What if. . . You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days a person.

What if. . . You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days a person misses class) You would simply do a two-sample t-test two-tailed Easy!

But, what if. . . You were asked to determine if psychology, sociology, and biology majors have significantly different class attendance You would do a one-way ANOVA

But, what if. . . You were asked to determine if psychology majors had significantly different class attendance than sociology and biology majors. You would do an ANOVA with contrast codes

But, what if. . . You were asked to determine the effects of both college major (psychology, sociology, and biology) and gender (male and female) on class attendance You now have 2 IVs and 1 DV You could do two separate analyses Problem: “Throw away” information that could explain some of the “error” Problem: Will not be able to determine if there is an interaction

Factorial Analysis of Variance
Factor = IV Factorial design is when every level of every factor is paired with every level of every other factor Psychology Sociology Biology Male X Female

Currently Different people in each cell Equal n in each cell Psychology Sociology Biology Male X Female

2 X 3 Factorial “2” is because one IV has 2 levels (male and female) “3” because one IV has 3 levels (psychology, biology, sociology) Psychology Sociology Biology Male X Female

2 X 3 4 X 5 2 X 2 X 7

Notation One factor is A and the other is B Psychology Sociology
Biology Male Female

Notation One factor is A and the other is B B B1 B2 B3 A1 A2 A

Notation One factor is A and the other is B
Any combination of A and B is called a cell B B1 B2 B3 A1 A2 A

Any combination of A and B is called a cell The number of subjects in each cell = n B B1 B2 B3 A1 A2 A

Any combination of A and B is called a cell The number of subjects in each cell = n Each subject in a cell = Xij i = level of A; j = level of B B B1 B2 B3 A1 A2 A

Any combination of A and B is called a cell The number of subjects in each cell = n Each subject in a cell = Xijk i = level of A; j = level of B; k = observation in that cell B B1 B2 B3 A1 X111,X112 X113 X121,X122 X123 X131,X132 X133 A2 X211,X212 X213 X221,X222 X223 X231,X232 X233 A

Notation n = 3; N = nab = 18 The means for each of the cells are: B1
X111,X112 X113 X121,X122 X123 X131,X132 X133 A2 X211,X212 X213 X221,X222 X223 X231,X232 X233 A

Notation n = 3; N = nab = 18 The means for each of the cells are: B1
X11 X12 X13 A2 X21 X22 X23 A

Notation n = 3; N = nab = 18 The means for row (level of A): B1 B2 B3
X11 X12 X13 X1. A2 X21 X22 X23 X2.

Notation n = 3; N = nab = 18 The means for column (level of B): B1 B2
X11 X12 X13 X1. A2 X21 X22 X23 X2. X.1 X.2 X.3

Notation n = 3; N = nab = 18 The grand mean: B1 B2 B3 A1 X11 X12 X13

Sociology Psychology Biology Female 2.00 1.00 3.00 .00 Males 4.00
n = 3 N = 18

Sociology Psychology Biology Female 2.00 1.00 3.00 .00 Mean1j 2.67 1.67 Males 4.00 Mean2j 3.67 0.33

Sociology Psychology Biology Female 2.00 1.00 3.00 .00 Mean1j 2.67 1.67 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33

Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j 2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33

Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j 2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06

Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 Main effect of gender

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 Main effect of major

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 Interaction between gender and major

Sum of Squares SS Total Computed the same way as before
The total deviation in the observed scores Computed the same way as before

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SStotal = (2-2.06)2+ (3-2.06) (1-2.06)2 = 30.94 *What makes this value get larger?

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SStotal = (2-2.06)2+ (3-2.06) (1-2.06)2 = 30.94 *What makes this value get larger? *The variability of the scores!

