Logistic Regression

Chapter 9
Math 215

Discrimination in Hiring

Does perceived race or sex of an applicant affect job application callback rates?
Data from an experiment (Bertrand and Mullainathan, 2003)
Researchers generated fake resumes with different characteristics

Randomly assigned a name to each resume
Name implied applicant’s race (Black or White) and sex (male or female)
Study preceded by separate survey to confirm association between names and race/sex

resume ¹ data from 4,870 applications
30 variables, including

Variable	Description
`received_callback`	Whether applicant received call from employer
`job_city`	Location of job (Boston or Chicago)
`college_degree`	Indicator: whether resume listed college degree
`years_experience`	Number of years of experience listed on resume
`honors`	Indicator: whether resume listed some sort of honors (e.g., employee of the month)
`military`	Indicator: whether resume listed military experience
`has_email_address`	Indicator: whether resume listed applicant’s email address
`race`	Race of applicant (implied by first name)
`sex`	Sex of applicant (implied by first name)

Let’s look at the data.

resume

EDA

Sample sizes

race	female	male
black	1,886	549
white	1,860	575

Proportions of applicants receiving calls back from employer

race	female	male
black	0.0663	0.0583
white	0.0989	0.0887

Regression with a Categorical Response?

We would like to build a model to predict whether an applicant will receive a call back from an employer
The response variable, received_callback, is categorical with two levels: 0 (no) and 1 (yes)
We could treat the response as numeric (it already has indicator coding) and fit a linear regression model
This doesn’t make much sense, because the linear model will predict some values for the response that are between 0 and 1, and others that are less than 0 or greater than 1. How do we interpret those predictions?

With a binary (2 levels) categorical response, we can use logistic regression to construct a model
Logistic regression predicts the probability of success, \(p\), instead of predicting the value of the response
In the hiring discrimination example, we consider receiving a call back (received_callback = 1) to be a success

A logistic model would then predict the probability of receiving a call back
Often this probability would then be used to make a prediction of the value of the response (“yes/1” if \(\hat{p}_i\geq0.5\), “no/0” if \(\hat{p}_i<0.5\))

We can think of a logistic model as fitting a linear model to the relationship between a transformation of the probability and the predictors \[\log\left(\frac{\hat{p}}{1-\hat{p}}\right)=b_0+b_1x_1+b_2x_2+\cdots+b_kx_k\]
The logarithm is the natural logarithm
The transformation \(\log\left(\frac{p}{1-p}\right)\) is referred to as the logit transformation or the log-odds

The quantity \(\frac{p}{1-p}\) is the odds (often encountered in betting)
E.g., in a basketball game the probability that team 1 will win is 3/4 and the probability that team 2 will win is 1/4. The odds that team 1 will win are (3/4)/(1/4) = 3/1 (“3 to 1”)
In the employment discrimination problems the odds are the odds of receiving a call back
Unlike \(p\), the odds can take on values in the interval \([0,\infty)\), and the log-odds can take on values in the interval \((-\infty, \infty)\)

We can solve the following relationship for \(\hat{p}\): \[\log\left(\frac{\hat{p}}{1-\hat{p}}\right)=b_0+b_1x_1+b_2x_2+\cdots+b_kx_k\]
We obtain \[\hat{p}=\frac{e^{b_0+b_1x_1+b_2x_2+\cdots+b_kx_k}}{1+e^{b_0+b_1x_1+b_2x_2+\cdots+b_kx_k}}\]
Once we have fit the model, this allows us to predict the probability of success for an observation

Heart Patient Survival

Before we fit a logistic model to the employee discrimination data, let’s consider a simpler example
The heart_transplant ¹ dataset is from a study that tracked 5-year survival rates of heart transplant candidates
We will explore how age (a single predictor) affects survival rate for these patients

Scatterplot
Jittered scatterplot

The original data has a lot of overlapping values

ggplot(data = heart_transplant, aes(x = age, y = is_alive)) + 
  geom_point()+
  theme_minimal()

Scatter plot showing survival vs. age. Points are *jittered*.

We added vertical “jitter” for better visualization

ggplot(data = heart_transplant, aes(x = age, y = is_alive)) + 
  geom_jitter(width = 0, height = 0.05, alpha = 0.5) +
  theme_minimal()

We can fit a logistic model to the data

mod <- glm(is_alive ~ age, family = binomial, data = heart_transplant)

tidy(mod)

The result is the model \[\log\left(\frac{\hat{p}}{1-\hat{p}}\right) = 1.6 - 0.058\times age\]
We can see from the model that the odds of survival are predicted to decrease with age
Solving for the predicted probability of survival yields \[\hat{p}=\frac{e^{1.6 - 0.058\times age}}{1+e^{1.6 - 0.058\times age}}\]

Scatter plot (jittered) showing survival vs. age. Curve shows predicted probability of survival using logistic model..

Fitting a Logistic Model to the Employment Discrimination Data

We fit a model to predict whether an applicant received a call back using all of the other variables in the table as predictors.

mod <- glm(received_callback ~ job_city + college_degree + years_experience +
             honors + military + has_email_address + race + sex,
           family = binomial, data = resume)

tidy(mod)

The resulting model is

\[\begin{array}{rcl}\log\left(\frac{\hat{p}}{1-\hat{p}}\right) &=& -2.66 \\ & - & 0.44\times job\_cityChicago \\ & - & 0.07 \times college\_degree \\ & + & 0.020 \times years\_experience \\ & + & 0.77 \times honors \\ & - & 0.34 \times military \\ & + & 0.22 \times has\_email\_address \\ & + & 0.44 \times racewhite \\ & - & 0.18 \times sexm\end{array} \]

Using a Logistic Model to Make Predictions

Use the model to predict the probability of a call back for an application with the following characteristics:

Variable	Value
`job_city`	Boston
`college_degree`	has college degree
`years_experience`	3
`honors`	No honors
`military`	No military experience
`has_email_address`	Resume has email address
`race`	Black
`sex`	Female

\[\begin{array}{rcl}\log\left(\frac{\hat{p}}{1-\hat{p}}\right) &=& -2.66 \\ & - & 0.44\times 0 \\ & - & 0.07 \times 1 \\ & + & 0.020 \times 3 \\ & + & 0.77 \times 0 \\ & - & 0.34 \times 0 \\ & + & 0.22 \times 1 \\ & + & 0.44 \times 0 \\ & - & 0.18 \times 0 \\ & = & -2.45\end{array} \]

Since \(\log\left(\frac{\hat{p}}{1-\hat{p}}\right)=-2.45\), the predicted probability that the applicant will receive a call back is \[\hat{p}=\frac{e^{-2.45}}{1+e^{-2.45}}=0.079\]