Logistic Regression

Chapter 9
Math 219

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)

1. resume is from the openintro package

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 \(\textbf{log}\) function 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

1. heart_transplant is from the openintro package

• Variable is_alive
• Scatterplot
• Jittered scatterplot

• We create an indicator variable is_alive based on the variable survived

glimpse(heart_transplant)

Rows: 103
Columns: 8
$ id         <int> 15, 43, 61, 75, 6, 42, 54, 38, 85, 2, 103, 12, 48, 102, 35,…
$ acceptyear <int> 68, 70, 71, 72, 68, 70, 71, 70, 73, 68, 67, 68, 71, 74, 70,…
$ age        <int> 53, 43, 52, 52, 54, 36, 47, 41, 47, 51, 39, 53, 56, 40, 43,…
$ survived   <fct> dead, dead, dead, dead, dead, dead, dead, dead, dead, dead,…
$ survtime   <int> 1, 2, 2, 2, 3, 3, 3, 5, 5, 6, 6, 8, 9, 11, 12, 16, 16, 16, …
$ prior      <fct> no, no, no, no, no, no, no, no, no, no, no, no, no, no, no,…
$ transplant <fct> control, control, control, control, control, control, contr…
$ wait       <int> NA, NA, NA, NA, NA, NA, NA, 5, NA, NA, NA, NA, NA, NA, NA, …

heart_transplant <- heart_transplant |> mutate(is_alive = ifelse(survived=="dead",0,1))

• The original data has a lot of overlapping values

heart_transplant <- heart_transplant |> mutate(is_alive = ifelse(survived=="dead",0,1))
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()

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

• 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 \[\hat{p}=\frac{e^{-2.45}}{1+e^{-2.45}}=0.079\]