분산분석 예제

1. 완전확률화설계

1.1. 반복수 동일한 경우group = c(rep(1,4), rep(2,4),rep(3,4), rep(4,4), rep(5,4))y = c(2.4, 2.7, 3.1, 3.1, 0.7, 1.6, 1.7, 1.8, 2.4, 3.1, 5.4, 6.1, 0.3, 0.3, 2.4, 2.7, 0.5, 0.9, 1.4, 2.0)sol = cbind(group, y)group = as.factor(group)aov1 = aov(y~group)summary(aov1)

## Df Sum Sq Mean Sq F value Pr(>F) ## group 4 27.01 6.752 5.966 0.00444 **## Residuals 15 16.98 1.132 ## ---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

plot(y~group) # 그룹별상자그림

summary.lm(aov1)

## ## Call:## aov(formula = y ~ group)## ## Residuals:## Min 1Q Median 3Q Max ## -1.8500 -0.7125 0.1750 0.4625 1.8500 ## ## Coefficients:## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 2.8250 0.5319 5.311 8.72e-05 ***## group2 -1.3750 0.7522 -1.828 0.0875 . ## group3 1.4250 0.7522 1.894 0.0776 . ## group4 -1.4000 0.7522 -1.861 0.0824 . ## group5 -1.6250 0.7522 -2.160 0.0473 * ## ---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## ## Residual standard error: 1.064 on 15 degrees of freedom## Multiple R-squared: 0.614, Adjusted R-squared: 0.5111 ## F-statistic: 5.966 on 4 and 15 DF, p-value: 0.004442

tapply(y, group, mean)

## 1 2 3 4 5 ## 2.825 1.450 4.250 1.425 1.200

plot(aov1)

bartlett.test(y~group) # 등분산성검정

## ## Bartlett test of homogeneity of variances

## ## data: y by group## Bartlett's K-squared = 8.7824, df = 4, p-value = 0.06677

• 분산분석표에서 p- 값 0.004442<0.05 이므로처리의평균이동일하다고할수없음

• 그룹별상자그림, 그룹별평균: 그룹 5<4<2<1<3

• 진단도표: (a) 잔차 vs 추정값 (b) 정규 QQ 도표 (c) 추정값 vs 표준화잔차의제곱근(0~2 범위) (d) 그룹별표준화잔차등에서특별한경향은발견하기어려움

• 등분산성검정: p 값 > 0.05 이므로등분산이라는귀무가설을기각할수없음

tapply(y, group, mean)

## 1 2 3 4 5 ## 2.825 1.450 4.250 1.425 1.200

pairwise.t.test(y, group, p.adjust="none", pool.sd=TRUE) # LSD

## ## Pairwise comparisons using t tests with pooled SD ## ## data: y and group ## ## 1 2 3 4 ## 2 0.0875 - - - ## 3 0.0776 0.0020 - - ## 4 0.0824 0.9739 0.0019 - ## 5 0.0473 0.7442 0.0010 0.7690## ## P value adjustment method: none

pairwise.t.test(y, group, p.adjust="bonferroni", pool.sd=FALSE) # Bonferroni

## ## Pairwise comparisons using t tests with non-pooled SD ## ## data: y and group ## ## 1 2 3 4 ## 2 0.057 - - - ## 3 1.000 0.465 - - ## 4 1.000 1.000 0.463 - ## 5 0.085 1.000 0.349 1.000## ## P value adjustment method: bonferroni

a.tukey = TukeyHSD(aov1, ordered=TRUE) # Tukey 의 HSDa.tukey

## Tukey multiple comparisons of means## 95% family-wise confidence level## factor levels have been ordered## ## Fit: aov(formula = y ~ group)## ## $group## diff lwr upr p adj## 4-5 0.225 -2.0977953 2.547795 0.9980497## 2-5 0.250 -2.0727953 2.572795 0.9970635## 1-5 1.625 -0.6977953 3.947795 0.2466201## 3-5 3.050 0.7272047 5.372795 0.0078279## 2-4 0.025 -2.2977953 2.347795 0.9999997## 1-4 1.400 -0.9227953 3.722795 0.3776700## 3-4 2.825 0.5022047 5.147795 0.0139855## 1-2 1.375 -0.9477953 3.697795 0.3944760## 3-2 2.800 0.4772047 5.122795 0.0149150## 3-1 1.425 -0.8977953 3.747795 0.3612849

plot(a.tukey)

1.2. 반복수가 다른 경우 눈동자색깔이갈색, 초록, 파랑인 19 명에대하여분동자깜박거림횟수데이터로광원을

감지했을때횟수로측정하며실험순서는눈동자색깔과랜덤하게배치됨

Color = c(rep("Brown", 8), rep("Green", 5), rep("Blue", 6))Color = as.factor(Color)Flicker = c(26.8, 27.9, 23.7, 25, 26.3, 24.8, 25.7, 24.5, 26.4, 24.2, 28.0, 26.9, 29.1, 25.7, 27.2, 29.9, 28.5, 29.4, 28.3)plot(Flicker~Color)

