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c:ms:2025:w07.2_factorial_anova [2025/04/30 03:59] – [output interpretation] hkimscilc:ms:2025:w07.2_factorial_anova [2026/04/20 04:04] (current) – hkimscil
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-====== Two-way ANOVA ====== +~~REDIRECT>:Factorial Anova~~
-Factorial ANOVA +
-<code> +
-rm(list=ls(all=TRUE))  +
- +
-################################################# +
-# two-way anova +
-# subject = factor(paste('sub', 1:30, sep='')) +
-################################################# +
- +
-n.a.group <- 3 # a treatment 숫자 e.g: drug a1, a2, a3 +
-n.b.group <- 2 # b 그룹 숫자 e.g.: exercise no(b1), yes(b2) +
-n.sub <- 30 # 총 샘플 숫자 +
-n.sub/n.a.group +
- +
-# 데이터 생성 +
-set.seed(9) +
-a <- gl(n.a.group,  +
-        n.sub/n.a.group,  +
-        n.sub,  +
-        labels=c('drg1', 'drg2', 'drg3')) +
-b <- gl(n.b.group,  +
-        (n.sub/n.a.group)/2,  +
-        n.sub,  +
-        labels=c('noex', 'exerc')) +
-a +
-b +
-y <- rnorm(30, mean=10) +  +
-  3.14 * (a=='drg1' & b=='exerc') +  +
-  1.43 * (a=='drg3' & b=='exerc')  +
-y +
- +
-dat <- data.frame(a, b, y) +
-dat +
-# aov.dat <- aov(y~a*b) # anova test +
-# summary(aov.dat) # summary of the test output +
- +
-# hand calculation +
-table(a,b) +
-tapply(y, list(a,b), mean) # 각 셀에서의 평균 +
-tapply(y, a, mean) +
-tapply(y, b, mean) +
-n.within.each <- tapply(y, list(a,b), length) # the same as table(a, b) +
-n.within.each +
-df.within.each <- n.within.each - 1  # 각 셀에서의 샘플숫자 +
-df.within.each +
-df.within <- sum(df.within.each) # df within +
-df.within +
- +
-var.within <- tapply(y, list(a,b), var) # var.within +
-var.within +
-ss.within.each <- tapply(y, list(a,b), var) * df.within.each +
-ss.within.each +
-ss.within <- sum(ss.within.each) # ss.within +
-ss.within +
- +
-ms.within <- ss.within / df.within +
-ms.within +
- +
-# interaction.plot(a,b,y) +
- +
-mean.a <- tapply(y, list(a), mean) +
-mean.b <- tapply(y, list(b), mean) +
-mean.a +
-mean.b +
- +
-var.a <- tapply(y, list(a), var) +
-var.b <- tapply(y, list(b), var) +
-var.a +
-var.b +
- +
-mean.tot <- mean(dat$y) +
-var.tot <- var(dat$y) +
-df.tot <- n.sub - 1  +
-ss.tot <- var.tot * df.tot +
-ms.tot <- ss.tot/df.tot +
- +
-mean.tot +
-var.tot +
-df.tot +
-ss.tot +
-ms.tot +
-ss.tot/df.tot +
- +
- +
-## between +
-mean.each <- tapply(y, list(a,b), mean) +
-mean.each  +
-mean.tot <- mean(y) +
-mean.tot +
-n.each <- tapply(y, list(a,b), length) +
-n.a.each <- tapply(y, list(a), length) +
-n.b.each <- tapply(y, list(b), length) +
-n.each +
-n.a.each +
-n.b.each +
- +
- +
- +
-ss.bet <- sum(n.each*(mean.each-mean.tot)^2) +
-ss.bet +
- +
-ss.tot +
-ss.within +
-ss.bet +
-ss.bet + ss.within +
- +
-ss.a <- sum(n.a.each * ((mean.tot - mean.a)^2)) +
