1.55–2.50) | 1.94 (1.51–2.48) | 1.93 (1.46–2.56) | 1.91 (1.38–2.64) |
ORG | 1.99 (1.49–2.65) | 1.98 (1.48–2.64) | 1.97 (1.48–2.62) | 1.94 (1.44–2.58) | 1.93 (1.43–2.62) | 1.91 (1.36–2.72) |
Study period | 168 days | 336 days | ||||
---|---|---|---|---|---|---|
Days in 1 period | 1 day | 14 days | 84 days | 1 days | 14 days | 168 days |
M | 167 | 11 | 1 | 335 | 23 | 1 |
ORSCL | 4.19 (3.16–5.55) | 3.39 (2.58–4.45) | 2.13 (1.54–2.95) | 7.16 (5.27–9.74) | 5.97 (4.43–8.05) | 3.73 (2.42–5.75) |
ORMH | 2.45 (1.94–3.09) | 2.38 (1.87–3.03) | 2.13 (1.54–2.95) | 2.95 (2.28–3.82) | 2.92 (2.24–3.79) | 3.73 (2.42–5.75) |
ORG | 2.45 (1.87–3.25) | 2.38 (1.82–3.15) | 2.13 (1.57–2.99) | 2.95 (2.14–4.09) | 2.92 (2.12–4.03) | 3.73 (2.49–6.06) |
Study period | 180 days | |||||
---|---|---|---|---|---|---|
Days in 1 period | 1 day | 5 days | 20 days | 30 days | 60 days | 90 days |
M | 179 | 35 | 8 | 5 | 2 | 1 |
ORSCL | 2.02 (1.71–2.37) | 1.85 (1.58–2.17) | 1.76 (1.49–2.06) | 1.72 (1.46–2.02) | 1.59 (1.33–1.91) | 1.51 (1.24–1.83) |
ORMH | 1.67 (1.46–1.92) | 1.60 (1.39–1.84) | 1.59 (1.37–1.84) | 1.61 (1.38–1.88) | 1.55 (1.30–1.85) | 1.51 (1.24–1.83) |
ORG | 1.67 (1.49–1.90) | 1.60 (1.40–1.83) | 1.59 (1.39–1.83) | 1.61 (1.41–1.87) | 1.55 (1.30–1.81) | 1.51 (1.24–1.85) |