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--- repeated_measure_anova [2016/06/23 19:04] – hkimscil
+++ repeated_measure_anova [2017/05/10 09:08] – hkimscil
@@ Line 3: / Line 3: @@
 ====== Repeated Measures ANOVA ======
+Introduction
+  * one-way ANOVA for //**related, not-independent groups**//
+  * extension of the dependent t-test (one group t-test, repeated measure t-test)
+  * also, within-subjects ANOVA or ANOVA for correlated samples
+  * the simplest one is __one-way repeated measures ANOVA__
+  * which requires one independent and one dependent variable
+  * the independent variable is categorical (either nominal or ordinal)
+  * the dependent variable is continuous (interval or ratio)
+Test Circumstances
+  * one subject with repeated measures across a time period (differences of mean scores across three or more time periods)
+  * one subject with repeated measures in different situation (treatments; differences of mean scores under three or more different conditions)
+LOGICS
+  * $\text{independent ANOVA: } F = \displaystyle \frac{MS_{between}}{MS_{within}} = \frac{MS_{between}}{MS_{error}}$
+  * $\text{rep measures ANOVA: } F = \displaystyle \frac{MS_{between}}{MS_{within}} = \displaystyle \frac{MS_{conditions}}{MS_{error}}$
+  * but, $\text{SS}_\text{{within}}$ can be partitioned as
+    * $\text{SS}_{\text{ subjects}}$ and $\text{SS}_{\text{ error}}$
+    * Among the two, we can exclude the first from SS<sub>within</sub>
+    * and solely use the latter as SS<sub>error</sub>
+    * This is to say:
+      * in $\text{independent ANOVA: } \text{SS}_\text{{within}} = \text{SS}_{\text{error}} $
+      * in $\text{rep measures ANOVA: } \text{SS}_\text{{within}} = \text{SS}_{\text{subjects}} + \text{SS}_{\text{error}}$
 ^  Headache Analysis  ^^^^^^^
-| ||  base  |||  treatment | average per case |
+| | base   treatment  ||||| average \\ per case \\ (subject, \\ participant)  |
-|  ser  | w1  |  w2  |  w3  |  w4  |  w5  |  $\overline{X}_{participants}$ = average per case  |
+|  ser  | w1  |  w2  |  w3  |  w4  |  w5  | $\overline{X}_{part}$ \\ = average \\ per case \\ (subject, \\ participant)  |
 |  1  |  21  |  22  |  8  |  6  |  6  |  12.6  |
 |  2  |  20  |  19  |  10  |  4  |  9  |  12.4  |
@@ Line 15: / Line 43: @@
 |  8  |  13  |  4  |  8  |  1  |  5  |  6.2  |
 |  9  |  26  |  24  |  14  |  8  |  17  |  17.8  |
-|  average per week  |  20.78  |  20.00  |  9.00  |  5.78  |  6.78  |  $\overline{X}$ = 12.47  |
+|  average \\ per week  |  20.78  |  20.00  |  9.00  |  5.78  |  6.78  |  $\overline{X}$ = 12.47  |
 ^  Stats  ^^
@@ Line 24: / Line 52: @@
 |  # of case (n) | 9  |
-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>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>weeks</sub> = $n\Sigma{(\overline{X}_{week} - \overline{X})}$ = 1934.5 \\
-SS<sub>residual</sub>
+SS<sub>residual</sub>  \\
-= SS<sub>error</sub>
+= SS<sub>error</sub> \\
-= SS<sub>total</sub> - SS<sub>participants</sub> - SS<sub>weeks</sub>
+= SS<sub>total</sub> - SS<sub>participants</sub> - SS<sub>weeks</sub>  \\
-= 721.1
+= 721.1 \\
+\\
-df<sub>total</sub> = N - 1 = 45 - 1 = 44
+df<sub>total</sub> = N - 1 = 45 - 1 = 44 \\
 df<sub>week</sub> = 5 - 1 = 4 = df<sub>between</sub> \\
 df<sub>participants</sub> = 9 - 1 = 8 = df<sub>subjects</sub> \\