Statistical interactions: what are they and what do they mean, anyway?

Quentin D. Read

Who is this talk for?

  • Scientists who do experiments or collect observational data
  • People who want to learn how different causal factors interact to explain the world

Quiz

Which of these represents an interaction between treatment and sex? Why or why not?

A: NO INTERACTION. Treatment has the same effect on both sexes.

B: INTERACTION. Effect of treatment depends on sex. no effect in females, positive effect in males.

C: NO INTERACTION. Males are different than females, but treatment has no effect regardless of sex.

D: INTERACTION. Effect of treatment depends on sex. small effect in females, large effect in males.

What is a statistical interaction?

  • When a predictor variable affects the response, and that effect depends on the value of another predictor variable.

Basic recap of a linear model

  • Predict the mean of a response variable y
  • Linear combination of predictor variables x
    • We’ll ignore fixed vs. random effects for now. Anyway, they both are different kinds of variables that help us predict and explain variation in y.
  • Error term (model residuals)
    • Simplest linear model assumes error is normally distributed
    • We’ll ignore generalized linear models with link functions and non-normal error distributions for now. The interpretation of interactions isn’t changed by this.

\[y = \beta_0 + \beta_1 x + \epsilon\]

\[ \hat{y} = \beta_0 + \beta_1 x\]

Model with more than one predictor

  • In the linear model framework, when we include multiple explanatory variables in a model, their effects are added

\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon\]

  • This does not allow for interactions

Model with more than one predictor: Example

  • Effect of blood pressure \(x_1\) on heart rate \(y\) in coffee drinkers and non coffee drinkers \(x_2\)
  • Continuous outcome variable, one continuous predictor and one categorical predictor with two possible values
  • The categorical predictor takes values of either 0 or 1. In this case no coffee = 0 and coffee = 1

  • In this case it’s still a good model but only because the effect of \(x_1\) does not depend on \(x_2\) and vice versa!
  • Here we show marginal trends for the relationship of \(y\) and \(x_1\) for each value of \(x_2\)
  • Slopes for coffee drinkers and non coffee drinkers are the same

Model with an interaction between a continuous and categorical predictor

\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2 + \epsilon\]

  • This is probably the easiest type of interaction to conceptualize and interpret
  • \(x_1\) is a continuous variable, \(x_2\) is a binary variable that can take values of 0 or 1
  • The effect of \(x_1\) of \(y\) depends on \(x_2\) and vice versa
  • Still a linear combination of variables, we just have a new variable defined by taking the product of \(x_1\) and \(x_2\)

Model with an interaction between a continuous and categorical predictor: Example

  • Income (\(y\)) depends on years of education (\(x_1\)) in males and females (\(x_2\))
  • Males have higher average income and steeper slope of income vs. education
  • Thus the effect of education on income depends on sex — interaction!

Model with an interaction between two continuous predictors

\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2 + \epsilon\]

  • Slightly tougher to conceptualize but still the effect of \(x_1\) of \(y\) depends on \(x_2\) and vice versa
  • Same equation as before (does not matter if predictors are continuous or categorical)

Model with an interaction between two continuous predictors: Example

  • How do calories consumed per day (\(x_1\)) and minutes of exercise per day (\(x_2\)) affect body mass index (\(y\))?
  • Visualizing continuous interactions often requires:
    • picking a few values for one of the predictor variables
    • plotting predicted trendlines between \(y\) and the other predictor at each of those levels

Model with an interaction between categorical predictors

\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2 + \epsilon\]

  • Same equation as previously, now both \(x\) variables are binary with values of 0 or 1
  • “Dummy variables” can be used if the categorical variables have more than two possibilities but the math is the same
  • Easy to plot and visualize interactions between categorical variables

Three-way interactions

Time to blow your minds!

Quiz number two

Which of these represents a three-way interaction between treatment, sex, and age? Why or why not?

A: NO 3-WAY INTERACTION.Two-way treatment by age interaction, and main effect of sex, but sex does not interact with either treatment or age

B: 3-WAY INTERACTION. The treatment is most effective in young males (treatment by sex by age). There is also a two-way treatment by age interaction

Three-way interactions and more

  • The same logic applies for three-way interactions, and higher, as for two-way
  • The effect of \(x_3\) depends on the combination of \(x_2\) and \(x_1\)
  • Or equivalently, the effect of the combination of \(x_3\) and \(x_2\) depends on \(x_1\), etc.

Issues with interactions

  • Discussing main effects in the presence of an interaction
  • Incorrectly diagnosing interactions
  • Nonlinear interactions
  • The data scale matters

Discussing main effects if there’s an interaction

  • Trying to make conclusions about the main effect in the presence of an interaction is not “always wrong” but can be dangerous.
  • Same goes for trying to make conclusions about two-way interactions when a three-way or even higher order interaction exists.
  • It is tempting to try to oversimplify but sometimes it isn’t possible.

How would you discuss each of these scenarios?

Can you say “there is evidence for a positive treatment effect” in any of these cases? How do interactions, or lack thereof, affect how you would describe them?

Incorrectly diagnosing interactions

  • We can incorrectly say there’s an interaction if the explanatory variables are correlated with each other
  • This occurs more in observational data and is not as much of an issue if the treatments are experimentally randomized

Incorrectly diagnosing interactions: Example

Candidate G×E research testing the hypothesis that an environmental exposure in childhood interacts with genetic factors to determine an outcome (e.g., depression)

  • Even if you include covariates like age, gender, ethnicity in the model, it will not fully control for the effect
  • You also need to account for the interactions of covariate with environment and covariate with gene
  • Many studies find G×E effects when it is really something like a covariate × environment effect (for example, certain ethnic groups may be less likely to report depression)
    • Keller 2014, Biological Psychiatry

Nonlinear relationships and nonlinear interactions

  • Some relationships may be nonlinear, and interactions too
  • For instance, the interactive effect of age and flu exposure on mortality
  • Babies and elderly are vulnerable; young adults are not as vulnerable
  • If age is a continuous variable, the effect of flu on mortality does not increase linearly with age
  • Just throwing an age × exposure interaction term into your model will not give good results
  • You might need to add a quadratic interaction term

The scale of the data matters

\[\log y = \beta_0 + \beta_1 x_1 + \beta_2 x_2\]

  • No interaction on log scale = interaction on linear scale

\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2\]

  • Interaction on log scale = no interaction on linear scale

  • If \(x_1\) and \(x_2\) don’t interact on the linear scale, they do interact on the log scale, and vice versa
  • It’s always important to have a good reason to transform your data (not just to “make it normal”), especially when you have interactions in your model
    • Duncan & Kefford 2021, Methods in Ecology & Evolution

Questions?