PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH

Lesson 13Two-factor Analysis of Variance

(Independent Measures) 

Two-factor ANOVA

ANOVA is a hypothesis-testing procedure that is used to evaluate mean differences between two or more treatments (or populations)

A research study with two independent variables: The effects of two different teaching methods and three different class size are evaluated. DV: math achievement test score

IV 1: Class Size

Small class

Medium Class

Large Class

IV 2: Teaching Methods

Method A Sample 1 Sample 2 Sample 3

Method B Sample 4 Sample 5 Sample 6

Two-factor ANOVA

A two-by-three factorial design

2X3= 6 different treatments

IV 1: Class Size

Small class

Medium Class

Large Class

IV 2: Teaching Methods

Method A Sample 1 Sample 2 Sample 3

Method B Sample 4 Sample 5 Sample 6

levels

levels

Two-factor ANOVA

Two factor ANOVA will allow researcher to test for mean differences in the experiment:

1. Mean difference between teaching methods.2. Mean differences between the three class sizes.3. Any other mean differences that may result for unique combinations of a specific

teaching method and a specific class size.

Two-factor ANOVA

Main effect:The mean differences among the levels of one factor are referred to as main effect of that factor.

Main effect for class size (factor B)

Main effect for methods (factor A)

0 1 2 3

1

: ..

:B B BH

H

0 1 2

1

:

:A AH

H

Two-factor ANOVA

variance (differences) between the means for factor A

variance (differences) expected from sampling errorF

variance (differences) between the means for factor B

variance (differences) expected from sampling errorF

Two-factor ANOVA

Interaction effect:There is an interaction between factors if the effect of one factor depends on the levels of the second

factor. The interaction is identified as the AXB interaction.

Two-factor ANOVA

0

20

40

60

80

100

1 2 3

Class Size

Mea

n M

ath

Sco

res

Series1

Series2

0102030405060708090

1 2 3

Class Size

Mea

n M

ath

Tes

t S

core

s

Series1

Series2

Two-factor ANOVA

variance (differences) not explained by main effects

variance (differences) expected from sampling errorF

There is no interaction between factors A and B. The effect of factor A does not depend on the levels of factor B (and B does not depend on A)0 :H

Two-factor ANOVA

It is composed of three distinct hypothesis tests:

1. The main effect of factor A (The A-effect)2. The main effect of factor B (The B-effect)3. The interaction (AXB interaction)

Two-factor ANOVA

Total variability

Between-treatments variability Within-treatments variability

1.Treatment effects2. Individual differences

3. Experimental error

1. Individual differences2. Experimental error

Two-factor ANOVA

variance between treatments

variance within treatmentsF

1. Treatment effect (factor A, factor B and AXB)

2. Individual differences (there are different subjects for

each trwatment condition)3. Experimental error

Two-factor ANOVA

df totalN-1

df between ab-1

df withinN-ab

Factor Adf=a-1

Factor Bdf=b-1

Interactiondf=df of A X df of B

Two-factor ANOVA

SS total

SS between SS within

within

betweenMean squared deviation (MS) =Variance between treatments = MS = ,

between

withinVariance within treatments =MS =

within

Variance betwen treatments

Variance within treatm

between

SS

df

SS

df

F ents

between

within

MS

MS

Distribution of F-ratios

Table B.4 The F-Distribution

Example (Do these data indicate that the size of the class and /or programs has a significant effect on test performance?)

Class size

18 Students 24 students 30 students

programs Program 1 5

3

3

8

6

9

9

13

6

8

3

8

3

3

3

Program 2 0

2

0

0

3

0

0

0

5

0

0

3

7

5

5

Assumptions

1. The observations within each sample must be independent.

2. The populations from which the samples are selected must be normal.

3. The populations from which the samples are selected must have equal variances.

Example

In 1968, Schachter published an article in Science reporting a series of experience on obesity and eating behavior. One of these studies examined the hypothesis. One of these studies examined the hypothesis that these individuals do not respond to internal , biological signals of hunger. In simple terms, this hypothesis says that obese individuals tend to eat whether or not their bodies are actually hungery.

In Shachter’s study, subjects were led to believe that they were taking part in a “taste test.” All subjects were told to come to the experiment wthout eating for several hours beforehand.

Example

The study used two indepedent variables or factors:

1. Weights (obese versus normal subjects)

2. Full stomach versus empty stomach

All subjects were then invited to taste and rate five different types crackers. The dependent variables was the number of crackers eaten by each subject.

Example

Factor B: Fullness

Empty Stomach Full Stomach

Factor A: Weight

Normal n=20

Mean=22

Cell Sum =440

SS=1540

n=20

Mean=15

Cell Sum =300

SS=1270

Obese n=20

Mean=17

Cell Sum =340

SS=1320

n=20

Mean=18

Cell Sum =360

SS=1266