Logistic regression

Lecture 5

2025-02-10

Warm up

Goals

Recap simple linear regression
Recap linear regression with multiple predictors
Logistic regression

Linear regression

Simple linear regression

Use simple linear regression to model the relationship between a quantitative outcome (\(Y\)) and a single quantitative predictor (\(X\)): \[\Large{Y = \beta_0 + \beta_1 X + \epsilon}\]

\(\beta_1\): True slope of the relationship between \(X\) and \(Y\)
\(\beta_0\): True intercept of the relationship between \(X\) and \(Y\)
\(\epsilon\): Error (residual)

Multiple linear regression

Use multiple linear regression to model the relationship between a quantitative outcome (\(Y\)) and a multiple quantitative predictors and factors (\(X\)): \[\Large{Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon}\]

\(\beta_1\): True slope of the relationship between \(X_1\) and \(Y\)
\(\beta_2\): True slope of the relationship between \(X_2\) and \(Y\)
\(\beta_0\): True intercept of the relationship between \(X\) and \(Y\)
\(\epsilon\): Error (residual)

Logistic regression

What is logistic regression?

Similar to linear regression…. but
Modeling tool when our response is categorical

Modelling binary outcomes

Variables with binary outcomes follow the Bernouilli distribution:
- \(y_i \sim Bern(p)\)
- \(p\): Probability of success
- \(1-p\): Probability of failure
We can’t model \(y\) directly, so instead we model \(p\)

Linear model

\[ p_i = \beta_o + \beta_1 \times X_1 + \cdots + \epsilon \]

But remember that \(p\) must be between 0 and 1
We need a link function that transforms the linear model to have an appropriate range

Logit link function

The logit function take values between 0 and 1 (probabilities) and maps them to values in the range negative infinity to positive infinity:

\[ logit(p) = log \bigg( \frac{p}{1 - p} \bigg) \]

This isn’t exactly what we need though…..

Recall, the goal is to take values between -\(\infty\) and \(\infty\) and map them to probabilities.
We need the opposite of the link function… or the inverse
Taking the inverse of the logit function will map arbitrary real values back to the range [0, 1]

Generalized linear model

We model the logit (log-odds) of \(p\) :

\[ logit(p) = log \bigg( \frac{p}{1 - p} \bigg) = \beta_o + \beta_1 \times X1_i + \cdots + \epsilon \]

Then take the inverse to obtain the predicted \(p\):

\[ p_i = \frac{e^{\beta_o + \beta_1 \times X1_i + \cdots + \epsilon}}{1 + e^{\beta_o + \beta_1 \times X1_i + \cdots + \epsilon}} \]

A logistic model visualized

Takeaways

Generalized linear models allow us to fit models to predict non-continuous outcomes
Predicting binary outcomes requires modeling the log-odds of success, where p = probability of success

Example

Read the data

df_raw_inseminations <- read.csv('df_inseminations.csv')

Data prep

Drop any columns with missing values using the drop_na() function
Convert FALSE/NEGATIVE to 0/1
Relevel to make Parity 1 the default

df_inseminations <- df_raw_inseminations %>% 
  dplyr::mutate(
    Pregnant = factor(case_when(
      ResultedInPregnancy == TRUE ~ 1, 
      ResultedInPregnancy == FALSE ~ 0
      )),
    Parity=factor(Parity),
    Parity=fct_relevel(Parity,c("1","2",">2"))
    ) %>% 
  drop_na() %>%
  dplyr::arrange(Parity)

Data overview

df_inseminations %>%
  select(InseminationNumber , Parity, Pregnant)

      InseminationNumber Parity Pregnant
1                      1      1        0
2                      1      1        0
3                      1      1        0
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 [ reached 'max' / getOption("max.print") -- omitted 174801 rows ]

Data visualization

Logistic regression model

How to interpret

cows_fit <- glm(Pregnant ~ Parity, data = df_inseminations, family = "binomial")
summary(cows_fit)


Call:
glm(formula = Pregnant ~ Parity, family = "binomial", data = df_inseminations)

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept) -0.860655   0.007856 -109.55   <2e-16 ***
Parity2     -0.124067   0.012381  -10.02   <2e-16 ***
Parity>2    -0.217551   0.011478  -18.95   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 244757  on 208133  degrees of freedom
Residual deviance: 244393  on 208131  degrees of freedom
AIC: 244399

Number of Fisher Scoring iterations: 4

Logistic regression interpretation 1

exp(coef(cows_fit))

(Intercept)     Parity2    Parity>2 
  0.4228850   0.8833209   0.8044869

Logistic regression interpretation 2

exp(confint(cows_fit))

                2.5 %    97.5 %
(Intercept) 0.4164167 0.4294401
Parity2     0.8621356 0.9050095
Parity>2    0.7865882 0.8227859