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repeated_measure_anova [2015/10/13 08:57] – hkimscilrepeated_measure_anova [2024/05/01 08:27] (current) – 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 ====== 
 +Introduction 
 +  * one-way ANOVA for //**related, not-independent groups**// 
 +  * extension of the dependent t-test (one group t-test, repeated measure t-test) 
 +  * also, it is called "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) 
 +    * 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) 
 +    * 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 
 +  * $\text{independent ANOVA: } F = \displaystyle \frac{MS_{between}}{MS_{within}} = \frac{MS_{between}}{MS_{error}}$ 
 + 
 +  * $\text{rep measures ANOVA: } F = \displaystyle \frac{MS_{between}}{MS_{within}} = \displaystyle \frac{MS_{conditions}}{MS_{error}}$ 
 + 
 +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: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> 
 +    * This is to say: 
 +      * in $\text{independent ANOVA: } \text{SS}_\text{{within}} = \text{SS}_{\text{error}} $   
 +      * in $\text{rep measures ANOVA: } \text{SS}_\text{{within}} = \text{SS}_{\text{subjects}} + \text{SS}_{\text{error}}$  
 +    * 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}}.  
 +We also require {{:ftable.pdf|fdistribution table}} to determine the null hypothesis test.
  
-====== Repeated Measures ANOVA ====== 
 ^  Headache Analysis  ^^^^^^^ ^  Headache Analysis  ^^^^^^^
-| ||  base  |||  treatment | average per case | +| | base   treatment  ||||| average \\ per case \\ (subject, \\ participant)  | 
-|  ser  | w1  |  w2  |  w3  |  w4  |  w5  |  avePerCase  |+|  ser  | w1  |  w2  |  w3  |  w4  |  w5  | $\overline{X}_{part}$ \\ = average \\ per case \\ (subject, \\ participant)  |
 |  1  |  21  |  22  |  8  |  6  |  6  |  12.6  | |  1  |  21  |  22  |  8  |  6  |  6  |  12.6  |
 |  2  |  20  |  19  |  10  |  4  |  9  |  12.4  | |  2  |  20  |  19  |  10  |  4  |  9  |  12.4  |
Line 15: Line 119:
 |  8  |  13  |  4  |  8  |  1  |  5  |  6.2  | |  8  |  13  |  4  |  8  |  1  |  5  |  6.2  |
 |  9  |  26  |  24  |  14  |  8  |  17  |  17.8  | |  9  |  26  |  24  |  14  |  8  |  17  |  17.8  |
-|  avePerW  |  20.78  |  20.00  |  9.00  |  5.78  |  6.78  |  12.47  |+|  average \\ per week  |  20.78  |  20.00  |  9.00  |  5.78  |  6.78  |  $\overline{X}$ = 12.47  |
  
 ^  Stats  ^^ ^  Stats  ^^
Line 24: Line 128:
 |  # of case (n) | 9  | |  # 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>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>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>error</sub> 
-= SS<sub>total</sub> - SS<sub>participants</sub> - SS<sub>weeks</sub> += SS<sub>within</sub> - SS<sub>participants</sub> 
 += 1554.7 - 833.6
 = 721.1 = 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 \\
 +\\
 +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> \\
 +df<sub>error</sub>= (n - 1)(k - 1) = 8 * 4 = 32 = 40 - 8 = 32 \\
 +df<sub>within</sub> = N - k = 45 - 5 = 40
  
 ====== ie ====== ====== ie ======
Line 40: Line 168:
 |  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 =====
 +<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
 +
 +# 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    1       30
 +2        1    2       28
 +3        1    3       16
 +4        1    4       34
 +5        2    1       14
 +6        2    2       18
 +7        2    3       10
 +8        2    4       22
 +9        3    1       24
 +10       3    2       20
 +11       3    3       18
 +12       3    4       30
 +13       4    1       38
 +14       4    2       34
 +15       4    3       20
 +16       4    4       44
 +17       5    1       26
 +18       5    2       28
 +19       5    3       14
 +20       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 ‘***’ 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")
 +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)
 +
 +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
 +    * {{:ezdata.sav|ezdata: SPSS Data file}}
   * 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.1444696032.txt.gz · Last modified: 2015/10/13 08:57 by hkimscil
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