Factorial Within Subjects. Psy 420 Ainsworth

Transcription

1 Factorial Within Subjects Psy 40 Ainsworth

2 Factorial WS Designs Analysis Factorial deviation and computational Power, relative efficiency and sample size Effect size Missing data Specific Comparisons

3 Example for Deviation Approach b 1 : Science Fiction b : Mystery Case Means a 1 : Month 1 a : Month a 3 : Month 3 a 1 : Month 1 a : Month a 3 : Month 3 S S 1 = S S = S S 3 = S S 4 = S S 5 = Treatment Means a 1 b 1 =.4 a b 1 = 3.8 a 3 b 1 = 6.4 a 1 b = 5 a b = 5 a 3 b =. GM = 4.133

4 Analysis Deviation What effects do we have? A B AB S AS BS ABS T

5 Analysis Deviation DFs DF A = a 1 DF B = b 1 B DF AB = (a 1)(b 1) DF S = (s 1) DF AS = (a 1)(s 1) DF BS = (b 1)(s 1) DF ABS = (a 1)(b 1)(s 1) DF T = abs - 1

6 Sums of Squares - Deviation The total variability can be partitioned into A, B, AB, Subjects and a Separate Error Variability for Each Effect SS = SS + SS + SS + SS + SS + SS + SS Total A B AB S AS BS ABS ( Y ) ( ) ( ) ijk Y... nj Y. j. Y... nk Y.. k Y... = n ( ) ( ) ( ) jk Y. jk Y... nj Y. j. Y... nk Y.. k Y... ( ) ( ) ( ) ( ) + jk Yi.. Y... + k Yij. Y. j. jk Yi.. Y... + j Yi. k Y. j. jk ( Yi.. Y ) ( ) ( ) Y ( ) ( ) ijk Y. jk jk Yi.. Y... k Yij. Y. j. j Yi. k Y. j.

7 Analysis Deviation Before we can calculate the sums of squares we need to rearrange the data When analyzing the effects of A (Month) you need to AVERAGE over the effects of B (Novel) and vice versa The data in its original form is only useful for calculating the AB and ABS interactions

8 For the A and AS effects Remake data, averaging over B a 1 : Month 1 a : Month a 3 : Month 3 Case Means S S 1 = S S = S S 3 = S S 4 = S S 5 = Treatment Means a 1 = 3.7 a = 4.4 a 3 = 4.3 GM = 4.133

9 (..... ) ( ) ( ) ( ) ( i..... ) (3*)*[( ) ( ) ( ) ( ) ( ) SS = n Y Y = 10*[ ] =.867 A j j S SS = jk Y Y = = (... ) (..... ) ( ij.. j. ) = *[ SS = k Y Y jk Y Y = AS ij j i ( 3 3.7) ( ) ( ) ( 4 3.7) ( ) ( ) ( 6 4.4) ( 4 4.4) ( ) ( 5 4.4) k Y Y ( ) ( ) ( ) ( ) ( ) ] = 0.6 SS AS = = 4.133

10 For B and BS effects Remake data, averaging over A b 1 : Science Fiction b : Mystery Case Means S S 1 = S S = S S 3 = S S 4 = S S 5 = Treatment Means b 1 = 4.00 b = GM = 4.133

11 B k.. k... ( ) ( ) ( ) SS = n Y Y = 15*[ ] =.133 ( ) ( ) SS = j Y Y jk Y Y = BS i k k i ( ) i. k.. k 3*[ ( ) ( ) ( 4 4.) j Y Y = ( ) ( ) ( ) ( ) ( ) + ( ) +( ) ( ) i..... SS = jk Y Y = S ] = 50 SS BS = =

12 For AB and ABS effects Use the data in its original form b 1 : Science Fiction b : Mystery Case Means a 1 : Month 1 a : Month a 3 : Month 3 a 1 : Month 1 a : Month a 3 : Month 3 S S 1 = S S = S S 3 = S S 4 = S S 5 = Treatment Means a 1 b 1 =.4 a b 1 = 3.8 a 3 b 1 = 6.4 a 1 b = 5 a b = 5 a 3 b =. GM = 4.133

