TRM Practical Data Analysis - Basic level

Lecture 6. Analysis of difference - Categorical data

Task 1. Determine whether professors’ ranks differed between male and female professors

1.1 Study design:

  • The cross-sectional investigation of 397 professors to determine whether male and female professors had different ranks.
  • Null hypothesis: professors’ ranks did not differ between male and female professors.
  • Alternative hypothesis: professors’ ranks differed between male and female professors.

1.2 Import the “Professorial Salaries” and name this dataset “salary”

salary = read.csv("C:\\Thach\\UTS\\Teaching\\TRM\\Practical Data Analysis\\2024_Autumn semester\\Data\\Professorial Salaries.csv")

1.3 Describe characteristics of the study sample

library(table1)
## 
## Attaching package: 'table1'
## The following objects are masked from 'package:base':
## 
##     units, units<-
table1(~ Rank + Discipline + Yrs.since.phd + Yrs.service + NPubs + Ncits + Salary | Sex, data = salary)
Female
(N=39)		 Male
(N=358)		 Overall
(N=397)
Rank
AssocProf 10 (25.6%) 54 (15.1%) 64 (16.1%)
AsstProf 11 (28.2%) 56 (15.6%) 67 (16.9%)
Prof 18 (46.2%) 248 (69.3%) 266 (67.0%)
Discipline
A 18 (46.2%) 163 (45.5%) 181 (45.6%)
B 21 (53.8%) 195 (54.5%) 216 (54.4%)
Yrs.since.phd
Mean (SD) 16.5 (9.78) 22.9 (13.0) 22.3 (12.9)
Median [Min, Max] 17.0 [2.00, 39.0] 22.0 [1.00, 56.0] 21.0 [1.00, 56.0]
Yrs.service
Mean (SD) 11.6 (8.81) 18.3 (13.2) 17.6 (13.0)
Median [Min, Max] 10.0 [0, 36.0] 18.0 [0, 60.0] 16.0 [0, 60.0]
NPubs
Mean (SD) 20.2 (14.4) 17.9 (13.9) 18.2 (14.0)
Median [Min, Max] 18.0 [1.00, 50.0] 13.0 [1.00, 69.0] 13.0 [1.00, 69.0]
Ncits
Mean (SD) 40.7 (16.2) 40.2 (17.0) 40.2 (16.9)
Median [Min, Max] 36.0 [14.0, 70.0] 35.0 [1.00, 90.0] 35.0 [1.00, 90.0]
Salary
Mean (SD) 101000 (26000) 115000 (30400) 114000 (30300)
Median [Min, Max] 104000 [62900, 161000] 108000 [57800, 232000] 107000 [57800, 232000]

1.4 Chi-square test to determine whether professors’ ranks differed between male and female professors

rank.sex = chisq.test(salary$Rank, salary$Sex)
rank.sex
## 
##  Pearson's Chi-squared test
## 
## data:  salary$Rank and salary$Sex
## X-squared = 8.5259, df = 2, p-value = 0.01408

Interpretation: there is evidence (P= 0.014) that professors’ ranks differed significantly between male and female professors.

1.5 Post-hoc analyses

#install.packages("chisq.posthoc.test")
library(chisq.posthoc.test)
## Warning: package 'chisq.posthoc.test' was built under R version 4.3.3
chisq.tab = as.table(rbind(c(10, 11, 18), c(54, 56, 248)))
dimnames(chisq.tab) = list(sex = c("Female", "Male"),
                           rank = c("AssocProf", "AsstProf", "Prof"))
chisq.tab
##         rank
## sex      AssocProf AsstProf Prof
##   Female        10       11   18
##   Male          54       56  248
chisq.posthoc.test(chisq.tab)
##   Dimension     Value AssocProf  AsstProf      Prof
## 1    Female Residuals  1.702577  1.989100 -2.915939
## 2    Female  p values  0.531883  0.280141  0.021277
## 3      Male Residuals -1.702577 -1.989100  2.915939
## 4      Male  p values  0.531883  0.280141  0.021277

Interpretation: there is evidence (P= 0.02) that professors differed between males and females; whereas there is no evidence that assistant and associate professors differed between males and females (P= 0.28 and 0.53, respectively)

Task 2. Determine whether the disciplines differed between male and female associate professors

2.1 Study design:

  • The cross-sectional investigation of 64 associate professors to determine whether the disciplines differed between male and female professors.
  • Null hypothesis: the disciplines did not differ between male and female professors.
  • Alternative hypothesis: the disciplines differed between male and female professors.

2.2 Select a subgroup of associate professors

assoc = subset(salary, Rank == "AssocProf")
dim(assoc)
## [1] 64  9

2.3 Describe characteristics of the study sample

table1(~ Rank + Discipline + Yrs.since.phd + Yrs.service + NPubs + Ncits + Salary | Sex, data = assoc)
Female
(N=10)		 Male
(N=54)		 Overall
(N=64)
Rank
AssocProf 10 (100%) 54 (100%) 64 (100%)
Discipline
A 4 (40.0%) 22 (40.7%) 26 (40.6%)
B 6 (60.0%) 32 (59.3%) 38 (59.4%)
Yrs.since.phd
Mean (SD) 15.5 (5.80) 15.4 (10.2) 15.5 (9.65)
Median [Min, Max] 13.0 [10.0, 26.0] 11.5 [6.00, 49.0] 12.0 [6.00, 49.0]
Yrs.service
Mean (SD) 11.5 (6.26) 12.0 (10.7) 12.0 (10.1)
Median [Min, Max] 9.50 [6.00, 24.0] 8.00 [1.00, 53.0] 8.00 [1.00, 53.0]
NPubs
Mean (SD) 13.8 (11.5) 19.6 (13.7) 18.7 (13.5)
Median [Min, Max] 11.0 [1.00, 38.0] 16.0 [1.00, 50.0] 16.0 [1.00, 50.0]
Ncits
Mean (SD) 42.9 (14.9) 41.9 (15.8) 42.0 (15.6)
Median [Min, Max] 47.5 [19.0, 60.0] 36.0 [14.0, 83.0] 36.5 [14.0, 83.0]
Salary
Mean (SD) 88500 (18000) 94900 (12900) 93900 (13800)
Median [Min, Max] 90600 [62900, 110000] 95600 [70000, 126000] 95600 [62900, 126000]

2.4 Fisher’s exact test to determine whether the disciplines differed between male and female associate professors

As the expected number of female professors in the Theoretical discipline (Discipline = A) is 4.06 (the observed number = 4)< 5, a Fisher’s exact test is used.

fisher = fisher.test(assoc$Sex, assoc$Discipline)
fisher
## 
##  Fisher's Exact Test for Count Data
## 
## data:  assoc$Sex and assoc$Discipline
## p-value = 1
## alternative hypothesis: true odds ratio is not equal to 1
## 95 percent confidence interval:
##  0.1795097 4.6576966
## sample estimates:
## odds ratio 
##  0.9701586

Interpretation: there is no evidence (P~ 1.0) that the disciplines differed between male and female professors.

Task 3. Save your work and upload it to your Rpubs account