Research Design - - Topic 13a Split Plot Design with Either a Continuous or Categorical Between Subjects Factor 2008 R.C. Gardner, Ph.D.

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1 esearch Design - - Topic 3a Split Plot Design with Either a ontinuous or ategorical etween Subjects Factor 8.. Gardner Ph.D. General Description Example Purpose Two ategorical Factors Using Multiple egression Using SPSS GLM epeated One ontinuous and One ategorical Factor Using Multiple egression Using SPSS GLM epeated

2 General Description This analysis is similar to Topics and except that in this case one of the factors is based on correlated observations as is the case with repeated Measures and randomized blocks. That is there will be separate error terms for the between subject and within subject variability. s before we will see how to do the analysis using either SPSS multiple regression or SPSS GLM repeated. The following table presents data for a two-level etween Subjects factor () and a three level epeated Measures factor ().

3 Two ategorical Factors and

4 Effect coding for the first subject in each level of. Note that is a etween Subjects factor and is epeated Measures a b b X S ab ab

5 Purpose The analysis is concerned with assessing the: Main Effect of the factor. Does the mean differ significantly from? Main Effects of the factor. Do the means vary more than can be reasonably attributed to chance? Interaction Effects. Do the means in the cells differ in ways that are not consistent with the variation in the means and the means? We will consider only Model I using Effect oding.

6 To perform this analysis it is necessary to compute 7 squared multiple correlations in order to calculate the squared semi-partial multiple correlations. To compute the values for the effects of interest we need the ² values for () () ( ) and (). To compute the values for the two error terms we need the values for (S) (S) and () as follows: S S

7 7 omputing squared semi-partial multiple correlations: / S S / S S Error Terms 9 ] ) [( ] ) [( + + S k k k df 8 ) ( 33 ) ( S k k k N df

8 omputing the F-ratios etween Subjects Factor F S / /.796 /.6/ Within Subjects Factor F / S /.64 /.6338/ F / S /.77/.6338/

9 Interpreting the F-ratios If the F-ratio for was significant it would indicate that the main effect means for differed significantly from each other. The significant F-ratio for indicates that the main effect means for vary more than can be reasonably attribute to chance. If the F-ratio for the interaction was significant it would indicate that the means in the cells vary in ways that are different from those of the main effects. 9

10 omputing Sums of Squares The total Sum of Squares is: SS Total ( X G) abi The sums of squares for the effects are: SS SS Total (.796)( ) 9.7 (.64)( ) 77.3 SSTotal SSTotal (.77)( ) S / S / SSTotal (.6)( ) S / S / SSTotal (.6338)( ) SS SS SS SS

11 The regression coefficients for the full model are: Note. The regression equation for the main and interaction effects does not include the subjects vector(s).

12 alculation of Means The means can be calculated using the regression equation and the relevant codes for each condition. Using Effect coding we can estimate the cell means and the marginal means. Following are the codes for the six cells: Groups Effect odes a b

13 egression Equation Sample calculation of means X 6.6+ (.77) + (.4444) + (.96) alculating cell means ab X 6.6+ (.77)() + (.4444)() ()() ab3 X 6.6+ (.77)( ) + (.4444)( ) + (.96) ( )( ) +.7( )( ).4 alculating main effect means a X 6.6+ (.77)( ) b X 6.6+ (.96)(). 3

14 Table of Means b b b3 a means a a b means Note. The n s differ from to but are equal across within. Thus the main effect means for are the actual means while those for are unweighted means. 4

15 If the data were run in SPSS GLM Univariate instead the analysis of variance summary table would be:

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17 oncluding Observations. The results obtained with multiple regression using effect coding and Model I are identical to those obtained with SPSS GLM epeated in terms of the Sums of squares the F-ratios and the means etc.. ecause effect coding was used the regression coefficients do not provide information about any contrasts involving means. 3. ecause the F-ratio for was significant it would generally be the case that post hoc tests of the main effect means would be computed. Note that since the sample sizes were different for and these means are unweighted means. 4. The significant F-ratio for indicates that the two means differ significantly. The F-ratio for is not equal to the square of the t- value for the regression coefficient because they have different error terms. 7

18 ontinuous etween Subjects factor () and a ategorical epeated Measures Factor () These are the same data as in the previous example except that has been replaced by a centred continuous variable 8

19 Effect coding for the first subject c b b X S bc bc

20 Purpose The analysis is concerned with assessing the: Main Effect of the factor. Does the mean slope differ significantly from? Main Effects of the factor. Do the intercepts for the groups vary more than can be reasonably attributed to chance? Interaction Effects. Do the slopes for the groups differ more than can be reasonably attributed to chance? We will consider only Model I using Effect oding.

21 To perform this analysis it is necessary as before to compute 7 squared multiple correlations in order to calculate the squared semi-partial multiple correlations. They are: S.737 S.79

22 omputing squared semi-partial multiple correlations: / S S / S S Error Terms Effects of Interest

23 omputing the F-ratios etween Subjects Factor F S / /.693/.988 / Within Subjects Factor F / S /.347 /.948/ F S / /.468/.948/

24 Interpreting the F-ratios The significant F-ratio for indicates that the mean slope differs significantly from. The significant F-ratio for indicates that the intercepts for the three levels of differ more than can be reasonably attributed to chance. Post hoc tests would involve comparing intercepts across the levels of. If the F-ratio for the interaction was significant that would indicate that the three slopes vary more than can be reasonably attributed to chance. If significant post hoc tests would involve comparing slopes across the levels of. 4

25 omputing Sums of Squares The total Sum of Squares is: SS Total ( X G) abi The sums of squares for the effects are: SS SS SS SS Total Total (.693)( ). (.347)( ) SSTotal S / S / SSTotal (.988)( ) S / S / SSTotal (.948)( ) SS SS SS (.468)( )

26 egression oefficients for the Effects of Interest Note. The regression equation for the main and interaction effects does not include the Subjects vector(s) 6

27 Table of Intercepts and Slopes INTEEPT (.369) + (.9466) SLOPE (.3938) + (.387) + (.3389) b b b3 means Intercept Slope

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29 If the data were run on SPSS GLM epeated the analysis of variance summary table would be as follows: 9

30 The following table presents the Parameter Estimates yielded by SPSS GLM if requested. Note that the values in the table are in fact the intercepts and slopes presented in slide 7. They are not the regression coefficients you would obtain if you were to use dummy coding and multiple regression. I show those values (obtained with multiple regression) on the next slide. 3

31 These are the regression coefficients that are obtained if you use dummy coding and run the full model in multiple regression. s before these can be used to compute the intercepts and slopes using standard dummy coding with group 3 given all s. 3