Accounting for Variation

(Lesson 9)

Packages

Make sure these are attached:

library(LSTbook)
library(mosaicData)
library(tigerData)

Categorical Explanatory Variables

CPS85 Data

CPSSmall <- 
  CPS85 |> 
  filter(wage < 40)

Wage Modelled by Sex

Compute the Model Values

Model Values as a New Column

Find Variances

R^2

\(R^2\) is defined as follows:

\[R^2 = \frac{\text{Model Variance}}{\text{Response Variance}}.\]

Wage Modelled by Sex and Sector

Find Variances

R^2

Facts

\[\text{Response Var} = \text{Model Var} + \text{Residual Var}\]

Also, the more explanatory variables you have, the higher \(R^2\) will be.

Numerical Explanatory Variables

Wage Modelled by Age

Model Values as a New Column

Plot of Model Values

Find Variances

R^2

Numerical and Categorical Explanatory Variables

Wage Modelled by Sex and Sector

New Column with Model Values

Find Variances

R^2

Categorical Response Variable

Categorical Response?

So far in this lesson, the response variable has been numerical.

But we can handle a categorical response variable if it has just two levels.

The Death Penalty

?deathpenalty
View(deathpenalty)

    defrace vicrace death
1     white   white   yes
2     white   white   yes
3     white   white   yes
4     white   white   yes
5     white   white   yes
6     white   white   yes
7     white   white   yes
8     white   white   yes
9     white   white   yes
10    white   white   yes
11    white   white   yes
12    white   white   yes
13    white   white   yes
14    white   white   yes
15    white   white   yes
16    white   white   yes
17    white   white   yes
18    white   white   yes
19    white   white   yes
20    white   white    no
21    white   white    no
22    white   white    no
23    white   white    no
24    white   white    no
25    white   white    no
26    white   white    no
27    white   white    no
28    white   white    no
29    white   white    no
30    white   white    no
31    white   white    no
32    white   white    no
33    white   white    no
34    white   white    no
35    white   white    no
36    white   white    no
37    white   white    no
38    white   white    no
39    white   white    no
40    white   white    no
41    white   white    no
42    white   white    no
43    white   white    no
44    white   white    no
45    white   white    no
46    white   white    no
47    white   white    no
48    white   white    no
49    white   white    no
50    white   white    no
51    white   white    no
52    white   white    no
53    white   white    no
54    white   white    no
55    white   white    no
56    white   white    no
57    white   white    no
58    white   white    no
59    white   white    no
60    white   white    no
61    white   white    no
62    white   white    no
63    white   white    no
64    white   white    no
65    white   white    no
66    white   white    no
67    white   white    no
68    white   white    no
69    white   white    no
70    white   white    no
71    white   white    no
72    white   white    no
73    white   white    no
74    white   white    no
75    white   white    no
76    white   white    no
77    white   white    no
78    white   white    no
79    white   white    no
80    white   white    no
81    white   white    no
82    white   white    no
83    white   white    no
84    white   white    no
85    white   white    no
86    white   white    no
87    white   white    no
88    white   white    no
89    white   white    no
90    white   white    no
91    white   white    no
92    white   white    no
93    white   white    no
94    white   white    no
95    white   white    no
96    white   white    no
97    white   white    no
98    white   white    no
99    white   white    no
100   white   white    no
101   white   white    no
102   white   white    no
103   white   white    no
104   white   white    no
105   white   white    no
106   white   white    no
107   white   white    no
108   white   white    no
109   white   white    no
110   white   white    no
111   white   white    no
112   white   white    no
113   white   white    no
114   white   white    no
115   white   white    no
116   white   white    no
117   white   white    no
118   white   white    no
119   white   white    no
120   white   white    no
121   white   white    no
122   white   white    no
123   white   white    no
124   white   white    no
125   white   white    no
126   white   white    no
127   white   white    no
128   white   white    no
129   white   white    no
130   white   white    no
131   white   white    no
132   white   white    no
133   white   white    no
134   white   white    no
135   white   white    no
136   white   white    no
137   white   white    no
138   white   white    no
139   white   white    no
140   white   white    no
141   white   white    no
142   white   white    no
143   white   white    no
144   white   white    no
145   white   white    no
146   white   white    no
147   white   white    no
148   white   white    no
149   white   white    no
150   white   white    no
151   white   white    no
152   white   black    no
153   white   black    no
154   white   black    no
155   white   black    no
156   white   black    no
157   white   black    no
158   white   black    no
159   white   black    no
160   white   black    no
161   black   white   yes
162   black   white   yes
163   black   white   yes
164   black   white   yes
165   black   white   yes
166   black   white   yes
167   black   white   yes
168   black   white   yes
169   black   white   yes
170   black   white   yes
171   black   white   yes
172   black   white    no
173   black   white    no
174   black   white    no
175   black   white    no
176   black   white    no
177   black   white    no
178   black   white    no
179   black   white    no
180   black   white    no
181   black   white    no
182   black   white    no
183   black   white    no
184   black   white    no
185   black   white    no
186   black   white    no
187   black   white    no
188   black   white    no
189   black   white    no
190   black   white    no
191   black   white    no
192   black   white    no
193   black   white    no
194   black   white    no
195   black   white    no
196   black   white    no
197   black   white    no
198   black   white    no
199   black   white    no
200   black   white    no
201   black   white    no
202   black   white    no
203   black   white    no
204   black   white    no
205   black   white    no
206   black   white    no
207   black   white    no
208   black   white    no
209   black   white    no
210   black   white    no
211   black   white    no
212   black   white    no
213   black   white    no
214   black   white    no
215   black   white    no
216   black   white    no
217   black   white    no
218   black   white    no
219   black   white    no
220   black   white    no
221   black   white    no
222   black   white    no
223   black   white    no
224   black   black   yes
225   black   black   yes
226   black   black   yes
227   black   black   yes
228   black   black   yes
229   black   black   yes
230   black   black    no
231   black   black    no
232   black   black    no
233   black   black    no
234   black   black    no
235   black   black    no
236   black   black    no
237   black   black    no
238   black   black    no
239   black   black    no
240   black   black    no
241   black   black    no
242   black   black    no
243   black   black    no
244   black   black    no
245   black   black    no
246   black   black    no
247   black   black    no
248   black   black    no
249   black   black    no
250   black   black    no
251   black   black    no
252   black   black    no
253   black   black    no
254   black   black    no
255   black   black    no
256   black   black    no
257   black   black    no
258   black   black    no
259   black   black    no
260   black   black    no
261   black   black    no
262   black   black    no
263   black   black    no
264   black   black    no
265   black   black    no
266   black   black    no
267   black   black    no
268   black   black    no
269   black   black    no
270   black   black    no
271   black   black    no
272   black   black    no
273   black   black    no
274   black   black    no
275   black   black    no
276   black   black    no
277   black   black    no
278   black   black    no
279   black   black    no
280   black   black    no
281   black   black    no
282   black   black    no
283   black   black    no
284   black   black    no
285   black   black    no
286   black   black    no
287   black   black    no
288   black   black    no
289   black   black    no
290   black   black    no
291   black   black    no
292   black   black    no
293   black   black    no
294   black   black    no
295   black   black    no
296   black   black    no
297   black   black    no
298   black   black    no
299   black   black    no
300   black   black    no
301   black   black    no
302   black   black    no
303   black   black    no
304   black   black    no
305   black   black    no
306   black   black    no
307   black   black    no
308   black   black    no
309   black   black    no
310   black   black    no
311   black   black    no
312   black   black    no
313   black   black    no
314   black   black    no
315   black   black    no
316   black   black    no
317   black   black    no
318   black   black    no
319   black   black    no
320   black   black    no
321   black   black    no
322   black   black    no
323   black   black    no
324   black   black    no
325   black   black    no
326   black   black    no

