参考:佐藤信『統計的官能検査法』第20章 (日科技連)
シェッフェの一対比較法(中屋の変法)は、比較順序を考慮しないもの。
参考文献通りに計算し、(p)値を求める部分だけ追加した。
## -*- coding: utf-8 -*-
## Scheffe's Pairwise Comparison
## (Nakaya's Variation, not considering stimulus order effect)
##
## 2012-09-12 by MARUI Atsushi (marui@ms.geidai.ac.jp)
##
## Reference: Chapter 20, Satoh "Statistical Methods in Sensory Tests"
## Nikkagiren Publishing (1985)
scheffe.nakaya <- function(Y, numStim, numSubj) {
## data preparation (Stim1 x Stim2 x Subj)
X <- array(data=0, dim=c(numStim, numStim, numSubj))
k <- 1
for (r in 1:(numStim-1)) {
for (c in (r+1):numStim) {
X[r,c,] <- Y[k,]
X[c,r,] <- -Y[k,]
k <- k+1
}
}
## calculate statistics
# average preference
Xi.. <- apply(X, 1, sum)
alphai <- Xi.. / (numStim*numSubj)
# individual differences
Xi.k <- apply(X, c(1,3), sum)
alphaik <- Xi.k / numStim - alphai
# combination effect
Xij. <- apply(X, c(1,2), sum)
gammaij <- Xij. / numSubj
- (matrix(alphai, nrow=length(alphai), ncol=length(alphai), byrow=FALSE)
- matrix(alphai, nrow=length(alphai), ncol=length(alphai), byrow=TRUE))
## calculate sum-of-squares
# main effect S_\alpha
S.alpha <- sum(Xi..^2) / (numStim*numSubj)
df.alpha <- numStim-1
# main x subject S_{\alpha(B)}
S.alphaB <- sum(Xi.k^2) / numStim - S.alpha
df.alphaB <- (numStim-1)*(numSubj-1)
# combination effect S_\gamma
tmp <- Xij.^2
S.gamma <- sum(tmp[upper.tri(tmp, diag=TRUE)]) / numSubj - S.alpha
df.gamma <- (numStim-1)*(numStim-2)/2
# total S_T
S.T <- sum(X^2)/2
df.T <- numSubj*numStim*(numStim-1)/2
# error S_e
S.e <- S.T - S.alpha - S.alphaB - S.gamma
df.e <- (numStim-1)*(numStim-2)*(numSubj-1)/2
## average preferences
cat(sprintf("\n"))
cat(sprintf("Average Preferences\n"))
cat(sprintf("---------------\n"))
for (r in 1:numStim) {
cat(sprintf("a%d = %9.4f\n", r, alphai[r]))
}
cat(sprintf("---------------\n"))
## ANOVA table
cat(sprintf("\n"))
cat(sprintf("ANOVA Table\n"))
cat(sprintf("--------------+---------------------------------------------\n"))
cat(sprintf("Source | SS df MS F p\n"))
cat(sprintf("--------------+---------------------------------------------\n"))
p <- pf((S.alpha/df.alpha)/(S.e/df.e), df.alpha, df.e, lower.tail=FALSE)
cat(sprintf("Main | %9.4f %4d %9.4f %9.4f %9.4f", S.alpha, df.alpha, S.alpha/df.alpha, (S.alpha/df.alpha)/(S.e/df.e), p))
if (0.05 < p && p <= 0.1) {
cat(sprintf(" .\n"))
} else if (0.01 < p && p <= 0.05) {
cat(sprintf(" *\n"))
} else if (0.001 < p && p <= 0.01) {
cat(sprintf(" **\n"))
} else if (p <= 0.001) {
cat(sprintf(" ***\n"))
} else {
cat(sprintf("\n"))
}
p <-pf((S.alphaB/df.alphaB)/(S.e/df.e), df.alphaB, df.e, lower.tail=FALSE)
cat(sprintf("Main x Indiv | %9.4f %4d %9.4f %9.4f %9.4f", S.alphaB, df.alphaB, S.alphaB/df.alphaB, (S.alphaB/df.alphaB)/(S.e/df.e), p))
if (0.05 < p && p <= 0.1) {
cat(sprintf(" .\n"))
} else if (0.01 < p && p <= 0.05) {
cat(sprintf(" *\n"))
} else if (0.001 < p && p <= 0.01) {
cat(sprintf(" **\n"))
} else if (p <= 0.001) {
cat(sprintf(" ***\n"))
} else {
cat(sprintf("\n"))
}
p <- pf((S.gamma/df.gamma)/(S.e/df.e), df.gamma, df.e, lower.tail=FALSE)
cat(sprintf("Combi | %9.4f %4d %9.4f %9.4f %9.4f", S.gamma, df.gamma, S.gamma/df.gamma, (S.gamma/df.gamma)/(S.e/df.e), p))
