Research Design - - Topic 12 MRC Analysis and Two Factor Designs: Completely Randomized and Repeated Measures 2010 R.C. Gardner, Ph.D.

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1 esearch Design - - Topic MC nalysis and Two Factor Designs: Completely andomized and epeated Measures C Gardner, PhD General overview Completely andomized Two Factor Designs Model I Effect Coding egression Equation and Means Model II Dummy Coding egression Equation and Means Model III Single Factor epeated Measures Designs The General Linear Model Using MC nalysis The Model (with levels of and levels of ) i b + b + b + b + b + b The general linear model is a least squares approach to the analysis of variance For a factorial design with equal sample sizes the results obtained are identical to those obtained with the Experimental Design model When sample sizes are not equal, different ways of expressing the general linear model will produce different models and different answers Overall and Spiegel (99) identified these as Model I (Unique Sums of Squares), Model II (General Experimental) and Model III (Hierarchical) They can be run on SP GLM Univariate by selecting SPTYPE, SPTYPE, and SPTYPE, respectively General Description Two factor analysis of variance permits you to study the simultaneous effects of two factors Consider the data for a design, in which there are an unequal number of observations in each cell, and each level of the factor appears in combination with each level of the factor Two Factor Designs Table of means Unweighted -means Weighted -means Unweighted -means Weighted -means 77 Questions to ask of the Data Main Effects of Do the -means vary more than you would expect on the basis of chance? Main Effects of Do the -means vary more than you would expect on the basis of chance? Interaction Effects of and Do the means vary from what you would expect given the values of the -means and the -means?

2 Dependent Variable: x Source Corrected Model Intercept a b a * b Error Corrected SP GLM Univariate Output Tests of etween-subjects Effects Type III Sum Partial Eta Noncent Observed of Squares df Mean Square F Sig Squared Parameter Power a b a Computed using alpha 77 9 b Squared 9 (djusted Squared 8) level Effect Coding of a (*) Factorial Design (Showing first observation for each cell) level Venn Diagrams and Models in nalysis of Variance Model I Unique 7 Model II Classical Experimental Model III Hierarchical Squared multiple correlations based on the Effect Coded variables needed to compute the relevant squared multiple semipartial correlations for the three models ²,, 9 ², 89 ², 99 ², ² 97 ² Note These two are not needed for Model Note Effect coding can be used for all three models, whereas Dummy coding can be used only for Models II and III 8

3 Semipartial ² estimates for the three models Model I,,, Model II Model III,,,,,,,,,,,,,,, Computing squared multiple semipartial correlations for Model I,,,,,,,,, Only Effect Coding can be used with Model I Dummy Coding will produce wrong values for ² 9 and ² F-ratios for Model I /( a ) 78/ F 99 / (,, ) /( N p ) /( b ) 99 / F 99 / (,, ) /( N p ) F /( a )( b ) 87/ 99 / (,, ) /( N p ) Note The general form of the F-ratio is: effect/ v F ( ) /( N p ) total Calculating the nalysis of Variance Summary Table If it were desired to construct the Summary Table for the nalysis of Variance, the Sums of Squares could be calculated by multiplying the ² values by the Sums of Squares as follows: 78* * * 77 S / ( total ) 99*77 8 Dividing by the appropriate degrees of freedom yields the Where: v number of vectors for the effect, N-p- degrees of freedom for error, and p is the number of vectors necessary to calculate ² total Mean Squares

4 egression Coefficients and egression Equation Coefficients a Unstandardized Standardized Coefficients Coefficients Model Std Error eta t Sig (Constant) 7 97 a b b ab ab a Dependent Variable: x egression Equation abi b + b + b + b + b + b abi + ( ) + ( ) + ( ) + ( 9) + 8 This equation will produce the cell means and marginal level Dummy Coding of a (*) Factorial Design (Showing first observation for each cell) level 8 unweighted means presented in Slide Squared multiple correlations based on the Dummy coded variables needed to compute the squared multiple semipartial correlations for Models II and III ²,, 9 ², 89 ², 9 ², ² 97 ² Note that ², and ², differ from the values obtained with Effect coding Squared semipartial multiple correlations and egression coefficients for Model II at Step Step Compute:,, egression Coefficients and egression Equation Model (Constant) a b b a Dependent Variable: x Unstandardized Coefficients Coefficients a Standardized Coefficients Std Error eta t Sig abi 7 + ( 7) + ( ) + 8

