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

– Picture about here –

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_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 (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.

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

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

SS_residual

= SS_error

= SS_total - SS_participants - SS_weeks

= 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

reference

http://rcompanion.org/handbook/I_09.html : This is an excellent example, but, difficult to swallow.

COMMunication
RESearch.NET

Table of Contents

Repeated Measures ANOVA

ie

reference

	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

	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

	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

	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

	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

	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