GLM notes of JKKs lecture I am using I for the incidence matrix, - - PDF document

GLM notes of JKK’s lecture

I am using I for the incidence matrix, while John used X in lecture. ℓ = C − 1 2 ln |V | − 1 2(Y − Iη)T V −1(Y − Iη) (1) Y =         y11 y12 y21 y22 y31 y32         (2) η =   µ1 µ2 µ3   (3) I =         1 1 1 1 1 1         (4) E[Y ] = Iη =         µ1 µ1 µ2 µ2 µ3 µ3         (5) (6) Diet C T Fam 1 yC11 yT11 yC12 yT12 2 yC21 yT21 yC22 yT22 Yijk = µ + τi + Fj + Cij + ǫijk (7) 1

Y =             yC11 yC12 yC21 yC22 yT11 yT12 yT21 yT22             (8) η =   µ τC τT   (9) Var[YC11] = ν = σ2

F + σ2 C + σ2 E

(10) Cov[YC11, YC12] = c1 = = Cov[F1 + CC1 + ǫC11, F1 + CC1 + ǫC12] (11) = σ2

F + σ2 C

(12) Cov[YT11, YT12] = Cov[F1 + CT1 + ǫT11, F1 + CT1 + ǫT12] (13) = σ2

F + σ2 C = c1

(14) Cov[YC11, YC21] = = Cov[F1 + CC1 + ǫC11, F2 + CT2 + ǫC22] (15) = (16) Cov[YC11, YT11] = = Cov[F1 + CC1 + ǫC11, F1 + CT1 + ǫT12] (17) = σ2

F

(18) V =             ν c1 σ2

F

σ2

F

c1 ν σ2

F

σ2

F

ν c1 σ2

F

σ2

F

c1 ν σ2

F

σ2

F

σ2

F

σ2

F

ν c1 σ2

F

σ2

F

c1 ν σ2

F

σ2

F

ν c1 σ2

F

σ2

F

c1 ν             (19) Reordering the order of data could give us a block diagonal. Which is nice because matrix inversion is O(N3): Y =             yC11 yC12 yT11 yT12 yC21 yC22 yT21 yT22             (20) 2

V =             ν c1 σ2

F

σ2

F

c1 ν σ2

F

σ2

F

σ2

F

σ2

F

ν c1 σ2

F

σ2

F

c1 ν ν c1 σ2

F

σ2

F

c1 ν σ2

F

σ2

F

σ2

F

σ2

F

ν c1 σ2

F

σ2

F

c1 ν             (21) 3