13 ( ) ( ) ( ) SS = n Y Y n Y Y n Y Y = AB jk jk j j k k jk ( ) = 5*[ ( ) + ( ) + ( ) n Y Y. jk... ( ) ( ) ( ) ] = ( ) n Y Y = SS = j. j.... A.867 ( ) k.. k... B n Y Y = SS =.133 SS AB = =

14 (. ) (..... ) (... ) (... ) SS ABS = Yijk Y jk jk Yi Y k Yij Y j j Yi k Y j = ( Yijk Y. jk ) ( 1.4) ( 1.4) ( 3.4) = ( 5.4) (.4) ( 3 3.8) ( 4 3.8) ( 3 3.8) ( 5 3.8) ( 4 3.8) ( 6 6.4) ( ) ( ) ( ) ( ) ( 6 5) + ( 8 5) + ( 4 5) + ( 3 5) ( 5 5) ( 4 5) ( 8 5) ( 5 5) ( 5) ( 6 5) ( 1.) ( 4.) ( ) ( ) ( ) = SS ABS = = 9.867

15 Total ( ) ( ) ( ) SS = ( ) ( 4.133) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( 4.133) ( ) ( ) ( ) ( ) ( ) ( ) =

16 Analysis Deviation DFs DF A = a 1 = 3 1 = DF B = b 1 = 1 = 1 B DF AB = (a 1)(b 1) = * 1 = DF S = (s 1) = 5 1 = 4 DF AS = (a 1)(s 1) = * 4 = 8 DF BS = (b 1)(s 1) = 1 * 4 = 4 DF ABS = (a 1)(b 1)(s 1) = * 1 * 4 = 8 DF T = abs 1 = [3*()*(5)] 1 = 30 1 = 9

17 Source Table - Deviation Source SS df MS F A AS B BS AB ABS S Total

18 Analysis Computational Example data re-calculated with totals b 1 : Science Fiction b : Mystery Case Totals a 1 : Month 1 a : Month a 3 : Month 3 a 1 : Month 1 a : Month a 3 : Month 3 S S 1 =0 S S =33 S S 3 =4 S S 4 = S S 5 =5 Treatment Totals a 1 b 1 = 1 a b 1 = 19 a 3 b 1 = 3 a 1 b = 5 a b = 5 a 3 b = 11 T = 14

19 Analysis Computational Before we can calculate the sums of squares we need to rearrange the data When analyzing the effects of A (Month) you need to SUM over the effects of B (Novel) and vice versa The data in its original form is only useful for calculating the AB and ABS interactions

20 Analysis Computational For the Effect of A (Month) and A x S Month a 1 : Month 1 a : Month a 3 : Month 3 Case Totals S S 1 =0 S S =33 S S 3 =4 S S 4 = S S 5 =5 Treatment Totals T = 14

21 SS SS SS SS A A S S ( a j ) ( si ) T = = = bs abs (5) (3)(5) = = = T = = = ab abs (3) 3174 = = = ( a jsi ) ( a j ) ( s i ) T SS AS = + = b bs ab abs SS AS = = = SS = = AS

22 Analysis Traditional For B (Novel) and B x S b 1 : Science Fiction Type of Novel b : Mystery Case Totals S S 1 =0 S 13 0 S =33 S S 3 =4 S S 4 = S S 5 =5 Treatment Totals T = 14

23 SS SS B B ( b ) k T = = = as abs 3(5) = = ( ) ( ) ( ) k i k i b s b s T SSBS = + = a as ab abs SSBS = = SS BS SS BS 1688 = = =

24 Analysis Computational Example data re-calculated with totals b 1 : Science Fiction b : Mystery Case Totals a 1 : Month 1 a : Month a 3 : Month 3 a 1 : Month 1 a : Month a 3 : Month 3 S S 1 =0 S S =33 S S 3 =4 S S 4 = S S 5 =5 Treatment Totals a 1 b 1 = 1 a b 1 = 19 a 3 b 1 = 3 a 1 b = 5 a b = 5 a 3 b = 11 T = 14

