QxQ Models: Main Effects and Full Models Using Regression
The purpose of the study was to examine the inter-relationships among social skills, the complexity of the social
situation, and performance in a social situation. Each participant considered their most recent interaction in a group of
10 or larger that included at least 50% strangers, and rated their social performance (perf) and the complexity of the
situation (sitcom), Then each participant completed a social skills inventory that provided a single index of this
construct (soskil). The researcher wanted to determine the contribution of the “person” and “situation” variables to
social performance, as well as to consider their interaction.
Descriptive Statistics
N
SITCOM
SOSKIL
Valid N (listwise)
60
60
60
Minimum
.00
27.00
Maximum
37.00
65.00
Mean
19.7000
49.4700
Std. Deviation
8.5300
8.2600
As before, we will want to center our quantitative variables by
subtracting the mean from each person’s scores.
We also need to compute an interaction term as the product of
the two centered variables.
Some prefer using a “full model” approach, others a
“hierarchical model” approach – remember they produce
the same results.
Using the hierarchical approach, the centered quantitative
variables (main effects) are entered on the first step and
the interaction is added on the second step
Be sure to check the “R-squared change” on the Statistics
window
SPSS Output:
Model 1 is the main effects model and Model 2 is the full model.
Model Summary
Change Statistics
Model
1
2
R
.374 a
.526 b
R Square
.140
.277
Adjusted
R Square
.110
.238
Std. Error of
the Estimate
11.3406
10.4899
R Square
Change
.140
.137
F Change
4.636
10.620
df1
df2
2
3
57
56
Sig. F Change
.014
.002
a. Predictors: (Constant), C_SITCOM, C_SOSKIL
b. Predictors: (Constant), C_SITCOM, C_SOSKIL, INT
ANOV Ac
Model
1
2
Regres sion
Residual
Total
Regres sion
Residual
Total
Sum of
Squares
1192.355
7330.742
8523.097
2360.961
6162.137
8523.098
df
2
57
59
3
56
59
Mean S quare
596.178
128.610
F
4.636
Sig.
.014a
786.986
110.038
7.152
.000b
a. Predic tors: (Constant), C_S ITCOM, C_S OSK IL
b. Predic tors: (Constant), C_S ITCOM, C_S OSK IL, INT
Both the main effects model and
the full model “work”.
R²Δ for Model 2 = .137. More
than ½ of the variance
accounted for by the full model is
due to the interaction
c. Dependent Variable: PE RF
We have 2 equivalent tests of
the interaction:
Coefficientsa
Model
1
2
(Constant)
C_SOSKIL
C_SITCOM
(Constant)
C_SOSKIL
C_SITCOM
INT
Unstandardized
Coefficients
B
Std. Error
29.102
1.464
.524
.180
.210
.174
29.651
1.365
.378
.172
.147
.162
7.120E-02
.022
Standardi
zed
Coefficien
ts
Beta
.360
.149
.260
.104
.385
The R²Δ F-test for Model 2 tells
us this R²Δ is significant.
t
19.878
2.914
1.204
21.728
2.196
.907
3.259
Sig.
.000
.005
.234
.000
.032
.368
.002
The t-test of the interaction b
weight in the full mode is
significant
t² (3.259²) = R²Δ F (10.62)
a. Dependent Variable: PERF
Interpreting the Regression Weights
Simple effect for social skills when situational complexity = 0 (its mean after centering)
C_SOSKIL -- The regression weight for social skills tells that for average situational complexity, there is a positive
relationship between social skills and performance. This positive slope is statistically significant.
Simple effect for the situational complexity when social skills = 0 (its mean after centering)
C_SITCOM -- The regression weight for situational complexity tells us that for those with average levels of social
skills, there is not a significant relationship between situational complexity and performance.
Interaction – each simple effect is different for different levels of the other variable
INT -- The interaction regression weight tells us that (both are true):
 For each 1-unit increase in social skills, the slope of the relationship between performance and situational
complexity increases by .007 (this increase in slope is statistically significant)
 For each 1-unit increase in situational complexity, the slope of the relationship between performance and
social skills increases by .007 (this increase in slope is statistically significant)
Note: Raw regression weights for interactions are often numerically small. Why? The interaction term is computed as
the product of the centered main effect terms. So it will have a relatively large standard deviation and , consequently a
relatively small regression weight compared to the main effects. Always use β weights to consider the “relative
contribution” of the main effects and interaction.
One Small Difference when Plotting the Interaction of 2 Quantitative Variables
When we plotted the model for the interaction between a quantitiative variable and a categorical variable, we plotted
the separate Y-X regression line for each of the different Z groups. Now, however, we have no groups – but we need
some lines! TheXLS plotting Computator follows the usual convention, which is to plot the Y-X regression line for the
mean of Z, for +1 std above the mean of Z and -1 std below the mean of Z.
Starting with the main effect model,,,
I’ve chosen to plot the
Performance –
Complexity regression
for diffferent valuesof
Social Skill,
Because there is no
interaction term, the
slope of the
PerformanceComplexity regression
line is the same for all
values of SocialSkill
Describing the Plot of the Interaction Model
Significant Interaction regression weights

