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repeated_measure_anova [2022/05/10 10:29] – hkimscilrepeated_measure_anova [2025/05/07 08:40] (current) – [Repeated Measure ANOVA] hkimscil
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-See also, [[ANOVA]], [[:Factorial Anova|Factorial ANOVA]], [[:t-test#동일집단_간의_차이에_대해서_알아볼_때|paired sample t-test]] [[:r:repeated_measure_anova]] +See also, [[ANOVA]], [[:Factorial Anova|Factorial ANOVA]], [[:t-test#동일집단_간의_차이에_대해서_알아볼_때|paired sample t-test]] [[:r:repeated_measures_anova]] 
 ====== Repeated Measure ANOVA ====== ====== Repeated Measure ANOVA ======
 Introduction Introduction
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 -- Picture about here -- -- Picture about here --
 +{{:pasted:20240501-082722.png}} 
 +---- 
 +{{:pasted:20240513-083858.png}} 
 +----
   * but, $\text{SS}_\text{{within}}$ can be partitioned as    * but, $\text{SS}_\text{{within}}$ can be partitioned as 
     * $\text{SS}_{\text{ subjects}}$ and $\text{SS}_{\text{ error}}$     * $\text{SS}_{\text{ subjects}}$ and $\text{SS}_{\text{ error}}$
 +    * that is, some of the "within variation" are carried along in each individual.  
     * Among the two, we can exclude the first from SS<sub>within</sub>     * Among the two, we can exclude the first from SS<sub>within</sub>
     * and solely use the latter as SS<sub>error</sub>     * and solely use the latter as SS<sub>error</sub>
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 |  **Grand mean: 45.9**      ||||| |  **Grand mean: 45.9**      |||||
  
-We do this (and the below example) with an excel {{:repeated_measures_anova_eg.xlsx|spreadsheet}}. +We do this (and the below example) with an excel {{:r:repeated_measures_anova_eg.xlsx|spreadsheet, repeated_measures_anova_eg.xlsx}}. 
 We also require {{:ftable.pdf|fdistribution table}} to determine the null hypothesis test. We also require {{:ftable.pdf|fdistribution table}} to determine the null hypothesis test.
  
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 SS<sub>total</sub> = $\Sigma{(X-\overline{X})^2} $ = 3489.2 \\ SS<sub>total</sub> = $\Sigma{(X-\overline{X})^2} $ = 3489.2 \\
-SS<sub>participants</sub> = $w\Sigma{(\overline{X}_{participants}-\overline{X})}$ = 833.6 \\ + 
-SS<sub>weeks</sub> = $n\Sigma{(\overline{X}_{week} - \overline{X})}$ = 1934.5 \\ +SS<sub>between</sub> 
-SS<sub>residual</sub>  \\ += SS<sub>conditions</sub>  
-= SS<sub>error</sub> \\ += SS<sub>weeks</sub>  
-= SS<sub>total</sub> - SS<sub>participants</sub> - SS<sub>weeks</sub>  \\+= $n\Sigma{(\overline{X}_{week} - \overline{X})^2}$ = 1934.5 \\ 
 + 
 +SS<sub>within</sub>  
 += $ \Sigma \Sigma{(X_{s_i.t_j} - \overline{X_{t_j}})^2}$  
 += $ \Sigma (411.6, 836.0, 78.0, 93.6, 135.6) $  
 += 1554.7  
 +\\ 
 + 
 +SS<sub>participants</sub> = $w\Sigma{(\overline{X}_{participants}-\overline{X})^2}$ = 833.6 \\ 
 + 
 +SS<sub>residual</sub> 
 += SS<sub>error</sub>  
 += SS<sub>within</sub> - SS<sub>participants</sub> 
 += 1554.7 - 833.6 
 += 721.1 
 + 
 +OR 
 +SS<sub>residual</sub> = 
 += SS<sub>error</sub>  
 += (SS<sub>total</sub> - SS<sub>weeks(between)</sub>) - SS<sub>participants</sub>  
 = 721.1 \\ = 721.1 \\
 \\ \\
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 </code> </code>
  
-<code> +see {{:r:repeated_measures_anova_eg.xlsx}} 
-demo1  <- read.csv("https://stats.idre.ucla.edu/stat/data/demo1.csv") +===== demo 2 ===== 
-demo1  +see [[:r:repeated measure anova]] 
-str(demo1) ## 모든 변인이 int이므로 (숫자) factor로 바꿔야 한다+===== Twoway repeated measure anova===== 
 +see [[:r:twoway repeated measure anova]]
  