# 그룹별평균표준편차by(Flicker, Color, FUN=function(x) {c(mean(x), sd(x))})

## Color: Blue## [1] 28.166667 1.527962## -------------------------------------------------------- ## Color: Brown## [1] 25.587500 1.365323## -------------------------------------------------------- ## Color: Green## [1] 26.920000 1.843095

# 분산분석aov1 = aov(Flicker~Color)summary(aov1)

## Df Sum Sq Mean Sq F value Pr(>F) ## Color 2 23.00 11.499 4.802 0.0232 *## Residuals 16 38.31 2.394 ## ---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary.lm(aov1)

## ## Call:## aov(formula = Flicker ~ Color)## ## Residuals:## Min 1Q Median 3Q Max ## -2.7200 -0.8771 0.1125 1.1462 2.3125 ## ## Coefficients:## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 28.1667 0.6317 44.588 < 2e-16 ***## ColorBrown -2.5792 0.8357 -3.086 0.00708 ** ## ColorGreen -1.2467 0.9370 -1.331 0.20200 ## ---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## ## Residual standard error: 1.547 on 16 degrees of freedom## Multiple R-squared: 0.3751, Adjusted R-squared: 0.297 ## F-statistic: 4.802 on 2 and 16 DF, p-value: 0.02325

# 추정값predict(aov1)

## 1 2 3 4 5 6 7 8 ## 25.58750 25.58750 25.58750 25.58750 25.58750 25.58750 25.58750 25.58750 ## 9 10 11 12 13 14 15 16 ## 26.92000 26.92000 26.92000 26.92000 26.92000 28.16667 28.16667 28.16667 ## 17 18 19 ## 28.16667 28.16667 28.16667

# 진단plot(aov1)

# 다중비교pairwise.t.test(Flicker, Color, p.adjust="none", pool.sd=TRUE) # LSD

## ## Pairwise comparisons using t tests with pooled SD ## ## data: Flicker and Color ## ## Blue Brown ## Brown 0.0071 - ## Green 0.2020 0.1504## ## P value adjustment method: none

pairwise.t.test(Flicker, Color, p.adjust="bonferroni", pool.sd=FALSE) # Bonferroni

## ## Pairwise comparisons using t tests with non-pooled SD ## ## data: Flicker and Color ## ## Blue Brown## Brown 0.025 - ## Green 0.789 0.621## ## P value adjustment method: bonferroni

a.tukey = TukeyHSD(aov1, ordered=TRUE) # Tukey 의 HSDa.tukey

## Tukey multiple comparisons of means## 95% family-wise confidence level## factor levels have been ordered## ## Fit: aov(formula = Flicker ~ Color)## ## $Color## diff lwr upr p adj## Green-Brown 1.332500 -0.9437168 3.608717 0.3124225## Blue-Brown 2.579167 0.4228360 4.735497 0.0183579## Blue-Green 1.246667 -1.1710626 3.664396 0.3994319

plot(a.tukey)

• 분산분석결과 p-값<0.05 이므로눈의색에따라눈동자깜박거림횟수의평균이같지않음

• 진단결과특별한문제는...

• 다중비교결과: Blue 와 Brown 은차이가남

2. 확률화블럭설계3 가지세제에대해박테리아성장의지연효과를비교. 하루에 3 번실험이가능하므로실

험일을블럭으로하여데이터를얻음

trt = c(rep(1,4), rep(2,4), rep(3,4))trt = as.factor(trt)block = c(rep(1:4, 3))block = as.factor(block)y = c(20, 22, 18, 25, 16, 18, 17, 19, 30, 34, 29, 27)

# 시각화plot(y~trt)

plot(y~block)

# 분산분석fit = aov(y~trt+block)summary(fit)

## Df Sum Sq Mean Sq F value Pr(>F) ## trt 2 329.2 164.58 26.810 0.00102 **## block 3 20.9 6.97 1.136 0.40722 ## Residuals 6 36.8 6.14 ## ---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# LSDpairwise.t.test(y, trt, p.adjust="none")

## ## Pairwise comparisons using t tests with pooled SD ## ## data: y and trt ## ## 1 2 ## 2 0.06580 - ## 3 0.00087 6.5e-05## ## P value adjustment method: none

plot(fit)

