TRM Practical Data Analysis - Basic level
Thach Tran
2024-02-10
TRM Practical Data Analysis - Basic level
Lecture 4. Analysis of difference - Numeric data
Task 1. Determine whether salaries differed between male and female professors
1.1 Study design:
- The cross-sectional investigation of 396 professors to determine whether salaries differed between male and female professors.
- Null hypothesis: salaries did not differ between male and female professors.
- Alternative hypothesis: salaries 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 by sex
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 Check the distribution of professors’ salaries
library(ggplot2)
p = ggplot(data = salary, aes(x = Salary))
p1 = p + geom_histogram(aes(y = ..density..), color = "white", fill = "blue")
p2 = p1 + geom_density(col="red")
p2 + ggtitle("Distribution of professors' salaries") + theme_bw()## Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
## ℹ Please use `after_stat(density)` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Describe the differences in salaries between male and female professors (optional)
p = ggplot(data = salary, aes(x = Sex, y = Salary, fill = Sex, col = Sex))
p1 = p + geom_boxplot(col = "black") + geom_jitter(alpha = 0.05)
p1 + labs(x = "Sex", y = "Professors' salaries (USD)") + ggtitle("Professors' salaries by sex") + theme_bw()
1.5 Student’s t-test to determine whether salaries were different between male and female professors
t.test(Salary ~ Sex, data = salary)##
## Welch Two Sample t-test
##
## data: Salary by Sex
## t = -3.1615, df = 50.122, p-value = 0.002664
## alternative hypothesis: true difference in means between group Female and group Male is not equal to 0
## 95 percent confidence interval:
## -23037.916 -5138.102
## sample estimates:
## mean in group Female mean in group Male
## 101002.4 115090.4Interpretation: There is evidence (P= 0.003) that the salaries of male professors were $14,088 higher than female professors, ranging from $5,138 to $23,037.
Task 2. Determine whether the number of publications differed between male and female professors in a subgroup of 26 associate professors in the Theoretical discipline
2.1 Study design:
- The cross-sectional investigation of 26 associate professors in the Theoretical discipline to determine whether the number of publications differed between male and female professors.
- Null hypothesis: the number of publications did not differ between male and female professors.
- Alternative hypothesis: the number of publications differed between male and female professors.
2.2 Select a subgroup of associate professors in the Theoretical discipline
Assoc.A = subset(salary, Rank == "AssocProf" & Discipline == "A")
dim(Assoc.A)## [1] 26 92.3 Describe characteristics of associate professors in Theoretical discipline by sex
library(table1)
table1(~ Rank + Discipline + Yrs.since.phd + Yrs.service + NPubs + Ncits + Salary | Sex, data = Assoc.A)| Female (N=4) |
Male (N=22) |
Overall (N=26) |
|
|---|---|---|---|
| Rank | |||
| AssocProf | 4 (100%) | 22 (100%) | 26 (100%) |
| Discipline | |||
| A | 4 (100%) | 22 (100%) | 26 (100%) |
| Yrs.since.phd | |||
| Mean (SD) | 18.5 (8.19) | 17.7 (12.2) | 17.8 (11.5) |
| Median [Min, Max] | 19.0 [10.0, 26.0] | 12.5 [8.00, 49.0] | 13.0 [8.00, 49.0] |
| Yrs.service | |||
| Mean (SD) | 15.5 (8.70) | 13.1 (12.3) | 13.5 (11.7) |
| Median [Min, Max] | 15.0 [8.00, 24.0] | 8.00 [1.00, 49.0] | 8.00 [1.00, 49.0] |
| NPubs | |||
| Mean (SD) | 10.0 (4.97) | 21.6 (14.2) | 19.8 (13.8) |
| Median [Min, Max] | 10.0 [4.00, 16.0] | 16.0 [3.00, 48.0] | 16.0 [3.00, 48.0] |
| Ncits | |||
| Mean (SD) | 38.5 (18.5) | 44.3 (15.2) | 43.4 (15.5) |
| Median [Min, Max] | 37.5 [19.0, 60.0] | 47.0 [24.0, 69.0] | 47.0 [19.0, 69.0] |
| Salary | |||
| Mean (SD) | 72100 (6400) | 85000 (10600) | 83100 (11100) |
| Median [Min, Max] | 74100 [62900, 77500] | 82400 [70000, 108000] | 81900 [62900, 108000] |
2.4 Check the distribution of number of publications among associate professors in Theoretical discipline
p = ggplot(data = Assoc.A, aes(x = NPubs))
p1 = p + geom_histogram(aes(y = ..density..), color = "white", fill = "blue")
p2 = p1 + geom_density(col="red")
p2 + ggtitle("Distribution of number of publications among associate professors in Theoretical discipline") + theme_bw()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Describe the differences in number of publications between male and female associate professors in Theoretical discipline (optional)
p = ggplot(data = Assoc.A, aes(x = Sex, y = NPubs, fill = Sex, col = Sex))
p1 = p + geom_boxplot(col = "black") + geom_jitter(alpha = 0.05)
p1 + labs(x = "Sex", y = "Number of publications") + ggtitle("Number of publications by sex") + theme_bw()
2.5 Mann-Whitney non-parametric test to determine whether number of publications differed between male and female associate professors in Theoretical discipline
wilcox.test(NPubs ~ Sex, data = Assoc.A)## Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot
## compute exact p-value with ties##
## Wilcoxon rank sum test with continuity correction
##
## data: NPubs by Sex
## W = 21.5, p-value = 0.1168
## alternative hypothesis: true location shift is not equal to 02.6 Bootstrap to determine whether number of publications differed between male and female associate professors in Theoretical discipline
Differences in the MEAN number of publications
library(simpleboot)## Warning: package 'simpleboot' was built under R version 4.3.2## Simple Bootstrap Routines (1.1-7)library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.3 ✔ readr 2.1.4
## ✔ forcats 1.0.0 ✔ stringr 1.5.0
## ✔ lubridate 1.9.3 ✔ tibble 3.2.1
## ✔ purrr 1.0.2 ✔ tidyr 1.3.0## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorsmale = Assoc.A %>% filter(Sex == "Male")
female = Assoc.A %>% filter(Sex == "Female")
set.seed(1234)
b.means = two.boot(male$NPubs, female$NPubs, mean, R = 1000)
hist (b.means$t, breaks = 20)
quantile(b.means$t, probs=c(0.025, 0.50, 0.975)) ## 2.5% 50% 97.5%
## 4.976136 11.568182 18.273295Differences in the MEDIAN number of publications
set.seed(1234)
b.medians = two.boot(male$NPubs, female$NPubs, median, R = 1000)
hist (b.medians$t, breaks = 20)
quantile(b.medians$t, probs=c(0.025, 0.50, 0.975)) ## 2.5% 50% 97.5%
## -0.5000 7.0000 19.5125