-ss.b <- sum(n.b.each * ((mean.tot - mean.b)^2)) +
-ss.a +
-ss.b +
-ss.ab <- ss.bet - (ss.a + ss.b) # ss.ab = ss.interaction +
-ss.ab +
- +
-ss.tot +
-ss.bet +
-ss.within +
-ss.a +
-ss.b +
-ss.ab +
- +
-df.tot <- n.sub - 1 +
-df.bet <- (n.a.group * n.b.group) - 1 +
-df.a <- n.a.group - 1 +
-df.b <- n.b.group - 1 +
-df.ab <- df.bet - (df.a + df.b) +
-df.within <- sum(df.within.each) +
- +
-df.tot +
-df.bet +
-df.a +
-df.b +
-df.ab +
-df.within +
- +
-ms.tot <- ss.tot / df.tot # we did it above  +
-ms.bet <- ss.bet / df.bet +
-ms.a <- ss.a / df.a +
-ms.b <- ss.b / df.b +
-ms.ab <- ss.ab / df.ab +
-ms.within <- ss.within / df.within +
- +
-ms.tot +
-ms.bet +
-ms.within +
-ms.a +
-ms.b +
-ms.ab +
- +
- +
-f.a <- ms.a / ms.within +
-f.b <- ms.b / ms.within +
-f.ab <- ms.ab / ms.within +
- +
-alpha <- .05 +
-# confidence interval +
-ci <- 1 - alpha +
- +
-f.a +
-# 봐야할 F분포표에서의 F-value +
-# qt 처럼 qf 사용 +
-# qf(alpha, df.between, df.within, lower.tail=F) 처럼 사용 +
-qf(ci, df.a, df.within) # or +
-qf(alpha, df.a, df.within, lower.tail = F) +
-# 혹은  +
-# qf(alpha, df.a, df.within, lower.tail = F) +
-# 도 마찬가지 +
-f.a.crit <- qf(ci, df.a, df.within)  +
-f.a > f.a.crit # 만약 f.a가 크다면  +
-# 혹은 f.a 값에 해당하는 percentage를 구하면 +
-pf(f.a, df.a, df.within, lower.tail = F) +
- +
-f.b +
-qf(ci, df.b, df.within) +
-f.b > qf(ci, df.b, df.within) +
-pf(f.b, df.b, df.within, lower.tail = F) +
- +
-f.ab +
-qf(ci, df.ab, df.within) +
-f.ab > qf(ci, df.ab, df.within) +
-pf(f.ab, df.ab, df.within, lower.tail = F) +
- +
-# aov result +
-aov.dat.all <- aov(y ~ a * b, data = dat) # anova test +
-summary(aov.dat.all) # summary of the test output +
-aov.dat.noint <- aov(y ~ a + b, data = dat) +
-# interaction이 갖는 SS와 df를 residuals이 갖음을 주목 +
-summary(aov.dat.noint)  +
-aov.dat.all2 <- aov(y ~ a + b + a:b, data = dat) +
-summary(aov.dat.all2) +
-aov.dat.all # what is standard error of residuals  +
- +
-# post-hoc test +
-TukeyHSD(aov.dat.all) +
-TukeyHSD(aov.dat.all, which = "a") +
-TukeyHSD(aov.dat.all, which = "b") +
-TukeyHSD(aov.dat.all, which = "a:b") +
-boxplot(dat$y~dat$a) +
-boxplot(dat$y~dat$b) +
-plot(TukeyHSD(aov.dat.all, which = "a:b"), +
-     las = 2 # rotate x-axis ticks +
-) +
- +
- +
-# +
-# +
-# see another e.g. at +
-# https://statsandr.com/blog/two-way-anova-in-r/ +
-# +
-# +
- +
-</code> +
- +
-====== Output, two-way ANOVA ====== +
-<code> +
-> rm(list=ls(all=TRUE))  +
->  +
-> ################################################# +
-> # two-way anova +
-> # subject = factor(paste('sub', 1:30, sep='')) +
-> ################################################# +
->  +
-> n.a.group <- 3 # a treatment 숫자 e.g: drug a1, a2, a3 +
-> n.b.group <- 2 # b 그룹 숫자 e.g.: exercise no(b1), yes(b2) +
-> n.sub <- 30 # 총 샘플 숫자 +
-> n.sub/n.a.group +
-[1] 10 +
->  +
-> # 데이터 생성 +
-> set.seed(9) +
-> a <- gl(n.a.group,  +
-+         n.sub/n.a.group,  +
-+         n.sub,  +
-+         labels=c('drg1', 'drg2', 'drg3')) +
-> b <- gl(n.b.group,  +
-+         (n.sub/n.a.group)/2,  +
-+         n.sub,  +
-+         labels=c('noex', 'exerc')) +
-> a +
- [1] drg1 drg1 drg1 drg1 drg1 drg1 drg1 drg1 drg1 drg1 drg2 drg2 drg2 drg2 drg2 drg2 drg2 drg2 drg2 drg2 drg3 +
-[22] drg3 drg3 drg3 drg3 drg3 drg3 drg3 drg3 drg3 +
-Levels: drg1 drg2 drg3 +
-> b +
- [1] noex  noex  noex  noex  noex  exerc exerc exerc exerc exerc noex  noex  noex  noex  noex  exerc exerc exerc +
-[19] exerc exerc noex  noex  noex  noex  noex  exerc exerc exerc exerc exerc +
-Levels: noex exerc +
-> y <- rnorm(30, mean=10) +  +
-+   3.14 * (a=='drg1' & b=='exerc') +  +
-+   1.43 * (a=='drg3' & b=='exerc')  +
-> y +
- [1]  9.233204  9.183542  9.858465  9.722395 10.436307 11.953127 14.331987 13.121810 12.891915 12.777063 +
-[11] 11.277571  9.531103 10.071054  9.733962 11.845257  9.160550  9.922552  7.382294 10.887884  9.292509 +
-[21] 11.756993 10.182252  9.733111 10.926422  9.306668 14.111990 11.652524 10.723328 11.847213 11.799557 +
->  +
-> dat <- data.frame(a, b, y) +
-> dat +
-      a     b         y +
-1  drg1  noex  9.233204 +
-2  drg1  noex  9.183542 +
-3  drg1  noex  9.858465 +
-4  drg1  noex  9.722395 +
-5  drg1  noex 10.436307 +
-6  drg1 exerc 11.953127 +
-7  drg1 exerc 14.331987 +
-8  drg1 exerc 13.121810 +
-9  drg1 exerc 12.891915 +
-10 drg1 exerc 12.777063 +
-11 drg2  noex 11.277571 +
-12 drg2  noex  9.531103 +
-13 drg2  noex 10.071054 +
-14 drg2  noex  9.733962 +
-15 drg2  noex 11.845257 +
-16 drg2 exerc  9.160550 +
-17 drg2 exerc  9.922552 +
-18 drg2 exerc  7.382294 +
-19 drg2 exerc 10.887884 +
-20 drg2 exerc  9.292509 +
-21 drg3  noex 11.756993 +
-22 drg3  noex 10.182252 +
-23 drg3  noex  9.733111 +
-24 drg3  noex 10.926422 +
-25 drg3  noex  9.306668 +
-26 drg3 exerc 14.111990 +
-27 drg3 exerc 11.652524 +
-28 drg3 exerc 10.723328 +
-29 drg3 exerc 11.847213 +
-30 drg3 exerc 11.799557 +
-> # aov.dat <- aov(y~a*b) # anova test +
-> # summary(aov.dat) # summary of the test output +
->  +
-> # hand calculation +
-> table(a,b) +
-      b +
-a      noex exerc +
-  drg1    5     5 +
-  drg2    5     5 +
-  drg3    5     5 +
-> tapply(y, list(a,b), mean) # 각 셀에서의 평균 +
-          noex     exerc +
-drg1  9.686782 13.015181 +
-drg2 10.491789  9.329158 +
-drg3 10.381089 12.026922 +
-> tapply(y, a, mean) +
-     drg1      drg2      drg3  +
-11.350981  9.910474 11.204006  +
-> tapply(y, b, mean) +
-    noex    exerc  +
-10.18655 11.45709  +
-> n.within.each <- tapply(y, list(a,b), length) # the same as table(a, b) +
-> n.within.each +
-     noex exerc +
-drg1    5     5 +
-drg2    5     5 +
-drg3    5     5 +
