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--- anova [2018/10/19 07:54] – [F and t value] hkimscil
+++ anova [2020/05/20 15:51] – [Example 2] hkimscil
@@ Line 375: / Line 375: @@
 ====== Post hoc test ======
 [[Post hoc test]]
+<code>> adata <- read.csv("https://datascienceplus.com/wp-content/uploads/2017/08/tyre.csv")
+> adata
+        Brands  Mileage
+       Apollo 32.99800
+       Apollo 36.43500
+       Apollo 32.77700
+       Apollo 37.63700
+       Apollo 36.30400
+       Apollo 35.91500
+       Apollo 34.70000
+       Apollo 32.37900
+       Apollo 33.63100
+      Apollo 36.41900
+      Apollo 36.43000
+      Apollo 34.83600
+      Apollo 38.32800
+      Apollo 30.62300
+      Apollo 32.57500
+Bridgestone 33.52300
+Bridgestone 31.99500
+Bridgestone 35.00600
+Bridgestone 27.87900
+Bridgestone 31.29700
+Bridgestone 31.06200
+Bridgestone 34.83800
+Bridgestone 33.97600
+Bridgestone 32.55200
+Bridgestone 30.88100
+Bridgestone 28.14400
+Bridgestone 29.18400
+Bridgestone 33.07500
+Bridgestone 32.36500
+Bridgestone 30.92500
+        CEAT 34.44565
+        CEAT 32.80658
+        CEAT 33.41499
+        CEAT 36.86118
+        CEAT 36.97277
+        CEAT 35.08145
+        CEAT 34.95412
+        CEAT 33.47516
+        CEAT 30.42748
+        CEAT 36.13392
+        CEAT 34.78336
+        CEAT 36.11675
+        CEAT 41.05000
+        CEAT 32.16845
+        CEAT 32.72624
+      Falken 39.59600
+      Falken 38.93700
+      Falken 36.12400
+      Falken 37.69500
+      Falken 36.58600
+      Falken 35.96700
+      Falken 36.73700
+      Falken 34.31000
+      Falken 40.25200
+      Falken 37.38200
+      Falken 40.66300
+      Falken 37.09500
+      Falken 38.00500
+      Falken 37.75600
+      Falken 37.26500
+>
+</code>
+<code>> fmod.tire <- aov(Mileage~Brands, data=adata)
+> summary(fmod.tire)
+            Df Sum Sq Mean Sq F value   Pr(>F)
+Brands       3  256.3   85.43   17.94 2.78e-08 ***
+Residuals   56  266.6    4.76
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+>
+</code>
+<code>
+> TukeyHSD(fmod.tire)
+  Tukey multiple comparisons of means
+% family-wise confidence level
+Fit: aov(formula = Mileage ~ Brands, data = adata)
+$Brands
+                          diff        lwr       upr     p adj
+Bridgestone-Apollo -3.01900000 -5.1288190 -0.909181 0.0020527
+CEAT-Apollo        -0.03792661 -2.1477456  2.071892 0.9999608
+Falken-Apollo       2.82553333  0.7157143  4.935352 0.0043198
+CEAT-Bridgestone    2.98107339  0.8712544  5.090892 0.0023806
+Falken-Bridgestone  5.84453333  3.7347143  7.954352 0.0000000
+Falken-CEAT         2.86345994  0.7536409  4.973279 0.0037424
+>
+> tapply(adata$Mileage, adata$Brands, mean)
+     Apollo Bridgestone        CEAT      Falken
+.79913    31.78013    34.76121    37.62467
+>
+</code>
 ====== F and t value ======
 $$ F = t^{2}$$
@@ Line 421: / Line 518: @@
 ====== Example ======
+가설. 단어맞히기 게임에서 첫글자를 힌트로 주거나, 마지막 글자를 힌트로 주거나, 힌트를 주지 않은 세 그룹 간에 틀린 단어의 숫자에 차이가 있을 것이다.
