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--- r:factorial_anova [2023/09/20 23:42] – hkimscil
+++ r:factorial_anova [2026/04/14 23:36] (current) – hkimscil
@@ Line 1: / Line 1: @@
-see [[:c/ma/2023/schedule/anova_note#two_way_anova_factorial_anova]]
+====== Two-way ANOVA ======
+Factorial ANOVA 라고도 부른다.
+<tabbox rs.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
+# 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) +
+.14 * (a=='drg1' & b=='exerc') +
+.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>
+<tabbox ro.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
+  drg1  noex  9.233204
+  drg1  noex  9.183542
+  drg1  noex  9.858465
+  drg1  noex  9.722395
+  drg1  noex 10.436307
+  drg1 exerc 11.953127
+  drg1 exerc 14.331987
+  drg1 exerc 13.121810
+  drg1 exerc 12.891915
+drg1 exerc 12.777063
+drg2  noex 11.277571
+drg2  noex  9.531103
+drg2  noex 10.071054
+drg2  noex  9.733962
+drg2  noex 11.845257
+drg2 exerc  9.160550
+drg2 exerc  9.922552
+drg2 exerc  7.382294
+drg2 exerc 10.887884
+drg2 exerc  9.292509
+drg3  noex 11.756993
+drg3  noex 10.182252
+drg3  noex  9.733111
+drg3  noex 10.926422
+drg3  noex  9.306668
+drg3 exerc 14.111990
+drg3 exerc 11.652524
+drg3 exerc 10.723328
+drg3 exerc 11.847213
+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
+.350981  9.910474 11.204006
+> tapply(y, b, mean)
+    noex    exerc
+.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
+.350981  9.910474 11.204006
+> mean.b
+    noex    exerc
+.18655 11.45709
+>
+> var.a <- tapply(y, list(a), var)
+> var.b <- tapply(y, list(b), var)
+> var.a
+    drg1     drg2     drg3
+.521366 1.567182 1.871915
+> var.b
+     noex     exerc
+.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
+> n.b.each
+ noex exerc
+    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
+% 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
+$`a:b`
+                            diff        lwr        upr     p adj
+drg2:noex-drg1:noex    0.8050068 -1.1827224  2.7927360 0.8069478
+drg3:noex-drg1:noex    0.6943068 -1.2934224  2.6820359 0.8845124
+drg1:exerc-drg1:noex   3.3283980  1.3406689  5.3161272 0.0003437
+drg2:exerc-drg1:noex  -0.3576246 -2.3453538  1.6301046 0.9929417
+drg3:exerc-drg1:noex   2.3401400  0.3524108  4.3278691 0.0145532
+drg3:noex-drg2:noex   -0.1107000 -2.0984292  1.8770291 0.9999759
+drg1:exerc-drg2:noex   2.5233913  0.5356621  4.5111204 0.0074152
+drg2:exerc-drg2:noex  -1.1626314 -3.1503606  0.8250978 0.4792630
+drg3:exerc-drg2:noex   1.5351332 -0.4525960  3.5228624 0.2000613
+drg1:exerc-drg3:noex   2.6340913  0.6463621  4.6218205 0.0048987
+drg2:exerc-drg3:noex  -1.0519314 -3.0396605  0.9357978 0.5839905
+drg3:exerc-drg3:noex   1.6458332 -0.3418960  3.6335624 0.1464756
+drg2:exerc-drg1:exerc -3.6860226 -5.6737518 -1.6982935 0.0000874
+drg3:exerc-drg1:exerc -0.9882581 -2.9759873  0.9994711 0.6448824
+drg3:exerc-drg2:exerc  2.6977646  0.7100354  4.6854937 0.0038521
+> TukeyHSD(aov.dat.all, which = "a")
+  Tukey multiple comparisons of means
+% family-wise confidence level
+Fit: aov(formula = y ~ a * b, data = dat)
+$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
+% family-wise confidence level
+Fit: aov(formula = y ~ a * b, data = dat)
+$b
+               diff       lwr     upr     p adj
+exerc-noex 1.270533 0.5044869 2.03658 0.0022274
+> TukeyHSD(aov.dat.all, which = "a:b")
+  Tukey multiple comparisons of means
+% family-wise confidence level
+Fit: aov(formula = y ~ a * b, data = dat)
+$`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>
+</tabbox>
+====== 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:pasted:20250430-130525.png}}