What if we have two continuous variables we allow to interact?
Meaning of parameters change when you add interaction:
The coefficients are now non-linear
coefficients in model without interaction:
change in outcome per unit change in predictor
e.g., when age increase 1 year, MaxHR increase 0.50 BPM
Coefficients within interactions:
Change in outcome per unit change in predictor when other predictor is zero
e.g., change in MaxHR per unit change in age when HD is zero
Interpreting continuous interaction
α: when x and z = 0
z: change in y per unit change z, when x = 0.
x: change in y per unit change x, when z = 0.
xz: interaction. Don’t even try… Just plot!
We can center the variables, but it’s not enough:
output of summary of non-center
output of summary of center
Triptych plotting
explain
Water on shade
Distribution of BMI.
Number of those with glycemia.
BMI and glycemia, side by side.
Shade on water
Distribution of BMI.
Number of those with glycemia.
BMI and glycemia, side by side.
Non-linear model from interaction only
Show example where at a certain point, the variable doesn’t have an effect anymore… E.g., when patient has too low functioning, therapy doesn’t work anymore!
What about logistic regression?
To determine what the interaction means for your outcome, evaluate the interaction coefficient’s mathematical sign:
Positive Interaction (β₃ > 0):
The combined effect of both variables is greater than the sum of their individual effects. In terms of odds, the effect of X₁ is amplified as X₂ increases.
Negative Interaction (β₃ < 0):
The combined effect is less than the sum of their individual effects. The effect of X₁ is dampened or diminishes as X₂ increases.
What about logistic regression?
Differing Interpretations
Log-Odds Scale: An interaction term here assesses whether the multiplicative effect of the odds of an outcome changes based on another variable
Probability Scale: An interaction term here assesses whether the combined effect of two variables is greater or smaller than the sum of their individual effects.
Interaction madness
3-way interaction ?
4-way interaction ?
10-way interaction ?
50-way interaction ?
Example of interaction madness
Genetics x environment
We believe that genes are expressed conditional on lived experiences:
