College major on Income
Hung Dinh
May 4, 2018
Introduction
We’ll look at a College data and my assignment is to study how income varies across college major categories.
A codebook for the dataset is given below:
- rank: Rank by median earnings
- major_code: Major code
- major: Major description
- major_category: Category of major
- total: Total number of people with major
- sample_size: Sample size of full-time, year-round individuals used for income/earnings estimates: p25th, median, p75th
- p25th: 25th percentile of earnings
- median: Median earnings of full-time, year-round workers
- p75th: 75th percentile of earnings
- perc_men: % men with major (out of total)
- perc_women: % women with major (out of total)
- perc_employed: % employed (out of total)
- perc_employed_fulltime: % employed 35 hours or more (out of employed)
- perc_employed_parttime: % employed less than 35 hours (out of employed)
- perc_employed_fulltime_yearround: % employed at least 50 weeks and at least 35 hours (out of employed and full-time)
- perc_unemployed: % unemployed (out of employed)
- perc_college_jobs: % with job requiring a college degree (out of employed)
- perc_non_college_jobs: % with job not requiring a college degree (out of employed)
- perc_low_wage_jobs: % in low-wage service jobs (out of total)
The specifically question for this project is: “Is there an association between college major category and income?”
Based on your analysis, would you conclude that there is a significant association between college major category and income?
Load data
library(collegeIncome)
data(college)Some exploratory analysis
head(college,5)## rank major_code major major_category
## 1 1 2419 Petroleum Engineering Engineering
## 2 2 2416 Mining And Mineral Engineering Engineering
## 3 3 2415 Metallurgical Engineering Engineering
## 4 4 2417 Naval Architecture And Marine Engineering Engineering
## 5 5 2405 Chemical Engineering Engineering
## total sample_size perc_women p25th median p75th perc_men
## 1 2339 36 0.9109326 25000 40000 50000 0.08906743
## 2 756 7 0.5154064 26000 37000 40000 0.48459355
## 3 856 3 0.5942076 26700 45000 60000 0.40579235
## 4 1258 16 0.6521298 26000 35000 45000 0.34787018
## 5 32260 289 0.4179248 31500 62000 109000 0.58207520
## perc_employed perc_employed_fulltime perc_employed_parttime
## 1 0.9115044 0.9206524 0.1774785
## 2 0.7980501 0.7110092 0.3623853
## 3 0.7871943 0.8833498 0.3387257
## 4 0.8465608 0.9366337 0.1673267
## 5 0.8515625 0.8086363 0.4020061
## perc_employed_fulltime_yearround perc_unemployed perc_college_jobs
## 1 0.7704431 0.08849558 0.6702970
## 2 0.7093101 0.20194986 0.3867764
## 3 0.7738366 0.21280567 0.7289116
## 4 0.6527853 0.15343915 0.2460902
## 5 0.6852821 0.14843750 0.5867515
## perc_non_college_jobs perc_low_wage_jobs
## 1 0.1821782 0.05544554
## 2 0.5158761 0.21560172
## 3 0.1759983 0.03014828
## 4 0.4107636 0.04323827
## 5 0.3860437 0.11801062str(college)## 'data.frame': 173 obs. of 19 variables:
## $ rank : int 1 2 3 4 5 6 7 8 9 10 ...
## $ major_code : int 2419 2416 2415 2417 2405 2418 6202 5001 2414 2408 ...
## $ major : chr "Petroleum Engineering" "Mining And Mineral Engineering" "Metallurgical Engineering" "Naval Architecture And Marine Engineering" ...
