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Nonlinear Interaction Components -- 2-group Example
Here are data from a 2-group design in which participants 90
were assigned to two different feedback conditions (1 =
intermittent feed, 2 = continuous feedback) and completed
an assigned number of practices with that type of feedback
before performance testing.
80
Both of the groups show a quadratic component to their
practice-performance function.
Below are analyses of the relationship between #practice, 70
feedback type and their interaction with performance -- with
GRP
Mean PERF
and without the quadratic component.
1.00
60 2.00
2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00
PRACTICE
Coding needed to run the linear model includes:
Centering the quantitative practice (X) variable
(mean = 6.5, std = 2.89)
Dummy coding the grouping variable (here the highest-
coded group – continuous feedback) was set as the
comparison group)
Interaction term computed as the product of the dummy
code and the centered quantitative variable
Additional coding needed for the quadratic model includes:
Quadratic term computed as the square of the centered
practice (X) variable (nonlinear main effect)
Quadratic interaction term computed as the product of
the dummy code and the quadratic term (nonlinear
interaction)
Results from this model…
Model Summary ANOVAc
Sum of
Change Statistics Model Squares df Mean Square F Sig.
Adjusted Std. Error of R Square 1 Regression 3138.655 3 1046.218 156.091 .000 a
Model R R Square R Square the Estimate Change F Change df1 df2 Sig. F Change Residual 375.345 56 6.703
1 .945 a .893 .887 2.58894 .893 156.091 3 56 .000 Total 3514.000 59
2 .972 b .945 .939 1.89998 .051 24.988 2 54 .000 2 Regression 3319.064 5 663.813 183.885 .000 b
a. Predictors: (Constant), INT, DC, PRAC_C Residual 194.936 54 3.610
b. Predictors: (Constant), INT, DC, PRAC_C, PRAC_CSQ, INTSQ Total 3514.000 59
a. Predictors: (Constant), INT, DC, PRAC_C
b. Predictors: (Constant), INT, DC, PRAC_C, PRAC_CSQ, INTSQ
c. Dependent Variable: PERF
Coefficientsa
Unstandardized Standardized
Coefficients Coefficients
Model B Std. Error Beta t Sig.
1 (Constant) 75.700 .473 160.153 .000
DC -7.400 .668 -.483 -11.070 .000
PRAC_C 2.164 .165 .812 13.148 .000
INT -1.02E-15 .233 .000 .000 1.000
2 (Constant) 77.669 .525 147.989 .000
DC -11.337 .742 -.741 -15.275 .000
PRAC_C 2.164 .121 .812 17.915 .000
INT -1.02E-15 .171 .000 .000 1.000
PRAC_CSQ -.239 .048 -.227 -4.999 .000
INTSQ .477 .068 .411 7.069 .000
a. Dependent Variable: PERF
The full model is …
Perf’ = b0 + b1*DC + b2*prac_c + b3*prac_csq + b4*int + b5*intsq
b0 – constant – expected performance for those in comparison group with the mean (0) amount of practice
b1 - the simple effect of feedback for the mean (0) amount of practice
- expected direction and extent of change in performance for those in the target group, compared to those in the
comparison group, holding all other predictors constant at the value 0
b2 - the simple linear effect of practice for those in the comparison group (continuous feedback)
- expected direction and extent of change in performance for a 1-unit increase in practice holding all other predictors
constant at 0
b3 - simple quadratic effect of practice for those in the comparison group (continuous feedback)
- expected direction and extent of change in performance for a 1-unit change in performance, holding all other
predictors constant at 0
b4 - linear interaction - how the linear effect of practice for the target (intermittent feedback) differs from the linear effect of
practice for the comparison group (continuous feedback)
- how the difference between target and comparison group performances changes for different
amounts of practice
- expected direction and extent of change in effect of one predictor for a 1-unit increase in the value of the other
predictor, holding all other predictors constant at 0
b5 - quadratic interaction - how the quadratic effect of practice for the target (intermittent feedback) differs from the
quadratic effect of practice for the comparison group (continuous feedback)
- how how the difference between target and comparison group performances changes for
different amounts of practice, for different amounts of practice
- difference in expected direction and extent of change in effect of one predictor for a 1-unit increase in the value of
the other predictor, holding all the other predictors constant, for a 1-unit change in practice
