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