STAT 3690 Lecture 14
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STAT 3690 Lecture 14
2022
2-way MANOVA (J&W Sec. 6.7)
● Model: Xijk = u + ri + 8j + yij + Eijk with Eijk MVNp (0, Σ), i = 1, . . . , m, j = 1, . . . , b, k = 1, . . . , n
_ ri : the main effect of factor 1 at level i
_ 8j : the main effect of factor 2 at level j
_ yij : the interaction of factors 1 and 2 whose levels are i and j, respectively
_ Identifiability: i ri = j 8j = i yij = j yij = 0
● Sum of squares and cross products matrix (SSP) _ Total corrected SSP
m b n
SSPcor = (Xijk - )(Xijk - )T
i=1 j=1 k=1
* = (mbn)_ 1 i,j,k Xijk
_ SSP for factor 1
* i . = (bn)_ 1 j,k Xijk
_ SSP for factor 2
b
SSPf2 = mn( .j - )( .j - )T
j=1
* .j = (mn)_ 1 i,k Xijk
_ SSP for interaction
m b
SSPint = n(ij - i . - .j + )(ij - i . - .j + )T
i=1 j=1
* ij = n_ 1 k Xijk
_ SSP for residual
m b n
SSPres = (Xijk - ij )(Xijk - ij )T
i=1 j=1 k=1
_ SSPcor = SSPf1 + SSPf2 + SSPint + SSPres
● Testing interaction
_ Hypotheses H0 : y 11 = . . . = ymb = 0 v.s. H1 : otherwise
_ Wilk’s lambda test statistic
* Under H0 , by Bartlett’s approximation
[{p + 1 - (m - 1)(b - 1)}/2 - mb(n - 1)] ln Λ s χ2 ((m - 1)(b - 1)) _ Rejection H0 at level α when
[{p + 1 - (m - 1)(b - 1)}/2 - mb(n - 1)] ln Λ 2 χ1(2) _α,(m _ 1)(b _ 1)
_ p-value
1 - Fχ2 ((m _ 1)(b _ 1)) ([{p + 1 - (m - 1)(b - 1)}/2 - mb(n - 1)] ln Λ)
● Testing main effects
_ Testing factor 1 main effects
* Hypotheses H0 : r 1 = . . . = rm = 0 v.s. H1 : otherwise
* Wilk’s lambda test statistic
Λ = det(SSPres + SSPf1 )
' Under H0 , by Bartlett’s approximation
[{p + 1 - (m - 1)}/2 - mb(n - 1)] ln Λ s χ2 (m - 1)
* Rejection H0 at level α when
[{p + 1 - (m - 1)}/2 - mb(n - 1)] ln Λ 2 χ1(2) _α,m _ 1
* p-value
1 - Fχ2 (m _ 1) ([{p + 1 - (m - 1)}/2 - mb(n - 1)] ln Λ)
_ Testing factor 2 main effects
* Hypotheses H0 : 8 1 = . . . = 8b = 0 v.s. H1 : otherwise
* Wilk’s lambda test statistic
Λ = det(SSPres + SSPf2 )
' Under H0 , by Bartlett’s approximation
[{p + 1 - (b - 1)}/2 - mb(n - 1)] ln Λ s χ2 (b - 1)
* Rejection H0 at level α when
[{p + 1 - (b - 1)}/2 - mb(n - 1)] ln Λ 2 χ1(2) _α,b _ 1
* p-value
1 - Fχ2 (b _ 1) ([{p + 1 - (b - 1)}/2 - mb(n - 1)] ln Λ)
● Exercise: factors in producing plastic film (continued)
_ One more factor ADDITIVE (amount of an additive, 2-level, low or high) in the production test
options(digits = 4)
tear <- c (
6.5 , 6.2 , 5.8 , 6.5 , 6.5 , 6.9 , 7.2 , 6.9 , 6.1 , 6.3 ,
6.7 , 6.6 , 7.2 , 7.1 , 6.8 , 7.1 , 7.0 , 7.2 , 7.5 , 7.6
)
gloss <- c (
9.5 , 9.9 , 9.6 , 9.6 , 9.2 , 9.1 , 10.0 , 9.9 , 9.5 , 9.4 ,
9.1 , 9.3 , 8.3 , 8.4 , 8.5 , 9.2 , 8.8 , 9.7 , 10.1 , 9.2
)
opacity <- c (
4.4 , 6.4 , 3.0 , 4.1 , 0.8 , 5.7 , 2.0 , 3.9 , 1.9 , 5.7 ,
2.8 , 4.1 , 3.8 , 1.6 , 3.4 , 8.4 , 5.2 , 6.9 , 2.7 , 1.9
)
(X <- cbind(tear, gloss, opacity))
(rate <- factor(gl(2 ,10 ,length=nrow (X)), labels=c ( "Low" , "High")))
(additive <- factor(gl(2 ,5 ,length=nrow (X)), labels=c ( "Low" , "High")))
summary (manova (X ~ rate*additive), test = 'Wilks ')
summary (car ::Manova(lm(X ~ rate*additive)), test.statistic= 'Wilks ')
2022-03-18