mba_starting_salaries
Kunal Bhattacharya
23 January 2018
Import the dataset:
mba.df <- read.csv(paste("F:/Data Analytics for Managerial Applications/MBA Starting Salaries Data.csv", sep = ""))
head(mba.df)## age sex gmat_tot gmat_qpc gmat_vpc gmat_tpc s_avg f_avg quarter work_yrs
## 1 23 2 620 77 87 87 3.4 3.00 1 2
## 2 24 1 610 90 71 87 3.5 4.00 1 2
## 3 24 1 670 99 78 95 3.3 3.25 1 2
## 4 24 1 570 56 81 75 3.3 2.67 1 1
## 5 24 2 710 93 98 98 3.6 3.75 1 2
## 6 24 1 640 82 89 91 3.9 3.75 1 2
## frstlang salary satis
## 1 1 0 7
## 2 1 0 6
## 3 1 0 6
## 4 1 0 7
## 5 1 999 5
## 6 1 0 6library(psych)
describe(mba.df[,c(1,3:10,12,13)]) ##Summarizing the data## vars n mean sd median trimmed mad min max
## age 1 274 27.36 3.71 27 26.76 2.97 22 48
## gmat_tot 2 274 619.45 57.54 620 618.86 59.30 450 790
## gmat_qpc 3 274 80.64 14.87 83 82.31 14.83 28 99
## gmat_vpc 4 274 78.32 16.86 81 80.33 14.83 16 99
## gmat_tpc 5 274 84.20 14.02 87 86.12 11.86 0 99
## s_avg 6 274 3.03 0.38 3 3.03 0.44 2 4
## f_avg 7 274 3.06 0.53 3 3.09 0.37 0 4
## quarter 8 274 2.48 1.11 2 2.47 1.48 1 4
## work_yrs 9 274 3.87 3.23 3 3.29 1.48 0 22
## salary 10 274 39025.69 50951.56 999 33607.86 1481.12 0 220000
## satis 11 274 172.18 371.61 6 91.50 1.48 1 998
## range skew kurtosis se
## age 26 2.16 6.45 0.22
## gmat_tot 340 -0.01 0.06 3.48
## gmat_qpc 71 -0.92 0.30 0.90
## gmat_vpc 83 -1.04 0.74 1.02
## gmat_tpc 99 -2.28 9.02 0.85
## s_avg 2 -0.06 -0.38 0.02
## f_avg 4 -2.08 10.85 0.03
## quarter 3 0.02 -1.35 0.07
## work_yrs 22 2.78 9.80 0.20
## salary 220000 0.70 -1.05 3078.10
## satis 997 1.77 1.13 22.45str(mba.df)## 'data.frame': 274 obs. of 13 variables:
## $ age : int 23 24 24 24 24 24 25 25 25 25 ...
## $ sex : int 2 1 1 1 2 1 1 2 1 1 ...
## $ gmat_tot: int 620 610 670 570 710 640 610 650 630 680 ...
## $ gmat_qpc: int 77 90 99 56 93 82 89 88 79 99 ...
## $ gmat_vpc: int 87 71 78 81 98 89 74 89 91 81 ...
## $ gmat_tpc: int 87 87 95 75 98 91 87 92 89 96 ...
## $ s_avg : num 3.4 3.5 3.3 3.3 3.6 3.9 3.4 3.3 3.3 3.45 ...
## $ f_avg : num 3 4 3.25 2.67 3.75 3.75 3.5 3.75 3.25 3.67 ...
## $ quarter : int 1 1 1 1 1 1 1 1 1 1 ...
## $ work_yrs: int 2 2 2 1 2 2 2 2 2 2 ...
## $ frstlang: int 1 1 1 1 1 1 1 1 2 1 ...
## $ salary : int 0 0 0 0 999 0 0 0 999 998 ...
## $ satis : int 7 6 6 7 5 6 5 6 4 998 ...Including Plots
Constructing a box plot to understand how the GMAT scores and fall semester grades are distributed by gender:
par(mfrow = c(2,1))
boxplot(mba.df$gmat_tot ~ mba.df$sex, horizontal = TRUE, ylab = "Sex", xlab = "GMAT scores", yaxt = "n", main = "Boxplot of GMAT scores by sex")
axis(side = 2, las = 2, at = c(1:2), labels = c("Male","Female"))
boxplot(mba.df$f_avg ~ mba.df$sex, horizontal = TRUE, ylab = "Sex", xlab = "Fall average grades", yaxt = "n", main = "Boxplot of fall average grades by sex")
axis(side = 2, las = 2, at = c(1:2), labels = c("Male","Female"))
Note that the median GMAT scores and IQR are very similar across the two genders while there is a considerable difference between the fall average grades between the two genders.
