Announcements this week: factorial ANOVAs (ch 14), correlation (ch 15) HW 5 due today, HW 5-R due Thurs 11/6 Quiz 6 ﬁll-in question regraded today/ tomorrow (Quiz 5 almost done) Prelim 2 next week: Wed., Nov 12 focus on t-tests, one-way ANOVAs (both types), two-way ANOVAs (this week), some estimation and signal detection 1 1 Monday, November 3, 14
34

# d28 11-3-14 Factorial ANOVA 1

Sep 28, 2015

## Documents

Jeanette Si

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#### variablesthe variables

Welcome message from author
Transcript
• Announcements this week: factorial ANOVAs (ch 14), correlation (ch

15)

HW 5 due today, HW 5-R due Thurs 11/6 Quiz 6 fill-in question regraded today/

tomorrow (Quiz 5 almost done)

Prelim 2 next week: Wed., Nov 12 focus on t-tests, one-way ANOVAs (both types),

two-way ANOVAs (this week), some estimation and signal detection

1

1Monday, November 3, 14

• Test multiple independent variables (factors) at multiple levels

Factor 1: Dosage (2 levels: 1 mg, 3 mg)Factor 2: Age (2 levels: students and faculty)

2

What is a factorial design?

What is the effect of caffeine on reaction time?

Caffeine might influence people differently (age, lifetime exposure): we can test dosage in combination

with age

2Monday, November 3, 14

• Test multiple independent variables (factors) at multiple levels

Factor 1: Dosage (2 levels: 1 mg, 3 mg)Factor 2: Age (2 levels: students and faculty)

3

What is a factorial design?

0

15

30

45

60

75

90

1 mg 3 mg

Effect of caffeine and age on reaction timestudentsfaculty

Dosage

Rea

ctio

n tim

e

3Monday, November 3, 14

• can look for relationships between the variables

The variables interact:As caffeine dosage increases, reaction time decreases more for faculty than for students

4

Dosage

Rea

ctio

n tim

e

0

15

30

45

60

75

90

1 mg 3 mg

Effect of caffeine and age on reaction timestudentsfaculty

4Monday, November 3, 14

• Sampling from how many populations?

Dosage

Age

Student Faculty

Low

High

Factorial designs

5

5Monday, November 3, 14

• Sampling from how many populations?

Student Faculty

Dosage

Age

Student Faculty

Factorial designs

6

6Monday, November 3, 14

• Sampling from how many populations?

Low High

Dosage

Age

Low

High

Factorial designs

7

7Monday, November 3, 14

• 0.7 0.6

0.8 0.2Dosage

Age

Student Faculty

Low

High

Multiple populations?

8

To see if there is an effect of just one variable, collapse (average) across the other variable

8Monday, November 3, 14

• Main effect of Dosage? Collapse across Age

0.70 0.60

0.80 0.20

.65

.5Dosage

Age

Student Faculty

Low

High

Main effects

9

9Monday, November 3, 14

• Main effect of Age? Collapse across Dosage

0.70 0.60

0.80 0.20

.75 .4

Dosage

Age

Student Faculty

Low

High

Main effects

10

10Monday, November 3, 14

• Identifying relationships among variables Interaction: when the effect of one variable

depends on the level of another variable

Does the relationship between the reaction times observed in high and low caffeine dosages depend on age?

Interactions

11

11Monday, November 3, 14

• Interaction?

12

Relationship between variables:Moving from 1mg to 3mg, the student reaction time increases slightly, but faculty reaction time decreases.

0

15

30

45

60

75

90

1 mg 3 mg

Effect of caffeine and age on reaction timestudentsfaculty

Dosage

Rea

ctio

n tim

e

12Monday, November 3, 14

• No Can have an interaction without main effects Can also have main effects without an

interaction

Are main effects a prerequisite for an interaction?

13

13Monday, November 3, 14

• 2882

Dosage

AgeStudent Faculty

LowHigh

Interaction without main effects

5

55 5

14

Relationship between variables:Moving from low to high, reaction time for students increases, while faculty reaction time decreases.

0

2

4

6

8

low high

studentsfaculty

Dosage

Rea

ctio

n tim

e

14Monday, November 3, 14

• 10662

Dosage

AgeStudent Faculty

Low

High

Main effects without interaction

4

84 8

15

Relationship between variables:Moving from 1 mg to 3mg, reaction time for students and faculty increases by the same amount.

