Data Science for Business Applications

Class 02 - Categorical Variables

Categorical predictors with 2 categories

What’s the impact of gender on student evaluations?

Incorporating gender into the multiple regression model

The model is given by: \[ \texttt{eval} = \beta_0 + \beta_1 \cdot \texttt{beauty} + \beta_2 \cdot \texttt{gender} + \epsilon \] Where we have that:

\(\texttt{eval}\): is the response variable - (numerical)
\(\texttt{beauty}\): is a predictor - (numerical)
\(\texttt{gender}\): is a predictor - (categorical) - two groups:
- female
- male

Incorporating gender into the multiple regression model

Gender is a categorical variable (male or female in this data set) so we can’t use it as-is as a predictor.
Idea: Recode gender into the quantitative variable 1 = male, 0 = female. In practice this choice is arbitrary.
R does this for us!
If you put a categorical variable into a model, R will alphabetically select the group that will associated with “0”, and the next to “1”.

Run the Regression Model

options(scipen = 999)
model1 <- lm(eval ~ beauty + gender, data=profs)
summary(model1)


Call:
lm(formula = eval ~ beauty + gender, data = profs)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.87196 -0.36913  0.03493  0.39919  1.03237 

Coefficients:
            Estimate Std. Error t value             Pr(>|t|)    
(Intercept)  3.88377    0.03866  100.47 < 0.0000000000000002 ***
beauty       0.14859    0.03195    4.65           0.00000434 ***
gendermale   0.19781    0.05098    3.88              0.00012 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5373 on 460 degrees of freedom
Multiple R-squared:  0.0663,    Adjusted R-squared:  0.06224 
F-statistic: 16.33 on 2 and 460 DF,  p-value: 0.0000001407

Regression a model using gender

A multiple regression predicting evaluation score from beauty and gender effectively fits two parallel regression lines:

Categorical predictors with 3+ categories

Is there a generation gap?

The variable generation is either silent (born before 1945), boomer (born 1945-1964), or genx (born after 1965).
Is generation a significant predictor of evaluations above and beyond gender and beauty?
To answer this, we need a model that includes as predictors all of gender, beauty, and generation.
But we can’t just create a variable that is 0 for the silent generation, 1 for baby boomers, and 2 for gen X—why not?
Solution is to pick a “reference category” and create dummy variables for the other categories.

OK boomer

Let’s arbitrarily pick boomers as a reference category:

Category	`genx`	`silent`
Boomers	0	0
Gen Xers	1	0
Silent Gens	0	1

R will do this automatically when you add a categorical variable with 3+ categories to a regression (it will arbitrarily pick a reference category)!

Run the Regression Model

model2 <- lm(eval ~ beauty + generation, data=profs)
summary(model2)


Call:
lm(formula = eval ~ beauty + generation, data = profs)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.82584 -0.36581  0.06317  0.42027  1.07809 

Coefficients:
                 Estimate Std. Error t value             Pr(>|t|)    
(Intercept)       4.02537    0.03201 125.742 < 0.0000000000000002 ***
beauty            0.13491    0.03360   4.015            0.0000694 ***
generationgenx   -0.06807    0.06181  -1.101                0.271    
generationsilent -0.08225    0.07889  -1.043                0.298    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.5455 on 459 degrees of freedom
Multiple R-squared:  0.03983,   Adjusted R-squared:  0.03355 
F-statistic: 6.346 on 3 and 459 DF,  p-value: 0.0003192

Analysis

All else equal (i.e., among professors of the same gender and beauty):

Gen X professors are predicted to get scores that are 0.07 points below those of boomers.
Silent gen professors are predicted to get scores that are 0.08 points below those of boomers.
Only the boomer/silent generation difference is statistically significant; Gen X professors are not significantly different than boomers.
In other words: age only seems to matter if you are really old.