Measurement error in linear model variables is an important consideration, and through simulation we demonstrated the importance for correcting measurement error in linear models. Of particular interest was using multiple imputation (MI) for measurement error correction for the Redden et al. [7
] SAT model. Although the Redden SAT model requires individual ancestry estimates to control for admixture confounding, individual admixture estimates were used because individual ancestry estimates are rarely known, so admixture estimates can be used as a surrogate for the ancestry estimates. We then describe how to use MI for measurement correction. Like Divers et al. [19
], we also used Cronbach's alpha [35
] as a component of our measurement error correction procedure. We also described three different methods for imputing probable true scores for admixture: Rubin, Bootstrap, Cole.
In the linear SAT model, of the three different methods for imputing probable admixture scores, the Rubin and Cole methods appear to work best. Although at first it looks like the Bootstrap method controls the Type I error correctly whereas the Rubin and Cole methods are slightly conservative, as the marker informativeness begins to decrease it is the Rubin and Cole methods that control Type I error rate and the Bootstrap method becomes liberal. Consistently, the Rubin and Cole method provided better control of the Type I error rate than the Bootstrap method. This same pattern was observed in Divers et al. [19
], in that measurement error correction only appears to be required when the informativeness of the markers is of intermediate value. The reason for this is that when markers are highly informative, the measurement correction method provides little improvement. On the other hand, when marker informativeness is low, the measurement correction method has poor information to borrow for measurement correction. MI for measurement correction as presented uses the existing data to accomplish this goal and require no external information.
In the quadratic SAT model, of the three different methods for imputing probable admixture scores, the Rubin and Cole methods again appear to work best. The Bootstrap method did not consistently provide reasonable control of the Type I error rate. One interesting point is that the type I error rates of the Bootstrap method, in all models, are very similar to the type I error rates of the model without measurement error correction, suggesting that the Bootstrap method is not providing much measurement error correction. Notably, none of the methods works particularly well for a quadratic SAT model with admixture reliability of 0.70. Because of this result the linear SAT model corrected for measurement error may be considered, yet it too can have problems if the genetic effects are markedly non-additive (e.g., overdominance).
There is now much agreement that population admixture and/or population stratification can confound association studies when not taken into account. However, it should also be mentioned that accuracy with which admixture is measured will have an influence on Type I error. When admixture or any other continuous variable are contaminated with error, MI for measurement error correction can help control the specified Type I error rate. However, this method is only useful if the data are of reasonably good quality with respect to marker information, which means that much care should still be taken when designing association studies, and in particular when measuring variables that will be used for analysis in a statistical model.