For those with average social skills,
performance increases with increases in
situational complexity (tested by
situational complexity regression term ttest)

For those with social skills +1 std,
performance increases with increases in
situational complexity (significantly
different from slope of “hard”, and we know
it is different 0 because the slope is
significantly more positive than that for the
average social skills group, which was
significantly positive)

For those with social skills -1 std,
performance appears to decrease with
increases in situational complexity
(significantly different from slope of “hard”,
but we don’t know if it is different from 0 )
Whatever the regression weight for situational
complexity (remember it is a simple effect) the
main effect is potentially misleading

the relationship between situational
complexity and the DV depends upon the
level of social skills you look at

when you have an interaction, always
check main effects for
descriptive/misleading
Whatever the regression weights for
social skills (each is a simple effect)
the main effect is potentially
misleading

the relationship between social
skills and the DV depends upon
the level of situational
complexity you look at

when you have an interaction,
always check main effects for
descriptive/misleading
Using SPSS to get & test Separate Regression Lines
At this point we know that the Performance-Situational Complexity regression line is not different from 0 for
those at the mean level of Social Skills (sitcom regression weight). We also know that the slope of the PerformanceSituational Complexity regression line varies with the level of Social Skills (i.e, the significant interaction). However, do
not have significance tests to tells us if the slopes of the Performance-Situational Complexity regression lines are
significantly different from 0 for +1 std social skills or for -1 std social skills values. When we want that formal test, we
can apply the following procedure.
Obtaining the simple regression line for complexity 1 std ABOVE the mean of social skills
-- we need to “re-center” the scores around a point one standard deviation above the mean
-- this means, in effect, that we need to “lower” all the scores by one standard deviation
-- then we need to compute an interaction term specific to these “re-centered” values
-- these new variables are used to obtain the simple regression line
SPSS: Transform  Compute
Compute the variable abvskil by subtracting the standard
deviation (8.26) from c_soskil.
Then compute the recentered interaction term abvint by
multiplying abvskil by c_sitcom.
Enter perf as the dependent variable and c_sitcom,
abvskil and abvint as the Independents.
SPSS Output:
The R² & F-test results will be the same as the original model. We did this analysis just to get the regression weight for
Situational Complexity with Social Skills re-centered.
Coeffi cientsa
Model
1
Unstandardized
Coeffic ients
B
St d. Error
(Const ant)
32.775
1.930
C_SITCOM
.735
.228
ABVSKIL
.378
.172
ABVINT
7.120E-02
.022
St andardi
zed
Coeffic ien
ts
Beta
.522
.260
.513
t
16.896
3.225
2.196
3.259
Sig.
.000
.002
.032
.002
a. Dependent Variable: PERF
The b-weight and its t-test situational complexity variable tells us that there is a significant, positive relationship
between performance and situational complexity for those 1 std above the mean on social skills (same value as was
obtained from the computator above).
One could interpret this as meaning that, for those with particularly good social skills, increased social
complexity “brings out their best”.
Obtaining the simple regression line for complexity 1 std BELOW the mean of social skills
-- we need to “recenter” the scores around a point one standard deviation below the mean
-- this means, in effect, that we need to “raise” all the scores by one standard deviation
-- then we need to compute an interaction term specific to these “recentered” values
-- these new variables are used to obtain the simple regression line
SPSS: Transform  Compute
Compute the variable belskil by adding the standard
deviation (8.26) to c_soskil.
Then compute the recentered interaction term belint by
multiplying belskil by c_sitcom.
Enter perf as the dependent variable and c_sitcom, belskil
and belint as the Independents.
SPSS Output:
The R² & F-test results will be the same as the original model. We did this analysis just to get the regression weight for
Situational Complexity with Social Skills re-centered.
Coeffi cientsa
Model
1
Unstandardized
Coeffic ients
B
St d. Error
(Const ant)
26.529
2.002
C_SITCOM
-.441
.257
BELSKIL
.378
.172
BELINT
7.120E-02
.022
St andardi
zed
Coeffic ien
ts
Beta
-.313
.260
.592
t
13.250
-1. 719
2.196
3.259
Sig.
.000
.091
.032
.002
a. Dependent Variable: PERF
The b-weight and its t-test situational complexity variable tells us that there is a significant, negative
relationship between performance and situational complexity for those 1 std below the mean on social skills (same
value as was obtained from the computator above).
One could interpret this as meaning that, for those with particularly poor social skills, do not “handle”
increased social complexity well, and their performance suffers.
You will probably notice that the significance test of the simple regression weight we have just interpreted does
not reject H0:. Is this a problem? Remember that the choice of +/- 1 standard deviation is arbitrary, so many folks
who do this sort of analysis suggest re-computing the simple slopes at +/- 1.5 standard deviations. It is likely that this
more extreme set of simple slopes will be “more different” from the slope for the mean of social skills.