1 R ID 1. ID Before After 1 X 1 Y 1 2 X 2 Y n 1 X n 1 Y n 1 n X n Y n. ID Group Measure. 1 1 Y 1... n 1 1 Y n1 n Y n n 0 Y n 1 E

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1 2010 R 0 C626 R 2 t Welch t Wilcoxon 3 Fisher McNemar Box-Muller p- Excel R 1 B USB tomo-statim i.softbank.jp R WWW D3 C626 d hiroshima-u.ac.jp url R 1

2 1 R ID 1. ID Before After 1 X 1 Y 1 2 X 2 Y n 1 X n 1 Y n 1 n X n Y n. ID Group Measure. 1 1 Y 1... n 1 1 Y n1 n Y n n 0 Y n 1 Excel 2 1 Data0415.csv (N) Data0415.csv (T) CSV( ).csv) R Data <- read.csv( Data0415.csv ) R Data Enter Data0415.csv C: Documents and Settings Administrator My Documents / Data0415.csv Data ID, GROUP, SCORE SCORE hist(data$score) #SCORE mean(data$score) #SCORE var(data$score) #SCORE 3 Data 2010 R 2

3 2 6, 2, 2, 4, 3, 5,... x n+1 = ax n + b modm x 1, x 2,... ModM M R for for(i in 1:100){ # C rand a = , b = 12345, M = 2 32 X M <- 2^(32) # a < ; b < X <- 1 # for(i in 1:100){ # 100 X <- c(x, (a*x[i]+b)%%m) # X (a*x[i]+b)%%m # X[i] X i %%M M (MT) 2 {x n x j+n := x j+m + x j+1 B + x j C (j = 0, 1,...) R MT 5 R runif(n) [0, 1] N [0, 0.5] [0.5, 1] coin p <- 0.5 # coin <- (NULL) # for(i in 1:20){ # 20 coin <- c(coin, ifelse(runif(1)<p,1,0)) # p 1 # R 3

4 R if-else ifelse(, 1, 2) # 1 2 Box-Muller R rnorm(n) N Box-Muller U 1, U 2 [0, 1] X 1 = 2 log U 1 cos(2πu 2 ), X 2 = 2 log U 1 sin(2πu 2 ) 6 U 1, U 2 X = 2 log U 1 cos(2πu 2 ) X NData Box-Muller Data1 <- runif(1000) # U Data2 <- runif(1000) # U R1 <- sqrt(-2*log(data1)) # sqrt(-2*log(u1)) R2 <- cos(2*pi*data2) # cos(2*pi*u2) NData <- R1*R2 # R1 R2 curve(500*dnorm(x,0,1),col= red,add=t) X 1, X 2 N(µ 1, σ 2 1), N(µ 2, σ 2 2) X 1 +X 2 N(µ 1 + µ 2, σ σ 2 2) NData1,NData2 NData1+NData2 NData1 <- rnorm(1000,3,2) # 3, NData2 <- rnorm(1000,5,4) NDataU <- NData1+NData2 # Data1 Data2 X 1, X 2,..., X n N(µ, σ 2 ) X = 1 n X i N(µ, σ2 n n ) i= R 4

5 3 N(µ, 1) µ H 0 : µ = 0 v.s. H 1 : µ > 0 X 1, X 2,..., X n X = n 1 n i=1 X i H 1 8 H 0 X X H 0 Data <- (NULL) AveData <- (NULL) N <- 10 for(i in 1:10000){ Data <- rnorm(n) Ave <- mean(data) AveData <- c(avedata, Ave) hist(avedata) # # N 10 # # 0 N # Ave # Ave # 8 H 0 H 1 X 1 H 0 H 0 α 9 5% X > 1.64 σ 2 /n H Data <- (NULL) B < C <- 0 N <- 20 for(i in 1:B){ Data <- rnorm(n) if(mean(data)>1.64/sqrt(n)) C <- C+1 C/B # # # 0 # # B # N # 1.64/sqrt(N) # 1 # 2010 R 5

6 , Power H 0 H 0 H 1 H rnorm(n) rnorm(n)+0.3 µ = 0.3 H 0 µ N µ 11 µ 0 Data <- (NULL) x <- (1:10)* y <- (NULL) B < N <- 100 for(j in 1:10){ C <- 0 for(i in 1:B){ Data <- rnorm(n)+0.05*(j-1) if(mean(data)>1.64/sqrt(n)) C <- C+1 y <- c(y, C/B) plot(x,y,type="l") p- # # x ( ) # # # j=1,2,..,10 # 0 # B # 0.05*(j-1) N # 1.64/sqrt(N) # 1 # y H0 ( ) # y P ( X z 0.05 H 0 ) = 0.05 z 0.05 m = x m z 0.05 H 0 p- P ( X m H 0 ) p % p- 2 (1) /sqrt(N) 0.35 (2) n X N(0, 1) n x R pnorm 1-pnorm(sqrt(20)*0.35) 1 α 2010 R 6

