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:

Levels = related groups of the independent variable “time”

in general, the data should look

You should discern the above from normal ANOVA situation.

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 $\text{SS}_{\text{ error}}$
  • 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 (n-1)(k-1)

We do this (and the below example) with an excel spreadsheet.
We also require fdistribution table to determine the null hypothesis test.

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

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

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               
> 

see repeated_measures_anova_eg.xlsx

see repeated measure anova

see twoway repeated measure anova

• Repeated measures one-way ANOVA by Akkelin
  • ezdata: SPSS Data file
• http://www.psych.utoronto.ca/courses/c1/chap14/chap14.html
• 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.