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 -0.75306930  -1.7993007  0.1797179  0.06032903          -0.32336662         1
## 2  1.10189574   1.4074719 -0.3185415  0.44888842          -0.44833826         1
## 3  0.38025544  -0.2554504 -0.8691029  0.79091279           0.22201270         1
## 4 -0.50416601  -1.3475335  0.9115643 -0.22670489          -1.22836340         1
## 5 -0.02845418   2.3573010 -0.3703654 -1.18330134          -0.87306263         1
## 6  0.35353883  -0.1211901  0.7102531  0.19245041          -0.08607567         1

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

##          (Intercept)          temperature             rainfall 
##           0.01766038           0.48214243           0.32101602 
## temperature:rainfall 
##          -0.08422462

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) 
##      -0.02261543       0.50498361      -0.30214780