Repeated measures ANOVA and Two-Factor (Factorial) ANOVA

1
Repeated measures ANOVA and Two-Factor
(Factorial) ANOVA

• A. Repeated measures All participants
  experience all of the k levels of the independent
  variable.
• Compare to the t-test for paired samples
• B. Factorial ANOVA Treatment combinations are
  applied to different participants
• Compare to independent-samples t- test and
  one-way ANOVA

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Repeated Measures ANOVA

• Here, we partition the within sum of squares and
  the within degrees of freedom.
• In a repeated measures design, differences
  between treatment conditions cannot be due to
  individual differences, so we subtract the
  variance due to participants from the within sum
  of squares, leaving us with a smaller error term
  and, as with the paired samples t-test, more
  power.

3
A repeated-measures version of the dating study

• Number of dates
• Participant Soph Jr Sr Person total
• Shane 2 4 6 12
• Eric 1 4 8 13
• Ryan 0 3 9 12
• Zachary 4 1 2 7
• Mathias 3 5 6 14
• Totals

10 17 31 58
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The F ratio in a repeated measures design

• As always, the F ratio compares the variance
  due to treatments error to the variance due to
  error.
• Therefore, we will compute SS for the total set
  of scores (SSTot), within groups (SSW), and
  between treatments (SSB).

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Partitioning or analyzing the within sum of
squares

• SSW SSBetweenSubj SSError
• And SSBetweenSubj S(P2/ k)- (SX)2 / N
• Then, subtract to find SSError
• SSError SSW - SSBetweenSubj

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The repeated-measures ANOVA summary table

• Source SS df MS or s2 F pBetween
  TreatmentsWithin
• Between subjects
• ErrorTotal

7
Post hoc tests with repeated-measures ANOVA

• Use Tukeys HSD or Scheffes test, but with
  MSerror and dferror rather than MSwithin and
  dfwithin.

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Two-way factorial ANOVA

• Partitioning the between-groups Sum of Squares
• The interaction Sum of Squares

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The ANOVA summary table

• Source SS df MS F p
  
• Between
• Within
• Between participants/subjects
• Error
• Total

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Partitioning the between-groups Sum of Squares

• Cell notation Rows, columns, and interactions
• Factorial design Fully crossed
• Set up the data so that the groups of one
  variable form rows and the groups of the other
  variable form columns.

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Setting up the data

• COLUMN_Variable
• 1 2 3_
• 1 R1C1 R1C2 R1C3
• ROW
• Variable 2 R2C1 R2C2 R2C3

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An example

• Number of dates/person this semester
• COLUMN___
• 1(So) 2(Jr) 3(Sr)_
• 1 7 2 9
• (Men) 6 3 11
• 7 0 10
• ROW
• 4 12 5
• 2 2 14 6
• (Women) 1 15 7

49 36 49
4 9 0
81 121 100
20 134 5 13 30 302
25 36 49
16 4 1
144 196 225
7 21 41 565 18 110
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The factorial ANOVA table

• Source SS df MS or s2 F p
• Between cells (Treatment)
• Row (A)
• Column (B)
• R x C (A x B)
• Within
• Total

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SStotal

• Calculate SStotal the same way as for the one-way
  ANOVA
• SStotal SX2 - (SX)2 / N 1145 - 1212/ 18
• 1145 - 14641/18 1145 - 813.389
• 331.611
• Total df N - 1 18 - 1 17

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SSw

• SSw is also computed the same as it was for the
  one-way ANOVA, this time computing SS for each R
  x C cell and adding them all together.
• SSR1C1 134 - 202 / 3 134 - 400/3 0.667
• SSR1C2 13 - 52 / 3 13 - 25/3 4.667
• SSR1C3 302 - 302 / 3 302 - 900/3 2.000

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SSw...

