先日の浦の変法に続き、シェッフェの一対比較法（中屋の変法）を実装してみました。佐藤信『統計的官能検査法』（日科技連出版社）に書かれている方法をそのままコードにして、今度はp値の計算結果も表示するようにしただけです。詳しいことは同書を参照して下さい。
使用例は以下の通り。
numStim <- 3 numSubj <- 6 Y <- matrix(data=c( ## comparison result of pairs 1-2, 1-3, 2-3 -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)
すると平均嗜好度・分散分析表・信頼区間がこんな感じで出てきます。
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
------------------+-----------+-----------+-----------+----------コードは以下の通り。
## Scheffe's Pairwise Comparison ## (Nakaya's Variation, not considering stimulus order effect) ## ## 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")) cat(sprintf("Main | %9.4f %4d %9.4f %9.4f %9.4f\n", S.alpha, df.alpha, S.alpha/df.alpha, (S.alpha/df.alpha)/(S.e/df.e), pf((S.alpha/df.alpha)/(S.e/df.e), df.alpha, df.e, lower.tail=FALSE))) cat(sprintf("Main x Indiv | %9.4f %4d %9.4f %9.4f %9.4f\n", S.alphaB, df.alphaB, S.alphaB/df.alphaB, (S.alphaB/df.alphaB)/(S.e/df.e), pf((S.alphaB/df.alphaB)/(S.e/df.e), df.alphaB, df.e, lower.tail=FALSE))) cat(sprintf("Combi | %9.4f %4d %9.4f %9.4f %9.4f\n", S.gamma, df.gamma, S.gamma/df.gamma, (S.gamma/df.gamma)/(S.e/df.e), pf((S.gamma/df.gamma)/(S.e/df.e), df.gamma, df.e, lower.tail=FALSE))) 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")) }
