How They Differ & What They Share

• Logistic regression is used when the outcome is dichotomous
  • Has / doesn’t have a disorder, recovers / dies, etc.
• General linear regression is used when the outcome is continuous
  • Or at least can be assumed to be so, even if we have only a few levels

The Problem of a Dichotomous Outcome

• Ma, He, & Ouyang (2022) investigated pneumonia deaths in ICUs from patients’ age, etc.
• Found (among other things¹):

• I.e., as one ages, they become significantly less likely to survive pneumonia in an ICU
• Let’s predict survival by age…

Predicting Survival by Age

• The mean age of those who died was ~90 years
• SDs for age were roughly 4 years
• And β = -0.07
• So:

Dichotomous Variables Wholly Violate the Normality Assumption

library(ggplot2)

# Calculate mean values for Outcome = 0 and Outcome = 1
mean_outcome_0 <- mean(df$Age[df$Outcome == 0])
mean_outcome_1 <- mean(df$Age[df$Outcome == 1])

# Create the plot
plot <- ggplot(df, aes(x = Age, y = Outcome)) +
  geom_point(size = 3, shape = 1, color = "blue", alpha = 0.8) +
  scale_x_continuous(name = "Patient Age in Years", breaks = seq(80, 100, by = 5)) +
  scale_y_continuous(name = "Outcome of Pneumonia (0 = Died, 1 = Lived)", breaks = seq(-0.2, 1.2, by = 0.2), limits = c(-0.2, 1.2)) +
  theme_minimal(base_family = "serif") +
  theme(
    axis.title = element_text(face = "plain", size = 12),  # Set face to "plain" for non-italic
    axis.text = element_text(size = 10),
    legend.position = "none"
  )

ggsave("images/outcome_plot.svg", plot, width = 6, height = 5)

Plot of Simulated Data from Ma et al.

The Solution

• The solution to using linear regression for dichotomous data is … to not use dichotomous data
• Instead, we essentially transform the outcome into a probability
  • “Essentially”

Using Logs

• We use odds instead of probabilities because the math is easier

• But the odds themselves can be computed from the probabilities:
  \[\text{Odds of Surviving} = \frac{P_{Surviving}}{1 - P_{Surviving}}\]

• We then take the natural log of that odds:
  \[\text{ln}({\text{Odds}}) = \text{ln} \left( \frac{P_{Outcome}}{1 - P_{Outcome}} \right)\]

• This natural log of an odds is called a logit

  • This logit is computed for every row in the data
  • And is used instead of the dichotomous outcome

Actual Equation

• Therefore, the actual equation tested in logistic regression is:

• It follows the same form as other linear regressions
  • It just transforms the outcome into a different value

Assumptios

• The assumptions of logistic regression are the same as for other linear regression models (using OLS or MLE):
  • Observations are independent
  • There is no severe multicollinearity among predictors
  • Data and error are roughly normally distributed
  • The relationship between each predictor and the logit of the outcome is roughly linear
    • (This assumption can be tested using the Box-Tidwell test)
  • The sample size is sufficiently large
    • As a rule of thumb, one should have at least 10 cases with the least frequent outcome for each predictor

Types of Tests

• As noted above, with logistic regression, we can test all of the things we can with general linear regression models
• However, the names are sometimes different
  • Viz., we use the Wald χ² test instead of t-tests

Three Main Types of Logistic Regression

• Binary logistic regression
  • The outcome is dichotomous
• Multinomial logistic regression
  • The outcome can include three or more categories
  • There is no natural ordering among the categories
• Ordinal logistic regression
  • The outcome can belong to one of three or more categories
  • And there is a natural ordering among the categories

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