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SPSS Problem # 7
• Page 467 – 13.5
• Page 416– 12.2
Cookbook due WednesdayMay 4th!!
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 X X
Female X X X
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
Main effect of gender
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
Main effect of major
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
Interaction between gender and major
Sum of Squares
• SS Total
– The total deviation in the observed scores
• Computed the same way as before
2..)( XXSSTotal
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06SStotal = (2-2.06)2+ (3-2.06)2+ . . . . (1-2.06)2 = 30.94
*What makes this value get larger?
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06SStotal = (2-2.06)2+ (3-2.06)2+ . . . . (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. ..)( XXnbSS iA
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06SSA = (3*3) ((1.78-2.06)2+ (2.33-2.06)2)=1.36
*Note: it is multiplied by nb because that is the number of scores each mean is based on
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06SSA = (3*3) ((1.78-2.06)2+ (2.33-2.06)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. ..)( XXnaSS jB
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06SSB = (3*2) ((3.17-2.06)2+ (2.00-2.06)2+ (1.00-2.06)2)= 14.16
*Note: it is multiplied by na because that is the number of scores each mean is based on
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06SSB = (3*2) ((3.17-2.06)2+ (2.00-2.06)2+ (1.00-2.06)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..)( XXnSS ijCells
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06SSCells = (3) ((2.67-2.06)2+ (1.00-2.06)2+. . . + (0.33-2.06)2)= 24.35
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06SSCells = (3) ((2.67-2.06)2+ (1.00-2.06)2+. . . + (0.33-2.06)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 = 24.35 - 14.16 - 1.36
Sum of Squares
• SSWithin
• 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 = 30.94 – (14.16 +1.36 + 8.83)
Sum of Squares
• SSWithin
2)( ijWithin XXSS
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
SSWithin = ((2-2.67)2+(3-2.67)2+(3-2.67)2) + . .. + ((1-.33)2 + (0-.33)2 + (0-2..33)2 = 6.667
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
SSWithin = ((2-2.67)2+(3-2.67)2+(3-2.67)2) + . .. + ((1-.33)2 + (0-.33)2 + (0-2..33)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 2 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 2 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 2 8.83
Within 12 6.59
Total 17 30.94
Compute MS
Source df SS MS
A 1 1.36 1.36
B 2 14.16 7.08
AB 2 8.83 4.42
Within 12 6.59 .55
Total 17 30.94
What does each MS mean?
Source df SS MS
A 1 1.36 1.36
B 2 14.16 7.08
AB 2 8.83 4.42
Within 12 6.59 .55
Total 17 30.94
22)( naMSE eB
22)( nbMSE eA 2)( eBMSE
22)( nMSE eA
Compute F
Source df SS MS
A 1 1.36 1.36
B 2 14.16 7.08
AB 2 8.83 4.42
Within 12 6.59 .55
Total 17 30.94
Source df SS MS F
A 1 1.36 1.36 2.47
B 2 14.16 7.08 12.87
AB 2 8.83 4.42 8.03
Within 12 6.59 .55
Total 17 30.94
2
22
e
e nb
Compute F
2
22
e
e na
2
22
e
e n
Test each F value for significance
Source df SS MS F
A 1 1.36 1.36 2.47
B 2 14.16 7.08 12.87
AB 2 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.
Test each F value for significance
Source df SS MS F
A 1 1.36 1.36 2.47
B 2 14.16 7.08 12.87
AB 2 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
Test each F value for significance
Source df SS MS F
A 1 1.36 1.36 2.47
B 2 14.16 7.08 12.87*
AB 2 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
15.500 3 5.167 9.300 .002
1.389 1 1.389 2.500 .140
14.111 2 7.056 12.700 .001
8.778 2 4.389 7.900 .006
24.278 5 4.856 8.740 .001
6.667 12 .556
30.944 17 1.820
(Combined)
GENDER
MAJOR
Main Effects
GENDER *MAJOR
2-Way Interactions
Model
Residual
Total
DAYS
Sum ofSquares df
MeanSquare F Sig.
Unique Method
ANOVAa,b
DAYS by GENDER, MAJORa.
All effects entered simultaneouslyb.
Interpreting the Results
• Main Effects
• Easy – just like a one-way ANOVA
15.500 3 5.167 9.300 .002
1.389 1 1.389 2.500 .140
14.111 2 7.056 12.700 .001
8.778 2 4.389 7.900 .006
24.278 5 4.856 8.740 .001
6.667 12 .556
30.944 17 1.820
(Combined)
GENDER
MAJOR
Main Effects
GENDER *MAJOR
2-Way Interactions
Model
Residual
Total
DAYS
Sum ofSquares df
MeanSquare F Sig.
Unique Method
ANOVAa,b
DAYS by GENDER, MAJORa.
All effects entered simultaneouslyb.
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
Interpreting the Results
• Interaction– Does the effect of one IV on the DV depend on the
level of another IV?
15.500 3 5.167 9.300 .002
1.389 1 1.389 2.500 .140
14.111 2 7.056 12.700 .001
8.778 2 4.389 7.900 .006
24.278 5 4.856 8.740 .001
6.667 12 .556
30.944 17 1.820
(Combined)
GENDER
MAJOR
Main Effects
GENDER *MAJOR
2-Way Interactions
Model
Residual
Total
DAYS
Sum ofSquares df
MeanSquare F Sig.
Unique Method
ANOVAa,b
DAYS by GENDER, MAJORa.
All effects entered simultaneouslyb.
