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SPSS Problem 7

• Page 467
• 13.5
• Page 416
• 12.2

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Cookbook due WednesdayMay 4th!!
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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 misses class)
• You would simply do a two-sample t-test
• two-tailed
• Easy!

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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

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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

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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

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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
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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
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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
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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
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Sum of Squares

• SS Total
• The total deviation in the observed scores
• Computed the same way as before

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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
SStotal (2-2.06)2 (3-2.06)2 . . . . (1-2.06)2
30.94 What makes this value get larger?
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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
SStotal (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!
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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

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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
SSA (33) ((1.78-2.06)2 (2.33-2.06)2)1.36 No
te it is multiplied by nb because that is the
number of scores each mean is based on
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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
SSA (33) ((1.78-2.06)2 (2.33-2.06)2)1.36 Wh
at makes these means differ? Error and the
effect of A
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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

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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
SSB (32) ((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
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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
SSB (32) ((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
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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)

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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
SSCells (3) ((2.67-2.06)2 (1.00-2.06)2. . .
(0.33-2.06)2) 24.35
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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
SSCells (3) ((2.67-2.06)2 (1.00-2.06)2. . .
(0.33-2.06)2) 24.35 What makes the cell means
differ?
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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

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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

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Sum of Squares

• SSAB SScells - SSA SSB
• 8.83 24.35 - 14.16 - 1.36

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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)

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Sum of Squares

• SSWithin

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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
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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
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Compute df
Source df SS
A 1.36
B 14.16
AB 8.83
Within 6.59
Total 30.94
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Source df SS
A 1.36
B 14.16
AB 8.83
Within 6.59
Total 17 30.94
dftotal N - 1
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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
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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
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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)
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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
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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
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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
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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
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Compute F
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
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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.
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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
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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
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Interpreting the Results

• Main Effects
• Easy just like a one-way ANOVA

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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
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Interpreting the Results

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

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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
Want to plot out the cell means
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Sociology Psychology Biology
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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

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Practice
A NO B NO AB NO
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Practice
A YES B NO AB NO
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Practice
A NO B YES AB NO
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Practice
A YES B YES AB NO
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Practice
A YES B YES AB YES
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Practice
A YES B NO AB YES
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Practice
A NO B YES AB YES
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Practice
A NO B NO AB YES
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Practice

• Page 467
• 13.5
• 13.6

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Why is this important?

• Requirement
• Understand research articles
• Do research for yourself
• Real world

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The Three Goals of this Course

• 1) Teach a new way of thinking
• 2) Teach factoids

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Mean









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r
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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

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What you have learned!

• Chapter 2 Learned ways to describing data
• Measures of central tendency
• Mean
• Median
• Mode
• Variability
• Range
• IQR
• Standard Deviation
• Variance

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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

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What you have learned!

• Chapter 4 Logic of hypothesis testing
• Is this quarter fair?
• Sampling distribution
• CLT
• The probability of a given score occuring

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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

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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

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What you have learned!

• Chapter 9 Ways to analyze two continuous
  variables
• Correlation
• Regression

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What you have learned!

• Chapter 10 Other methods for correlations
• Pearson Formulas
• Point-Biserial
• Phi Coefficent
• Spearmans rho
• Non-Pearson Formulas
• Kendalls Tau

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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

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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

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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

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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
• Dunnetts Test

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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

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The Three Goals of this Course

• 1) Teach a new way of thinking
• 2) Teach factoids
• 3) Self-confidence in statistics

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