• SSR2C1 21 - 72 / 3 21 - 49/3 4.667
• SSR2C2 565 - 412 /3 565 - 1681/3 4.667
• SSR2C3110 - 182 / 3 110 - 324/3 2.000
• SSW 0.667 4.667 2.000 4.667 4.667
  2.000 18.668
• Within df N - k 18 - 6 12

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The factorial ANOVA table

• Source SS df MS or s2 F p
• Betweencells
• Row
• Column
• R x C
• Within 18.668 12
• Total 331.611 17

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SS between cells

• Compute SSbetween cells the same way you computed
  SSbetween in the one-way ANOVA
• SSbetween cells S(SXcell)2/ncell - (SXtotal)2/
  N
• 202 52 302 72 412 182 - 1212/18
• 3 3 3 3 3 3
• 40025900491681324 - 813.389
• 3

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SS between cells

• 3379 / 3 - 813.389 1126.333-813.389
• 312.944
• Between cells df k - 1 6 - 1 5

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The factorial ANOVA table

• Source SS df MS or s2 F p
• Betweencells 312.944 5
• Row
• Column
• R x C
• Within 18.668 12
• Total 331.611 17
• SPSS and everyone else in the world uses MS.

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SS rows

• Compute SSrows in the same way as SSBetween,
  using the rows as the only groups (pretend there
  are no columns)
• SSrows S(SXrow)2/nrow - (SXtotal)2/ N
• 552 662 - 813.389
• 9 9
• 3025 4356 - 813.389 6.722
• 9

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SS columns

• Similarly, find SScolumns using the SSBetween
  formula, using columns as the only groups
• SScolumns S(SXcolumns)2/ncolumns - (SXtotal)2/
  N
• 272 462 482 - 813.389
• 6 6 6
• 729 2116 2304 - 813.389 44.778
• 6

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SS row by column interaction

• Compute the SSR x C interaction by subtracting
  both the SSRows and the SScolumns from the
  SSBetween cells
• SSR x C SSBetween cells - SSRows - SSColumns
• 312.944 - 6.722 - 44.778 261.444
• dfRows r - 1 (number of rows - 1) 2-11
• dfColumns c - 1 (number of columns - 1) 2
• dfR x C (r - 1)(c - 1) (1)(2) 2

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The factorial ANOVA table

• Source SS df MS or s2 F p
• Betweencells 312.944 5
• Row 6.722 1
• Column 44.778 2
• R x C 261.444 2
• Within 18.668 12
• Total 331.611 17

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Computing MS or sW2

• Divide each SS by its df
• MSRows SSRows / dfRows 6.722 / 1 6.722
• MSCols SSCols / dfCols 44.778 / 2 22.389
• MSR x C SSRxC / dfRxC 261.444/2 130.722
• MSW SSW / dfW 18.668 / 12 1.556

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The factorial ANOVA table

• Source SS df MS or s2 F p
• Betweencells 312.944 5
• Row 6.722 1 6.722
• Column 44.778 2 22.389
• R x C 261.444 2 130.722
• Within 18.668 12 1.556
• Total 331.611 17

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F ratios

• To compute F ratios, divide each MSBetween by
  MSW
• FRows MSRows / MSW 6.722 / 1.556 4.32
• FCols MSCols / MSW 22.389 / 1.55614.39
• FRxC MSRxC / MSW 130.722/1.55684.01

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The factorial ANOVA table

• Source SS df MS or s2 F p
• Betweencells 312.944 5
• Row 6.722 1 6.722 4.32 gt.05
• Column 44.778 2 22.389 14.39 lt.05
• R x C 261.444 2 130.722 84.01 lt.05
• Within 18.668 12 1.556
• Total 331.611 17

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Interpretation of main effects

• The main effect for rows (gender) was not
  significant. We retain the null hypothesis the
  difference is due to chance.
• The main effect for columns (class) was
  significant. We reject the null hypothesis at
  least one difference is not due to chance. Post
  hoc comparisons are needed next.

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Interpretation of interaction effect

• The interaction between gender (rows) and class
  (columns) was significant. The effect of class
  on number of dates is different for the two
  genders.
• A graph of the means shows that the most frequent
  dating for men occurred among the seniors, while
  for women, the most frequent dating was among the
  juniors.

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Interpreting the interaction...

• The two lines are clearly not parallel, showing
  the interaction.
• When there is a significant interaction,
  interpret the main effects cautiously.

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Group comparisons

• Main effect comparisons
• Interaction comparisons
• By row variable
• By column variable