PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH
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Transcript of PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH
PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH
Lesson 12Repeated-Measures Analysis of
Variance (ANOVA)
Repeated-Measures ANOVA
ANOVA is a hypothesis-testing procedure that is used to evaluate mean differences between two or more treatments.
Repeated-measures designs are used commonly to examine development (over time), to chart the course of learning (at different levels of practice), or simply to examine performance under different conditions.
Repeated-Measures ANOVA
Independent variable: Practice
Practice Session 1
Practice Session 2
Practice Session 3
Number of errors made
Number of errors made
Number of errors made
Rats are tested in the maze in one daily session for three days
Repeated-Measures ANOVA
Statistical Hypothesis for ANOVA:
0 1 2 3
1
: ..
:
H
H
At least one treatment mean is different from others
variance (differences) between treatments
variance (differences) expected by sampling errorF
Logic of Repeated-Measures ANOVA
treatment effect+individual differences+experimental error
individual differences+experimental errorF
Logic of Repeated-Measures ANOVA
Total variability
Between-treatments variability Within-treatments variability
1.Treatment effects2. Experimental error
1. Individual differences2. Experimental error
Between subjects variabilty
Error variability
Logic of Repeated-Measures ANOVA
SS total
SS between SS within=
SS between subjects +SS error
error
betweenMean squared deviation (MS) =Variance between treatments = MS = ,
between
errorMS =
error
between
between
error
SS
df
SS
df
MSF
MS
df totalN-1
df between k-1
df withinN-k
df between subjectsn-1
df error(N-k)-(n-1)
Example
A school psychologist would like to test the effectiveness of behavior-modification technique in controlling classroom outbursts of unruly children. A teacher is instructed to use the response-cost technique. Every time a child disrupts the class, he or she is told that the behavior has cost him or her 10 minutes of free time. That is, the free-time period is shortened for each unruly act. For a sample of n=4 children, the number of outburst is measured for a day before treatment is initiated and again one week, one month, and six months after the response-cost technique began (Note that the measurements taken after the response-cost technique is administered serve as a long term follow-up on the effectiveness of the treatment).
Logic of Repeated-Measures ANOVA
Subjects Before treatment
One week later
One month later
Six months later
A
B
C
D
8
4
6
8
2
1
2
3
1
1
0
4
1
0
2
1
Distribution of F-ratios
Table B.4 The F-Distribution
Post hoc tests
In ANOVA, when you reject the null hypothesis, you conclude that the means are not all the same.
Post hoc test are done after ANOVA when 1. You reject the null hypothesis AND, 2. There are 3 or more treatments
Post hoc tests
Method: Tukey’s Honestly Significant (HSD) Test
The value of q is found in Table B.5n is the number of scores in each treatment.Use df error instead of df within (Table)
HSD allows you to compute a single value that determines the minimum difference between treatment means that is necessary for significance.
errorMSHSD q
n
Advantage & Disadvantage
If there is large variability due to individual differences, treatment effect might be masked by this variability.
Repeated measures ANOVA eliminates this drawback.
Carry-over effects & Progressive errors…
Assumptions
1. The observations within each sample must be independent.
2. The population distribution within each treatment must be normal.
3. The variances of the population distributions for each treatment should be equivalent.
Example
The following data were obtained to compare three experimental treatments.
a. If these data were obtained from an independent measures design, then could youconclude that there is a significant differenceamong the treatment conditions?
b. If these data were obtained from an repeated measures design so that each row scoresrepresents data from a single subject, then could youconclude that there is a significant difference among the treatment conditions?
Treatments
1 2 3
0
2
1
5
2
1
5
2
4
8
2
5
6
9
8