Now our ultimate quest is to find the factors that are most correlated with getting a job or NOT getting a job. So we need to split our data frame into 2 parts according to who got jobs and who didn’t:
placed <- mba.df[which(mba.df$salary > 999),]
notplaced <- mba.df[which(mba.df$salary == 0),]
View(placed)
View(notplaced)Boxplots to check the median gmat scores of the students who got jobs and who didn’t
par(mfrow = c(2,1))
boxplot(placed$gmat_tot ~ placed$sex, horizontal = TRUE, ylab = "Sex", xlab = "GMAT scores", yaxt = "n", main = "Boxplot of GMAT scores by sex for placed candidates")
axis(side = 2, las = 2, at = c(1:2), labels = c("Male","Female"))
boxplot(notplaced$gmat_tot ~ notplaced$sex, horizontal = TRUE, ylab = "Sex", xlab = "GMAT scores", yaxt = "n", main = "Boxplot of GMAT scores by sex for unplaced candidates")
axis(side = 2, las = 2, at = c(1:2), labels = c("Male","Female"))
Interestingly, we see that the median GMAT score in each case in almost identical while there seems to be an approx 15 points difference in the median GMAT scores between the placed and unplaced candidates.
Plotting a scatterplot matrix for various variables
library(car)##
## Attaching package: 'car'## The following object is masked from 'package:psych':
##
## logitscatterplotMatrix(mba.df[("salary" != 998) & ("salary" != 999),c("salary","gmat_tot","s_avg","work_yrs")], spread = FALSE, smoother.args = list(lty = 2), main = "Scatter Plot Matrix")
Plotting a corrgram for all numeric variables in the dataset
library(corrgram)
corrgram(mba.df[("salary" != 998) & ("salary" != 999),c(1,3:10,12)], order=FALSE, lower.panel=panel.shade, upper.panel=panel.pie, text.panel=panel.txt, main="Corrgram of MBA data intercorrelations")
Contingency Tables
jobs_per_sex <- xtabs(~ sex+got_job, data = all_sal)
addmargins(jobs_per_sex)## got_job
## sex 0 1 Sum
## 1 67 72 139
## 2 23 31 54
## Sum 90 103 193jobs_by_lang <- xtabs(~ frstlang+got_job, data = all_sal)
addmargins(jobs_by_lang)## got_job
## frstlang 0 1 Sum
## 1 82 96 178
## 2 8 7 15
## Sum 90 103 193Propertion of jobs by gender and language:
prop.table(jobs_per_sex,1)*100## got_job
## sex 0 1
## 1 48.20144 51.79856
## 2 42.59259 57.40741prop.table(jobs_by_lang,1)*100## got_job
## frstlang 0 1
## 1 46.06742 53.93258
## 2 53.33333 46.66667Running chi-square and t-tests of independence
Case-1:
Null Hypothesis: There is no significant difference in starting salaries and sex i.e. there is no correlation between starting salaries and sex. Alternative Hypothesis: There is a significant difference in starting salaries and sex indicating there is a correlation between the two variables.
chisq.test(jobs_per_sex)##
## Pearson's Chi-squared test with Yates' continuity correction
##
## data: jobs_per_sex
## X-squared = 0.29208, df = 1, p-value = 0.5889Thus, as the p-value = 0.5889 > 0.05, we fail to reject the null hypothesis that there is no correlation between starting salaries and sex.
Case-2:
Null Hypothesis: There is no significant difference in starting salaries and language i.e. there is no correlation between starting salaries and language. Alternative Hypothesis: There is a significant difference in starting salaries and language indicating there is a correlation between the two variables.
chisq.test(jobs_by_lang)##
## Pearson's Chi-squared test with Yates' continuity correction
##
## data: jobs_by_lang
## X-squared = 0.074127, df = 1, p-value = 0.7854Here again, since p-value = 0.7854 > 0.05, we fail to reject the null hypothesis that there is no correlation between starting salaries and language.