-> df.within.each <- n.within.each - 1  # 각 셀에서의 샘플숫자 +
-> df.within.each +
-     noex exerc +
-drg1    4     4 +
-drg2    4     4 +
-drg3    4     4 +
-> df.within <- sum(df.within.each) # df within +
-> df.within +
-[1] 24 +
->  +
-> var.within <- tapply(y, list(a,b), var) # var.within +
-> var.within +
-          noex     exerc +
-drg1 0.2628787 0.7362999 +
-drg2 1.0308917 1.6504481 +
-drg3 0.9510727 1.5677577 +
-> ss.within.each <- tapply(y, list(a,b), var) * df.within.each +
-> ss.within.each +
-         noex    exerc +
-drg1 1.051515 2.945200 +
-drg2 4.123567 6.601793 +
-drg3 3.804291 6.271031 +
-> ss.within <- sum(ss.within.each) # ss.within +
-> ss.within +
-[1] 24.7974 +
->  +
-> ms.within <- ss.within / df.within +
-> ms.within +
-[1] 1.033225 +
->  +
-> # interaction.plot(a,b,y) +
->  +
-> mean.a <- tapply(y, list(a), mean) +
-> mean.b <- tapply(y, list(b), mean) +
-> mean.a +
-     drg1      drg2      drg3  +
-11.350981  9.910474 11.204006  +
-> mean.b +
-    noex    exerc  +
-10.18655 11.45709  +
->  +
-> var.a <- tapply(y, list(a), var) +
-> var.b <- tapply(y, list(b), var) +
-> var.a +
-    drg1     drg2     drg3  +
-3.521366 1.567182 1.871915  +
-> var.b +
-     noex     exerc  +
-0.7773781 3.7300196  +
->  +
-> mean.tot <- mean(dat$y) +
-> var.tot <- var(dat$y) +
-> df.tot <- n.sub - 1  +
-> ss.tot <- var.tot * df.tot +
-> ms.tot <- ss.tot/df.tot +
->  +
-> mean.tot +
-[1] 10.82182 +
-> var.tot +
-[1] 2.593465 +
-> df.tot +
-[1] 29 +
-> ss.tot +
-[1] 75.21048 +
-> ms.tot +
-[1] 2.593465 +
-> ss.tot/df.tot +
-[1] 2.593465 +
->  +
->  +
-> ## between +
-> mean.each <- tapply(y, list(a,b), mean) +
-> mean.each  +
-          noex     exerc +
-drg1  9.686782 13.015181 +
-drg2 10.491789  9.329158 +
-drg3 10.381089 12.026922 +
-> mean.tot <- mean(y) +
-> mean.tot +
-[1] 10.82182 +
-> n.each <- tapply(y, list(a,b), length) +
-> n.a.each <- tapply(y, list(a), length) +
-> n.b.each <- tapply(y, list(b), length) +
-> n.each +
-     noex exerc +
-drg1    5     5 +
-drg2    5     5 +
-drg3    5     5 +
-> n.a.each +
-drg1 drg2 drg3  +
-  10   10   10  +
-> n.b.each +
- noex exerc  +
-   15    15  +
->  +
->  +
->  +
-> ss.bet <- sum(n.each*(mean.each-mean.tot)^2) +
-> ss.bet +
-[1] 50.41308 +
->  +
-> ss.tot +
-[1] 75.21048 +
-> ss.within +
-[1] 24.7974 +
-> ss.bet +
-[1] 50.41308 +
-> ss.bet + ss.within +
-[1] 75.21048 +
->  +
-> ss.a <- sum(n.a.each * ((mean.tot - mean.a)^2)) +
-> ss.b <- sum(n.b.each * ((mean.tot - mean.b)^2)) +
-> ss.a +
-[1] 12.5663 +
-> ss.b +
-[1] 12.10691 +
-> ss.ab <- ss.bet - (ss.a + ss.b) # ss.ab = ss.interaction +
-> ss.ab +
-[1] 25.73987 +
->  +
-> ss.tot +
-[1] 75.21048 +
-> ss.bet +
-[1] 50.41308 +
-> ss.within +
-[1] 24.7974 +
-> ss.a +
-[1] 12.5663 +
-> ss.b +
-[1] 12.10691 +
