Gaining Strength from Correlation among Endpoints in Multiple Contrast Tests

(mat <- matrix(c(1, 0.9, 0.9, 1), nrow=2))

##      [,1] [,2]
## [1,]  1.0  0.9
## [2,]  0.9  1.0

Comparing Multiple Treatments at Multiple Occasions

2 Treatment Groups

(mat2 <- diag(2) %x% mat)

##      [,1] [,2] [,3] [,4]
## [1,]  1.0  0.9  0.0  0.0
## [2,]  0.9  1.0  0.0  0.0
## [3,]  0.0  0.0  1.0  0.9
## [4,]  0.0  0.0  0.9  1.0

(C2a <- rbind(c(-1, 0, 1, 0), c(0, -1, 0, 1)))

##      [,1] [,2] [,3] [,4]
## [1,]   -1    0    1    0
## [2,]    0   -1    0    1

(Sigma2a <- C2a %*% mat2 %*% t(C2a))

##      [,1] [,2]
## [1,]  2.0  1.8
## [2,]  1.8  2.0

cov2cor(Sigma2a)

##      [,1] [,2]
## [1,]  1.0  0.9
## [2,]  0.9  1.0

cov2cor(C2a %*% diag(4) %*% t(C2a))

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

3 Treatment Groups

(mat3 <- diag(3) %x% mat)

##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]  1.0  0.9  0.0  0.0  0.0  0.0
## [2,]  0.9  1.0  0.0  0.0  0.0  0.0
## [3,]  0.0  0.0  1.0  0.9  0.0  0.0
## [4,]  0.0  0.0  0.9  1.0  0.0  0.0
## [5,]  0.0  0.0  0.0  0.0  1.0  0.9
## [6,]  0.0  0.0  0.0  0.0  0.9  1.0

(C3 <- rbind(c(-1, 0, 1, 0, 0, 0), c(-1, 0, 0, 0, 1, 0), c(0, -1, 0, 1, 0, 0), c(0, -1, 0, 0, 0, 1)))

##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]   -1    0    1    0    0    0
## [2,]   -1    0    0    0    1    0
## [3,]    0   -1    0    1    0    0
## [4,]    0   -1    0    0    0    1

(Sigma3 <- C3 %*% mat3 %*% t(C3))

##      [,1] [,2] [,3] [,4]
## [1,]  2.0  1.0  1.8  0.9
## [2,]  1.0  2.0  0.9  1.8
## [3,]  1.8  0.9  2.0  1.0
## [4,]  0.9  1.8  1.0  2.0

cov2cor(Sigma3)

##      [,1] [,2] [,3] [,4]
## [1,] 1.00 0.50 0.90 0.45
## [2,] 0.50 1.00 0.45 0.90
## [3,] 0.90 0.45 1.00 0.50
## [4,] 0.45 0.90 0.50 1.00

cov2cor(C3 %*% diag(6) %*% t(C3))

##      [,1] [,2] [,3] [,4]
## [1,]  1.0  0.5  0.0  0.0
## [2,]  0.5  1.0  0.0  0.0
## [3,]  0.0  0.0  1.0  0.5
## [4,]  0.0  0.0  0.5  1.0

Comparing Multiple Occasions within Multiple Treatments

2 Treatment Groups

(C2b <- rbind(c(-1, 1, 0, 0), c(0, 0, -1, 1)))

##      [,1] [,2] [,3] [,4]
## [1,]   -1    1    0    0
## [2,]    0    0   -1    1

(Sigma2b <- C2b %*% mat2 %*% t(C2b))

##      [,1] [,2]
## [1,]  0.2  0.0
## [2,]  0.0  0.2

cov2cor(Sigma2b)

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

cov2cor(C2b %*% diag(4) %*% t(C2b))

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

Comparing Multiple Treatments at Multiple Occasions AND Multiple Occasions within Multiple Treatments

2 Treatment Groups

(C2c <- rbind(C2a, C2b))

##      [,1] [,2] [,3] [,4]
## [1,]   -1    0    1    0
## [2,]    0   -1    0    1
## [3,]   -1    1    0    0
## [4,]    0    0   -1    1

(Sigma2c <- C2c %*% mat2 %*% t(C2c))

##      [,1] [,2] [,3] [,4]
## [1,]  2.0  1.8  0.1 -0.1
## [2,]  1.8  2.0 -0.1  0.1
## [3,]  0.1 -0.1  0.2  0.0
## [4,] -0.1  0.1  0.0  0.2

round(cov2cor(Sigma2c), 3)

##        [,1]   [,2]   [,3]   [,4]
## [1,]  1.000  0.900  0.158 -0.158
## [2,]  0.900  1.000 -0.158  0.158
## [3,]  0.158 -0.158  1.000  0.000
## [4,] -0.158  0.158  0.000  1.000

cov2cor(C2c %*% diag(4) %*% t(C2c))

##      [,1] [,2] [,3] [,4]
## [1,]  1.0  0.0  0.5 -0.5
## [2,]  0.0  1.0 -0.5  0.5
## [3,]  0.5 -0.5  1.0  0.0
## [4,] -0.5  0.5  0.0  1.0