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Contents

1 SE, Standard Error
2 蠏 觜蟲 蟆 覦覯
3 t 蟆 螳
4 One Sample t-test
5 Two Sample t-test
6 Paired t-test
7 覿磯
8 殊覿磯
9 伎覿磯
10 螻給磯(ANCOVA; Analysis of Covariance)
11 谿伎 譟壱
12 谿瑚


1 SE, Standard Error #

覈暑レ 蠍語企ゼ 5 豸′.
x <- c(76.2, 76.3, 76.1, 76,3, 76.4)
se <- sd(x)/sqrt(length(x)) #0.6009252
se #1.643168


2,000覈 覈螻 煙 .
set.seed(1000)
x <- rnorm(2000, mean = 70, sd = 10)
summary(x)
蟆郁骸
> summary(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  36.38   63.33   69.89   69.94   76.54   99.19 

譴れ姶..
mu <- c()
for(i in 1:100){
    mu <- c(mean(sample(x, 5)), mu)
}

se <- sd(mu) / sqrt(5)
mean(mu) #69.82971
se #1.935552


2 蠏 觜蟲 蟆 覦覯 #

蠏覿 覿蟆覯
Yt 蟆, 覿磯
NWilcoxon 蟆, Kurskal-Wallis 蟆

  • Two-sample t-test <-> Wilcoxonrank sum test
  • Paired t-test <-> Wilcoxonsigned rank test

3 t 蟆 螳 #

  • 蠏 觜蟲 企 讌 螳朱 螻 る 蟆 覩誤.
  • 伎豺(outlier)螳 蠏覿螳 一危磯 t 蟆 覃 .

蠍一 蠍一
  • 蠏覿覿 覦覯
  • 伎豺 蟇 覦覯

4 One Sample t-test #

2 1覦 2011 襭 蠏 蟆螳 2.1螳 伎. 2012 10覈 覓伎襦 覦 蟆 螳 譟一. 2011螻 2012 るジ螳?
x <- c(3.3, 2.8, 3.0, 2.7, 2.7, 2.0, 1.9, 3.4, 1.4, 1.4)
t.test(x, mu=2.1)

蠏 蟆
> shapiro.test(x)

	Shapiro-Wilk normality test

data:  x 
W = 0.9085, p-value = 0.2711

    • 蠏覓願: 蠏覿 谿願 .
    • 襴所: 蠏覿 谿願 .
  • 譴 0.05手 , p-value螳 0.2711企襦 蠏覓願 蠍郁讌 覈詩. 讀, 蠏覿.

> t.test(x, mu=2.1)

	One Sample t-test

data:  x 
t = 1.5454, df = 9, p-value = 0.1567
alternative hypothesis: true mean is not equal to 2.1 
95 percent confidence interval:
 1.933026 2.986974 
sample estimates:
mean of x 
     2.46 

    • 蠏覓願: 蠏 谿願 .
    • 襴所: 蠏 谿願 .
  • 譴 0.05手 , p-value螳 0.1567企襦 蠏覓願 蠍郁讌 覈詩.

螳 蟲螳豢 .
> t.test(x)

	One Sample t-test

data:  x 
t = 10.6, df = 9, p-value = 2.269e-06
alternative hypothesis: true mean is not equal to 0 
95 percent confidence interval:
 1.93 2.99 
sample estimates:
mean of x 
     2.46 

5 Two Sample t-test #

  • 讌 蠏 觜蟲

    • 讌 蠏覿
    • 讌 覿一 螳


x1 <- c(15,10,13,7,9,8,21,9,14,8)
x2 <- c(15,14,12,8,14,7,16,10,15,12)

蠏 蟆 --> x1, x2螳 譴 0.05 蠏覿.
> shapiro.test(x1)

	Shapiro-Wilk normality test

data:  x1 
W = 0.8666, p-value = 0.09131

> shapiro.test(x2)

	Shapiro-Wilk normality test

data:  x2 
W = 0.9125, p-value = 0.2986

覿一 狩螳?
> var.test(x1, x2)

	F test to compare two variances

data:  x1 and x2 
F = 1.9791, num df = 9, denom df = 9, p-value = 0.3237
alternative hypothesis: true ratio of variances is not equal to 1 
95 percent confidence interval:
 0.491579 7.967821 
sample estimates:
ratio of variances 
          1.979094 

    • 蠏覓願: x1螻 x2 覿一 谿願 .
    • 襴所: x1螻 x2 覿一 谿願 .
  • 譴 0.05手 , p-value螳 0.3237企襦 蠏覓願 蠍郁讌 覈詩. 讀, 覿一 谿願 .