Recode the death Variable

d2 <-
  deathpenalty |> 
  mutate(death_num = zero_one(death, one = "yes"))
d2

    defrace vicrace death death_num
1     white   white   yes         1
2     white   white   yes         1
3     white   white   yes         1
4     white   white   yes         1
5     white   white   yes         1
6     white   white   yes         1
7     white   white   yes         1
8     white   white   yes         1
9     white   white   yes         1
10    white   white   yes         1
11    white   white   yes         1
12    white   white   yes         1
13    white   white   yes         1
14    white   white   yes         1
15    white   white   yes         1
16    white   white   yes         1
17    white   white   yes         1
18    white   white   yes         1
19    white   white   yes         1
20    white   white    no         0
21    white   white    no         0
22    white   white    no         0
23    white   white    no         0
24    white   white    no         0
25    white   white    no         0
26    white   white    no         0
27    white   white    no         0
28    white   white    no         0
29    white   white    no         0
30    white   white    no         0
31    white   white    no         0
32    white   white    no         0
33    white   white    no         0
34    white   white    no         0
35    white   white    no         0
36    white   white    no         0
37    white   white    no         0
38    white   white    no         0
39    white   white    no         0
40    white   white    no         0
41    white   white    no         0
42    white   white    no         0
43    white   white    no         0
44    white   white    no         0
45    white   white    no         0
46    white   white    no         0
47    white   white    no         0
48    white   white    no         0
49    white   white    no         0
50    white   white    no         0
51    white   white    no         0
52    white   white    no         0
53    white   white    no         0
54    white   white    no         0
55    white   white    no         0
56    white   white    no         0
57    white   white    no         0
58    white   white    no         0
59    white   white    no         0
60    white   white    no         0
61    white   white    no         0
62    white   white    no         0
63    white   white    no         0
64    white   white    no         0
65    white   white    no         0
66    white   white    no         0
67    white   white    no         0
68    white   white    no         0
69    white   white    no         0
70    white   white    no         0
71    white   white    no         0
72    white   white    no         0
73    white   white    no         0
74    white   white    no         0
75    white   white    no         0
76    white   white    no         0
77    white   white    no         0
78    white   white    no         0
79    white   white    no         0
80    white   white    no         0
81    white   white    no         0
82    white   white    no         0
83    white   white    no         0
84    white   white    no         0
85    white   white    no         0
86    white   white    no         0
87    white   white    no         0
88    white   white    no         0
89    white   white    no         0
90    white   white    no         0
91    white   white    no         0
92    white   white    no         0
93    white   white    no         0
94    white   white    no         0
95    white   white    no         0
96    white   white    no         0
97    white   white    no         0
98    white   white    no         0
99    white   white    no         0
100   white   white    no         0
101   white   white    no         0
102   white   white    no         0
103   white   white    no         0
104   white   white    no         0
105   white   white    no         0
106   white   white    no         0
107   white   white    no         0
108   white   white    no         0
109   white   white    no         0
110   white   white    no         0
111   white   white    no         0
112   white   white    no         0
113   white   white    no         0
114   white   white    no         0
115   white   white    no         0
116   white   white    no         0
117   white   white    no         0
118   white   white    no         0
119   white   white    no         0
120   white   white    no         0
121   white   white    no         0
122   white   white    no         0
123   white   white    no         0
124   white   white    no         0
125   white   white    no         0
126   white   white    no         0
127   white   white    no         0
128   white   white    no         0
129   white   white    no         0
130   white   white    no         0
131   white   white    no         0
132   white   white    no         0
133   white   white    no         0
134   white   white    no         0
135   white   white    no         0
136   white   white    no         0
137   white   white    no         0
138   white   white    no         0
139   white   white    no         0
140   white   white    no         0
141   white   white    no         0
142   white   white    no         0
143   white   white    no         0
144   white   white    no         0
145   white   white    no         0
146   white   white    no         0
147   white   white    no         0
148   white   white    no         0
149   white   white    no         0
150   white   white    no         0
151   white   white    no         0
152   white   black    no         0
153   white   black    no         0
154   white   black    no         0
155   white   black    no         0
156   white   black    no         0
157   white   black    no         0
158   white   black    no         0
159   white   black    no         0
160   white   black    no         0
161   black   white   yes         1