-> ss.ab +
-[1] 25.73987 +
->  +
-> df.tot <- n.sub - 1 +
-> df.bet <- (n.a.group * n.b.group) - 1 +
-> df.a <- n.a.group - 1 +
-> df.b <- n.b.group - 1 +
-> df.ab <- df.bet - (df.a + df.b) +
-> df.within <- sum(df.within.each) +
->  +
-> df.tot +
-[1] 29 +
-> df.bet +
-[1] 5 +
-> df.a +
-[1] 2 +
-> df.b +
-[1] 1 +
-> df.ab +
-[1] 2 +
-> df.within +
-[1] 24 +
->  +
-> ms.tot <- ss.tot / df.tot # we did it above  +
-> ms.bet <- ss.bet / df.bet +
-> ms.a <- ss.a / df.a +
-> ms.b <- ss.b / df.b +
-> ms.ab <- ss.ab / df.ab +
-> ms.within <- ss.within / df.within +
->  +
-> ms.tot +
-[1] 2.593465 +
-> ms.bet +
-[1] 10.08262 +
-> ms.within +
-[1] 1.033225 +
-> ms.a +
-[1] 6.283151 +
-> ms.b +
-[1] 12.10691 +
-> ms.ab +
-[1] 12.86993 +
->  +
->  +
-> f.a <- ms.a / ms.within +
-> f.b <- ms.b / ms.within +
-> f.ab <- ms.ab / ms.within +
->  +
-> alpha <- .05 +
-> # confidence interval +
-> ci <- 1 - alpha +
->  +
-> f.a +
-[1] 6.081107 +
-> # 봐야할 F분포표에서의 F-value +
-> # qt 처럼 qf 사용 +
-> # qf(alpha, df.between, df.within, lower.tail=F) 처럼 사용 +
-> qf(ci, df.a, df.within) # or +
-[1] 3.402826 +
-> qf(alpha, df.a, df.within, lower.tail = F) +
-[1] 3.402826 +
-> # 혹은  +
-> # qf(alpha, df.a, df.within, lower.tail = F) +
-> # 도 마찬가지 +
-> f.a.crit <- qf(ci, df.a, df.within)  +
-> f.a > f.a.crit # 만약 f.a가 크다면  +
-[1] TRUE +
-> # 혹은 f.a 값에 해당하는 percentage를 구하면 +
-> pf(f.a, df.a, df.within, lower.tail = F) +
-[1] 0.007302552 +
->  +
-> f.b +
-[1] 11.7176 +
-> qf(ci, df.b, df.within) +
-[1] 4.259677 +
-> f.b > qf(ci, df.b, df.within) +
-[1] TRUE +
-> pf(f.b, df.b, df.within, lower.tail = F) +
-[1] 0.00222738 +
->  +
-> f.ab +
-[1] 12.45608 +
-> qf(ci, df.ab, df.within) +
-[1] 3.402826 +
-> f.ab > qf(ci, df.ab, df.within) +
-[1] TRUE +
-> pf(f.ab, df.ab, df.within, lower.tail = F) +
-[1] 0.0001947745 +
->  +
-> # aov result +
-> aov.dat.all <- aov(y ~ a * b, data = dat) # anova test +
-> summary(aov.dat.all) # summary of the test output +
-            Df Sum Sq Mean Sq F value   Pr(>F)     +
-a            2  12.57   6.283   6.081 0.007303 **  +
-b            1  12.11  12.107  11.718 0.002227 **  +
-a:b          2  25.74  12.870  12.456 0.000195 *** +
-Residuals   24  24.80   1.033                      +
---- +
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 +
-> aov.dat.noint <- aov(y ~ a + b, data = dat) +
-> # interaction이 갖는 SS와 df를 residuals이 갖음을 주목 +
-> summary(aov.dat.noint)  +
-            Df Sum Sq Mean Sq F value Pr(>F)   +
-a            2  12.57   6.283   3.233 0.0558 . +
-b            1  12.11  12.107   6.229 0.0192 * +
-Residuals   26  50.54   1.944                  +
---- +
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 +
-> aov.dat.all2 <- aov(y ~ a + b + a:b, data = dat) +
-> summary(aov.dat.all2) +
