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--- repeated_measure_anova [2022/05/10 10:29] – hkimscil
+++ repeated_measure_anova [2024/05/01 08:27] (current) – hkimscil
@@ Line 1: / Line 1: @@
-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 ======
 Introduction
@@ Line 79: / Line 79: @@
 -- Picture about here --
+{{:pasted:20240501-082722.png}}
+----
+{{:pasted:20240501-082738.png}}
+----
   * but, $\text{SS}_\text{{within}}$ can be partitioned as
     * $\text{SS}_{\text{ subjects}}$ 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>
     * and solely use the latter as SS<sub>error</sub>
@@ Line 100: / Line 104: @@
 |  **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}}.
 We also require {{:ftable.pdf|fdistribution table}} to determine the null hypothesis test.
@@ Line 125: / Line 129: @@
 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 \\
 \\
@@ Line 236: / Line 259: @@
 </code>
+see {{:r:repeated_measures_anova_eg.xlsx}}
+===== demo 2 =====
+<code>
+# create data
+df <- data.frame(patient=rep(1:5, each=4),
+                 drug=rep(1:4, times=5),
+                 response=c(30, 28, 16, 34,
+, 18, 10, 22,
+, 20, 18, 30,
+, 34, 20, 44,
+, 28, 14, 30))
+# view data
+df
+# within sujbect anova
+within.aov.mod <- aov(response~drug+Error(patient), data=df)
+</code>
+<code>
+> #create data
+> df <- data.frame(patient=rep(1:5, each=4),
++                  drug=rep(1:4, times=5),
++                  response=c(30, 28, 16, 34,
++                             14, 18, 10, 22,
++                             24, 20, 18, 30,
++                             38, 34, 20, 44,
++                             26, 28, 14, 30))
+>
+> #view data
+> df
+   patient drug response
+        1    1       30
+        1    2       28
+        1    3       16
+        1    4       34
+        2    1       14
+        2    2       18
+        2    3       10
+        2    4       22
+        3    1       24
+       3    2       20
+       3    3       18
+       3    4       30
+       4    1       38
+       4    2       34
+       4    3       20
+       4    4       44
+       5    1       26
+       5    2       28
+       5    3       14
+       5    4       30
+>
+> #within sujbect anova
+> within.aov.mod <- aov(response~drug+Error(patient), data=df)
+>
+>
+> summary(within.aov.mod)
+Error: patient
+          Df Sum Sq Mean Sq F value Pr(>F)
+Residuals  1   67.6    67.6
+Error: Within
+          Df Sum Sq Mean Sq F value Pr(>F)
+drug       1   11.6   11.56   0.139  0.714
+Residuals 17 1412.6   83.10
+</code>
+The above is <fc #ff0000>**WRONG**</fc>.
+<code>
+within.aov.mod <- aov(response~factor(drug)+Error(factor(patient)), data=df)
+summary(within.aov.mod)
+</code>
+<code>
+> within.aov.mod <- aov(response~factor(drug)+Error(factor(patient)), data=df)
+>
+>
+> summary(within.aov.mod)
+Error: factor(patient)
+          Df Sum Sq Mean Sq F value Pr(>F)
+Residuals  4  680.8   170.2
+Error: Within
+             Df Sum Sq Mean Sq F value   Pr(>F)
+factor(drug)  3  698.2   232.7   24.76 1.99e-05 ***
+Residuals    12  112.8     9.4
+---
+Signif. codes:
+‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+>
+</code>
+====== two way ======
 <code>
 demo1  <- read.csv("https://stats.idre.ucla.edu/stat/data/demo1.csv")