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Trace: • 2016 • aaron_swartz • grounded_theory
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c:ms:2025:w07.2_factorial_anova [2025/04/20 23:02] – hkimscilc:ms:2025:w07.2_factorial_anova [2025/05/06 22:25] (current) – hkimscil
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 +====== Two-way ANOVA ======
 +Factorial ANOVA
 <code> <code>
 +rm(list=ls(all=TRUE)) 
 +
 ################################################# #################################################
 # two-way anova # two-way anova
Line 5: Line 9:
 ################################################# #################################################
  
-n.a.group <- 3 # a treatment 숫자 +n.a.group <- 3 # a treatment 숫자 e.g: drug a1, a2, a3 
-n.b.group <- 2 # b 그룹 숫자+n.b.group <- 2 # b 그룹 숫자 e.g.: exercise no(b1), yes(b2)
 n.sub <- 30 # 총 샘플 숫자 n.sub <- 30 # 총 샘플 숫자
 n.sub/n.a.group n.sub/n.a.group
Line 12: Line 16:
 # 데이터 생성 # 데이터 생성
 set.seed(9) set.seed(9)
-a <- gl(3, 10, n.sub, labels=c('a1', 'a2', 'a3')) +a <- gl(n.a.group,  
-b <- gl(2, 5, n.sub, labels=c('b1', 'b2'))+        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 a
 b b
 y <- rnorm(30, mean=10) +  y <- rnorm(30, mean=10) + 
-  3.14 * (a=='a1' & b=='b2') +  +  3.14 * (a=='drg1' & b=='exerc') +  
-  1.43 * (a=='a3' & b=='b2') +  1.43 * (a=='drg3' & b=='exerc') 
 y y
  
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 table(a,b) table(a,b)
 tapply(y, list(a,b), mean) # 각 셀에서의 평균 tapply(y, list(a,b), mean) # 각 셀에서의 평균
-n.within.each <- tapply(y, list(a,b), length)+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 n.within.each
 df.within.each <- n.within.each - 1  # 각 셀에서의 샘플숫자 df.within.each <- n.within.each - 1  # 각 셀에서의 샘플숫자
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 ss.tot <- var.tot * df.tot ss.tot <- var.tot * df.tot
 ms.tot <- ss.tot/df.tot ms.tot <- ss.tot/df.tot
 +
 +mean.tot
 +var.tot
 +df.tot
 +ss.tot
 +ms.tot
 +ss.tot/df.tot
 +
  
 ## between ## between
Line 114: Line 134:
 df.within df.within
  
-ms.tot <- ss.tot/df.tot # we did it above  +ms.tot <- ss.tot / df.tot # we did it above  
-ms.bet <- ss.bet - df.bet+ms.bet <- ss.bet / df.bet
 ms.a <- ss.a / df.a ms.a <- ss.a / df.a
 ms.b <- ss.b / df.b ms.b <- ss.b / df.b
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 # qf(alpha, df.a, df.within, lower.tail = F) # qf(alpha, df.a, df.within, lower.tail = F)
 # 도 마찬가지 # 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) pf(f.a, df.a, df.within, lower.tail = F)
  
 f.b f.b
 qf(ci, df.b, df.within) qf(ci, df.b, df.within)
 +f.b > qf(ci, df.b, df.within)
 pf(f.b, df.b, df.within, lower.tail = F) pf(f.b, df.b, df.within, lower.tail = F)
  
 f.ab f.ab
 qf(ci, df.ab, df.within) qf(ci, df.ab, df.within)
 +f.ab > qf(ci, df.ab, df.within)
 pf(f.ab, df.ab, df.within, lower.tail = F) pf(f.ab, df.ab, df.within, lower.tail = F)
  