7 R boxplot( ) Data1 Data2 2 R boxplot(data1$before, Data1$After) # Data1 2 Data2 Score 2 Data2$Score[Data$Group==1] boxplot 2 1 Measure (Student )t- Welch t- Wilcoxon Wilcoxon (Student )t- Welch t QQ rnorm(100), rchisq(100,2) QQ 2010 R 7

8 Data <- rchisq(100,2) qqnorm(data) # Data # Data QQ Student t- 2 N(µ 1, σ 2 ), N(µ 2, σ 2 ) H 0 : µ 1 = µ 2 v.s. H 1 : µ 1 µ 2 Ȳ1 Ȳ2 0 H 1 15 F - var.test F - p-value 0.20 Student t- Welch t- p-value 2 Data$ F - var.test(score[group==1],score[group==0]) Student t- t.test(score[group==1],score[group==0],var.equal=t) Welch t- t.test(score[group==1],score[group==0],var.equal=f) F - t- α 4 t- 1% F - 4% Wilcoxon median(score[group==1]) # 1 summary(data2$score[group==1]) 17 Wilcoxon p- Wilcoxon Mann-Whitney wilcox.test(score[group==1],score[group==0]) 2010 R 8

9 5 θ θ ˆθ E(ˆθ) = θ 18 σ 2 V ˆσ 2 V = 1 n (x i x) 2, ˆσ 2 = 1 n (x i x) 2, n 1 n i=1 i=1 ( ˆσ 2 = n 1 ) n V R var(x) X V ˆσ 2 X <- (NULL); V <- (NULL) # X,V N <- 20 ; B < # for(i in 1:B){ # B X <- rnorm(n) # X N V <- c(v,var(x)) # V X # X 1, X 2,..., X n ˆθ n = ˆθ n (X 1, X 2,..., X n ) ˆθ n p θ (n ) ˆθ n 19 n 0 X <- (NULL) x <- (1:100)*10 y <- (NULL) for(i in 1:100){ X <- c(x,rnorm(10)) y <- c(y,mean(x)) plot(x,y,type="l") abline(h=0) # # x # n # 100 # 10 X # X y # y # y= R 9

10 N(µ, σ 2 ) µ µ 20 Data <- (NULL) # Mean <- (NULL) # Median <- (NULL) # for(i in 1:10000){ # Data <- rnorm(20) # 20 Data Mean <- c(mean,mean(data)) # Data Median <- c(median,median(data)) # Data x 20 0 E((ˆθ θ) 2 ) 21 µ = 0 mean(mean*mean) Mean Mean-θ % 22 hist(rchisq(1000,2)) N 2, 5, 10, 20, 50 N S <- (NULL) N <- 100 Data <- (NULL) for(i in 1:1000){ Data <- rchisq(n,2) S <- c(s,mean(data)) # # # Data # 1000 # 2 N Data # Data S 2010 R 10

11 6 95% N(µ, σ 2 ) X 1, X 2,..., X n 95% ( ) σ 2 σ X 1.96 n, 2 X n 23 N(0, 1) 20 95% 95% n <- 20 Data <- rnorm(n) u <- mean(data)+1.96/sqrt(n) l <- mean(data)-1.96/sqrt(n) c(l,u) 95% # # N(0,1) n # u ( =1 ) # l # 95% 95% 95% 95% 24 % 0 ( ) n <- 20; B < ; C <- 0 # for(i in 1:B){ # B Data <- rnorm(n) # N(0,1) n u <- mean(data)+1.96/sqrt(n) ; l <- mean(data)-1.96/sqrt(n) if(u>0&&l<0) C <- C+1 # (l,u) 0 # 1 C/B # n <- 50; B <- 100 # d <- 5.5/sqrt(n) # plot(1:b, ylim=c(-d,d),type="n") # abline(h=0, col="red") # 0 for(i in 1:B){ Data <- rnorm(data) u <- mean(data)+1.96/sqrt(n) ; l <- mean(data)-1.96/sqrt(n) segments(i,l,i,u) # if(u<0 l>0) text(i,d,"x") # x 2010 R 11

