Many X Variables

Module 6.4: The Linear Probability Model

Author
Affiliation

Alex Cardazzi

Old Dominion University

All materials can be found at alexcardazzi.github.io.

Binary Outcomes

So far in this module, we have learned about binary explanatory variables. However, what if your outcome variable is binary? We can still use OLS for this!

Let’s consider some data on police treatment of individuals arrested in Toronto for simple possession of small quantities of marijuana (data, documentation). In these data, we can observe whether the arrestee was released, their race, age, sex, employment status, citizen status and previous police interactions.

Let’s build a model to explain whether the arrestee was released given some of these controls.

Code 
df <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/carData/Arrests.csv")
df$released <- ifelse(df$released == "Yes", 1, 0)
df$black <- ifelse(df$colour == "Black", 1, 0)
df$female <- ifelse(df$sex == "Female", 1, 0)
df$citizen <- ifelse(df$citizen == "Yes", 1, 0)
df$employed <- ifelse(df$employed == "Yes", 1, 0)
summary(r_lpm <- lm(released ~ female + citizen + employed + age + black + checks, df))
Output 

Call:
lm(formula = released ~ female + citizen + employed + age + black + 
    checks, data = df)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.98272  0.03631  0.09415  0.19252  0.60548 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.7405159  0.0238808  31.009  < 2e-16 ***
female      -0.0034265  0.0179731  -0.191    0.849    
citizen      0.0892898  0.0143901   6.205 5.89e-10 ***
employed     0.1236378  0.0125926   9.818  < 2e-16 ***
age          0.0004879  0.0006049   0.807    0.420    
black       -0.0573499  0.0119889  -4.784 1.77e-06 ***
checks      -0.0499783  0.0033967 -14.714  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.3581 on 5219 degrees of freedom
Multiple R-squared:  0.09551,   Adjusted R-squared:  0.09447 
F-statistic: 91.85 on 6 and 5219 DF,  p-value: < 2.2e-16

Since the outcome is either 0 or 1, the average of the outcome is the probability of being released. This means that a coefficient is the change in that probability when the explanatory variable increases by one unit. Because probabilities are already percentages, we describe these changes in percentage points. A model like this is called a linear probability model.

Coefficient Interpretations
  1. The coefficient for female suggests that females are 0.3 percentage points less likely than men to be released following an arrest for small amounts of marijuana. However, this coefficient is not statistically different from zero, so we do not have evidence that this difference is meaningful.
  2. The estimate for citizen suggests that, relative to non-citizens, citizens are 8.9 percentage points more likely to be released.
  3. Similarly, those who are employed, relative to those who are unemployed, are 12.4 percentage points more likely to be released.
  4. An increase in age of one year results in an (insignificant) 0.05 percentage point increase in the probability of being released.
  5. Individuals who are black, controlling for other factors, are less likely by 5.7 percentage points to be released following an arrest.
  6. Each additional previous police check (checks) reduces the probability of being released by 5 percentage points.
  7. The intercept suggests a baseline probability of being released of 74.1%. Note that this applies when all other variables are equal to zero. Specifically, this says that the probability of being released for unemployed white male non-citizens with no previous arrest history and who are age zero is 74.1%.
Are these results causal?

An interesting, albeit unsurprising, result of this regression is that black Canadians are less likely to be released than white Canadians. Can we interpret this coefficient as causal? Recall the four reasons why we might find a correlation:

  1. The causal interpretation: race causes release. To settle on this one, we need to rule out the other three.
  2. The reverse causality interpretation: release causes race. This one is clearly implausible, and can be promptly ruled out.
  3. The third variable interpretation: is there some third variable that is causing both race and release? Unfortunately, given this setting/regression, we cannot rule out there being some third factor that is correlated with race and whether an individual is released following an arrest.
  4. The random chance interpretation: we found this correlation by “accident”. This is unlikely given the number of observations we have and our statistical tests. We can probably rule this out.
If we could figure out a clever way (e.g. a Randomized Control Trial) to rule out all possible third variables, we could make causal claims. Unfortunately, we will not cover that in this course. However, if you want to stick around and take ECON 400, we’ll get to things like this!