7. Interactions: Continuous*Continuous

Question 1

What is the definition of a continuous*continuous interaction?

Answer 1

The slope of the regression line between a continuous predictor and the outcome changes as the values of a second continuous predictor change

Question 2

What is mean centering?

Answer 2

Shifting the scale to the average, doesn’t change the units

Changes inferences of the effect values referred to as average

Question 3

What coefficient won’t change when centering a model?

Answer 3

The interaction term/Highest-order term is invariant to lower order scaling

Question 4

Does mean centering differ between continuouscontinuous interactions and categoricalcontinuous interactions?

Answer 4

Typically produces a larger change in continuouscontinuous than categoricalcontinuous

Question 5

What type of effects are B1 and B2 in continuous*continuous interactions?

Answer 5

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.

Question 6

What is a simple slope?

Answer 6

Regression of the outcome Y on a predictor X at specific values of an interacting variable Z.

Question 7

How does a simple slope differ in concon to concat?

Answer 7

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)

Question 8

How do we visualise simple slopes?

Answer 8

Via Johnson-Neyman Plot

Question 9

What are the different features of a Johnson-Neyman plot?

Answer 9

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

Question 10

What is a region of significance analysis?

Answer 10

Threshold of Z at which the simple slopes of y on x become significant

Question 11

How do you locate the crossing point for x and z?

Answer 11

x = -B2/B3
z = -B1/B3

Clear cross point is dependent on magnitudes of first-order effect to high-order effect

Question 12

What is an ordinal interaction?

Answer 12

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.

Question 13

What is a Disordinal interaction?

Answer 13

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.

Question 14

What is a synergistic interaction and how does it impact coefficients?

Answer 14

Enhancing effect.

Interaction has a bigger change than expected

All coefficients will be the same (all positive or all negative)

Question 15

What is an antagonistic interaction and how does it impact coefficients?

Answer 15

Diminishing returns.

Strength of combined effect weakens as level of variable increases

B1 = +ve B2 = +ve B3 = -ve
B1 = -ve B2 = -ve B3 = +ve

Question 16

What is a buffering interaction and how does it impact coefficients?

Answer 16

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 = +/-

Question 17

What is the equation for a continuous*continuous interaction when the term for x is non-linear?

Answer 17

Note B3xx

yi = B0 + B1xi + B2zi + B3xxi + Ei

or

yi = B0 + B1xi + B2zi + B3x2i+ Ei

Question 18

How do we know to include non-linear?

Answer 18

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

Question 18

Why can including higher order terms be useful?

Answer 18

It can help resolve issues of violated assumptions such as linearity and heteroscedasticity

Question 19

What is the statistical power for identifying interactions normally like?

Answer 19

Low

Question 20

What does it mean if there is low power?

Answer 20

With low power there is also a tendency for effects to be over estimated.

Must be careful when interpreting coefficients

Question 21

What coefficient won’t change when centering a model?

Answer 21

The interaction term/Highest-order term is invariant to lower order scaling

Type I and Type 2 error rates are increased

Question 22

How does the way that interactions are referred to change once they are mean centered?

Answer 22

Referred to as the average