Statistical Inferences

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 samplesB. Factorial ANOVA: Treatment combinations are applied to different participants Compare to independent-samples t- test and one-way ANOVA

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.

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

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).

Partitioning or analyzing the within sum of squares

SSW = SSBetweenSubj + SSError

And SSBetweenSubj = P2/ k)- (X)2 / N

Then, subtract to find SSError:

SSError = SSW - SSBetweenSubj

The repeated-measures ANOVA summary table

Source SS df MS or s2 Fp

Between TreatmentsWithin

Between subjectsError

Total

Post hoc tests with repeated-measures ANOVA

Use Tukey’s HSD or Scheffe’s test, but with MSerror and dferror rather than MSwithin

and dfwithin.

Two-way factorial ANOVA

Partitioning the between-groups Sum of Squares

The interaction Sum of Squares

The ANOVA summary table

Source SS df MS F p

Between

Within

Between participants/subjects

Error

Total

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.

Setting up the data

COLUMN_Variable

1 2 3_

| 1 | R1C1 R1C2 R1C3

ROW |

Variable| 2 | R2C1 R2C2 R2C3

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 10ROW

4 12 5 2 2 14 6 (Women) 1 15 7

493649

490

81121100

16 4 1

144196225

253649

20 134 5 13 30 302

7 21 41 565 18 110

The factorial ANOVA table

Source SS df MS or s2 F pBetween cells (Treatment)

Row (A) Column (B)

R x C (A x B)WithinTotal

SStotal

Calculate SStotal the same way as for the one-way ANOVA:

SStotal = X2 - (X)2 / N = 1145 - 1212/ 18

= 1145 - 14641/18 = 1145 - 813.389

= 331.611 Total df = N - 1 = 18 - 1 = 17

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

SSw...

SSR2C1= 21 - 72 / 3 = 21 - 49/3 = 4.667

SSR2C2= 565 - 412 /3 = 565 - 1681/3 =4.667

SSR2C3=110 - 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

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

SS between cells

Compute SSbetween cells the same way you computed SSbetween in the one-way ANOVA:

SSbetween cells= (Xcell)2/ncell] - (Xtotal)2/ N

= 202 + 52 + 302 + 72 + 412 + 182 - 1212/18

3 3 3 3 3 3

= 400+25+900+49+1681+324 - 813.389

3

SS between cells

= 3379 / 3 - 813.389 = 1126.333-813.389

= 312.944 Between cells df = k - 1 = 6 - 1 = 5

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.

SS rows

Compute SSrows in the same way as SSBetween, using the rows as the only groups (pretend there are no columns):

SSrows= (Xrow)2/nrow] - (Xtotal)2/ N= 552 + 662 - 813.389 9 9= 3025 + 4356 - 813.389 = 6.722

9

SS columns

Similarly, find SScolumns using the SSBetween formula, using columns as the only groups:

SScolumns= (Xcolumns)2/ncolumns] - (Xtotal)2/ N= 272 + 462 + 482 - 813.389 6 6 6= 729 + 2116 + 2304 - 813.389 = 44.778

6

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) = 2-1=1

dfColumns = c - 1 (number of columns - 1)= 2

dfR x C = (r - 1)(c - 1) = (1)(2) = 2

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

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

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

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 / 1.556=14.39

FRxC = MSRxC / MSW = 130.722/1.556=84.01

The factorial ANOVA table

Source SS df MS or s2 F p

Betweencells 312.944 5

Row 6.722 1 6.722 4.32 >.05

Column 44.778 2 22.389 14.39 <.05

R x C 261.444 2 130.722 84.01 <.05

Within 18.668 12 1.556

Total 331.611 17

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.

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.

Interpreting the interaction...

0

2

4

6

8

10

12

14

16

So Jr Sr

MenWomen

The two lines are clearly not parallel, showing the interaction.

When there is a significant interaction, interpret the main effects cautiously.

Group comparisons

Main effect comparisons

Interaction comparisons– By row variable– By column variable