 # aov result # aov result
-summary(aov.dat)+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> </code>
-====== output ======+ 
 +====== Output, two-way ANOVA ======
 <code> <code>
 +> rm(list=ls(all=TRUE)) 
 +> 
 > ################################################# > #################################################
 > # two-way anova > # two-way anova
Line 166: Line 221:
 > ################################################# > #################################################
 >  > 
-> n.a.group <- 3 # a treatment 숫자 +> n.a.group <- 3 # a treatment 숫자 e.g: drug a1, a2, a3 
-> n.b.group <- 2 # b 그룹 숫자+> n.b.group <- 2 # b 그룹 숫자 e.g.: exercise no(b1), yes(b2)
 > n.sub <- 30 # 총 샘플 숫자 > n.sub <- 30 # 총 샘플 숫자
 > n.sub/n.a.group > n.sub/n.a.group
Line 174: Line 229:
 > # 데이터 생성 > # 데이터 생성
 > set.seed(9) > set.seed(9)
-> a <- gl(3, 10, n.sub, labels=c('a1', 'a2', 'a3')) +> a <- gl(n.a.group,  
-> b <- gl(2, 5, n.sub, labels=c('b1', 'b2'))++         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 > a
- [1] a1 a1 a1 a1 a1 a1 a1 a1 a1 a1 a2 a2 a2 a2 a2 a2 a2 a2 + [1] drg1 drg1 drg1 drg1 drg1 drg1 drg1 drg1 drg1 drg1 drg2 drg2 drg2 drg2 drg2 drg2 drg2 drg2 drg2 drg2 drg3 
-[19] a2 a2 a3 a3 a3 a3 a3 a3 a3 a3 a3 a3 +[22] drg3 drg3 drg3 drg3 drg3 drg3 drg3 drg3 drg3 
-Levels: a1 a2 a3+Levels: drg1 drg2 drg3
 > b > b
- [1] b1 b1 b1 b1 b1 b2 b2 b2 b2 b2 b1 b1 b1 b1 b1 b2 b2 b2 + [1] noex  noex  noex  noex  noex  exerc exerc exerc exerc exerc noex  noex  noex  noex  noex  exerc exerc exerc 
-[19] b2 b2 b1 b1 b1 b1 b1 b2 b2 b2 b2 b2 +[19] exerc exerc noex  noex  noex  noex  noex  exerc exerc exerc exerc exerc 
-Levels: b1 b2+Levels: noex exerc
 > y <- rnorm(30, mean=10) +  > y <- rnorm(30, mean=10) + 
-+   3.14 * (a=='a1' & b=='b2') +  ++   3.14 * (a=='drg1' & b=='exerc') +  
-+   1.43 * (a=='a3' & b=='b2') ++   1.43 * (a=='drg3' & b=='exerc') 
 > y > y
- [1]  9.233204  9.183542  9.858465  9.722395 10.436307 + [1]  9.233204  9.183542  9.858465  9.722395 10.436307 11.953127 14.331987 13.121810 12.891915 12.777063 
- [6] 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 
-[11] 11.277571  9.531103 10.071054  9.733962 11.845257 +[21] 11.756993 10.182252  9.733111 10.926422  9.306668 14.111990 11.652524 10.723328 11.847213 11.799557
-[16]  9.160550  9.922552  7.382294 10.887884  9.292509 +
-[21] 11.756993 10.182252  9.733111 10.926422  9.306668 +
-[26] 14.111990 11.652524 10.723328 11.847213 11.799557+
 >  > 
 > dat <- data.frame(a, b, y) > dat <- data.frame(a, b, y)
 > dat > dat
-    a  b         y +      a     b         y 
-1  a1 b1  9.233204 +1  drg1  noex  9.233204 
-2  a1 b1  9.183542 +2  drg1  noex  9.183542 
-3  a1 b1  9.858465 +3  drg1  noex  9.858465 
-4  a1 b1  9.722395 +4  drg1  noex  9.722395 
-5  a1 b1 10.436307 +5  drg1  noex 10.436307 
-6  a1 b2 11.953127 +6  drg1 exerc 11.953127 
-7  a1 b2 14.331987 +7  drg1 exerc 14.331987 
-8  a1 b2 13.121810 +8  drg1 exerc 13.121810 
-9  a1 b2 12.891915 +9  drg1 exerc 12.891915 
-10 a1 b2 12.777063 +10 drg1 exerc 12.777063 
-11 a2 b1 11.277571 +11 drg2  noex 11.277571 
-12 a2 b1  9.531103 +12 drg2  noex  9.531103 