## $ major_category : chr "Engineering" "Engineering" "Engineering" "Engineering" ...
## $ total : int 2339 756 856 1258 32260 2573 3777 1792 91227 81527 ...
## $ sample_size : int 36 7 3 16 289 17 51 10 1029 631 ...
## $ perc_women : num 0.911 0.515 0.594 0.652 0.418 ...
## $ p25th : num 25000 26000 26700 26000 31500 23000 32500 37900 29200 23000 ...
## $ median : num 40000 37000 45000 35000 62000 44700 45000 57000 36000 32200 ...
## $ p75th : num 50000 40000 60000 45000 109000 50000 58000 67000 46000 47100 ...
## $ perc_men : num 0.0891 0.4846 0.4058 0.3479 0.5821 ...
## $ perc_employed : num 0.912 0.798 0.787 0.847 0.852 ...
## $ perc_employed_fulltime : num 0.921 0.711 0.883 0.937 0.809 ...
## $ perc_employed_parttime : num 0.177 0.362 0.339 0.167 0.402 ...
## $ perc_employed_fulltime_yearround: num 0.77 0.709 0.774 0.653 0.685 ...
## $ perc_unemployed : num 0.0885 0.2019 0.2128 0.1534 0.1484 ...
## $ perc_college_jobs : num 0.67 0.387 0.729 0.246 0.587 ...
## $ perc_non_college_jobs : num 0.182 0.516 0.176 0.411 0.386 ...
## $ perc_low_wage_jobs : num 0.0554 0.2156 0.0301 0.0432 0.118 ...We can see that the data has 173 observations of 19 variables which corresponds to the codebook.
The question asks about relationship between the major category and income, so I will only look at major_category and median. There are obviously other factors that may affect our analysis, for example: gender perc_men and perc_women, sample size (number of objects that provide income) perc_employed and total… I assume to omit all other variables.
Now let’s factorize the data and see the relationship between our two interested values:
college$major <- as.factor(college$major)
college$major_code <- as.factor(college$major_code)
college$major_category <- as.factor(college$major_category)boxplot(median/1000 ~ major_category, data = college, main = "Income vs. Major", ylab="Income (thousands of dollar)", las = 2)
We can see the distribution of the median of Income of each major is not normal, they’re skewed. However for the purpose of this project of practicing linear model, I assume they’re normal.
Analyze
Let’s have a look at the major_category:
unique(college$major_category)## [1] Engineering
## [2] Business
## [3] Physical Sciences
## [4] Law & Public Policy
## [5] Computers & Mathematics
## [6] Agriculture & Natural Resources
## [7] Industrial Arts & Consumer Services
## [8] Arts
## [9] Health
## [10] Social Science
## [11] Biology & Life Science
## [12] Education
## [13] Humanities & Liberal Arts
## [14] Psychology & Social Work
## [15] Communications & Journalism
## [16] Interdisciplinary
## 16 Levels: Agriculture & Natural Resources Arts ... Social ScienceThere are 16 of them. Let’s first reorder the category before doing regression model:
college <- college[order(college$major_category),]When we apply a linear model to this data, linking Income to all Majors, the default output intercept is the mean of the referenced major (alphabet sorted, with Agriculture first), the gradient coefficient of other majors is the difference of the mean of that major to the referenced one, and the p-value of those coefficients is the probability of a t-test if that mean and the referenced mean is different. For example, say we want to compare major Arts with others:
major_category_ref <- relevel(college$major_category, "Arts")
fit <- lm(median ~ major_category_ref, data = college)
summary(fit)$coef## Estimate
## (Intercept) 38050.000
## major_category_refAgriculture & Natural Resources 5450.000
## major_category_refBiology & Life Science 5814.286
## major_category_refBusiness 11103.846
## major_category_refCommunications & Journalism 3950.000
## major_category_refComputers & Mathematics -3331.818
## major_category_refEducation -112.500
## major_category_refEngineering 2343.103
## major_category_refHealth 2266.667
## major_category_refHumanities & Liberal Arts -2883.333
## major_category_refIndustrial Arts & Consumer Services 2378.571
## major_category_refInterdisciplinary -10550.000
## major_category_refLaw & Public Policy -250.000
## major_category_refPhysical Sciences 2350.000
## major_category_refPsychology & Social Work 1838.889
## major_category_refSocial Science 1016.667
## Std. Error
## (Intercept) 4014.658
## major_category_refAgriculture & Natural Resources 5386.228
## major_category_refBiology & Life Science 5032.640
## major_category_refBusiness 5102.541
## major_category_refCommunications & Journalism 6953.591
## major_category_refComputers & Mathematics 5276.294
## major_category_refEducation 