-            Df Sum Sq Mean Sq F value   Pr(>F)     +
-a            2  12.57   6.283   6.081 0.007303 **  +
-b            1  12.11  12.107  11.718 0.002227 **  +
-a:b          2  25.74  12.870  12.456 0.000195 *** +
-Residuals   24  24.80   1.033                      +
---- +
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 +
-> aov.dat.all # what is standard error of residuals  +
-Call: +
-   aov(formula = y ~ a * b, data = dat) +
- +
-Terms: +
-                       a        b      a:b Residuals +
-Sum of Squares  12.56630 12.10691 25.73987  24.79740 +
-Deg. of Freedom        2        1        2        24 +
- +
-Residual standard error: 1.016477 +
-Estimated effects may be unbalanced +
->  +
-> # post-hoc test +
-> TukeyHSD(aov.dat.all) +
-  Tukey multiple comparisons of means +
-    95% family-wise confidence level +
- +
-Fit: aov(formula = y ~ a * b, data = dat) +
- +
-$a +
-                diff       lwr        upr     p adj +
-drg2-drg1 -1.4405079 -2.575730 -0.3052857 0.0111258 +
-drg3-drg1 -0.1469757 -1.282198  0.9882466 0.9441388 +
-drg3-drg2  1.2935323  0.158310  2.4287545 0.0233996 +
- +
-$b +
-               diff       lwr     upr     p adj +
-exerc-noex 1.270533 0.5044869 2.03658 0.0022274 +
- +
-$`a:b` +
-                            diff        lwr        upr     p adj +
-drg2:noex-drg1:noex    0.8050068 -1.1827224  2.7927360 0.8069478 +
-drg3:noex-drg1:noex    0.6943068 -1.2934224  2.6820359 0.8845124 +
-drg1:exerc-drg1:noex   3.3283980  1.3406689  5.3161272 0.0003437 +
-drg2:exerc-drg1:noex  -0.3576246 -2.3453538  1.6301046 0.9929417 +
-drg3:exerc-drg1:noex   2.3401400  0.3524108  4.3278691 0.0145532 +
-drg3:noex-drg2:noex   -0.1107000 -2.0984292  1.8770291 0.9999759 +
-drg1:exerc-drg2:noex   2.5233913  0.5356621  4.5111204 0.0074152 +
-drg2:exerc-drg2:noex  -1.1626314 -3.1503606  0.8250978 0.4792630 +
-drg3:exerc-drg2:noex   1.5351332 -0.4525960  3.5228624 0.2000613 +
-drg1:exerc-drg3:noex   2.6340913  0.6463621  4.6218205 0.0048987 +
-drg2:exerc-drg3:noex  -1.0519314 -3.0396605  0.9357978 0.5839905 +
-drg3:exerc-drg3:noex   1.6458332 -0.3418960  3.6335624 0.1464756 +
-drg2:exerc-drg1:exerc -3.6860226 -5.6737518 -1.6982935 0.0000874 +
-drg3:exerc-drg1:exerc -0.9882581 -2.9759873  0.9994711 0.6448824 +
-drg3:exerc-drg2:exerc  2.6977646  0.7100354  4.6854937 0.0038521 +
- +
-> TukeyHSD(aov.dat.all, which = "a") +
-  Tukey multiple comparisons of means +
-    95% family-wise confidence level +
- +
-Fit: aov(formula = y ~ a * b, data = dat) +
- +
-$a +
-                diff       lwr        upr     p adj +
-drg2-drg1 -1.4405079 -2.575730 -0.3052857 0.0111258 +
-drg3-drg1 -0.1469757 -1.282198  0.9882466 0.9441388 +
-drg3-drg2  1.2935323  0.158310  2.4287545 0.0233996 +
- +
-> TukeyHSD(aov.dat.all, which = "b") +
