7. Interactions: Continuous*Continuous Flashcards
What is the definition of a continuous*continuous interaction?
The slope of the regression line between a continuous predictor and the outcome changes as the values of a second continuous predictor change
What is mean centering?
Shifting the scale to the average, doesn’t change the units
Changes inferences of the effect values referred to as average
What coefficient won’t change when centering a model?
The interaction term/Highest-order term is invariant to lower order scaling
Does mean centering differ between continuouscontinuous interactions and categoricalcontinuous interactions?
Typically produces a larger change in continuouscontinuous than categoricalcontinuous
What type of effects are B1 and B2 in continuous*continuous interactions?
Conditional effects, not as main effects
They are the effects at the value 0 of the interacting variable.
For any β associated with a variable not included in the interaction, interpretation does not change.
What is a simple slope?
Regression of the outcome Y on a predictor X at specific values of an interacting variable Z.
How does a simple slope differ in concon to concat?
Z isn’t binary as it is not categorical
Predicted y = (b1 + b3z)x + (b2z + b0)
Need to pick reasonable value for z
b3z = (Mean x +/- 1 SD)
How do we visualise simple slopes?
Via Johnson-Neyman Plot
What are the different features of a Johnson-Neyman plot?
y axis = conditional slopes of x effect
x axis = shows values of z
Horizontal black line = conditional slope (null)
Shaded area = Point-wise CI for simple slopes (if it crosses black line then CI includes 0)
Vertical dash line = Interval where 95% CIs include 0
What is a region of significance analysis?
Threshold of Z at which the simple slopes of y on x become significant
How do you locate the crossing point for x and z?
x = -B2/B3
z = -B1/B3
Clear cross point is dependent on magnitudes of first-order effect to high-order effect
What is an ordinal interaction?
Lines do not cross within the plausible range of measurement of x
Rank order of one predictor is maintained across levels of another.
More common in observational studies.
What is a Disordinal interaction?
Line cross within the plausible range of measurement of x
Rank order of one predictor is not maintained across levels of another.
More common in experimental work.
What is a synergistic interaction and how does it impact coefficients?
Enhancing effect.
Interaction has a bigger change than expected
All coefficients will be the same (all positive or all negative)
What is an antagonistic interaction and how does it impact coefficients?
Diminishing returns.
Strength of combined effect weakens as level of variable increases
B1 = +ve B2 = +ve B3 = -ve
B1 = -ve B2 = -ve B3 = +ve
What is a buffering interaction and how does it impact coefficients?
One variable weakens the effect of the other. Direction drive by sign on coefficient for interaction (cancelling out idea)
B1 = -ve B2 = +ve B3 = +/-
B1 = +ve B2 = -ve B3 = +/-
What is the equation for a continuous*continuous interaction when the term for x is non-linear?
Note B3xx
yi = B0 + B1xi + B2zi + B3xxi + Ei
or
yi = B0 + B1xi + B2zi + B3x2i+ Ei
How do we know to include non-linear?
Power for non-linear (higher order, interaction) effects is usually low.
This, and other features of data (e.g. skew), can lead to spurious interactions.
So the best plan is not to go looking for them unless there is solid theory
Why can including higher order terms be useful?
It can help resolve issues of violated assumptions such as linearity and heteroscedasticity
What is the statistical power for identifying interactions normally like?
Low
What does it mean if there is low power?
With low power there is also a tendency for effects to be over estimated.
Must be careful when interpreting coefficients
What coefficient won’t change when centering a model?
The interaction term/Highest-order term is invariant to lower order scaling
Type I and Type 2 error rates are increased
How does the way that interactions are referred to change once they are mean centered?
Referred to as the average
