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# ANOVA N-Way Factorial

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```									   Factorial Analysis of
Variance

46-511
Between Groups Fixed Effects Designs

1
Two-Way ANOVA Example:
(Yerkes – Dodson Law)
Factor B: Arousal
Low               Medium   High

3                   5      9
1                   5      9
1                   9      13
Easy
6                   7      6
4                   7      8

0                   3      0
2                   8      0
0                   3      0
Difficult
0                   3      5
3                   3      0

2
Partitioning Variance
Factor B: Arousal
Low          Medium            High

3              5                  9
Variation
…              …                  …
among
4              7                  8
Easy                                                         means on A
represent
0              3                  0             effect of A
…              …                  …
3              3                  0
Difficult

Variation among
people treated the
Variation among means on B
same = error
represents effect of B

Leftover variation = interaction                            3
Partitioning Variance: Interaction
Factor B: Arousal
Low        Medium           High         Total

Easy           3.00         6.00           9.00         6.00

Difficult      1.00         4.00           1.00         2.00

Total          2.00         3.00           5.00         4.00

Dependence of means on levels of both A & B represents the
effect of an interaction.                                       4
Or Graphically…
10
9
8
7
6
Easy
5
Difficult
4
3
2
1
0
Low   Medium    High

5
In words
 Types of Effects vs. 1-way
 Main Effect for A
 Main Effect for B
 Interaction (A x B)

 Structural Model: XIJK = μ++++IJK
 Partitioning Variance/Sums of Squares

   First, total variance:     SSTOT  SS BG  SSW
   Between Groups:          SS BG  SS A  SS B  SS AXB
   Thus Total is:   SSTOT  SS A  SS B  SS AXB  SSW

6
Sums of Squares Between
Definitional Formula
___
SS BG  n ( AB ij  G )
Variation of cell
2
means around grand
mean, weighted by n.

Computational Formula
2      2
T    G                Computational formulae:
SS BG                         •More accurate for hand
n   N                calculation
•Easier to work
•Less intuitive

7
Sums of Squares A
Definitional Formula              Variation of row
_   _
SS A  nq  ( Ai  G )
means around grand
2
mean, weighted by n
times the number of
levels of B, or q.
Computational Formula
2
T     G2
SS A       
ROW
nROW N

8
Sums of Squares B
Definitional Formula            Variation of column
_     _
SS B  np  ( B j  G )
means around grand
2
mean, weighted by n
times the number of
levels of A, or p.
Computational Formula
2
TCOL G 2
SS B       
nCOL N

9
Sums of Squares AxB
Definitional Formula
SS AxB  n( ABij  Ai  B j  G)        2

Computational Formula

SS AB  SS Between  SS A  SS B

SSAxB = Variation of cell means around grand mean, that
cannot be accounted for by effects of A or B alone.
10
Sums of Squares Within (Error)
Definitional Formula

SSW  ( X ijk  ABij )        2

Computational Formula
2
T
SSW  X     2
ijk
n

SSW = Variation of individual scores around cell mean.

11
Numerical Example
Effect of Task Difficulty and Anxiety Level on Performance
Low                Medium           High Marginals for B
3                     2             9
1                     5             9
Easy                  1                     9            13 T                  90
6                     7             6 Mean                6
4                     7             8 SS                162
T                   15                     30            45
Mean                  3                     6             9
SS                  18                     28            26

0                     3              0
2                     8              0
Difficult            0                     3              0T                  30
0                     3              5 Mean               2
3                     3              0 SS                78
T                    5                    20              5
Mean                 1                     4              1
SS                   8                    20             20

Marginal for A
T                   20                    50             50
12
Mean                 2                     5              5
SS                  36                    58            206
Degrees of Freedom
 df between = k – 1; or, (kA x kB – 1)

 df A = kA – 1
 df B = kB – 1

 df A x B = dfbetween – dfA – dfB
 dfW = k(n-1)

13
Source Table

Source     SS    df   MS   F
Between
A
B
AxB
Within

Total

14
More Digression on Interactions
 Ways to talk about interactions
   Scores on the DV depend upon levels of both
A and B
   The effect of A is moderated by B
   The effect of B is moderated by A
   There is a multiplicative effect for A and B

15
More Digresions (cont’d)
No effect whatsoever…
No Significant Effects
Interaction Effect: Cell & Marginal Means
B: Anxiety
A: Task Difficulty          Low      Medium     High   Totals
Easy                          4             4      4        4
Hard                          4             4      4        4
Totals                        4             4      4        4

Deviations: cell mean - row mean - column mean + grand mean
Anxiety
Easy                         0          0        0
Hard                         0          0        0

Interaction Sum of Squares:                0
Main Effect for A                          0
Main Effect for B                          0
16
Main effects for A and B…
Only Main Effects Significant
Interaction Effect: Cell & Marginal Means
B: Anxiety
A: Task Difficulty          Low      Medium     High   Totals
Easy                          3             6      9        6
Hard                          1             4      7        4
Totals                        2             5      8        5

Deviations: cell mean - row mean - column mean + grand mean
Anxiety
Easy                         0          0        0
Hard                         0          0        0

Interaction Sum of Squares:                0
Main Effect for A                         30
Main Effect for B                        180

17
Graphically…
2-Way ANOVA Anxiety by Task Difficulty: Main Effects, No Interaction

10

9

8

7

6
Performance

Easy
5
Hard

4

3

2

1

0
Low                        Medium                      High
Anxiety Level

18
Interaction significant also…
Significant Interaction
Interaction Effect: Cell & Marginal Means
B: Anxiety
A: Task Difficulty          Low      Medium     High   Totals
Easy                          3             6      9        6
Hard                          1             4      1        2
Totals                        2             5      5        4

