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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
Factor A: Task Difficulty




                                         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
Factor A: Task Difficulty




                                         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
 Factor A: Task Difficulty




                             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
     Task Difficulty           Low      Medium     High
     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
    Task Difficulty           Low      Medium     High
    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
    Task Difficulty           Low      Medium     High
    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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