讌 蠏 狩讌 蟆
> t.test(x1, x2, var.equal=T)

	Two Sample t-test

data:  x1 and x2 
t = -0.5331, df = 18, p-value = 0.6005
alternative hypothesis: true difference in means is not equal to 0 
95 percent confidence interval:
 -4.446765  2.646765 
sample estimates:
mean of x mean of y 
     11.4      12.3 

    • 蠏覓願: x1螻 x2 蠏 谿願 .
    • 襴所: x1螻 x2 蠏 谿願 .
  • 譴 0.05手 , p-value螳 0.6005企襦 蠏覓願 蠍郁讌 覈詩. 讀, 讌螳 蠏 谿願 .

豸 蟆 蟆曙 (殊 讀, x1 る 蟆)
> t.test(x1, x2, alternative="less", var.equal=T)

	Two Sample t-test

data:  x1 and x2
t = -0.5331, df = 18, p-value = 0.3002
alternative hypothesis: true difference in means is less than 0
95 percent confidence interval:
     -Inf 2.027436
sample estimates:
mean of x mean of y 
     11.4      12.3 

6 Paired t-test #

  • Paired -> 11 30覈 豢豢 覈碁願襯 豸′螻, 12 11 豢豢 30覈 覈碁願襯 れ 豸′ 蠏谿願 讌 蟆 ..
  • 襾語 Two Sample t-test
  • Two Sample t-test 一危郁 Paired 一危磯朱..

> t.test(x1, x2, var.equal=T, paired=T)

	Paired t-test

data:  x1 and x2 
t = -0.9612, df = 9, p-value = 0.3616
alternative hypothesis: true difference in means is not equal to 0 
95 percent confidence interval:
 -3.018069  1.218069 
sample estimates:
mean of the differences 
                   -0.9 

    • 蠏覓願: x1螻 x2 蠏 谿願 .
    • 襴所: x1螻 x2 蠏 谿願 .
  • 譴 0.05手 , p-value螳 0.3616企襦 蠏覓願 蠍郁讌 覈詩. 讀, 讌螳 蠏 谿願 .

7 覿磯 #

  • 3螳 伎 讌 蠏 谿 蟆(two-sample test )
  • 覿磯
    • 殊覿磯(one-way ANOVA) - 1螳 蠏碁9覲
    • 伎覿磯(two-way ANOVA) - 2螳 蠏碁9覲
    • 螻給磯(ANCOVA; Analysis of Covariance) - 殊覿磯 一 覲 螻給(covariate)豢螳
  • 蠍磯蓋 螳
    • 讌螳 襴
    • 蠏覿襯 磯殊
    • 狩 覿
  • 蠍磯蓋 螳 襷讌 蟆曙
    • 覿磯 Kurskal-Wallis Test


8 殊覿磯 #

#http://code.google.com/p/sonya/source/browse/trunk/r-project/sample/PlantGrowth.csv
plantGrowth = read.csv("c:\\data\\PlantGrowth.csv")
head(plantGrowth)
boxplot(weight ~ group, data=plantGrowth)
out <- lm(weight ~ group, data=plantGrowth)
summary(out)
anova(out)

> summary(out)

Call:
lm(formula = weight ~ group, data = plantGrowth)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.0710 -0.4180 -0.0060  0.2627  1.3690 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   5.0320     0.1971  25.527   <2e-16 ***
grouptrt1    -0.3710     0.2788  -1.331   0.1944    
grouptrt2     0.4940     0.2788   1.772   0.0877 .  
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1 

Residual standard error: 0.6234 on 27 degrees of freedom
Multiple R-squared: 0.2641,	Adjusted R-squared: 0.2096 
F-statistic: 4.846 on 2 and 27 DF,  p-value: 0.01591 

> anova(out)
Analysis of Variance Table

Response: weight
          Df  Sum Sq Mean Sq F value  Pr(>F)  
group      2  3.7663  1.8832  4.8461 0.01591 *
Residuals 27 10.4921  0.3886                  
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1

    • 蠏覓願: 3蠏碁9 蠏 螳.
    • 襴所: 3蠏碁9 蠏 るゴ.
  • 譴 0.05手 , p-value螳 0.01591企襦 蠏覓願 蠍郁. 讀, 3蠏碁9 蠏 谿願 .
  • 蠏碁, 蠏讌 企 讌 .

par(mfrow=c(2,2))
plot(out)
1.png

蠏 -> p-value = 0.4379企襦 蠏覿.
> shapiro.test(resid(out))