-  Tukey multiple comparisons of means +
-    95% family-wise confidence level +
- +
-Fit: aov(formula = y ~ a * b, data = dat) +
- +
-$b +
-               diff       lwr     upr     p adj +
-exerc-noex 1.270533 0.5044869 2.03658 0.0022274 +
- +
-> TukeyHSD(aov.dat.all, which = "a:b") +
-  Tukey multiple comparisons of means +
-    95% family-wise confidence level +
- +
-Fit: aov(formula = y ~ a * b, data = dat) +
- +
-$`a:b` +
-                            diff        lwr        upr     p adj +
-drg2:noex-drg1:noex    0.8050068 -1.1827224  2.7927360 0.8069478 +
-drg3:noex-drg1:noex    0.6943068 -1.2934224  2.6820359 0.8845124 +
-drg1:exerc-drg1:noex   3.3283980  1.3406689  5.3161272 0.0003437 +
-drg2:exerc-drg1:noex  -0.3576246 -2.3453538  1.6301046 0.9929417 +
-drg3:exerc-drg1:noex   2.3401400  0.3524108  4.3278691 0.0145532 +
-drg3:noex-drg2:noex   -0.1107000 -2.0984292  1.8770291 0.9999759 +
-drg1:exerc-drg2:noex   2.5233913  0.5356621  4.5111204 0.0074152 +
-drg2:exerc-drg2:noex  -1.1626314 -3.1503606  0.8250978 0.4792630 +
-drg3:exerc-drg2:noex   1.5351332 -0.4525960  3.5228624 0.2000613 +
-drg1:exerc-drg3:noex   2.6340913  0.6463621  4.6218205 0.0048987 +
-drg2:exerc-drg3:noex  -1.0519314 -3.0396605  0.9357978 0.5839905 +
-drg3:exerc-drg3:noex   1.6458332 -0.3418960  3.6335624 0.1464756 +
-drg2:exerc-drg1:exerc -3.6860226 -5.6737518 -1.6982935 0.0000874 +
-drg3:exerc-drg1:exerc -0.9882581 -2.9759873  0.9994711 0.6448824 +
-drg3:exerc-drg2:exerc  2.6977646  0.7100354  4.6854937 0.0038521 +
- +
-> boxplot(dat$y~dat$a) +
-> boxplot(dat$y~dat$b) +
-> plot(TukeyHSD(aov.dat.all, which = "a:b"), +
-+      las = 2 # rotate x-axis ticks +
-+ ) +
->  +
->  +
-> # +
-> # +
-> # see another e.g. at +
-> # https://statsandr.com/blog/two-way-anova-in-r/ +
-> # +
-> # +
->  +
-> +
-</code> +
-====== output interpretation ====== +
- +
-{{:c:ms:2025:pasted:20250421-082701.png}} +
-{{:c:ms:2025:pasted:20250421-082710.png}} +
-{{:c:ms:2025:pasted:20250421-084815.png}} +
-{{:c:ms:2025:pasted:20250430-125904.png}} +
-{{:c:ms:2025:pasted:20250430-124435.png}} +
- +
-상호작용효과 포함 해석 +
-  * drug1의 효과에 운동의 역할은 아주 중요한 것으로 밝혀졌다. +
-  * 반면에 drug2, 3에 한해서는 운동의 역할은 제한적이었다.  +
-  * 특히 drg3의 경우 운동이 효과를 증대시키기는 하였지만 그것이 통계학적으로 의미있는 것은 아니었다. +
-  * 또한 drg2의 경우 운동이 오히려 효과를 감소하는 것으로 나타났다. 그렇지만 이 또한 통계학적으로 의미가 있는 것은 아니었다.  +
-  * 이를 통해서 drg1을 처치받는 환자의 경우에는 운동을 꼭 하도록 하여야 함을 알 수 있고 +
-  * drg2의 경우에는 그 효과가 미미하므로 권장을 하는 정도가 필요함을 알 수 있었다.  +
-혹은 +
-  * drg1과 운동이 병행 된 경우에는 대부분의 다른 처방보다 월등한 효과가 있는 것으로 밝혀졌다.  +
-  * 운동이 병행된 drg1과 drg3의  효과에는 차이가 없었다.  +
-  * drg2의 운동유무, 그리고 drg3의 운동을 하지않음과 통계학적으로 유의미한 차이가 있었다. +
- +
c/ms/2025/w07.2_factorial_anova.1745985558.txt.gz · Last modified: by hkimscil
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