# 닭의 목아지를 비틀어도 새벽은 온다. - 1979, 김영삼 -

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repeated_measures_anova

# 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
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 의 비교를 의미하므로, 반복측정(repeated measure)의 경우에는 conditions라는 용어를 사용.

• but, $\text{SS}_\text{{within}}$ can be partitioned as
• $\text{SS}_{\text{ subjects}}$ and $\text{SS}_{\text{ error}}$
• Among the two, we can exclude the first from SSwithin
• and solely use the latter as SSerror
• 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 SSerror will be smaller
• But, with this SSerror, 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 spreadsheet.
We also require fdistribution table to determine the null hypothesis test.

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

SStotal = $\Sigma{(X-\overline{X})^2}$ = 3489.2

SSparticipants = $w\Sigma{(\overline{X}_{participants}-\overline{X})}$ = 833.6

SSweeks = $n\Sigma{(\overline{X}_{week} - \overline{X})}$ = 1934.5

SSresidual

= SSerror

= SStotal - SSparticipants - SSweeks

= 721.1

dftotal = N - 1 = 45 - 1 = 44

dfweek = 5 - 1 = 4 = dfbetween

dfparticipants = 9 - 1 = 8 = dfsubjects

dferror= (n - 1)(k - 1) = 8 * 4 = 32 = 40 - 8 = 32

dfwithin = 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

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
> 
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)
> 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


## demo2

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)

> 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
> 

## demo 3

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)

> 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
>
>