Simulation of how simple logistic regresion can fail to model binomial response when the underlying function is more complicated.

Logistic Simulations

# Simulation of how simple logistic regression can fail to model
# binomial response when the underlying function is more complicated
install.packages(c('data.table', 'plyr', 'ggplot2', 'drc'))
library(data.table)
library(plyr)
library(ggplot2)
library(drc)

# Generalised logistic function
logistic_function <- function(x) {
  0.25+0.5/(1+exp(-(1*(x-12)+0.1)))
}

# Range to explore function
numeric_variable <- 1:20

# Conduct sampling
sample_frame <- adply(
  .data=numeric_variable
  , .margins=1
  , .id=NULL
  , .fun=function(x) {
    p <- logistic_function(x)
    data.frame(
      outcome=sample(c(0,1),100,replace=TRUE,prob=(c(1-p, p)))
      , numeric_variable=x
    )
  }
)

# Calculate conversion at each level of numeric variable
sample_summary <- data.table(sample_frame)[
  ,list(
    outcome=sum(outcome)/.N
    , N=.N
  )
  ,keyby=numeric_variable
]

# Actual generating function
sample_summary[,'actual_function'] <- logistic_function(numeric_variable)

# 2-parameter logistic regression
sample_summary[,'logistic_prediction_2p'] <- predict(
  glm(outcome ~ numeric_variable, data=sample_frame, family="binomial")
  , data.frame(numeric_variable=numeric_variable)
  , type="response"
)

# 4-parameter logistic regression
sample_summary[,'logistic_prediction_4p'] <- predict(
  drm(outcome ~ numeric_variable, weights=N, fct=LL.4(), data=sample_summary, type="binomial")
  , data.frame(numeric_variable=numeric_variable)
)

# Comparison plot
ggplot(
  data=sample_frame
  , aes(x=numeric_variable, y=outcome)
)+
geom_point(position='jitter')+
geom_line(data=sample_summary, aes(color="Observed Conversion"), size=1)+
geom_line(data=sample_summary, aes(y=logistic_prediction_2p, color="2-Parameter Logistic"), size=1)+
geom_line(data=sample_summary, aes(y=logistic_prediction_4p, color="4-Parameter Logistic"), size=1)+
geom_line(data=sample_summary, aes(y=actual_function, color="Generating Function"), size=1)+
scale_y_continuous("Outcome")+
scale_x_continuous("Numeric Variable")+
guides(color=guide_legend("Models"))