• Visualization: good and bad graphs
• Break
• R exercise on gender discrimination
• Discussion + suggested solutions

Each team has 5 mins to discuss pros and cons of assigned graph (can be found on the canvas):

• Team 1: work_life.png
• Team 2: military_spending.png
• Team 3: titanic_casualties.png

Poor numerical reasoning and misleading presentations are common problems

Always consider:

• Is the evidence misrepresented?
• Was there an agenda being pursued?

• What does it mean, “is there discrimination?”
  
  Suppose overall acceptance rate is 30%

• 50 men apply

• 30 women apply

Suppose overall acceptance rate is 30%

• 50 men apply 15 are accepted

• 30 women apply 9 are accepted

  Is this evidence of discrimination?

Suppose overall acceptance rate is 30%

• 50 men apply 23 are accepted

• 30 women apply 1 are accepted

  Is this evidence of discrimination?

Suppose overall acceptance rate is 30%

• 50 men apply 18 are accepted

• 30 women apply 6 are accepted

  Is this evidence of discrimination?
  

  Where is the threshold for us to say there is discrimination happening?

Question #1: What was the overall acceptance rate for male vs. female applicants?

Question #2: Is there a significant difference or not?

Question #3: What are the male and female admission rates for each department?

Question #4: Is there evidence for sex-based discrimination? Is there evidence against discrimination?

x <- read.csv("AdmissionsData.csv")
dim(x)

## [1] 4425    3

head(x)

##   Department Sex Admitted
## 1    Biology   M      Yes
## 2    Biology   M      Yes
## 3    Biology   M      Yes
## 4    Biology   M      Yes
## 5    Biology   M      Yes
## 6    Biology   M      Yes

table(x$Department)

## 
##     Biology     English     History Mathematics  Philosophy  Psychology 
##         613         584         792         933         585         918

table(x$Department, x$Sex)

##              
##                 F   M
##   Biology     341 272
##   English     393 191
##   History     375 417
##   Mathematics 108 825
##   Philosophy   25 560
##   Psychology  593 325

• What was the overall acceptance rate for men vs. women?
• You have 5 min to try out

Female <- x[x$Sex == "F", ] 
Male <- x[x$Sex == "M", ]
nrow(Female[Female$Admitted == "Yes", ]) / nrow(Female)

## [1] 0.3035422

nrow(Male[Male$Admitted == "Yes", ]) / nrow(Male)

## [1] 0.4602317

sum(x$Sex == "M" & x$Admitted == "Yes") / sum(x$Sex == "M") 

## [1] 0.4602317

sum(x$Sex == "F" & x$Admitted == "Yes") / sum(x$Sex == "F")

## [1] 0.3035422

table(x$Sex, x$Admitted)

##    
##       No  Yes
##   F 1278  557
##   M 1398 1192

1192 / (1398 + 1192)

## [1] 0.4602317

557 / (557 + 1278)

## [1] 0.3035422

• Is there a significant difference or not? What does it mean by being “significant”?
• Think for a min

• Is there a significant difference or not? What does it mean by being “significant”?
• Difference seems large but how “reliable” this evidence can convince us there is a systematic difference.
• Significance is a concept that is outside the scope of this lesson (light touch on it here).
• The idea behind is the following: hard to judge since there is so much uncertainty in the admission. However, if the uncertainty is very unlikely to generate a pattern that we get, then “most likely” a systematic difference exists.

• Is/are there particular department(s) that seems especially problematic in terms of differential admission rates?

• What are the male and female admission rates for each department?

• Demo the admission rates by sex in Biology.

Bio <- x[x$Department == "Biology", ] 
rate.M <- sum(Bio$Sex == "M" & Bio$Admitted == "Yes") / sum(Bio$Sex == "M")
rate.F <- sum(Bio$Sex == "F" & Bio$Admitted == "Yes") / sum(Bio$Sex == "F")
rate.M

## [1] 0.05882353

rate.F

## [1] 0.07038123

Acceptance rate by sex:

• Team 1: English

• Team 2: History

• Team 3: Psychology

• You have 5 min

table(x[x$Admitted == "Yes",]$Sex, x[x$Admitted == "Yes",]$Department)

##    
##     Biology English History Mathematics Philosophy Psychology
##   F      24      94     131          89         17        202
##   M      16      53     139         511        353        120

table(x$Sex, x$Department)

##    
##     Biology English History Mathematics Philosophy Psychology
##   F     341     393     375         108         25        593
##   M     272     191     417         825        560        325

table(x[x$Admitted == "Yes",]$Sex, x[x$Admitted == "Yes",]$Department)/
         table(x$Sex, x$Department)

##    
##        Biology    English    History Mathematics Philosophy Psychology
##   F 0.07038123 0.23918575 0.34933333  0.82407407 0.68000000 0.34064081
##   M 0.05882353 0.27748691 0.33333333  0.61939394 0.63035714 0.36923077

• Is there evidence for sex-based discrimination? Is there evidence against discrimination?
• Visualizing the data should help.
• Which plot do you think is appropriate?

barplot(table(x[x$Admitted == "Yes",]$Sex, x[x$Admitted == "Yes",]
    $Department)/table(x$Sex, x$Department), beside=TRUE)

barplot(table(x[x$Admitted == "Yes",]$Sex, x[x$Admitted == "Yes",]
    $Department)/table(x$Sex, x$Department), beside=TRUE, legend=TRUE)

• The admission rate by sex in every department seems similar.
• Why this finding contradicts our previous observation?
• Think for two mins

• Every departments have different admission rates
• Every departments have different number of applicants by gender.
• Examing the overall addmission rate by gender may overlook these underlying differences.

barplot(table(x$Admitted, x$Department), legend=TRUE)

barplot(table(x$Sex, x$Department), legend=TRUE)

• Some departments are harder to be admitted.
• More female applicants apply to competitive departments.
• Fewer female applicants are admitted.
• As a result, it seems men are more likely to be admitted.

• Overall statistics give us a picture
  • but we need to look more closely
• Women apply more to competitive departments, so overall it looks like there is more discrimination.
• We do not find evidence of discrimination once we look department by department
  • Simpson’s Paradox

• This does not mean there is no gender discrimination
• There’s ample evidence of gender-biased discrimination in many countries, fields and industries - just not in this one
• No evidence against discrimination != no discrimination
• What factors might have resulted in women applying in such different ways to different departments?
• Next class: displaying and summarizing data