Background
Methods
Overview
Simple Product-Based ESR
Variance and Confidence Interval for the Simple Product-Based ESR
Revised Product-Based ESR
Variance and Confidence Interval for the Revised Product-Based ESR
Simulation study details
Parameter | Possible values |
---|---|
Exposure probability | .05, .20 |
Probability of disease among unexposed | .02, .09 |
RR | 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0 |
Sample size combinations for the overall risk and RR (N1/N2) | 250/250*, 1,000/1,000, 1,000/5,000, 5,000/1,000, 5,000/5,000 |
Evaluation of ESR Estimators
Evaluation of Confidence Intervals
Case Study
Results
Scenario 1: Low exposure probability (.05)/Low disease probability among unexposed (.02)
Low exposure probability (.05)/Low disease probability in unexposed (.02) | ||||
---|---|---|---|---|
N1 = 1,000, N2 = 1,000 | N1 = 5,000, N2 = 5,000 | |||
RR/ESR | ESRS
| ESRR
| ESRS
| ESRR
|
1.0/.02 | 9.5% | 3.9% | -1.4% | -2.3% |
2.0/.04 | 7.8% | -2.6% | 4.5% | -1.4% |
3.0/.06 | 18.0% | 1.8% | 12.2% | 0.8% |
4.0/.08 | 21.1% | 0.0% | 17.6% | 1.1% |
5.0/.10 | 31.4% | 3.4% | 22.6% | 1.0% |
Low exposure probability (.05)/Moderate disease probability in unexposed (.09)
| ||||
N1 = 1,000, N2 = 1,000 | N1 = 5,000, N2 = 5,000 | |||
RR/ESR | ESRS
| ESRR
| ESRS
| ESRR
|
1.0/.09 | 0.1% | -0.9% | 0.9% | 0.6% |
2.0/.18 | 6.1% | 0.1% | 5.5% | 0.3% |
3.0/.27 | 9.2% | -1.4% | 10.4% | 0.2% |
4.0/.36 | 16.6% | 0.4% | 15.9% | 0.5% |
5.0/.45 | 22.0% | 0.7% | 21.2% | 0.8% |
High exposure probability (.20)/Low disease probability in unexposed (.02)
| ||||
N1 = 1,000, N2 = 1,000 | N1 = 5,000, N2 = 5,000 | |||
RR/ESR | ESRS
| ESRR
| ESRS
| ESRR
|
1.0/.02 | 8.7% | 1.1% | -0.2% | -1.3% |
2.0/.04 | 26.3% | -1.4% | 21.4% | -0.1% |
3.0/.06 | 45.0% | -1.8% | 41.7% | 0.1% |
4.0/.08 | 73.3% | 1.2% | 61.6% | 0.1% |
5.0/.10 | 93.9% | 0.3% | 82.5% | 0.0% |
High exposure probability (.20)/Moderate disease probability in unexposed (.09)
| ||||
N1 = 1,000, N2 = 1,000 | N1 = 5,000, N2 = 5,000 | |||
RR/ESR | ESRS
| ESRR
| ESRS
| ESRR
|
1.0/.09 | -0.8% | -0.4% | -1.1% | -1.0% |
2.0/.18 | 22.6% | 0.7% | 20.4% | 0.2% |
3.0/.27 | 40.9% | -0.2% | 40.3% | 0.0% |
4.0/.36 | 63.6% | 0.7% | 60.6% | 0.1% |
5.0/.45 | 82.6% | 0.2% | 81.0% | 0.2% |
Low exposure probability (.05)/Low disease probability in unexposed (.02) | ||||
---|---|---|---|---|
N1 = 1,000, N2 = 1,000 | N1 = 5,000, N2 = 5,000 | |||
RR/ESR | ESRS
| ESRR
| ESRS
| ESRR
|
1.0/.02 | 96.8 | 97.3 | 97.5 | 97.5 |
2.0/.04 | 96.4 | 98.1 | 95.9 | 97.2 |
3.0/.06 | 95.7 | 98.3 |
92.7
| 96.6 |
4.0/.08 |
94.1
| 98.2 |
90.3
| 98.0 |
5.0/.10 |
93.9
| 98.4 |
87.1
| 97.7 |
Low exposure probability (.05)/Moderate disease probability in unexposed (.09)
| ||||
N1 = 1,000, N2 = 1,000 | N1 = 5,000, N2 = 5,000 | |||
RR/ESR | ESRS
| ESRR
| ESRS
| ESRR
|
1.0/.09 | 96.8 | 97.2 | 95.9 | 96.2 |
2.0/.18 | 95.8 | 96.8 |
93.4
| 96.3 |
3.0/.27 |
94.3
| 97.0 |
87.8
| 96.4 |
4.0/.36 |
89.3
| 96.1 |
70.8
| 97.0 |
5.0/.45 |
83.2
| 96.6 |
45.0
| 96.6 |
High exposure probability (.20)/Low disease probability in unexposed (.02)
| ||||
N1 = 1,000, N2 = 1,000 | N1 = 5,000, N2 = 5,000 | |||
RR/ESR | ESRS
| ESRR
| ESRS
| ESRR
|
1.0/.02 | 97.6 | 98.5 |
94.5
| 96.6 |
2.0/.04 |
94.2
| 98.9 |
85.6
| 98.0 |