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06Want to plot out the cell means
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
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
0123456789
10
B1 B2
A1
A2
A: NO
B: NO
AB: NO
Practice
0123456789
10
B1 B2
A1
A2
A: YES
B: NO
AB: NO
Practice
0123456789
10
B1 B2
A1
A2
A: NO
B: YES
AB: NO
Practice
0123456789
10
B1 B2
A1
A2
A: YES
B: YES
AB: NO
Practice
0123456789
10
B1 B2
A1
A2
A: YES
B: YES
AB: YES
Practice
0123456789
10
B1 B2
A1
A2
A: YES
B: NO
AB: YES
Practice
0123456789
10
B1 B2
A1
A2
A: NO
B: YES
AB: YES
Practice
0123456789
10
B1 B2
A1
A2
A: NO
B: NO
AB: YES
Practice
• Page 467
– 13.5– 13.6
544.622 4 136.156 4.645 .004
188.578 2 94.289 3.217 .052
356.044 2 178.022 6.074 .005
371.956 4 92.989 3.172 .025
916.578 8 114.572 3.909 .002
1055.200 36 29.311
1971.778 44 44.813
(Combined)
DELAY
AREA
Main Effects
DELAY *AREA
2-Way Interactions
Model
Residual
Total
TIME
Sum ofSquares df
MeanSquare F Sig.
Unique Method
ANOVAa,b
TIME by DELAY, AREAa.
All effects entered simultaneouslyb.
Neutral Area A Area B Mean50 28.6 16.8 24.4 23.27
100 28 23 16 22.33150 28 26.8 26.4 27.07
Mean 28.2 22.2 22.27 24.22
544.622 4 136.156 4.645 .004
188.578 2 94.289 3.217 .052
356.044 2 178.022 6.074 .005
371.956 4 92.989 3.172 .025
916.578 8 114.572 3.909 .002
1055.200 36 29.311
1971.778 44 44.813
(Combined)
DELAY
AREA
Main Effects
DELAY *AREA
2-Way Interactions
Model
Residual
Total
TIME
Sum ofSquares df
MeanSquare F Sig.
Unique Method
ANOVAa,b
TIME by DELAY, AREAa.
All effects entered simultaneouslyb.
0
5
10
15
20
25
30
35
50 100 150
Neutral
Area A
Area B
0
5
10
15
20
25
30
35
Neutral Area A Area B
50
100
150
Why is this important?
• Requirement
• Understand research articles
• Do research for yourself
• Real world
The Three Goals of this Course
• 1) Teach a new way of thinking
• 2) Teach “factoids”
Mean
r =
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22
e
e nb
2)( ijWithin XXSS
YX
XY
SS
COVr
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)1(2
2
Rp
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aa
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1e b c a
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What you have learned!
• Chapter 1 – Introduced to statistics and learned key words – Scales of measurement– Populations vs. Samples
• Learned how to organize scores of one variable using:
– frequency distributions– graphs
What you have learned!
• Chapter 2 – Learned ways to describing data
• Measures of central tendency– Mean– Median– Mode
• Variability– Range– IQR– Standard Deviation– Variance
What you have learned!
• Chapter 3 – Learned about issues related to the normal curve:
– Z Scores
– Find the percentile of a give score– Find the score for a given percentile
What you have learned!
• Chapter 4 – Logic of hypothesis testing
– Is this quarter fair?– Sampling distribution
• CLT
– The probability of a given score occuring
What you have learned!
• Chapter 5 – Basic issues related to probability
– Joint probabilities– Conditional probabilities
– Different ways events can occur• Permutations• Combinations
– The probability of winning the lottery
– Binomial Distributions• Probability of winning the next 4 out of 10 games of Blingoo
What you have learned!
• Chapter 6 – Ways to analyze categorical data
– Chi square as a measure of independence• Phi coefficient
– Chi square as a measure of goodness of fit
What you have learned!
• Chapter 9 – Ways to analyze two continuous variables
– Correlation
– Regression
What you have learned!
• Chapter 10 – Other methods for correlations
– Pearson Formulas• Point-Biserial• Phi Coefficent• Spearman’s rho
– Non-Pearson Formulas• Kendall’s Tau
What you have learned!
• Chapter 15 – How to analyze continuous data with two or more IVs
– Multiple Regression• Causal Models• Standardized vs. unstandarized • Multiple R• Semipartical correlations
– Common applications• Mediator Models• Moderator Mordels
What you have learned!
• Chapter 7 – Significance testing applied to means
– One Sample t-tests
– Two Sample t-tests• Equal N• Unequal N• Dependent samples
What you have learned!
• Chapter 11 – Significance testing applied to two or more means
– ANOVA
– Computation of ANOVA
– Logic of ANOVA• Variance• Expected Mean Square• Sum of Squares
What you have learned!
• Chapter 12 – Extending ANOVA
– What to do with an omnibus ANOVA• Multiple t-tests• Linear Contrasts• Orthogonal Contrasts• Trend Analysis
– Controlling for Type I errors• Bonferroni t• Fisher Least Significance Difference• Studentized Range Statistic• Dunnett’s Test
What you have learned!
• Chapter 13 – How to analyze catagorical data with two or more IVs
– Factorial ANOVA
– Computation and logic of Factorial ANOVA
– Interpreting Results• Main Effects• Interactions
The Three Goals of this Course
• 1) Teach a new way of thinking
• 2) Teach “factoids”
• 3) Self-confidence in statistics