Sum of Squares SS A Represents the SS deviations of the treatment means around the grand mean Its multiplied by nb to give an estimate of the population variance (Central limit theorem) Same formula as SSbetween in the one-way

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSA = (3*3) (( )2+ ( )2)=1.36 *Note: it is multiplied by nb because that is the number of scores each mean is based on

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSA = (3*3) (( )2+ ( )2)=1.36 *What makes these means differ? *Error and the effect of A

Sum of Squares SS B Represents the SS deviations of the treatment means around the grand mean Its multiplied by na to give an estimate of the population variance (Central limit theorem) Same formula as SSbetween in the one-way

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSB = (3*2) (( )2+ ( )2+ ( )2)= 14.16 *Note: it is multiplied by na because that is the number of scores each mean is based on

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSB = (3*2) (( )2+ ( )2+ ( )2)= 14.16 *What makes these means differ? *Error and the effect of B

Sum of Squares SS Cells Represents the SS deviations of the cell means around the grand mean Its multiplied by n to give an estimate of the population variance (Central limit theorem)

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSCells = (3) (( )2+ ( ) ( )2)= 24.35

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSCells = (3) (( )2+ ( ) ( )2)= 24.35 What makes the cell means differ?

Sum of Squares SS Cells What makes the cell means differ? 1) error
2) the effect of A (gender) 3) the effect of B (major) 4) an interaction between A and B

Sum of Squares Have a measure of how much cells differ
SScells Have a measure of how much this difference is due to A SSA Have a measure of how much this difference is due to B SSB What is left in SScells must be due to error and the interaction between A and B

Sum of Squares SSAB = SScells - SSA – SSB 8.83 =

Sum of Squares SSWithin SSWithin = SSTotal – (SSA + SSB + SSAB)
The total deviation in the scores not caused by 1) the main effect of A 2) the main effect of B 3) the interaction of A and B SSWithin = SSTotal – (SSA + SSB + SSAB) 6.59 = – ( )

Sum of Squares SSWithin

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSWithin = ((2-2.67)2+(3-2.67)2+(3-2.67)2) ((1-.33)2 + (0-.33)2 + ( )2 = 6.667

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSWithin = ((2-2.67)2+(3-2.67)2+(3-2.67)2) ((1-.33)2 + (0-.33)2 + ( )2 = 6.667 *What makes these values differ from the cell means? *Error

Compute df Source df SS A 1.36 B 14.16 AB 8.83 Within 6.59 Total 30.94

Source df SS A 1.36 B 14.16 AB 8.83 Within 6.59 Total 17 30.94
dftotal = N - 1

Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 6.59 Total 17 30.94
dftotal = N – 1 dfA = a – 1 dfB = b - 1

Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 6.59 Total 17 30.94
dftotal = N – 1 dfA = a – 1 dfB = b – 1 dfAB = dfa * dfb

Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 12 6.59 Total 17 30.94
dftotal = N – 1 dfA = a – 1 dfB = b – 1 dfAB = dfa * dfb dfwithin= ab(n – 1)

Compute MS Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 12 6.59
Total 17 30.94

Compute MS Source df SS MS A 1 1.36 B 2 14.16 7.08 AB 8.83 4.42 Within
12 6.59 .55 Total 17 30.94

Compute F Source df SS MS A 1 1.36 B 2 14.16 7.08 AB 8.83 4.42 Within
12 6.59 .55 Total 17 30.94

Test each F value for significance
Source df SS MS F A 1 1.36 2.47 B 2 14.16 7.08 12.87 AB 8.83 4.42 8.03 Within 12 6.59 .55 Total 17 30.94 F critical values (may be different for each F test) Use df for that factor and the df within.

Source df SS MS F A 1 1.36 2.47 B 2 14.16 7.08 12.87 AB 8.83 4.42 8.03 Within 12 6.59 .55 Total 17 30.94 F critical A (1, 12) = 4.75 F critical B (2, 12) = 3.89 F critical AB (2, 12) = 3.89

Source df SS MS F A 1 1.36 2.47 B 2 14.16 7.08 12.87* AB 8.83 4.42 8.03* Within 12 6.59 .55 Total 17 30.94 F critical A (1, 12) = 4.75 F critical B (2, 12) = 3.89 F critical AB (2, 12) = 3.89

In SPSS

In SPSS Main Effects Easy – just like a one-way ANOVA

Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j 2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06

In SPSS Interaction Does the effect of one IV on the DV depend on the level of another IV?