Dosage

Rea

ctio

n tim

e

0

2

4

6

8

10

low high

studentsfaculty

15Monday, November 3, 14

• ANOVA answers two questions: Do the different levels of a factor represent

real differences in the dependent variable?

Is there an interaction between the factors?

ANOVA: a statistical test of main effects and interactions

16

16Monday, November 3, 14

• Sum of squares Sum of the variances from the grand mean for each level of a factor Degrees of freedom Number of independent observations Mean square Mean deviation from the grand mean of each observation in a factor Error Tendency for scores to vary from the overall mean

Essential parts of an ANOVA

17

17Monday, November 3, 14

• Variance (within a factor)

errorF-ratio =

Main effects

18

18Monday, November 3, 14

• F-ratio: If between-group differences equal within-group

differences: H0 true

If between-group differences are larger than within-group differences: H0 false

Main effects: Hypotheses

19

19Monday, November 3, 14

• All the variance must be accounted for Any variance not due to the main effects or

error is due to an interaction of the factors

Interactions: the variance left over

20

20Monday, November 3, 14

• Example:

Two-Factor ANOVA

70 degrees 80 degrees 90 degrees

30% humidity M = 85 M = 80 M = 75 M = 80

70% humidity M = 75 M = 70 M = 65 M = 70

M = 80 M = 75 M = 70

21

21Monday, November 3, 14

• Two-factor ANOVA will do three things:

- Examine differences in sample means for humidity (factor A)

- Examine differences in sample means for temperature (factor B)

- Examine differences in sample means for combinations of humidity and temperature (factor A and B).

Three F-ratios.

Two-Factor ANOVA

22

22Monday, November 3, 14

• Main effect for humidity (Factor A)Main effect for temperature (Factor B)

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

Main Effects and Interactions

An example: 70 degrees 80 degrees 90 degrees

30% humidity M = 85 M = 80 M = 75 M = 80

70% humidity M = 75 M = 70 M = 65 M = 70

M = 80 M = 75 M = 70

23

23Monday, November 3, 14

• Evaluation of main effects two out of three hypothesis tests in two-factor ANOVA.

Factor A (humidity - 2 levels):Hypotheses:

H0: A1 = A2

H1: A1 A2

Main Effects and Interactions

F = variance between means (Factor A)variance expected by chance/error

24

24Monday, November 3, 14

• Factor B (temperature - 3 levels):

Hypotheses:

H0: B1 = B2 = B3

H1: At least one is different.

F-ratio:

Main Effects and Interactions

F = variance between means (Factor B)variance expected by chance/error

25

25Monday, November 3, 14

• All the variance must be accounted for

Any variance not due to the main effects or error is due to an interaction of the factors

Interactions: the variance left over

26

26Monday, November 3, 14

• variance not explained by main effectsvariance expected by chance/error

Interaction Hypotheses

H0: There is no interaction between factors A and B. (all mean differences are explained by main effects)

H1: There is an interaction between factors A and B

Main Effects and Interactions

F = 27

27Monday, November 3, 14

• 28

In a graph, lines that are non-parallel indicate the presence of an interaction between two factors.

Main Effects and Interactions

28Monday, November 3, 14

• 29

Two-factor ANOVA consists of three hypothesis tests. The outcomes of these tests are totally independent.

All combinations of outcomes are possible:

2 main effects and interaction

1 main effect and interaction

2 main effects and no interaction

1 main effect and no interaction

interaction but no main effects

no main effects, no interaction

Main Effects and Interactions

29Monday, November 3, 14

• 30

Two-factor ANOVA hypothesis test

Step 1: State hypotheses

Step 2: Determine critical region (critical F values)

Step 3: Calculate F-ratios

Step 4: Make decision

30Monday, November 3, 14

• 31

Notation and Formulas

Three hypothesis tests three F-ratios four variances.

Schematic view:

31Monday, November 3, 14

• 32

Notation and Formulas

32Monday, November 3, 14

• 33

Stage 1:

Total variability:

SStotal = X2 -G2

N

dftotal = N-1

Notation and Formulas

33Monday, November 3, 14

• 34

Stage 1:

Between-treatments variability:

dfbetween treatments = number of cells -1

SSbetween treatments = T2

nG2

N-

Notation and Formulas

34Monday, November 3, 14

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