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repeated_measure_anova

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repeated_measure_anova [2017/05/10 00:38] – hkimscilrepeated_measure_anova [2025/05/06 23:40] (current) – [Repeated Measure ANOVA] hkimscil
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-See also, [[ANOVA]], [[:Factorial Anova|Factorial ANOVA]] +See also, [[ANOVA]], [[:Factorial Anova|Factorial ANOVA]], [[:t-test#동일집단_간의_차이에_대해서_알아볼_때|paired sample t-test]] [[:r:repeated_measures_anova]]  
-설명이 불충분하므로, [[:Repeated Measure Anova#s-3|아래 참고 사이트]](reference)를 숙지할 것. +====== Repeated Measure ANOVA ======
- +
-====== Repeated Measures ANOVA ======+
 Introduction Introduction
   * one-way ANOVA for //**related, not-independent groups**//   * one-way ANOVA for //**related, not-independent groups**//
   * extension of the dependent t-test (one group t-test, repeated measure t-test)   * extension of the dependent t-test (one group t-test, repeated measure t-test)
-  * also, within-subjects ANOVA or ANOVA for correlated samples+  * also, it is called "within-subjects ANOVA" or "ANOVA for correlated samples"
   * the simplest one is __one-way repeated measures ANOVA__   * the simplest one is __one-way repeated measures ANOVA__
   * which requires one independent and one dependent variable   * which requires one independent and one dependent variable
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 Test Circumstances  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 across a time period (differences of mean scores across three or more time periods)
 +    * participants being tested with headache drugs such as 
 +      * group A, B, C, placebo 
 +      * across the time periods j, k, l, m
 +    * testing the effect of a three-month exercise training program on blood sugar level
 +      * measure blood sugar level at 3 different points (pre-exercise, midway, post-exercise)
   * one subject with repeated measures in different situation (treatments; differences of mean scores under three or more different conditions)   * one subject with repeated measures in different situation (treatments; differences of mean scores under three or more different conditions)
 +    * e.g., participant (n=30) using and evaluating three web site UI (naver, daum, and google)
 +    * and rate its usefulness, usability and ease of use
 +  * data should look as follows:
 +
 +^ ^ pre-excerise \\ "sugar level"   ^ mid-term \\ "sugar level"   ^ post-exercise  \\ "sugar level"  ^
 +|  a  | 250  | 220  | 150  |
 +|  b  | 300  | 170  | 120  |
 +|  c  | 150  | 120  | 120  |
 +|  d  | 230  | 170  | 160  |
 +|  e  | 260  | 250  | 250  |
 +|     | level 1  | level 2  | level 3  |
 +
 +Levels = related groups of the independent variable "time"
 +
 +^ ^ treatment \\ condition \\ "naver"   ^ treatment \\ condition \\ "daum"   ^ treatment \\ condition \\ "google"   ^
 +|  a  | 70  | 60  | 80  |
 +|  b  | 50  | 70  | 50  |
 +|  c  | 40  | 50  | 60  |
 +|  d  | 30  | 40  | 60  |
 +|  e  | 60  | 50  | 40  |
 +|     | level 1  | level 2  | level 3  |
 +
 +in general, the data should look 
 +^ ^  time/condition  ^^^
 +| |  T1  |  T2  |  T3  |
 +|  s1  |  s1  |  s1  |  s1  |
 +|  s2  |  s2  |  s2  |  s2  |
 +|  s3  |  s3  |  s3  |  s3  |
 +|  s4  |  s4  |  s4  |  s4  |
 +|  s5  |  s5  |  s5  |  s5  |
 +|  ..  |  ..  |  ..  |  ..  |
 +|  sn  |  sn  |  sn  |  sn  |
 +
 +You should discern the above from normal ANOVA situation.
 +
 +^  ^  group  ^  treatment  ^
 +| a |  1  |  70  |
 +| b |  1  |  50  |
 +| c |  1  |  40  |
 +| d |  1  |  30  |
 +| e |  1  |  60  |
 +| f |  2  |  60  |
 +| g |  2  |  70  |
 +| h |  2  |  50  |
 +| i |  2  |  40  |
 +| j |  2  |  50  |
 +| k |  3  |  80  |
 +| l |  3  |  50  |
 +| m |  3  |  60  |
 +| n |  3  |  60  |
 +| o |  3  |  40  |
  
 LOGICS LOGICS
   * $\text{independent ANOVA: } F = \displaystyle \frac{MS_{between}}{MS_{within}} = \frac{MS_{between}}{MS_{error}}$   * $\text{independent ANOVA: } F = \displaystyle \frac{MS_{between}}{MS_{within}} = \frac{MS_{between}}{MS_{error}}$
- 
  
   * $\text{rep measures ANOVA: } F = \displaystyle \frac{MS_{between}}{MS_{within}} = \displaystyle \frac{MS_{conditions}}{MS_{error}}$   * $\text{rep measures ANOVA: } F = \displaystyle \frac{MS_{between}}{MS_{within}} = \displaystyle \frac{MS_{conditions}}{MS_{error}}$
  