-13 a2 b1 10.071054 +13 drg2  noex 10.071054 
-14 a2 b1  9.733962 +14 drg2  noex  9.733962 
-15 a2 b1 11.845257 +15 drg2  noex 11.845257 
-16 a2 b2  9.160550 +16 drg2 exerc  9.160550 
-17 a2 b2  9.922552 +17 drg2 exerc  9.922552 
-18 a2 b2  7.382294 +18 drg2 exerc  7.382294 
-19 a2 b2 10.887884 +19 drg2 exerc 10.887884 
-20 a2 b2  9.292509 +20 drg2 exerc  9.292509 
-21 a3 b1 11.756993 +21 drg3  noex 11.756993 
-22 a3 b1 10.182252 +22 drg3  noex 10.182252 
-23 a3 b1  9.733111 +23 drg3  noex  9.733111 
-24 a3 b1 10.926422 +24 drg3  noex 10.926422 
-25 a3 b1  9.306668 +25 drg3  noex  9.306668 
-26 a3 b2 14.111990 +26 drg3 exerc 14.111990 
-27 a3 b2 11.652524 +27 drg3 exerc 11.652524 
-28 a3 b2 10.723328 +28 drg3 exerc 10.723328 
-29 a3 b2 11.847213 +29 drg3 exerc 11.847213 
-30 a3 b2 11.799557+30 drg3 exerc 11.799557
 > # aov.dat <- aov(y~a*b) # anova test > # aov.dat <- aov(y~a*b) # anova test
 > # summary(aov.dat) # summary of the test output > # summary(aov.dat) # summary of the test output
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 > # hand calculation > # hand calculation
 > table(a,b) > table(a,b)
-    b +      b 
-a    b1 b2 +a      noex exerc 
-  a1  5  5 +  drg1    5     5 
-  a2  5  5 +  drg2    5     5 
-  a3  5  5+  drg3    5     5
 > tapply(y, list(a,b), mean) # 각 셀에서의 평균 > tapply(y, list(a,b), mean) # 각 셀에서의 평균
-          b1        b2 +          noex     exerc 
-a1  9.686782 13.015181 +drg1  9.686782 13.015181 
-a2 10.491789  9.329158 +drg2 10.491789  9.329158 
-a3 10.381089 12.026922+drg3 10.381089 12.026922
 > tapply(y, a, mean) > tapply(y, a, mean)
-       a1        a2        a3 +     drg1      drg2      drg3 
 11.350981  9.910474 11.204006  11.350981  9.910474 11.204006 
 > tapply(y, b, mean) > tapply(y, b, mean)
-      b1       b2 +    noex    exerc 
 10.18655 11.45709  10.18655 11.45709 
 > n.within.each <- tapply(y, list(a,b), length) # the same as table(a, b) > n.within.each <- tapply(y, list(a,b), length) # the same as table(a, b)
 > n.within.each > n.within.each
-   b1 b2 +     noex exerc 
-a1  5  5 +drg1    5     5 
-a2  5  5 +drg2    5     5 
-a3  5  5+drg3    5     5
 > df.within.each <- n.within.each - 1  # 각 셀에서의 샘플숫자 > df.within.each <- n.within.each - 1  # 각 셀에서의 샘플숫자
 > df.within.each > df.within.each
-   b1 b2 +     noex exerc 
-a1  4  4 +drg1    4     4 
-a2  4  4 +drg2    4     4 
-a3  4  4+drg3    4     4
 > df.within <- sum(df.within.each) # df within > df.within <- sum(df.within.each) # df within
 > df.within > df.within
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 > var.within <- tapply(y, list(a,b), var) # var.within > var.within <- tapply(y, list(a,b), var) # var.within
 > var.within > var.within
-          b1        b2 +          noex     exerc 
-a1 0.2628787 0.7362999 +drg1 0.2628787 0.7362999 
-a2 1.0308917 1.6504481 +drg2 1.0308917 1.6504481 
-a3 0.9510727 1.5677577+drg3 0.9510727 1.5677577
 > ss.within.each <- tapply(y, list(a,b), var) * df.within.each > ss.within.each <- tapply(y, list(a,b), var) * df.within.each
 > ss.within.each > ss.within.each
-         b1       b2 +         noex    exerc 
-a1 1.051515 2.945200 +drg1 1.051515 2.945200 