3.0/.06 |
89.5
| 99.0 |
55.1
| 98.6 |
4.0/.08 |
76.8
| 99.2 |
22.3
| 99.4 |
5.0/.10 |
65.4
| 99.6 |
4.2
| 99.4 |
High exposure probability (.20)/Moderate disease probability in unexposed (.09)
| ||||
N1 = 1,000, N2 = 1,000 | N1 = 5,000, N2 = 5,000 | |||
RR/ESR | ESRS
| ESRR
| ESRS
| ESRR
|
1.0/.09 |
94.7
| 96.8 |
94.7
| 96.5 |
2.0/.18 |
85.4
| 98.5 |
51.1
| 98.0 |
3.0/.27 |
55.9
| 98.5 |
1.8
| 98.6 |
4.0/.36 |
17.6
| 99.4 |
0
| 99.4 |
5.0/.45 |
2.9
| 99.5 |
0
| 99.4 |
Low exposure probability (.05)/Low disease probability in unexposed (.02) | |||||||
---|---|---|---|---|---|---|---|
RR/ESR | |||||||
Sample Size (N1/N2) | 1.0/.02 | 1.5/.03 | 2.0/.04 | 2.5/.05 | 3.0/.06 | 4.0/.08 | 5.0/.10 |
1,000/1,000 |
66.1
|
78.1
|
83.2
|
87.1
|
89.6
|
88.9
|
90.0
|
1,000/5,000 | 98.9 | 99.3 | 99.6 | 99.7 | 99.9 | 99.5 | 99.5 |
5,000/1,000 |
54.4
|
60.6
|
61.2
|
62.3
|
63.5
|
64.9
|
65.0
|
5,000/5,000 |
90.0
|
92.8
|
92.8
|
93.7
|
92.4
|
94.4
|
94.8
|
Low exposure probability (.05)/Moderate disease probability in unexposed (.09)
| |||||||
RR/ESR | |||||||
Sample Size (N1/N2) | 1.0/.09 | 1.5/.14 | 2.0/.18 | 2.5/.23 | 3.0/.27 | 4.0/.36 | 5.0/.45 |
1,000/1,000 |
90.4
|
91.6
|
93.2
|
92.4
|
92.2
|
92.0
|
91.5
|
1,000/5,000 | 99.9 | 100 | 100 | 100 | 99.9 | 99.7 | 99.3 |
5,000/1,000 |
62.8
|
63.8
|
63.9
|
63.7
|
62.1
|
61.6
|
62.7
|
5,000/5,000 |
93.8
|
93.4
|
94.2
|
93.5
|
93.7
|
93.6
|
91.6
|
High exposure probability (.20)/Low disease probability in unexposed (.02)
| |||||||
RR/ESR | |||||||
Sample Size (N1/N2) | 1.0/.02 | 1.5/.03 | 2.0/.04 | 2.5/.05 | 3.0/.06 | 4.0/.08 | 5.0/.10 |
1,000/1,000 |
89.1
|
91.2
|
91.8
|
92.6
|
91.5
|
92.5
|
93.8
|
1,000/5,000 | 99.7 | 99.1 | 99.4 | 99.0 | 98.0 | 98.8 | 98.1 |
5,000/1,000 |
63.9
|
68.0
|
64.7
|
69.3
|
70.7
|
70.5
|
71.9
|
5,000/5,000 |
92.1
|
94.0
| 95.1 |
94.8
|
94.3
|
94.5
|
93.5
|
High exposure probability (.20)/Moderate disease probability in unexposed (.09)
| |||||||
RR/ESR | |||||||
Sample Size (N1/N2) | 1.0/.09 | 1.5/.14 | 2.0/.18 | 2.5/.23 | 3.0/.27 | 4.0/.36 | 5.0/.45 |
1,000/1,000 |
92.5
|
93.2
| 95.1 |
94.5
|
92.6
|
90.7
|
86.7
|
1,000/5,000 | 99.9 | 99.7 | 98.9 | 98.8 | 98.2 | 97.3 | 95.5 |
5,000/1,000 |
66.0
|
68.3
|
69.7
|
69.0
|
64.4
|
66.6
|
66.0
|
5,000/5,000 |
94.0
|
94.0
|
93.2
|
94.0
|
91.9
|
92.0
|
88.9
|
Scenario 2: Low exposure probability (.05)/Moderate disease probability among unexposed (.09)
Scenario 3: High exposure probability (.20)/Low disease probability among unexposed (.02)
Scenario 4: High exposure probability (.20)/Moderate disease probability among unexposed (.09)
Results of the case study
Age | Overall risk of symptomatic knee OA in the Oliveria study | Risk of symptomatic knee OA for obese persons using the simple product-based method | 95% CI for ESRS using a log-based variance | Risk of symptomatic knee OA for obese persons using the revised product-based method | 95% CI for ESRR using a log-based variance | 95% CI for ESRR using a binomial variance |
---|---|---|---|---|---|---|
50-59 | 0.0040 | 0.0076 | 0.0052-0.0110 | 0.0057 | 0.0038-0.0085 | 0.0037-0.0077 |
60-69 | 0.0087 | 0.0167 | 0.0121-0.0230 | 0.0125 | 0.0089-0.0175 | 0.0095-0.0155 |
70-79 | 0.0147 | 0.0282 | 0.0207-0.0383 | 0.0211 | 0.0153-0.0289 | 0.0168-0.0253 |