12 95% B(n, p) p 95% = n ( ) p(1 p) p(1 p) P ˆp 1.96 p ˆp = 0.95 n n ˆp n p ˆp ( ) ˆp(1 ˆp) ˆp(1 ˆp) ˆp 1.96, ˆp n n 25 n < % n <- 50 p <- 0.4 B < ; C <- 0 for(i in 1:B){ Data <- rbinom(n,1,p) u <- mean(data)+1.96*sd(data)/sqrt(n) l <- mean(data)-1.96*sd(data)/sqrt(n) if(u>p&&l<p) C <- C+1 C/B # # # # B(1,p) n # 26 # 26 # 26 # n Y B(n, p) X = sin 1 ( Y/n) n V (X) 1/4n sin 1 ( p) ( sin 1 ( ˆp) n, sin 1 ( ˆp) ) n ˆp p n <- 10 u <- asin(sqrt(mean(data)))+1.96/sqrt(4*n) l <- asin(sqrt(mean(data)))-1.96/sqrt(4*n) if(u>asin(sqrt(p))&&l<asin(sqrt(p))) C <- C R 12

13 t- Wilcoxon 2 t- Wilcoxon QQ 27 (1) t- (2) Wilcoxon 27 4 D.S.Sladsburg(1970) Hamilton Scale Factor IV Bef <- c(1.83, 0.50, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.30) Aft <- c(0.878, 1.647, 0.598, 2.050, 1.060, 1.290, 1.060, 3.140, 1.290) Dif <- Aft- Bef # qqnorm(dif) # Dif QQ t- X i Y i Z i = Y i X i t- Z i N(θ, σ 2 ) H 0 : θ = 0 v.s. H 1 : θ 0 27 t- t.test(bef, Aft, paired=t) Wilcoxon # 2 Z 1, Z 2,..., Z n 0 Wilcoxon 27 Wilcoxon wilcox.test(bef, Aft, paired=t) # R 13

14 1 4 Data$SCORE attach 1 4 attach Data <- read.csv("data0415.csv") # Data attach(data) # Data hist(score) # Data Score boxplot(score[group==1],score[group==0]) # 1 0 t.test(score[group==1],score[group==0],var.equal=t) # t detach(data) # attach 4 alternative= greater alternative= less Student t- t.test(case, Cont, alternative="grater") # 2 3 n 2 t- α σ 2 d = µ B µ A 1 β n 2σ2 (z α/2 + z β ) 2 d 2, ( ) e t2 /2 dt = α 2π z α 1 σ 2 29 A B % 90% R qnorm qnorm( ) samplesize <- function(a,b,d,s){ # a,b,d,s samplesize return(2*s^2*(qnorm(1-a/2)+qnorm(b))^2/(d^2)) # samplesize samplesize(0.05,0.90,10,34.0) # power.t.test(power=0.9,delta=10,sd=34) 2010 R 14

15 A 1, A 2, A 3 2 SD A A A A 1, A 2, A 3 Excel R Group <- c(1,1,1,1,1,2,2,2,2,2,3,3,3,3) Data <- c(2.9, 3.2, 3.2,..., 3.2, 3.3) plot(group,data) # # ( ) # 3 N(µ 1, σ 2 ), N(µ 2, σ 2 ), N(µ 3, σ 2 ) H 0 : µ 1 = µ 2 = µ 3 v.s. H 1 : noth 0 i.i.d. i n i y i1, y i2,..., y ini N(µ i, σ 2 ) (i = 1, 2, 3) ȳ ȳ i 3 n i (y ij ȳ) 2 = i=1 j=1 3 n i (y ij ȳ i ) 2 + i=1 j=1 3 n i (ȳ i ȳ) 2 S 1 W 2 B H 0 ȳ i ȳ W S F = i=1 B/(3 1) W/(n 3) > F 3 1,n 3(0.05) H 0 31 R oneway.test aov R 15

16 oneway.test(data~group, var=t) # ~ Kruskal-Wallis 4 Student t 2 3 Kruskal-Wallis 32 N(µ i, σ 2 i ) H 0 : σ 2 1 = σ 2 2 = σ 2 3 v.s. H 1 : noth 0 Bartlett 30 Bartlett Bartlett (3 ) bartlett.test(data, Group) #, 33 Kruskal-Wallis kruskal.test Kruskal-Wallis Kruskal-Wallis kruskal.test(data~group) # ~ Kruskal-Wallis 34 Tukey HSD Bonferroni Holm 30 Tukey HSD Group2 <- factor(group2) TukeyHSD(aov(Data~Group2)) Holm # HSD # aov( ~ ) pairwise.t.test(data, Group, p.adj="holm") # p.adj pairwise.wilcox.test 30 pairwise.wilcox.test(data, Group, p.adj="holm") # p.adj 2010 R 16

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PowerPoint プレゼンテーション 40 6 Fischer 2 McNemar 2 No Yes No Yes Mann Whitney s U Wilcoxon 1 Kruskal Wallis 2 Friedman No Yes 2 Yes No Yes Bonferroni ANOVA No Scheffe Yes Student s T No Paired T Welch s T 40 6 Fischer 2 McNemar