25 SS SS SS SS AB AB AB AB ( j k ) ( j ) ( k ) a b a b T = + = s bs as abs = = = = 5 = = SS + SS ABS ( a jbk ) ( a jsi ) ( bk si ) ( a j ) ( k ) ( i ) ABS = Y + s b a b s T + + = bs as ab abs = = SS Total T abs = Y = =

26 DFs Analysis Computational DFs are the same DF A = a 1 = 3 1 = DF B = b 1 = 1 = 1 B DF AB = (a 1)(b 1) = * 1 = DF S = (s 1) = 5 1 = 4 DF AS = (a 1)(s 1) = * 4 = 8 DF BS = (b 1)(s 1) = 1 * 4 = 4 DF ABS = (a 1)(b 1)(s 1) = * 1 * 4 = 8 DF T = abs 1 = [3*()*(5)] 1 = 30 1 = 9

27 Source Table Computational Is also the same Source SS df MS F A AS B BS AB ABS S Total

28 Relative Efficiency One way of conceptualizing the power of a within subjects design is to calculate how efficient (uses less subjects) the design is compared to a between subjects design One way of increasing power is to minimize error, which is what WS designs do, compared to BG designs WS designs are more powerful, require less subjects, therefore are more efficient

29 Relative Efficiency Efficiency is not an absolute value in terms of how it s calculated In short, efficiency is a measure of how much smaller the WS error term is compared to the BG error term Remember that in a one-way WS design: SS AS =SS S/A - SS S

30 Relative Efficiency Relative Efficiency = MS S / A df AS + 1 dfs / A + 3 MS AS df AS + 3 dfs / A + 1 MS S/A can be found by either re-running the analysis as a BG design or for one-way: SS S + SS AS = SS S/A and df S + df AS = df S/A MS S / A = s 1 MS + a 1 s 1 MS ( ) ( )( ) S as 1 AS

31 Relative Efficiency From our one-way example: Source SS df MS F A S AxS Total SS = SS + SS = = 19. S / A S AS df = df + df = = 1 MS S / A S AS S / A = 19. /1 = 1.6

32 Relative Efficiency Relative Efficiency = = From the example, the within subjects design is roughly 6% more efficient than a between subjects design (controlling for degrees of freedom) For # of BG subjects (n) to match WS (s) efficiency: n = s(relative efficiency) = 5(1.6) = 6.3 7

33 Power and Sample Size Estimating sample size is the same from before For a one-way within subjects you have and AS interaction but you only estimate sample size based on the main effect For factorial designs you need to estimate sample size based on each effect and use the largest estimate Realize that if it (PC-Size, formula) estimates 5 subjects, that total, not per cell as with BG designs

34 Effects Size Because of the risk of a true AS interaction we need to estimate lower and upper bound estimates η ϖˆ SS = η = = SS A A L SSTotal SS A + SSS + SS AS partial L = partial η df ( MS MS ) df ( MS MS ) + as( MS ) + s( MS ) ϖˆ SS SS A SS + SS SS + SS A = = A error A AS A A AS A A AS AS S = df ( MS MS ) A A AS df ( MS MS ) + as( MS ) A A AS AS

35 Missing Values In repeated measures designs it is often the case that the subjects may not have scores at all levels (e.g. inadmissible score, drop- out, etc.) The default in most programs is to delete the case if it doesn t have complete scores at all levels If you have a lot of data that s fine If you only have a limited cases

36 Missing Values Estimating values for missing cases in WS designs (one method: takes mean of A j, mean for the case and the grand mean) Y * ij * ij ' i ' j ' = where : Y s S a A T = = = = ss + aa T ' ' ' i j ( a 1)( s 1) predicted score to replace missing score number of cases sum of known values for that case number of levels of A = sum of known values for A = sum of all known values

37 Specific Comparisons F-tests for comparisons are exactly the same as for BG F n ( w Y ) / w j j j = MS error Except that MS error is not just MS AS, it is going to be different for every comparison It is recommended to just calculate this through SPSS