statistical inferences
DESCRIPTION
Statistics, ANOVA, Central Limit TheoremTRANSCRIPT
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