1.3 Interactions and non-linear effects

1.3.1 Interactions

\[ y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \beta_3 x_{1i}x_{2i} + \epsilon_i \]

We can specify the interaction between two predictors by adding an interactions list to the parameters list. Interactions can then be specified between two named variables using “:”. Interactions can be specified between two predictors at the same level, or at different hierarchical levels.

squid_data <- simulate_population(
  n=2000,
  response_name = "body_mass",
  parameters=list(
    observation=list(
      names=c("temperature","rainfall"),
      beta = c(0.5,0.3)
    ),
    residual=list(
      vcov=0.3
    ),
    interactions=list(
      names=c("temperature:rainfall"),
      beta = c(-0.1)
    )
  )
)

data <- get_population_data(squid_data)
head(data)

##    body_mass temperature    rainfall    residual temperature:rainfall squid_pop
## 1  0.3684694 1.933232214  0.19203892 -0.61863284         0.3712558292         1
## 2 -0.4730299 0.081011530 -1.32092129 -0.12796029        -0.1070098551         1
## 3 -1.1576776 0.088942791 -0.70727428 -0.99625744        -0.0629069479         1
## 4  0.1311283 0.004136626 -0.08379699  0.15416441        -0.0003466369         1
## 5  0.1344398 0.296427406 -0.27937107  0.06175613        -0.0828132420         1
## 6 -1.4594139 0.555758164 -1.15559295 -1.45483810        -0.6422302156         1

coef(lm(body_mass ~ temperature * rainfall, data))

##          (Intercept)          temperature             rainfall 
##           0.01175390           0.50865026           0.33617972 
## temperature:rainfall 
##          -0.09533579

1.3.2 Non-linear effects

Polynomial (quadratic, cubic, etc) functions are essentially interactions with the same predictor. They can therefore be specified in the same way:

squid_data <- simulate_population(
  n=2000,
  response_name = "body_mass",
  parameters=list(
    observation=list(
      names=c("temperature"),
      beta = c(0.5)
    ),
    interactions=list(
      names=c("temperature:temperature"),
      beta = c(-0.3)
    ),
    residual=list(
      vcov=0.3
    )
  )
)
data <- get_population_data(squid_data)

plot(body_mass ~ temperature, data, pch=19, cex=0.5, col=alpha(1,0.5))

coef(lm(body_mass ~ temperature + I(temperature^2), data))

##      (Intercept)      temperature I(temperature^2) 
##     -0.002908256      0.516327302     -0.291477043