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 Want to plot out the cell means

Sociology Psychology Biology

Practice 2 x 2 Factorial Determine if 1) there is a main effect of A
2) there is a main effect of B 3) if there is an interaction between AB

Practice A: NO B: NO AB: NO

Practice A: YES B: NO AB: NO

Practice A: NO B: YES AB: NO

Practice A: YES B: YES AB: NO

Practice A: YES B: YES AB: YES

Practice A: YES B: NO AB: YES

Practice A: NO B: YES AB: YES

Practice A: NO B: NO AB: YES

Practice These are sample data from Diener et. al (1999). Participants were asked their marital status and how often they engaged in religious behavior. They also indicated how happy they were on a scale of 1 to 10. Examine the data

Frequency of religious behavior
Never Occasionally Often Married 6 3 7 2 8 4 5 9 Unmarried 1

Remember You were asked to determine the effects of both college major (psychology, sociology, and biology) and gender (male and female) on class attendance

Simple Effects The effect of one factor at one level on the other factor. Use to “tease apart” a significant interaction You wonder if college major has an effect on class attendance for just females? You wonder if male and female psychology students differ in their class attendance?

Simple Effects Why not just do 5 new analyses You will lose df!
2 ANOVAs 3 Independent Samples t-tests You will lose df! Type I error problem still exists

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 Are these means different?

Simple Effects SS Major at Female
Represents the SS deviations of the treatment means around the Female mean Its multiplied by n to give an estimate of the population variance (Central limit theorem) Same formula as SSbetween in the one-way

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SS M at Female = 3*((2.67 – 1.78)2 + (1.00 – 1.78)2 + (1.67 – 1.78)2 = 4.23

Simple Effects SS Major at Male
Same formula as SSbetween in the one-way

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SS M at Male = 3*((3.67 – 2.33)2 + (3.00 – 2.33)2 + (0.33 – 2.33)2 = 18.71

Simple Effects SS Gender at Socio

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SS G at Socio = 3*((2.67 – 3.17)2 + (3.67 – 3.17)2 = 1.5

Simple Effects SS Gender at Psych

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SS G at Psych = 3*((1.00 – 2.00)2 + (3.00 – 2.00)2 = 6.00

Simple Effects SS Gender at Bio

2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SS G at Bio = 3*((1.67 – 1.00)2 + (0.33 – 1.00)2 = 2.69

Overall ANOVA Source df SS MS F Gender 1 1.36 2.47 Major 2 14.16 7.08
12.87* G x M 8.83 4.42 8.03* Within 12 6.59 .55 Total 17 30.94

Simple Effects Source df SS MS F Gender G at Socio G at Psych G at Bio
Major M at Female M at Male Within 12 6.59 .55

1.50 6.00 2.69 Major M at Female M at Male 4.23 18.71 Within 12 6.59 .55

dfA (Gender) = a – 1 dfB (Major) = b – 1 Simple Effects Source df SS MS F Gender G at Socio G at Psych G at Bio 1 1.50 6.00 2.69 Major M at Female M at Male 2 4.23 18.71 Within 12 6.59 .55

Compute MS Simple Effects Source df SS MS F Gender G at Socio G at Psych G at Bio 1 1.50 6.00 2.69 Major M at Female M at Male 2 4.23 18.71 2.12 9.36 Within 12 6.59 .55

Compute F Simple Effects Source df SS MS F Gender G at Socio G at Psych G at Bio 1 1.50 6.00 2.69 2.72 10.91 4.89 Major M at Female M at Male 2 4.23 18.71 2.12 9.36 3.85 17.02 Within 12 6.59 .55

F critical (1, 12) = 4.75 F critical B (2, 12) = 3.89 Simple Effects Source df SS MS F Gender G at Socio G at Psych G at Bio 1 1.50 6.00 2.69 2.72 10.91* 4.89* Major M at Female M at Male 2 4.23 18.71 2.12 9.36 3.85 17.02* Within 12 6.59 .55

Practice ANOVA is from exercise 13.5 1) Interpret the ANOVA results
2) Plot the cell means 3) Compute an entire set of simple effects

Source df SS MS F Delay D at Neutral D at Area A D at Area B 2 1.20 254.80 304.53 0.60 127.40 152.26 .02 4.34* 5.19* Area A at D 50 A at D 100 A at D 150 357.74 363.33 6.93 178.87 181.67 3.47 6.10* 6.19* .12 Within 36 29.31

What if. . . You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days a person.

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What if. . . You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days a person.

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Presentation on theme: "What if. . . You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days a person."— Presentation transcript:

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