Table of Contents
See also, ANOVA, Factorial ANOVA, paired sample ttest repeated_measures_anova
Repeated Measure ANOVA
Introduction
 oneway ANOVA for related, notindependent groups
 extension of the dependent ttest (one group ttest, repeated measure ttest)
 also, it is called “withinsubjects ANOVA” or “ANOVA for correlated samples”
 the simplest one is oneway 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 threemonth exercise training program on blood sugar level
 measure blood sugar level at 3 different points (preexercise, midway, postexercise)
 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:
preexcerise “sugar level”  midterm “sugar level”  postexercise “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 간의 비교를 의미하므로, 반복측정(repeated measure)의 경우에는 conditions라는 용어를 사용.
 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_{within}
 and solely use the latter as SS_{error}
 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_{error} will be smaller
 But, with this SS_{error}, the df is going to be (n1)(k1)
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 spreadsheet.
We also require fdistribution table to determine the null hypothesis test.
Headache Analysis  

base treatment  average per case (subject, participant) 

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 
3  7  5  5  4  5  5.2 
4  25  30  13  12  4  16.8 
5  30  33  10  8  6  17.4 
6  19  27  8  7  4  13 
7  26  16  5  2  5  10.8 
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 
Stats  

Mean Total  12.47 
$\Sigma{X_i}$  561 
$\Sigma{{X_i}^2}$  10483 
# of week  5 
# of case (n)  9 
SS_{total} = $\Sigma{(X\overline{X})^2} $ = 3489.2
SS_{between}
= SS_{conditions}
= SS_{weeks}
= $n\Sigma{(\overline{X}_{week}  \overline{X})^2}$ = 1934.5
SS_{within}
= $ \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_{participants} = $w\Sigma{(\overline{X}_{participants}\overline{X})^2}$ = 833.6
SS_{residual}
= SS_{error}
= SS_{within}  SS_{participants}
= 1554.7  833.6
= 721.1
OR
SS_{residual} =
= SS_{error}
= (SS_{total}  SS_{weeks(between)})  SS_{participants}
= 721.1
df_{total} = N  1 = 45  1 = 44
df_{week} = 5  1 = 4 = df_{between}
df_{participants} = 9  1 = 8 = df_{subjects}
df_{error}= (n  1)(k  1) = 8 * 4 = 32 = 40  8 = 32
df_{within} = N  k = 45  5 = 40
ie
시각적 인지점수  

참가자  No visual distraction  Visual distraction  Sound Distraction 
A  47  22  41 
B  57  31  52 
C  38  18  40 
D  45  32  43 
in r
demo1
https://rcompanion.org/handbook/I_09.html
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)
demo1 data는 아래와 같다.
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
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 
demo1.within.only.aov < aov(pulse ~ time + Error(id), data = demo1) summary(demo1.within.only.aov)
> 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 >
demo 2
Twoway repeated measure anova
reference
 Repeated measures oneway ANOVA by Akkelin
 http://rcompanion.org/handbook/I_09.html : This is an excellent example, but, difficult to swallow.