-## Convert variables to factor 
-demo1 <- within(demo1, { 
-  group <- factor(group) 
-  time <- factor(time) 
-  id <- factor(id) 
-}) ## 이제 pulse만 제외하고 모두 factor로 변환된 데이터 
- 
-str(demo1) 
- 
-par(cex = .6) 
- 
-with(demo1, interaction.plot(time, group, pulse, 
-  ylim = c(5, 20), lty= c(1, 12), lwd = 3, 
-  ylab = "mean of pulse", xlab = "time", trace.label = "group")) 
- 
-demo1.aov <- aov(pulse ~ group * time + Error(id), data = demo1) 
-summary(demo1.aov) 
-</code> 
- 
-<code> 
-> summary(demo1.aov) 
- 
-Error: id 
-          Df Sum Sq Mean Sq F value  Pr(>F)     
-group      1 155.04  155.04    3721 1.3e-09 *** 
-Residuals  6   0.25    0.04                     
---- 
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
- 
-Error: Within 
-           Df Sum Sq Mean Sq F value Pr(>F) 
-time        2 0.0833 0.04167       1  0.397 
-group:time  2 0.0833 0.04167       1  0.397 
-Residuals  12 0.5000 0.04167                
- 
-</code> 
-{{:pasted:20200611-142331.png?350}} 
- 
-===== demo2 ===== 
-<code> 
-demo2 <- read.csv("https://stats.idre.ucla.edu/stat/data/demo2.csv") 
-## Convert variables to factor 
-demo2 <- within(demo2, { 
-    group <- factor(group) 
-    time <- factor(time) 
-    id <- factor(id) 
-}) 
-demo2 
- 
-with(demo2, interaction.plot(time, group, pulse, 
- ylim = c(10, 40), lty = c(1, 12), lwd = 3, 
- ylab = "mean of pulse", xlab = "time", trace.label = "group")) 
- 
-demo2.aov <- aov(pulse ~ group * time + Error(id), data = demo2) 
-summary(demo2.aov) 
-</code> 
- 
-{{:pasted:20200611-151520.png?350}} 
- 
-<code> 
-> demo2 <- read.csv("https://stats.idre.ucla.edu/stat/data/demo2.csv") 
-> ## Convert variables to factor 
-> demo2 <- within(demo2, { 
-+     group <- factor(group) 
-+     time <- factor(time) 
-+     id <- factor(id) 
-+ }) 
- 
-> demo2 
-   id group pulse time 
-1   1     1    14    1 
-2   1     1    19    2 
-3   1     1    29    3 
-4   2     1    15    1 
-5   2     1    25    2 
-6   2     1    26    3 
-7   3     1    16    1 
-8   3     1    16    2 
-9   3     1    31    3 
-10  4     1    12    1 
-11  4     1    24    2 
-12  4     1    32    3 
-13  5     2    10    1 
-14  5     2    21    2 
-15  5     2    24    3 
-16  6     2    17    1 
-17  6     2    26    2 
-18  6     2    35    3 
-19  7     2    19    1 
-20  7     2    22    2 
-21  7     2    32    3 
-22  8     2    15    1 
-23  8     2    23    2 
-24  8     2    34    3 
->  
-> with(demo2, interaction.plot(time, group, pulse, 
-+  ylim = c(10, 40), lty = c(1, 12), lwd = 3, 
-+  ylab = "mean of pulse", xlab = "time", trace.label = "group")) 
->  
-> demo2.aov <- aov(pulse ~ group * time + Error(id), data = demo2) 
-> summary(demo2.aov) 
- 
-Error: id 
-          Df Sum Sq Mean Sq F value Pr(>F) 
-group      1  15.04   15.04   0.836  0.396 
-Residuals  6 107.92   17.99                
- 
-Error: Within 
-           Df Sum Sq Mean Sq F value   Pr(>F)     
-time        2  978.2   489.1  53.684 1.03e-06 *** 
-group:time  2    1.1     0.5   0.059    0.943     
-Residuals  12  109.3     9.1                      
---- 
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
->  
-</code> 
-===== demo 3 ===== 
-<code> 
-demo3 <- read.csv("https://stats.idre.ucla.edu/stat/data/demo3.csv") 
-## Convert variables to factor 
-demo3 <- within(demo3, { 
-  group <- factor(group) 
-  time <- factor(time) 
-  id <- factor(id) 
-}) 
- 
-with(demo3, interaction.plot(time, group, pulse, 
-  ylim = c(10, 60), lty = c(1, 12), lwd = 3, 
-  ylab = "mean of pulse", xlab = "time", trace.label = "group")) 
- 
-demo3.aov <- aov(pulse ~ group * time + Error(id), data = demo3) 
-summary(demo3.aov) 
-</code> 
- 
-{{:pasted:20200611-151755.png?350}} 
- 
-<code> 
-> demo3 <- read.csv("https://stats.idre.ucla.edu/stat/data/demo3.csv") 
-> ## Convert variables to factor 
-> demo3 <- within(demo3, { 
-+     group <- factor(group) 
-+     time <- factor(time) 
-+     id <- factor(id) 
-+ }) 
->  
-> with(demo3, interaction.plot(time, group, pulse, 
-+  ylim = c(10, 60), lty = c(1, 12), lwd = 3, 
-+  ylab = "mean of pulse", xlab = "time", trace.label = "group")) 
->  
-> demo3.aov <- aov(pulse ~ group * time + Error(id), data = demo3) 
-> summary(demo3.aov) 
- 
-Error: id 
-          Df Sum Sq Mean Sq F value  Pr(>F)     
-group      1 2035.0  2035.0   343.1 1.6e-06 *** 
-Residuals  6   35.6     5.9                     
---- 
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
- 
-Error: Within 
-           Df Sum Sq Mean Sq F value   Pr(>F)     
-time        2 2830.3  1415.2   553.8 1.52e-12 *** 
-group:time  2  200.3   100.2    39.2 5.47e-06 *** 
-Residuals  12   30.7     2.6                      
---- 
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
->  
->  
- 
-</code> 
 ====== reference ====== ====== reference ======
   * [[http://wwwstage.valpo.edu/other/dabook/ch12/c12-1.htm|Repeated measures one-way ANOVA]] by Akkelin   * [[http://wwwstage.valpo.edu/other/dabook/ch12/c12-1.htm|Repeated measures one-way ANOVA]] by Akkelin
repeated_measure_anova.1652146181.txt.gz · Last modified: by hkimscil
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