4916.931
## major_category_refEngineering 4534.719
## major_category_refHealth 5182.901
## major_category_refHumanities & Liberal Arts 4971.264
## major_category_refIndustrial Arts & Consumer Services 5876.857
## major_category_refInterdisciplinary 12043.973
## major_category_refLaw & Public Policy 6473.441
## major_category_refPhysical Sciences 5386.228
## major_category_refPsychology & Social Work 5517.619
## major_category_refSocial Science 5517.619
## t value
## (Intercept) 9.47776950
## major_category_refAgriculture & Natural Resources 1.01183974
## major_category_refBiology & Life Science 1.15531531
## major_category_refBusiness 2.17614057
## major_category_refCommunications & Journalism 0.56805181
## major_category_refComputers & Mathematics -0.63146941
## major_category_refEducation -0.02288012
## major_category_refEngineering 0.51670312
## major_category_refHealth 0.43733553
## major_category_refHumanities & Liberal Arts -0.58000007
## major_category_refIndustrial Arts & Consumer Services 0.40473529
## major_category_refInterdisciplinary -0.87595680
## major_category_refLaw & Public Policy -0.03861934
## major_category_refPhysical Sciences 0.43629787
## major_category_refPsychology & Social Work 0.33327579
## major_category_refSocial Science 0.18425822
## Pr(>|t|)
## (Intercept) 3.919976e-17
## major_category_refAgriculture & Natural Resources 3.131715e-01
## major_category_refBiology & Life Science 2.497166e-01
## major_category_refBusiness 3.103954e-02
## major_category_refCommunications & Journalism 5.708113e-01
## major_category_refComputers & Mathematics 5.286520e-01
## major_category_refEducation 9.817749e-01
## major_category_refEngineering 6.060905e-01
## major_category_refHealth 6.624690e-01
## major_category_refHumanities & Liberal Arts 5.627460e-01
## major_category_refIndustrial Arts & Consumer Services 6.862230e-01
## major_category_refInterdisciplinary 3.823917e-01
## major_category_refLaw & Public Policy 9.692429e-01
## major_category_refPhysical Sciences 6.632200e-01
## major_category_refPsychology & Social Work 7.393708e-01
## major_category_refSocial Science 8.540487e-01From this result we can get some information:
- mean of median of Income from major Arts is 38,050
- difference of mean of median of Income of Agriculture & Natural Resources from Arts is 5,450, and p-value of this difference is 0.31, which implies that the difference is not significant
- the same interpretation can be done for coefficients of other variables
For this project, we ideally run linear regression models of income (median) vs. college major (major_catecory) for all majors as referenced. Given a referenced level, the model coefficients will indicate the difference of the mean of other variables and the probability if they are the same.
I will run regression model for each major as the reference. The similar probabilities are stored in a 2D matrix A.
A <- matrix(, nrow = 16, ncol = 16)
for (i in 1:16){
major_category_ref <- relevel(college$major_category, as.character(unique(college$major_category)[i]))
fit <- lm(median ~ major_category_ref, data = college)
tmp <- summary(fit)$coef[,4]
# swap the first element to the corresponding position in the diagonal matrix
tmp1 <- tmp[1:i]
tmp1 <- c(0,tmp1)
tmp1 <- c(tmp1[-2],tmp1[2])
tmp1 <- tmp1[-1]
# save to A
A[,i] <- c(tmp1,tmp[-(1:i)])
}Edit the matrix and plot.
library(reshape)## Warning: package 'reshape' was built under R version 3.4.4library(ggplot2)We should expect a square symmetric matrix, with diagonal values are very low.
B <- data.frame(A)
names(B) <- unique(college$major_category)
B$major <- unique(college$major_category)
Bmelt <- melt(B)## Using major as id variablesg = ggplot(data=Bmelt, aes(x=variable, y=major, fill=value))
g = g + geom_tile()
g = g + theme(axis.text.x = element_text(angle = 90, hjust = 1)) + ylab("Major") + xlab("Major")
g = g + ggtitle("Probability of difference in Income between Majors")
g = g + coord_fixed()
g
Assuming a confidence interval of 95%, I mark all probabilities smaller than 2.5% as Different and larger than or equal to 2.5% as Same.
g = ggplot(data=Bmelt, aes(x=variable, y=major, fill=value < 0.025))
g = g + geom_tile()
g = g + theme(axis.text.x = element_text(angle = 90, hjust = 1)) + ylab("Major") + xlab("Major")
g = g + ggtitle("Difference in Income between Majors")
g = g + coord_fixed()
g
Apparently, most majors have similar income, except the Business. Its income is significant different from Computers & Mathematics, Education, Engineering, and Humanities & Liberal Arts.
