Table 2.
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 difference in allele effect over strata (the interaction effect) and comparison of convergence rate between logistic and linear regression GEE models. Data were generated using different assumptions on effect size (β3 = (P11 − P10) − (P01 − P00)) from model (2) and the corresponding OR1/OR0) and correlation among repeated events (ρ) as specified in the table where event rates for exposed and nonexposed are P10 = 0.001 and P11, respectively for stratum 1 with n = 200 clusters and the proportion of exposure P(X = 1) = 40%; event rates for exposed and nonexposed are P00 = 0.0005 and P01 = 0.001, respectively for stratum 0 with n = 400 clusters and P(X = 1) = 30% based on 1000 simulations.
| Binary logistic regression GEE | Linear regression GEE | Proposed method | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| OR1/OR0a | Propb conv |
Propc sig among conv |
Prop sig among totalc |
Biasd in log(ÔR1/ÔR0) |
β3 | Prop conv |
Prop sig among conv |
Prop sig among total |
Bias in β̂3 |
Prop sig perm test |
Bias in β̂3 |
|
| ρ = 0.2 | 0.755 | 7% | 1.5% | 0.1% | 150% | 0.000e | 94% | 0.5% | 0.5% | 0.1% | 5.1% | 0.1% |
| 2.26 | 10% | 4.2% | 0.4% | −107% | 0.003 | 99% | 4.7% | 4.6% | 4.7% | 26.0% | 4.7% | |
| 5.81 | 11% | 3.5% | 0.4% | −36.0% | 0.010 | 100% | 27.9% | 27.8% | −2.6% | 64.6% | −2.6% | |
| 10.4 | 13% | 6.3% | 0.8% | 16.4% | 0.019 | 100% | 71.0% | 71.0% | 1.3% | 91.2% | 1.3% | |
| ρ = 0.5 | 0.75 | 1% | 0.0% | 0.0% | 317% | 0.000 | 78% | 0.0% | 0.0% | 0.0% | 3.6% | 0.0% |
| 2.26 | 2% | 0.0% | 0.0% | −108% | 0.003 | 85% | 0.2% | 0.2% | 14.1% | 15.4% | −2.8% | |
| 5.81 | 2% | 0.0% | 0.0% | −40.6% | 0.010 | 98% | 7.4% | 7.2% | 6.7% | 53.1% | 6.7% | |
| 10.4 | 3% | 5.9% | 0.2% | −40.2% | 0.019 | 100% | 32.1% | 32.0% | −1.4% | 76.8% | −1.4% | |
For the logistic regression model, the effect size is measured by ratio of odds ratio (OR1/OR0) and for the linear regression model, the effect size is measured by the difference in allele effect between two strata.
For each regression model, we also show the proportion of simulated datasets for which the model achieved convergence.
For each regression model, we show the proportion of rejecting the null hypothesis of no differential allele effects among the simulated datasets for which the model was converged as well as the proportion of rejection the null hypothesis of no differential allele effects at the nominal statistical significance level of 0.05 among all simulated datasets, while treating nonconvergence as a failure to demonstrate a significant interaction; for the proposed method, 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 propose 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%.
The null interaction in the linear regression model does not correspond to a null interaction in the logistic regression model.
prop = proportion, conv =convergence.