Statistical Methods for Psychology

614 Chapter 16 Analyses of Variance and Covariance as General Linear Models

Table 16.11 Hypothetical data on smoking and performance (modeled on Spilich et al., 1992)

Pattern Recognition

NS:Errors 9 8 12 10 7 10 9 11 8 10 8 10 8 11 10
Distract 107 133 123 94 83 86 112 117 130 111 102 120 118 134 97

DS:Errors 12 7 14 4 8 11 16 1756966716
Distract 101 75 138 94 138 127 126 124 100 103 120 91 138 88 118

AS:Errors889197161911221218810
Distract 64 135 130 106 123 117 124 141 95 98 95 103 134 119 123

Cognitive Task

NS:Errors 27 34 19 20 56 35 23 37 4 30 4 42 34 19 49
Distract 126 154 113 87 125 130 103 139 85 131 98 107 107 96 143

DS:Errors 48 29 34 6 18 63 9 54 28 71 60 54 51 25 49
Distract 113 100 114 74 76 162 80 118 99 146 132 135 111 106 96

AS:Errors 34 65 55 33 42 54 21 44 61 38 75 61 51 32 47
Distract 108 191 112 98 128 145 76 107 128 128 142 144 131 110 132

Driving Simulation

NS:Errors 15 2 2 14 5 0 16 14 9 17 15 9 3 15 13
Distract 110 96 112 114 137 125 168 102 109 111 137 106 117 101 116

DS:Errors70601217111443516511
Distract 93 102 108 100 123 131 99 116 81 103 78 103 139 101 102

AS:Errors320062064100623
Distract 130 83 91 92 109 106 99 109 136 102 119 84 68 67 114

Table 16.12 contains an abbreviated form of the design matrix, showing only the

entries for the first and last subject in each cell. Notice that the matrix contains a column

for the covariate (denoted C), the usual design matrix elements for the main effects of Task

(T) and Group (G), and the Task 3 Group interaction. I have also added columns repre-

senting the interaction of the covariate with the Task 3 Group interaction. The latter will

be used to test the hypothesis for all values of iand j, since the assumption of

homogeneity of regression applies to any analysis of covariance.

It is important to consider just what the interactions involving the covariate represent.

If I had included the terms and I could have used them to test the null hypothesis

that the regression lines of Errors as a function of Distract have equivalent slopes in the

three tasks. Similarly, if I had included and , I could have tested homogeneity of

regression in each of the three smoking groups. Since I am most interested in testing the

hypothesis of homogeneity of regression in each of the nine cells, I have included only the

terms.

The first regression is based on all predictors in X. From this analysis we obtain

MSresidualc,a,b,ab,cab=71.10333

SSregressionc,a,b,ab,cab=36728.42272

CTGij

CG 1 CG 2

CT 1 CT 2

H 0 : b*i =b*j