if (0.05 < p && p <= 0.1) {
cat(sprintf(" .\n"))
} else if (0.01 < p && p <= 0.05) {
cat(sprintf(" *\n"))
} else if (0.001 < p && p <= 0.01) {
cat(sprintf(" **\n"))
} else if (p <= 0.001) {
cat(sprintf(" ***\n"))
} else {
cat(sprintf("\n"))
}
cat(sprintf("Error | %9.4f %4d %9.4f\n", S.e, df.e, S.e/df.e))
cat(sprintf("Total | %9.4f %4d\n", S.T, df.T))
cat(sprintf("--------------+---------------------------------------------\n"))
## calculate yardsticks
Y001 <- qtukey(1-0.01, numStim, df.e) * sqrt(S.e/df.e / (numSubj*numStim))
Y005 <- qtukey(1-0.05, numStim, df.e) * sqrt(S.e/df.e / (numSubj*numStim))
cat(sprintf(" Y(0.05)=%.4f, Y(0.01)=%.4f\n", Y005, Y001))
cat(sprintf("\n"))
## confidence interval
cat(sprintf("Confidence Interval\n"))
cat(sprintf("------------------+-----------------------+----------------------\n"))
cat(sprintf(" | 95%%CI | 99%%CI \n"))
cat(sprintf(" ai - aj +-----------+-----------+-----------+----------\n"))
cat(sprintf(" | %+9.4f | %+9.4f | %+9.4f | %+9.4f \n",
Y005, -Y005, Y001, -Y001))
cat(sprintf("------------------+-----------+-----------+-----------+----------\n"))
for (r in 1:(numStim-1)) {
for (c in (r+1):numStim) {
z <- alphai[r]-alphai[c]
cat(sprintf("a%d-a%d = %9.4f | %9.4f | %9.4f | %9.4f | %9.4f\n",
r, c, z, z+Y005, z-Y005, z+Y001, z-Y001))
}
}
cat(sprintf("------------------+-----------+-----------+-----------+----------\n"))
cat(sprintf("\n"))
}
3つの試料A〜Cの場合、A-B、A-C、B-Cの3通りの一対比較が考えられる。6人による7段階評定((-3)〜(+3))が以下の表のようになったとする。
| A-B比較 | A-C比較 | B-C比較
––––|———|———|–––– 評定者1 | -2 | -1 | 1 評定者2 | -2 | -2 | 2 評定者3 | -3 | 0 | 3 評定者4 | -3 | 0 | 2 評定者5 | -1 | 1 | 2 評定者6 | -3 | -1 | 0
表と同様にデータを準備して計算すると、分散分析表と信頼区間の表が出力される。((p)値より、主効果に有意差があることがわかる)
numStim <- 3
numSubj <- 6
Y <- matrix(data=c(
## comparison result of pairs A-B, A-C, B-C
-2, -1, 1, # subject 1
-2, -2, 2, # subject 2
-3, 0, 3, # subject 3
-3, 0, 2, # subject 4
-1, 1, 2, # subject 5
-3, -1, 0), # subject 6
ncol=numSubj, byrow=FALSE)
scheffe.nakaya(Y, numStim, numSubj)
##
## Average Preferences
## ---------------
## a1 = -0.9444
## a2 = 1.3333
## a3 = -0.3889
## ---------------
##
## ANOVA Table
## --------------+---------------------------------------------
## Source | SS df MS F p
## --------------+---------------------------------------------
## Main | 50.7778 2 25.3889 43.1132 0.0007 ***
## Main x Indiv | 11.2222 10 1.1222 1.9057 0.2470
## Combi | 0.0556 1 0.0556 0.0943 0.7711
## Error | 2.9444 5 0.5889
## Total | 65.0000 18
## --------------+---------------------------------------------
## Y(0.05)=0.8323, Y(0.01)=1.2617
##
## Confidence Interval
## ------------------+-----------------------+----------------------
## | 95%CI | 99%CI
## ai - aj +-----------+-----------+-----------+----------
## | +0.8323 | -0.8323 | +1.2617 | -1.2617
## ------------------+-----------+-----------+-----------+----------
## a1-a2 = -2.2778 | -1.4454 | -3.1101 | -1.0160 | -3.5395
## a1-a3 = -0.5556 | 0.2768 | -1.3879 | 0.7062 | -1.8173
## a2-a3 = 1.7222 | 2.5546 | 0.8899 | 2.9840 | 0.4605
## ------------------+-----------+-----------+-----------+----------