-a2 4.123567 6.601793 +drg2 4.123567 6.601793 
-a3 3.804291 6.271031+drg3 3.804291 6.271031
 > ss.within <- sum(ss.within.each) # ss.within > ss.within <- sum(ss.within.each) # ss.within
 > ss.within > ss.within
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 > mean.b <- tapply(y, list(b), mean) > mean.b <- tapply(y, list(b), mean)
 > mean.a > mean.a
-       a1        a2        a3 +     drg1      drg2      drg3 
 11.350981  9.910474 11.204006  11.350981  9.910474 11.204006 
 > mean.b > mean.b
-      b1       b2 +    noex    exerc 
 10.18655 11.45709  10.18655 11.45709 
 >  > 
Line 299: Line 357:
 > var.b <- tapply(y, list(b), var) > var.b <- tapply(y, list(b), var)
 > var.a > var.a
-      a1       a2       a3 +    drg1     drg2     drg3 
 3.521366 1.567182 1.871915  3.521366 1.567182 1.871915 
 > var.b > var.b
-       b1        b2 +     noex     exerc 
 0.7773781 3.7300196  0.7773781 3.7300196 
 >  > 
Line 328: Line 386:
 > mean.each <- tapply(y, list(a,b), mean) > mean.each <- tapply(y, list(a,b), mean)
 > mean.each  > mean.each 
-          b1        b2 +          noex     exerc 
-a1  9.686782 13.015181 +drg1  9.686782 13.015181 
-a2 10.491789  9.329158 +drg2 10.491789  9.329158 
-a3 10.381089 12.026922+drg3 10.381089 12.026922
 > mean.tot <- mean(y) > mean.tot <- mean(y)
 > mean.tot > mean.tot
Line 339: Line 397:
 > n.b.each <- tapply(y, list(b), length) > n.b.each <- tapply(y, list(b), length)
 > n.each > n.each
-   b1 b2 +     noex exerc 
-a1  5  5 +drg1    5     5 
-a2  5  5 +drg2    5     5 
-a3  5  5+drg3    5     5
 > n.a.each > n.a.each
-a1 a2 a3  +drg1 drg2 drg3  
-10 10 10 +  10   10   10 
 > n.b.each > n.b.each
-b1 b2  + noex exerc  
-15 15 +   15    15 
 >  > 
 >  > 
Line 483: Line 541:
 Residuals   24  24.80   1.033                      Residuals   24  24.80   1.033                     
 --- ---
-Signif. codes:   +Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1+
 > aov.dat.noint <- aov(y ~ a + b, data = dat) > aov.dat.noint <- aov(y ~ a + b, data = dat)
 > # interaction이 갖는 SS와 df를 residuals이 갖음을 주목 > # interaction이 갖는 SS와 df를 residuals이 갖음을 주목
Line 493: Line 550:
 Residuals   26  50.54   1.944                  Residuals   26  50.54   1.944                 
 --- ---
-Signif. codes:   +Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1+
 > aov.dat.all2 <- aov(y ~ a + b + a:b, data = dat) > aov.dat.all2 <- aov(y ~ a + b + a:b, data = dat)
 > summary(aov.dat.all2) > summary(aov.dat.all2)
Line 503: Line 559:
 Residuals   24  24.80   1.033                      Residuals   24  24.80   1.033                     
 --- ---
-Signif. codes:   +Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1+
 > aov.dat.all # what is standard error of residuals  > aov.dat.all # what is standard error of residuals 
 Call: Call:
Line 517: Line 572:
 Estimated effects may be unbalanced 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> </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:pasted:20250430-130525.png}}
c/ms/2025/w07.2_factorial_anova.1745190156.txt.gz · Last modified: by hkimscil
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