Deviations: cell mean - row mean - column mean + grand mean
Anxiety
Easy                        -1          -1        2
Hard                         1           1       -2

Interaction Sum of Squares:               60
Main Effect for A                        120
Main Effect for B                         60
19
Graphically…
2-Way ANOVA Anxiety by Task Difficulty: Main Effects AND Interaction

10

9

8

7

6
Performance

Easy
5
Hard

4

3

2

1

0
Low                        Medium                      High
Anxiety Level

20
Further Analyses on Main Effects

 Contrasts

 Planned Comparisons

 Post-Hoc Methods

 In the presence of a significant interaction

21
Further Analyses on Interaction
 What it means

 Simple (Main) Effects

   Contrasts

 Partial Interactions

   Contrasts

 Simple Comparisons / Post-Hoc Methods

   How to get q

22
Simple Main Effects Analysis
Low    Medium   High   Total

Easy        3.00    6.00    9.00   6.00

Difficult   1.00    4.00    1.00   2.00

Total       2.00    3.00    5.00   4.00

23
Simple Main Effects

Tij2 at _ rj       T j2
Sum of Squares Formula:   SS J                    
n               nj
MS j
F Ratio:                     Fc 
MSW

df = dfj,dfw:

24
Partial Interaction Analysis
Low    Medium   High   Total

Easy        3.00    6.00    9.00   6.00

Difficult   1.00    4.00    1.00   2.00

Total       2.00    3.00    5.00   4.00

25
In Class Exercise
Drug             B1 (no dose)   B2 (low dose)   B3 (high dose)
A1 (no dose)                5               9                4
7              10                8
5               7                4
6               6                5
8               5                9

A2 (low dose)               5               7                5
5               8                4
4               4                3
6               5                9
8               6                7

A3 (high dose)              8              12              18
9              11              17
7              15              19
8              13              20
6              10              20
26
Based on two pieces of information

1)                    Des criptive Statis tics

Dependent Variable: anxiety
FactorA   FactorB    Mean       Std. Deviation   N
1.00      1.00        6.2000          1.30384         5
2.00        7.4000          2.07364         5
3.00        6.0000          2.34521         5
Total       6.5333          1.92230        15
2.00      1.00        5.6000          1.51658         5
2.00        6.0000          1.58114         5
3.00        5.6000          2.40832         5
Total       5.7333          1.75119        15
3.00      1.00        7.6000          1.14018         5
2.00       12.2000          1.92354         5
3.00       18.8000          1.30384         5
Total      12.8667          4.95504        15
Total     1.00        6.4667          1.50555        15
2.00        8.5333          3.24844        15
3.00       10.1333          6.63181        15
Total       8.3778          4.51406        45

27
Compute simple main effects

2)

Tes ts of Be tw ee n-Subje cts Effects

Dependent Variable: anx iety
Ty pe III Sum                                                     Partial Eta    Nonc ent.   Obs erved
a
Sourc e              of Squares        df        Mean Square       F          Sig.      Squared      Parameter     Pow er
Correc ted Model         781.378b            8        97.672      30.523        .000          .872     244.181        1.000
Intercept               3158.422             1      3158.422     987.007        .000          .965     987.007        1.000
FactorA                  458.178             2       229.089      71.590        .000          .799     143.181        1.000
FactorB                  101.378             2        50.689      15.840        .000          .468      31.681          .999
FactorA * FactorB        221.822             4        55.456      17.330        .000          .658      69.319        1.000
Error                    115.200            36         3.200
Total                   4055.000            45
Correc ted Total         896.578            44
a. Computed us ing alpha = .05
b. R Squared = .872 (Adjus ted R Squared = .843)

28
3-Way ANOVA
 Effects       A Vague Example
   A             DV = Treatment
   B              Outcome
   C             Factor A: Gender
   AxB           Factor B: Age (14 or 17)
   AxC           Factor C: Treatment
   BxC
   AxBxC

29
Results
Tests of Between-Subj ects Effects

Dependent Variable: SCORE
Type III Sum                                                           Partial Eta   Noncent.    Observed
a
Source             of Squares         df          Mean Square       F           Sig.      Squared      Parameter    Power
Corrected Model       221.556 b            11          20.141       2.224         .049          .505      24.466       .804
Intercept            2635.111               1       2635.111      290.994         .000          .924     290.994      1.000
SEX                        .111             1            .111        .012         .913          .001        .012       .051
AGE                    36.000               1          36.000       3.975         .058          .142       3.975       .482
TREAT                  24.889               2          12.444       1.374         .272          .103       2.748       .266
SEX * AGE                  .111             1            .111        .012         .913          .001        .012       .051
SEX * TREAT            80.889               2          40.444       4.466         .022          .271       8.933       .710
AGE * TREAT              4.667              2           2.333        .258         .775          .021        .515       .086
SEX * AGE * TREAT      74.889               2          37.444       4.135         .029          .256       8.270       .674
Error                 217.333              24           9.056
Total                3074.000              36
Corrected Total       438.889              35
a. Computed using alpha = .05
b. R Squared = .505 (Adjusted R Squared = .278)

30
Significant Two-Way Interaction

31
Significant Three-Way Interaction

32
Other Stuff
 Higher order models (4-way, 5-way, etc.)
 Unequal Cell Sizes and SS Type
 Use of contrast coefficients
 Short-Cuts using SPSS
 Custom Models in SPSS
 Observed Power

33

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