Models for cross-sectional data | |||
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Model | Independent subjects design | Case-parent design | Comments |
Main genetic effect | |||
(2) | E(Y i |X, Z) = β0 + β1X i + β2Z i | Ordinary linear regression model | |
(3) | E(Y i |X, Z) = β0 + β1(X i - E(X i |g im , g if )) + β2Z i | Adjusted version of (2). The model is adjusted by the expected value of the offspring's genotype conditional to the parental genotypes | |
(4) | E(Y i |X, Z) = β0M+ β1(X i - E(X i |g im , g if )) + β2Z i | Gauderman's model (QTDTM) adjusted for the covariate Z | |
(5) | E(Y i |X, Z) = β0M+ β1X i + β2Z i | (4) equivalent to (5) | |
(10) |
| FBAT statistic | |
Gene-environment interaction | |||
(1) | E(Y i |X, Z) = β0 + β1X i + β2Z i + β3X i Z i | Ordinary linear regression model | |
(6) | E(Y i |X, Z) = β0M+ β1X i + β2Z i + β3X i Z i | Gauderman's model (QTDTM) | |
(7) | E(Y i |X, Z) = β0M+ β1[X i - E(X i |g im , g if )] + β2Z i + β3Z i [X i - E(X i |g im , g if )] | (6) is not equivalent to (7) when the environment covariate (Z i ) is not constant within mating type. | |
(8) | E(Y i |X, Z) = β0M+ β1[X i - E(X i |g im , g if )] + β2MZ i + β3Z i [X i - E(X i |g im , g if )] | Adjusted QTDTM | |
(9) | E(Y i |X, Z) = β0M+ β1X i + β2MZ i + β3Z i X i | (8) equivalent to (9) | |
Main genetic effect | |||
(19) | E(Y ij |X, Y, t) = α0 + α1t ij + α2Z i + α3X i + α4X i Z i + α5X i t ij + α6Z i t ij | Ordinary linear mixed model (OLMM) | |
(20) | E(Y ij |X, Y, t) = α0M+ α1Mt ij + α2Z i + α3X i + α4X i Z i + α5X i t ij + α6MZ i t ij | Adjusted linear mixed model (ALMM) | |
Gene-environment interaction | |||
(11) | Y ij = α0 + α1t ij + α2Z i + α3X i + α4X i Z i + α5X i t ij + α6Z i t ij + α7X i Z i t ij + b1i+b2it ij + e ij | Ordinary linear mixed model | |
(12) | FEF2575ij= α0M+ α1Mt ij + α2Z i + α3[X i - E(X i |g im , g if )] + α4[X i - E(X i |g im , g if )]Z i + α5[X i - E(X i |g im , g if )]t ij + α6MZ i t ij + α7[X i - E(X i |g im , g if )]Z i t ij + b1i+b2it ij + e ij | Adjusted linear mixed model (ALMM) | |
(13) | FEF2575ij= α0M+ α1Mt ij + α2Z i + α3X i + α4X i Z i + α5X i t ij + α6MZ i t ij + α7X i Z i t ij + b1i+b2it ij + e ij | (13) is equivalent to (12) |