 +주>
 +  * "between" 이란 단어는 독립적인 그룹 **간**의 비교를 의미하므로, 반복측정(repeated measure)의 경우에는 conditions라는 용어를 사용.
 +
 +-- 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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       * in $\text{independent ANOVA: } \text{SS}_\text{{within}} = \text{SS}_{\text{error}} $         * in $\text{independent ANOVA: } \text{SS}_\text{{within}} = \text{SS}_{\text{error}} $  
       * in $\text{rep measures ANOVA: } \text{SS}_\text{{within}} = \text{SS}_{\text{subjects}} + \text{SS}_{\text{error}}$        * in $\text{rep measures ANOVA: } \text{SS}_\text{{within}} = \text{SS}_{\text{subjects}} + \text{SS}_{\text{error}}$ 
- +    * This means that the term SS<sub>error</sub> will be **__smaller__** 
 +    * But, with this SS<sub>error</sub>, the df is going to be (n-1)(k-1) 
 + 
 +^  subjects  ^  Pre  ^  1 Month  ^  3 Month  ^  Subject \\ Means  ^ 
 +|  1  |  45  |  50  |  55  |  **50**  | 
 +|  2  |  42  |  42  |  45  |  **43**  | 
 +|  3  |  36  |  41  |  43  |  **40**  | 
 +|  4  |  39  |  35  |  40  |  **38**  | 
 +|  5  |  51  |  55  |  59  |  **55**  | 
 +|  6  |  44  |  49  |  56  |  **49.7**  | 
 +|  **Monthly mean**  |  **42.8**  |  **45.3**  |  **49.97**  |   | 
 +|  **Grand mean: 45.9**      ||||| 
 + 
 +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.
  
 ^  Headache Analysis  ^^^^^^^ ^  Headache Analysis  ^^^^^^^
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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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 |  C  |  38  |  18  |  40  | |  C  |  38  |  18  |  40  |
 |  D  |  45  |  32  |  43  | |  D  |  45  |  32  |  43  |
 +====== in r ======
 +===== demo1 =====
 +
 +[[https://rcompanion.org/handbook/I_09.html]] 
 +<WRAP box info>
 +data files in e.gs:
 +{{:demo1.csv}}
 +{{:demo2.csv}}
 +{{:demo3.csv}}
 +{{:demo4.csv}}
 +{{:exer.csv}}
 +</WRAP>
 +
 +<code>
 +demo1  <- read.csv("https://stats.idre.ucla.edu/stat/data/demo1.csv")
 +demo1 
 +str(demo1) ## 모든 변인이 int이므로 (숫자) factor로 바꿔야 한다
 +
 +## 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)
 +</code>
 +
 +demo1 data는 아래와 같다.
 +<code>
 +id group pulse time
 +1 1 10 1
 +1 1 10 2
 +1 1 10 3
 +2 1 10 1
 +2 1 10 2
 +2 1 10 3
 +3 1 10 1
 +3 1 10 2
 +3 1 10 3
 +4 1 10 1
 +4 1 10 2
 +4 1 10 3
 +5 2 15 1
 +5 2 15 2
 +5 2 15 3
 +6 2 15 1
 +6 2 15 2
 +6 2 15 3
 +7 2 16 1
 +7 2 15 2
 +7 2 15 3
 +8 2 15 1
 +8 2 15 2
 +8 2 15 3
 +</code>
 +이를 정리해보면 
 +
 +||   || time  ||||||||
 +||   || t1  || t2  || t3  || mean \\ of the \\ same person's \\ measures  ||
 +|| 1  || 10  || 10  || 10  || 10  ||
 +|| 2  || 10  || 10  || 10  || 10  ||
 +|| 3  || 10  || 10  || 10  || 10  ||
 +|| 4  || 10  || 10  || 10  || 10  ||
 +|| 5  || 15  || 15  || 15  || 15  ||
 +|| 6  || 15  || 15  || 15  || 15  ||
 +|| 7  || 16  || 15  || 15  || 15.333  ||
 +|| 8  || 15  || 15  || 15  || 15  ||
 +|| mean \\ across \\ the time  || 12.625  || 12.5  || 12.5  || 12.542  ||
 +
 +
 +<code>
 +demo1.within.only.aov <- aov(pulse ~ time + Error(id), data = demo1)
 +summary(demo1.within.only.aov)
 +</code>
 +
 +<code>
 +> demo1.within.only.aov <- aov(pulse ~ time + Error(id), data = demo1)
 +> summary(demo1.within.only.aov)
 +
 +Error: id
 +          Df Sum Sq Mean Sq F value Pr(>F)
 +Residuals  7  155.3   22.18               
 +
 +Error: Within
 +          Df Sum Sq Mean Sq F value Pr(>F)
 +time       2 0.0833 0.04167       1  0.393
 +Residuals 14 0.5833 0.04167               
 +> 
 +</code>
 +
 +see {{:r:repeated_measures_anova_eg.xlsx}}
 +===== demo 2 =====
 +see [[:r:repeated measure anova]]
 +===== Twoway repeated measure anova=====
 +see [[:r:twoway repeated measure anova]]
  
 ====== reference ====== ====== reference ======
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   * http://www.psych.utoronto.ca/courses/c1/chap14/chap14.html   * http://www.psych.utoronto.ca/courses/c1/chap14/chap14.html
   * https://statistics.laerd.com/statistical-guides/repeated-measures-anova-statistical-guide.php   * https://statistics.laerd.com/statistical-guides/repeated-measures-anova-statistical-guide.php
 +
 +  * http://rcompanion.org/handbook/I_09.html : This is an excellent example, but, difficult to swallow.
  
repeated_measure_anova.1494376694.txt.gz · Last modified: by hkimscil
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