Table 1.
Comparison of logistic regression generalized estimating equation (GEE) models and linear regression GEE models and the proposed two-step method in detection and estimation of the main allele effect and comparison of convergence rate between logistic and linear regression GEE models. Data were generated using different assumptions on effect size (β1 = P1 − P0 from model (1) and its corresponding OR) and correlation among repeated events (ρ) as specified in the table where n = 400 and the event rate for non-exposed group P0 = 0.001 and the proportion of exposure P(X = 1) = 30% based on 1000 simulations.
| Binary logistic regression GEE | Linear regression GEE | Proposed method | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ORa | Propb conv |
Propc sig among conv |
Propc sig among total |
Biasd in log ÔR |
β1 | Prop conv |
Prop sig among conv |
Prop sig among total |
Bias in β̂1e |
Prop sig perm test |
Bias in β̂1 |
|
| ρ = 0.2 | 1.0 | 39% | 2.5% | 1.0% | 185% | 0.000 | 88% | 1.4% | 1.2% | 0.0% | 3.8% | 0.0% |
| 4.0 | 73% | 20.2% | 14.8% | −12.6% | 0.003 | 99% | 4.8% | 4.8% | 1.9% | 31.0% | 1.9% | |
| 10.0 | 80% | 63.4% | 50.6% | −5.5% | 0.009 | 100% | 45.1% | 45.0% | −3.8% | 75.6% | −3.8% | |
| ρ = 0.5 | 1.0 | 17% | 6.0% | 1.0% | 289% | 0.000 | 68% | 0.3% | 0.2% | −0.1% | 1.8% | −0.1% |
| 4.0 | 44% | 22.4% | 9.8% | −22.9% | 0.003 | 90% | 0.9% | 0.8% | 7.6% | 19.0% | −2.7% | |
| 10.0 | 52% | 45.0% | 23.4% | −13.4% | 0.009 | 99% | 11.1% | 11.0% | −1.4% | 60.4% | −1.4% | |
For the logistic regression model, the effect size is measured by odds ratio (OR) and for the linear regression model, the effect size is measured by the difference in proportion of events between two allele groups (β1).
For each regression model, we report the proportion of simulations which the model achieved convergence.
For each regression model, we report the proportion of rejecting the null hypothesis of no allele and disease association at the nominal statistical significance level of 0.05 among the simulations which the model was converged as well as the proportion of rejection the null hypothesis of no allele and disease association among all simulations, while treating non-convergence as a failure to demonstrate a significant exposure effect; for the permutation test, we show the proportion of rejecting the null hypothesis among all simulations; note when the null hypothesis is true, the proportion of rejecting the null hypothesis reflects the empirical significance value and when the null hypothesis is not true, the proportion of rejecting the null hypothesis is the empirical power.
For each regression model and the proposed approach, the bias of each parameter estimate was calculated as (estimated value-true value)/true value *100%; when the true effect is null, the bias = (exp(estimated value) − exp(true value))/exp(true value) *100%.
Although β̂1 from the linear the linear regression GEE model with an independent working correlation equals to the estimate of β1 from the proposed method, there is some difference in bias estimates because some linear regression GEE models failed to converge.
prop = proportion, conv = convergence.