5 The regression equation produces the following cell means Dummy coding does not permit calculation of marginal means, but they can be estimated as means of the cell means Cell Means for Step I of Model II -means means Note that these means do not correspond to any of the means in slide They are estimated assuming no interaction Thus, the main effects tested in slide 8 refer to variation in the marginal means assuming no interaction 7 F-ratios for Main Effects for Model II Cohen, Cohen, iken & West (, p 7) refer to two different error terms that can be used They define Model error as the residual at step (ie, -², ) and Model error as the residual for the full model The general form is: effect/ v F ( residual error) /( N p ) where: v number of vectors for the effect, N-p- degrees of freedom for error, and p is the number of vectors necessary to calculate residual error F F Model error / ( 89) / / 9 ( 89) / Model error / ( 9) / / ( 9) / 99 8 Step Model would be as computed in slide 9 and have the same value as in slide Moreover, the F-ratio would be the same as in slide egression coefficients for the full model using Dummy Coding (Constant) a b b ab ab a Dependent Variable: x The regression equation is: Coefficients a Unstandardized Standardized Coefficients Coefficients Std Error eta t Sig abi + ( 8) + ( ) + ( 8) This equation produces the cell means from slide s before, marginal means cannot be computed with Dummy coding but they can be calculated as the means of the cell means (ie, unweighted 9 means) Model III Step Compute the squared multiple correlation for one of the factors (eg, ) 97 The two F-ratios are: Model error 97 / F 79 ( 97) / 8 Model error 97 / ( 9) / 7 If you were to obtain the regression coefficients and solve for the means at this point, you would obtain the weighted means from Slide Step yields results for identical to those from Step for Model II Step yields results for identical to those from Step for Model I and step for Model II

6 Definition of egression Coefficients for Effect Coding and Dummy Coding Vector Constant Coefficient b b b b b b Effect coding G a G b G b G a b a b + G a b a b + G Dummy coding ab ab ab ab ab ab ab + ab ab ab ab + ab ab ab ab Summary Points The three models differ in terms of how contrasts are defined Model I contrasts each set from all others in the study Model II contrasts each set from others at the same and lower levels Model III contrasts each set from others at the lower levels, and in a specified order in each set The type of coding does have an influence This is discussed by Cohen, Cohen, iken & West (, p) who refer to them as Type III, Type II, and Type I respectively, and by Gardner (8), who uses the above labelling In short: Effect coding can be used for all models Dummy coding can be used for models II and III Single Factor epeated Measures Designs The Model (with 8 subjects and treatments) i b + b + + b + bs + bs7 + b S + + b S Using the logic we used for the two factor design, this would require vectors for the treatment conditions, 7 for the 8 subjects, and for the product terms When using the multiple regression approach, it is not necessary to form the product vectors because if this were done, we would have accounted for all the variation In this type of analysis, the product terms are treated as residual variation 7 Means Variances n Example Using the Data from Kirk, p 7 P i Major Question to ask of the data: Do the -means vary more than can be reasonably attributed to chance? G S 8

7 nalysis of Variance for these data from Topic 7 Measure: MESUE_ Source Error() Sphericity ssumed Greenhouse-Geisser Huynh-Feldt Lower-bound Sphericity ssumed Greenhouse-Geisser Huynh-Feldt Lower-bound Measure: MESUE_ Transformed Variable: verage Type III Sum Source Intercept Error Tests of Within-Subjects Effects Type III Sum of Squares df Mean Square F Sig Tests of etween-subjects Effects of Squares df Mean Square F Sig The following slide shows the Effect coding for the first two subjects and the last subject for the sample data from Slide Note there are no vectors representing the 8 subjects ather there is Subject vector (P) which contains the sum of each subject s score on the dependent variable Pedhazur (977) showed that the correlation of this vector with the dependent variable was identical to the multiple correlation of the Subject vectors with the dependent variable It is viewed as a multiple correlation based on (n-) vectors Note too, that there are no product vectors They are not needed; they constitute the residual term P 8 7 elevant and F values S 8,S 78 F (, S /( a ) / ) /( N ( a ) ( n ) ( 78) /( 7 ) Note that this value is the same as that obtained in Slide 7 Dummy coding would yield the same ² values because there is no product term Of course, the regression coefficients for the Constant and vectors would be different and in both types of coding the S vector yields the within conditions regression coefficient 8 7

8 Calculating the nalysis of Variance Summary Table If it were desired to construct the Summary Table for the nalysis of Variance, the Sums of Squares could be calculated for etween Subjects,, and esidual by multiplying the ² values by the total Sums of Squares as follows: S S S 8* * 9 (, S ) 88* 9 The degrees of freedom would be (n-) 7, (a-), and (a-)(n-), respectively 9 eferences Cohen, J, Cohen, P, West, SG & iken, LS () pplied Multiple egression/correlation nalysis for the ehavioral Sciences (Third Edition) Mahwah, NJ: Lawrence Erlbaum Gardner, C (8) analysis of variance and multiple regression: Coding does make a difference Unpublished manuscript, University of Western Ontario vailable at Overall, JE & Spiegel, D K (99) Concerning least squares analysis of experimental data Psychological ulletin, 7, - Pedhazur, EJ (977) Coding subjects in repeated measures designs Psychological ulletin, 8, 98-8