	Shapiro-Wilk normality test

data:  resid(out) 
W = 0.9661, p-value = 0.4379

焔一 -> p-value = 0.1714襦 蠏覓願 讌讌. 讀, 焔
#library("lmtest")
> bptest(out)

	studentized Breusch-Pagan test

data:  out 
BP = 3.5273, df = 2, p-value = 0.1714

襴曙
> dwtest(out) #library("lmtest")

	Durbin-Watson test

data:  out 
DW = 2.704, p-value = 0.9502
alternative hypothesis: true autocorrelation is greater than 0 

  • d糾
    • 0 : 蠍一蟯, p-value = 1
    • 2 : 襴, p-value = 0
    • 4 : 蠍一蟯, p-value = -1
  • 旧朱 DW螳 1覲企 蟇磯 3覲企 覃 蠍一蟯 ろ り 朱, 1.5~2.5 伎 蟆曙 襴曙企手
  • 襴.

蠏, 焔一, 襴曙, Q-Q Plot 讌煙 誤 覈 蟆 誤. 3螳 蠏碁9 蠏 谿 蟆 蟆郁骸 谿願 . 企 蠏碁9朱Μ 谿願 覦讌 覲伎.

覦覯1: Dunnett -> 蠏 谿願 譟壱 覲伎譴. (control 觜 觜蟲覯)
install.packages("multcomp")
library("multcomp")
out <- lm(weight ~ group, data=PlantGrowth)
dunnett <- glht(out, linfct=mcp(group="Dunnett")) #蠍一 group plantGrowth$group 企.
summary(dunnett)
plot(dunnett)
2.png
95% 襤郁規螳 0

> summary(dunnett)

	 Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Dunnett Contrasts


Fit: lm(formula = weight ~ group, data = PlantGrowth)

Linear Hypotheses:
                 Estimate Std. Error t value Pr(>|t|)
trt1 - ctrl == 0  -0.3710     0.2788  -1.331    0.323
trt2 - ctrl == 0   0.4940     0.2788   1.772    0.153
(Adjusted p values reported -- single-step method)
  • trt1 - ctrl p-value = 0.323 -> 蠏覓願 讌讌, 蠏 谿
  • trt2 - ctrl p-value = 0.153 -> 蠏覓願 讌讌, 蠏 谿

覦覯2: Tukey -> 蠏 谿願 譟壱螻 蠏 谿願 譟壱 覈 覲伎譴.
install.packages("multcomp")
library("multcomp")
out <- lm(weight ~ group, data=PlantGrowth)
tukey <- glht(out, linfct=mcp(group="Tukey")) #蠍一 group PlantGrowth$group 企.
summary(tukey)
plot(tukey)
3.png

> summary(tukey)

	 Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts


Fit: lm(formula = weight ~ group, data = plantGrowth)

Linear Hypotheses:
                 Estimate Std. Error t value Pr(>|t|)  
trt1 - ctrl == 0  -0.3710     0.2788  -1.331    0.391  
trt2 - ctrl == 0   0.4940     0.2788   1.772    0.198  
trt2 - trt1 == 0   0.8650     0.2788   3.103    0.012 *
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1 
(Adjusted p values reported -- single-step method)
  • trt1 - ctrl p-value = 0.323 -> 蠏覓願 讌讌, 蠏 谿
  • trt2 - ctrl p-value = 0.153 -> 蠏覓願 讌讌, 蠏 谿
  • trt2 - trt1 p-value = 0.153 -> 蠏覓願 蠍郁, 蠏 谿

9 伎覿磯 #

#http://code.google.com/p/sonya/source/browse/trunk/r-project/sample/warpbreaks.csv?r=653
warpbreaks = read.csv("c:\\data\\warpbreaks.csv")

伎壱 覲 wool螻 tension
> levels(warpbreaks$wool)
[1] "A" "B"
> levels(warpbreaks$tension)
[1] "L" "M" "H"

  • wool -> 螳 襷.
  • tension -> 螳 襷讌 . L, M, H 伎伎 .

tension 襯 覦襦 ′.
> warpbreaks$tension = factor(warpbreaks$tension, level = c("L", "M", "H"))
> levels(warpbreaks$tension)
[1] "L" "M" "H"

覿磯
out <- lm(breaks ~ wool*tension, data = warpbreaks)
summary(out)

> summary(out)

Call:
lm(formula = breaks ~ wool * tension, data = warpbreaks)

Residuals:
     Min       1Q   Median       3Q      Max 
-19.5556  -6.8889  -0.6667   7.1944  25.4444 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)      44.556      3.647  12.218 2.43e-16 ***
woolB           -16.333      5.157  -3.167 0.002677 ** 
tensionM        -20.556      5.157  -3.986 0.000228 ***
tensionH        -20.000      5.157  -3.878 0.000320 ***
woolB:tensionM   21.111      7.294   2.895 0.005698 ** 
woolB:tensionH   10.556      7.294   1.447 0.154327    
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1 