162   black   white   yes         1
163   black   white   yes         1
164   black   white   yes         1
165   black   white   yes         1
166   black   white   yes         1
167   black   white   yes         1
168   black   white   yes         1
169   black   white   yes         1
170   black   white   yes         1
171   black   white   yes         1
172   black   white    no         0
173   black   white    no         0
174   black   white    no         0
175   black   white    no         0
176   black   white    no         0
177   black   white    no         0
178   black   white    no         0
179   black   white    no         0
180   black   white    no         0
181   black   white    no         0
182   black   white    no         0
183   black   white    no         0
184   black   white    no         0
185   black   white    no         0
186   black   white    no         0
187   black   white    no         0
188   black   white    no         0
189   black   white    no         0
190   black   white    no         0
191   black   white    no         0
192   black   white    no         0
193   black   white    no         0
194   black   white    no         0
195   black   white    no         0
196   black   white    no         0
197   black   white    no         0
198   black   white    no         0
199   black   white    no         0
200   black   white    no         0
201   black   white    no         0
202   black   white    no         0
203   black   white    no         0
204   black   white    no         0
205   black   white    no         0
206   black   white    no         0
207   black   white    no         0
208   black   white    no         0
209   black   white    no         0
210   black   white    no         0
211   black   white    no         0
212   black   white    no         0
213   black   white    no         0
214   black   white    no         0
215   black   white    no         0
216   black   white    no         0
217   black   white    no         0
218   black   white    no         0
219   black   white    no         0
220   black   white    no         0
221   black   white    no         0
222   black   white    no         0
223   black   white    no         0
224   black   black   yes         1
225   black   black   yes         1
226   black   black   yes         1
227   black   black   yes         1
228   black   black   yes         1
229   black   black   yes         1
230   black   black    no         0
231   black   black    no         0
232   black   black    no         0
233   black   black    no         0
234   black   black    no         0
235   black   black    no         0
236   black   black    no         0
237   black   black    no         0
238   black   black    no         0
239   black   black    no         0
240   black   black    no         0
241   black   black    no         0
242   black   black    no         0
243   black   black    no         0
244   black   black    no         0
245   black   black    no         0
246   black   black    no         0
247   black   black    no         0
248   black   black    no         0
249   black   black    no         0
250   black   black    no         0
251   black   black    no         0
252   black   black    no         0
253   black   black    no         0
254   black   black    no         0
255   black   black    no         0
256   black   black    no         0
257   black   black    no         0
258   black   black    no         0
259   black   black    no         0
260   black   black    no         0
261   black   black    no         0
262   black   black    no         0
263   black   black    no         0
264   black   black    no         0
265   black   black    no         0
266   black   black    no         0
267   black   black    no         0
268   black   black    no         0
269   black   black    no         0
270   black   black    no         0
271   black   black    no         0
272   black   black    no         0
273   black   black    no         0
274   black   black    no         0
275   black   black    no         0
276   black   black    no         0
277   black   black    no         0
278   black   black    no         0
279   black   black    no         0
280   black   black    no         0
281   black   black    no         0
282   black   black    no         0
283   black   black    no         0
284   black   black    no         0
285   black   black    no         0
286   black   black    no         0
287   black   black    no         0
288   black   black    no         0
289   black   black    no         0
290   black   black    no         0
291   black   black    no         0
292   black   black    no         0
293   black   black    no         0
294   black   black    no         0
295   black   black    no         0
296   black   black    no         0
297   black   black    no         0
298   black   black    no         0
299   black   black    no         0
300   black   black    no         0
301   black   black    no         0
302   black   black    no         0
303   black   black    no         0
304   black   black    no         0
305   black   black    no         0
306   black   black    no         0
307   black   black    no         0
308   black   black    no         0
309   black   black    no         0
310   black   black    no         0
311   black   black    no         0
312   black   black    no         0
313   black   black    no         0
314   black   black    no         0
315   black   black    no         0
316   black   black    no         0
317   black   black    no         0
318   black   black    no         0
319   black   black    no         0
320   black   black    no         0
321   black   black    no         0
322   black   black    no         0
323   black   black    no         0
324   black   black    no         0
325   black   black    no         0
326   black   black    no         0