Let’s try calculating how different they are from the Business:
major_category_ref <- relevel(college$major_category, "Business")
fit <- lm(median ~ major_category_ref, data = college)
summary(fit)$coef## Estimate
## (Intercept) 49153.846
## major_category_refAgriculture & Natural Resources -5653.846
## major_category_refArts -11103.846
## major_category_refBiology & Life Science -5289.560
## major_category_refCommunications & Journalism -7153.846
## major_category_refComputers & Mathematics -14435.664
## major_category_refEducation -11216.346
## major_category_refEngineering -8760.743
## major_category_refHealth -8837.179
## major_category_refHumanities & Liberal Arts -13987.179
## major_category_refIndustrial Arts & Consumer Services -8725.275
## major_category_refInterdisciplinary -21653.846
## major_category_refLaw & Public Policy -11353.846
## major_category_refPhysical Sciences -8753.846
## major_category_refPsychology & Social Work -9264.957
## major_category_refSocial Science -10087.179
## Std. Error t value
## (Intercept) 3149.357 15.607584
## major_category_refAgriculture & Natural Resources 4776.236 -1.183745
## major_category_refArts 5102.541 -2.176141
## major_category_refBiology & Life Science 4373.606 -1.209428
## major_category_refCommunications & Journalism 6492.565 -1.101852
## major_category_refComputers & Mathematics 4651.908 -3.103171
## major_category_refEducation 4239.951 -2.645395
## major_category_refEngineering 3790.072 -2.311498
## major_category_refHealth 4545.705 -1.944073
## major_category_refHumanities & Liberal Arts 4302.840 -3.250685
## major_category_refIndustrial Arts & Consumer Services 5323.384 -1.639047
## major_category_refInterdisciplinary 11783.813 -1.837592
## major_category_refLaw & Public Policy 5975.484 -1.900071
## major_category_refPhysical Sciences 4776.236 -1.832792
## major_category_refPsychology & Social Work 4923.931 -1.881618
## major_category_refSocial Science 4923.931 -2.048603
## Pr(>|t|)
## (Intercept) 9.444322e-34
## major_category_refAgriculture & Natural Resources 2.383031e-01
## major_category_refArts 3.103954e-02
## major_category_refBiology & Life Science 2.283166e-01
## major_category_refCommunications & Journalism 2.722123e-01
## major_category_refComputers & Mathematics 2.271210e-03
## major_category_refEducation 8.989341e-03
## major_category_refEngineering 2.210557e-02
## major_category_refHealth 5.367450e-02
## major_category_refHumanities & Liberal Arts 1.408831e-03
## major_category_refIndustrial Arts & Consumer Services 1.032059e-01
## major_category_refInterdisciplinary 6.801278e-02
## major_category_refLaw & Public Policy 5.925698e-02
## major_category_refPhysical Sciences 6.872781e-02
## major_category_refPsychology & Social Work 6.173891e-02
## major_category_refSocial Science 4.216615e-02and look at the lowest 5 majors
business_diff <- summary(fit)$coef[-1,]
business_diff[order(business_diff[,4])[1:5],]## Estimate Std. Error
## major_category_refHumanities & Liberal Arts -13987.179 4302.840
## major_category_refComputers & Mathematics -14435.664 4651.908
## major_category_refEducation -11216.346 4239.951
## major_category_refEngineering -8760.743 3790.072
## major_category_refArts -11103.846 5102.541
## t value Pr(>|t|)
## major_category_refHumanities & Liberal Arts -3.250685 0.001408831
## major_category_refComputers & Mathematics -3.103171 0.002271210
## major_category_refEducation -2.645395 0.008989341
## major_category_refEngineering -2.311498 0.022105573
## major_category_refArts -2.176141 0.031039539Clearly the 4 majors we pointed out above have lower p-value, and the fifth one (Arts) starts to have high enough p-value of 0.31.