 |  |First Letter \\ Condition 1 \\ X<sub>1</sub>|Last Letter \\ Condition 2 \\ X<sub>2</sub>|No Letter \\ Condition 3 \\ X<sub>3</sub> |
 | | 15 | 21 | 28 |
@@ Line 433: / Line 532: @@
 | | 41 | 21 | 15 |
 | T  | 180  | 240  | 270  |
-| n  | 10  |  10  | 10  |
+| n  | 10  | 10  | 10  |
-| Mean  | $\overline{X_1}$ = 18  | $\overline{X_2}$ = 24  | $\overline{X_3}$ = 27  |
+| Mean  | 18  | 24  | 27  |
-| Total  | $ T_1 = 180 $  | $ T_2 = 240 $ | $ T_3 = 270 $  |
+| Mean%%^%%2  | 324  | 576  | 729  |
-| $\sum{(X_i)^2}$  | 3888  | 5868  | 7614  |
+| b = $n*(mean)^2$  | 3240  | 5760  | 7290  |
+| a = $\sum{(X_i)^2}$  | 3888  | 5868  | 7614  |
+| a-b = Var.within  | 648  | 108  | 324  | $f = $1080   |
+\begin{eqnarray*}
+a & = & \sum (X_i)^2 = 17370 \\
+b & = & \overline{X_{total}} = 23 \\
+c & = & \overline{X}^2 = 529 \\
+d & = & N = 30 \\
+e & = & SS_{total} = \sum {(X_i)^2} = a - (d * c) = 1500 \\
+f & = & SS_{within} = 648 + 108+ 324 = 1080 \\
+g & = & SS_{between} = e - f = 420
+\end{eqnarray*}
-$\sum (X_i)^2 = 17370 $ \\
+| m.total = Mean.total  | $b = \; $ 23  |||
-$\overline{X_{total}} = 23 $ \\
+| m.within = Mean.each.group  | 18  | 24  | 27  |
-$\overline{X}^2 = 529 $ \\
+| o = m.total - m.within  | 5   | -1  | -4  |
+| o%%^%%2  | 25  | 1  | 16  |
+| n  | 10  | 10  | 10  |
+| o%%^%%2 * n | 250  | 10  | 160  |
+| sum  | $g = \; $ 420  |||
-$ G = ? $ \\
-$ G^2 = ? $ \\
-$ \sum {(X_i)^2} = 17370 $ \\
-\\
-$ k =  $ \\
-$ n =  $ \\
-$ N =  $ \\
-\\
-$ df_{total} = $ \\
-$ df_{between} = $ \\
-$ df_{within} = $ \\
-<code>x1 <- c(15, 20, 14, 13, 18, 16, 13, 12, 18, 41)
+\begin{eqnarray*}
+k & = & 3 \\
+n & = & 10 \\
+N & = & 30 \\
+\end{eqnarray*}
+\begin{eqnarray*}
+e' & = & df_{between} =  \\
+f' & = & df_{within} =   \\
+g' & = & df_{total} =  \\
+\end{eqnarray*}
+|          | SS   | df    | MS  | F   |
+| between  | $e = \;$ 420   | $e' = \;$ 2 | 210  |  210/40 = 5.25  |
+| within   | $f = \;$ 1080  | $f' = \;$ 27 | 40  |   |
+| total    | $g = \;$ 1500  | $g' = \;$ 29 |   |   |
+F<sub>crit</sub>(2, 27) =  3.35
+F<sub>cal</sub> = 5.25
+F<sub>cal</sub> > F<sub>crit</sub> 이므로 3집단 간의 평균은 통계학적으로 의미가 있는 차이를 가지고 있다.
+====== Example 2 ======
+<code>
+x1 <- c(15, 20, 14, 13, 18, 16, 13, 12, 18, 41)
 x2 <- c(21, 25, 29, 18, 26, 22, 26, 24, 28, 21)
 x3 <- c(28, 30, 32, 28, 26, 30, 25, 36, 20, 15)
 </code>
-<code>> data.frame(x1,x2,x3)
+<code>> xc <- data.frame(x1,x2,x3)
    x1 x2 x3
   15 21 28
@@ Line 509: / Line 635: @@
 <code>
-> colnames(xs[1]) <- "wrong"
+> colnames(xs)  <- c("wrong", "condition")
-> colnames(xs[2]) <- "cond"
 </code>
-<code># cf
+<code>
+# cf
 # lengthofelements <- length(x1)
 # varofvariable <- var(x1)</code>
 <code>
-df_x1
+df.total <- length(xs$wrong) - 1
-df_x2
+ss.total <- var(xs$wrong)*df_tot
-df_x3
+var.total <- ss.total/df.total
+var.total.r <- var(xs$wrong)
-ss_x1
+df.x1 <- length(x1)-1
-ss_x2
+df.x2 <- length(x2)-1
-ss_x3
+df.x3 <- length(x3)-1
+ss.x1 <- var(x1)*df.x1
+ss.x2 <- var(x2)*df.x2
+ss.x3 <- var(x3)*df.x3
-df_bet
+ss.within <- ss.x1 + ss.x2 + ss.x3
-ss_bet
+df.within <- df.x1 + df.x2 + df.x3
+ss.between <- ss.total - ss.within
+df.between <- df.total - df.within
-df_tot
+ms.between <- ss.between/df.between
-ss_tot
+ms.within <- ss.within/df.within
+f.value <- ms.between/ms.within
-df_with
+ss.between
-ss_with
+df.between
-df_bet
+ss.within
-ss_bet
+df.within
-</code>
+ms.between
+ms.within
+f.value
-<code>
-df_tot <- length(xs$ind) - 1
-ss_tot <- var(xs$values)*df_tot
-var_tot <- var(xs$values)
-df_x1 <- length(x1)-1
-df_x2 <- length(x2)-1
-df_x3 <- length(x3)-1
-ss_x1 <- var(x1)*df_x1
-ss_x2 <- var(x2)*df_x2
-ss_x3 <- var(x3)*df_x3
 </code>
 ====== E.G. 1 (R) ======