Model Death by Defendant’s Race

d2 |> 
  point_plot(
    death_num ~ defrace,
    annot = "model"
  )

A Numerical View

d2 |> 
  summarize(count = n(), .by = c(death_num, defrace)) |> 
  mutate(total_race = sum(count), .by = defrace) |> 
  mutate(proportion = count / total_race)

  death_num defrace count total_race proportion
1         1   white    19        160  0.1187500
2         0   white   141        160  0.8812500
3         1   black    17        166  0.1024096
4         0   black   149        166  0.8975904

Surprising, considering most people think the justice system in Florida was biased against black people in the mod-1970s.

But …

A Lurking Variable: Race of the Victim

d2 |> 
  point_plot(
    death_num ~ defrace + 1 + vicrace,
    annot = "model"
  )

Death Penalty Modelled on Race of Victim

d2 |> 
  point_plot(
    death_num ~ vicrace,
    annot = "model"
  )

Numerical Point of View

d2 |> 
  summarize(count = n(), .by = c(death_num, vicrace)) |> 
  mutate(total_race = sum(count), .by = vicrace) |> 
  mutate(proportion = count / total_race)

  death_num vicrace count total_race proportion
1         1   white    30        214 0.14018692
2         0   white   184        214 0.85981308
3         0   black   106        112 0.94642857
4         1   black     6        112 0.05357143

If there was bias, it was against black victims, not so much against black defendants.