Residual standard error: 10.94 on 48 degrees of freedom
Multiple R-squared: 0.3778,	Adjusted R-squared: 0.3129 
F-statistic: 5.828 on 5 and 48 DF,  p-value: 0.0002772 
  • 蠏 谿願 (讌 蠏讌 讌 )
  • 蟲語
    • woolB:tensionM -> p-value = 0.005698, 譴 0.05 蠏覓願 蠍郁 => 蟲語 誤!!
    • woolB:tensionH -> p-value = 0.154327, 譴 0.05 蠏覓願 讌讌

蠏 -> p-value = 0.8162企襦 蠏覿.
> shapiro.test(resid(out))

	Shapiro-Wilk normality test

data:  resid(out) 
W = 0.9869, p-value = 0.8162

焔一 -> p-value = 0.0006307襦 蠏覓願 蠍郁. 讀, 焔一 . 譬覲 breaks log() sqrt().
#library("lmtest")
> bptest(out)

	studentized Breusch-Pagan test

data:  out 
BP = 21.5744, df = 5, p-value = 0.0006307

襴曙
> dwtest(out)

	Durbin-Watson test

data:  out 
DW = 2.2376, p-value = 0.575
alternative hypothesis: true autocorrelation is greater than 0 

  • d糾
    • 0 : 蠍一蟯, p-value = 1
    • 2 : 襴, p-value = 0
    • 4 : 蠍一蟯, p-value = -1
  • 旧朱 DW螳 1覲企 蟇磯 3覲企 覃 蠍一蟯 ろ り 朱, 1.5~2.5 伎 蟆曙 襴曙企手
  • 襴.

譬覲 覲(log) れ 蟆
> out <- lm(log(breaks) ~ wool*tension, data = warpbreaks)
> shapiro.test(resid(out))

	Shapiro-Wilk normality test

data:  resid(out) 
W = 0.9729, p-value = 0.2583

> bptest(out)

	studentized Breusch-Pagan test

data:  out 
BP = 4.8045, df = 5, p-value = 0.4402

> dwtest(out)

	Durbin-Watson test

data:  out 
DW = 2.06, p-value = 0.3167
alternative hypothesis: true autocorrelation is greater than 0 
  • 蠏, 焔一, 襴曙煙 覈 螳豢.

企 譟壱 蠏 るジ螳?
library("multcomp")
out <- lm(log(breaks) ~ wool + tension, data=warpbreaks)
tukey1 <- glht(out, linfct=mcp(wool="Tukey"))
tukey2 <- glht(out, linfct=mcp(tension="Tukey"))
summary(tukey1)
summary(tukey2)

> summary(tukey1)

	 Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts


Fit: lm(formula = log(breaks) ~ wool + tension, data = warpbreaks)

Linear Hypotheses:
           Estimate Std. Error t value Pr(>|t|)
B - A == 0  -0.1522     0.1063  -1.431    0.159
(Adjusted p values reported -- single-step method)

> summary(tukey2)

	 Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts


Fit: lm(formula = log(breaks) ~ wool + tension, data = warpbreaks)

Linear Hypotheses:
           Estimate Std. Error t value Pr(>|t|)   
M - L == 0  -0.2871     0.1302  -2.205  0.08018 . 
H - L == 0  -0.4893     0.1302  -3.758  0.00133 **
H - M == 0  -0.2022     0.1302  -1.553  0.27550   
---
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1 
(Adjusted p values reported -- single-step method)

10 螻給磯(ANCOVA; Analysis of Covariance) #

  • lm(譬覲 ~ 螻給覲 + 蠏碁9覲)襦 覃 .
  • 螻給覲
    • 一 覲
    • 譬覲 レ 殊 企 れ れ 覲
    • : 覈碁願螳 殊襦 豺襭 蟯 豺襭 / 谿企 . -> 豺襭 覈碁願襯 螻給 覲襦 豢螳
  • 螻給 覲襯 豢螳 蟆 觜手 覿磯螻

11 谿伎 譟壱 #

out <- aov(lifetime ~ q, data=x8)
TukeyHSD(out)

12 谿瑚 #


Hey, that's the grseetat! So with ll this brain power AWHFY? -- Earng 2015-12-11 04:19:24
蠍 蠍郁鍵..
企: : るジ讓曙 襦螻豺 企Ν 譯殊語. 襦螻豺
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