• 그림에서처리별차이가있는것으로보임. 블럭별로는차이가유의하지않을듯

• 분산분석표: 처리효과는유의하지만블럭효과는유의하지않음

• 잔차분석결과큰문제발견하지못함

• LSD: 1-3, 2-3 은유의한차이

3. 이원배치 분산분석(two-way ANOVA)y i j k=μ+α i+β j+¿

• 주효과: α 는 I 개의수준, β 는 J 개의수준

• ni j: α 와 β 의수준이 i 와 j 일때관측값의갯수

PVC 생산에미치는요인을알아보기위해 3 명의작업자에게 8 가지다른장비를사용하여PVC 를생산하도록함. 24 가지조합에서두번반복

library(faraway)data(pvc)with(pvc, stripchart(psize~resin, xlab="Particle Size", ylab="Resin Railcar"))

with(pvc, stripchart(psize~operator, xlab="Particle Size", ylab="Operator"))

with(pvc, interaction.plot(operator, resin, psize))

with(pvc, interaction.plot(resin, operator, psize))

시각적으로볼때대략평행하므로교호작용이강한것같지는않음

# full modelg = lm(psize ~ operator*resin, pvc)anova(g)

## Analysis of Variance Table## ## Response: psize## Df Sum Sq Mean Sq F value Pr(>F) ## operator 2 20.718 10.359 7.0072 0.00401 ** ## resin 7 283.946 40.564 27.4388 5.661e-10 ***## operator:resin 14 14.335 1.024 0.6926 0.75987 ## Residuals 24 35.480 1.478 ## ---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

qqnorm(residuals(g))qqline(residuals(g))

plot(fitted(g), residuals(g), xlab="Fitted", ylab="Residuals")

# 잔차값이큰두값제거후g2 = lm(psize ~ operator*resin, pvc[-c(45,46),])anova(g2)

## Analysis of Variance Table## ## Response: psize## Df Sum Sq Mean Sq F value Pr(>F) ## operator 2 24.546 12.273 13.5063 0.000132 ***## resin 7 283.018 40.431 44.4936 6.945e-12 ***## operator:resin 13 11.020 0.848 0.9329 0.537167 ## Residuals 23 20.900 0.909 ## ---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# main effectsanova(lm(psize~operator+resin, pvc))

## Analysis of Variance Table## ## Response: psize## Df Sum Sq Mean Sq F value Pr(>F) ## operator 2 20.718 10.359 7.902 0.00135 ** ## resin 7 283.946 40.564 30.943 8.111e-14 ***## Residuals 38 49.815 1.311 ## ---## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

TukeyHSD(aov(psize~operator+resin, pvc))

## Tukey multiple comparisons of means## 95% family-wise confidence level## ## Fit: aov(formula = psize ~ operator + resin, data = pvc)## ## $operator## diff lwr upr p adj## 2-1 -0.26250 -1.249747 0.7247472 0.7943575## 3-1 -1.50625 -2.493497 -0.5190028 0.0018126## 3-2 -1.24375 -2.230997 -0.2565028 0.0106800## ## $resin## diff lwr upr p adj## 2-1 -1.0333333 -3.1522815 1.0856149 0.7683288## 3-1 -5.8000000 -7.9189482 -3.6810518 0.0000000## 4-1 -6.1833333 -8.3022815 -4.0643851 0.0000000## 5-1 -4.8000000 -6.9189482 -2.6810518 0.0000003## 6-1 -5.4500000 -7.5689482 -3.3310518 0.0000000## 7-1 -2.9166667 -5.0356149 -0.7977185 0.0019046## 8-1 -0.1833333 -2.3022815 1.9356149 0.9999924## 3-2 -4.7666667 -6.8856149 -2.6477185 0.0000003

## 4-2 -5.1500000 -7.2689482 -3.0310518 0.0000001## 5-2 -3.7666667 -5.8856149 -1.6477185 0.0000379## 6-2 -4.4166667 -6.5356149 -2.2977185 0.0000018## 7-2 -1.8833333 -4.0022815 0.2356149 0.1127668## 8-2 0.8500000 -1.2689482 2.9689482 0.8984776## 4-3 -0.3833333 -2.5022815 1.7356149 0.9989372## 5-3 1.0000000 -1.1189482 3.1189482 0.7958917## 6-3 0.3500000 -1.7689482 2.4689482 0.9994110## 7-3 2.8833333 0.7643851 5.0022815 0.0022073## 8-3 5.6166667 3.4977185 7.7356149 0.0000000## 5-4 1.3833333 -0.7356149 3.5022815 0.4375901## 6-4 0.7333333 -1.3856149 2.8522815 0.9507745## 7-4 3.2666667 1.1477185 5.3856149 0.0003909## 8-4 6.0000000 3.8810518 8.1189482 0.0000000## 6-5 -0.6500000 -2.7689482 1.4689482 0.9741405## 7-5 1.8833333 -0.2356149 4.0022815 0.1127668## 8-5 4.6166667 2.4977185 6.7356149 0.0000007## 7-6 2.5333333 0.4143851 4.6522815 0.0098978## 8-6 5.2666667 3.1477185 7.3856149 0.0000000## 8-7 2.7333333 0.6143851 4.8522815 0.0042481