Constructing a correlation matrix to determine correlation between starting salary and numeric variables such as GMAT scores, MBA grades and work experience
cor(all_sal[,c(3,4,7,8,10,12)])## gmat_tot gmat_qpc s_avg f_avg work_yrs
## gmat_tot 1.000000e+00 0.74309972 0.14356746 0.101082103 -0.17369086
## gmat_qpc 7.430997e-01 1.00000000 0.01903816 0.130285115 -0.24138468
## s_avg 1.435675e-01 0.01903816 1.00000000 0.520554250 0.15913663
## f_avg 1.010821e-01 0.13028512 0.52055425 1.000000000 -0.04795136
## work_yrs -1.736909e-01 -0.24138468 0.15913663 -0.047951357 1.00000000
## salary -5.685962e-05 0.02839164 0.09632412 0.008846655 -0.05326685
## salary
## gmat_tot -5.685962e-05
## gmat_qpc 2.839164e-02
## s_avg 9.632412e-02
## f_avg 8.846655e-03
## work_yrs -5.326685e-02
## salary 1.000000e+00Conducting a correlation test for the variables above:
library(psych)
corr.test(all_sal[,c(3,6,7,8,10,12)], use = "complete")## Call:corr.test(x = all_sal[, c(3, 6, 7, 8, 10, 12)], use = "complete")
## Correlation matrix
## gmat_tot gmat_tpc s_avg f_avg work_yrs salary
## gmat_tot 1.00 0.88 0.14 0.10 -0.17 0.00
## gmat_tpc 0.88 1.00 0.19 0.11 -0.17 0.06
## s_avg 0.14 0.19 1.00 0.52 0.16 0.10
## f_avg 0.10 0.11 0.52 1.00 -0.05 0.01
## work_yrs -0.17 -0.17 0.16 -0.05 1.00 -0.05
## salary 0.00 0.06 0.10 0.01 -0.05 1.00
## Sample Size
## [1] 193
## Probability values (Entries above the diagonal are adjusted for multiple tests.)
## gmat_tot gmat_tpc s_avg f_avg work_yrs salary
## gmat_tot 0.00 0.00 0.42 1.00 0.19 1
## gmat_tpc 0.00 0.00 0.11 1.00 0.23 1
## s_avg 0.05 0.01 0.00 0.00 0.27 1
## f_avg 0.16 0.13 0.00 0.00 1.00 1
## work_yrs 0.02 0.02 0.03 0.51 0.00 1
## salary 1.00 0.40 0.18 0.90 0.46 0
##
## To see confidence intervals of the correlations, print with the short=FALSE optionTherefore we see that: 1. Total GMAT score (gmat_tot), Fall average grades (f_avg) and Yrs of Work Experience (work_yrs) have p-values < 0.05 which means we can reject the null hypothesis that these variables are not correlated with starting salary. 2. Total GMAT percentile (gmat_tpc) and Spring average grades (s_avg) have p-values > 0.05 which means we fail to reject the null hypothesis that these variables are not correlated with the starting salary.
Linear regression on placed data
Applying the linear regression model to the problem: y = b0 + b1x1 + b2x2 + b3x3 + … + bnxn Here the dependent variable is starting salary and the independent variables are variables such as gmat_tot, s_avg, f_avg etc.
Salary = b0 + b1(gmat_tot) + b2(s_avg) + b3(f_avg) + b4(work_yrs)
lmodel <- lm(salary ~ gmat_qpc + gmat_vpc + s_avg + f_avg + work_yrs + sex + frstlang + satis, data = placed)
summary(lmodel)##
## Call:
## lm(formula = salary ~ gmat_qpc + gmat_vpc + s_avg + f_avg + work_yrs +
## sex + frstlang + satis, data = placed)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29800 -7822 -1742 4869 82341
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 86719.94 23350.43 3.714 0.000346 ***
## gmat_qpc 98.72 121.85 0.810 0.419884
## gmat_vpc -95.80 102.99 -0.930 0.354699
## s_avg 4659.05 5015.66 0.929 0.355320
## f_avg -1698.83 3834.70 -0.443 0.658773
## work_yrs 2331.12 585.99 3.978 0.000137 ***
## sex -5289.24 3545.91 -1.492 0.139140
## frstlang 13994.76 6641.66 2.107 0.037770 *
## satis -1671.20 2070.62 -0.807 0.421643
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15740 on 94 degrees of freedom
## Multiple R-squared: 0.285, Adjusted R-squared: 0.2241
## F-statistic: 4.683 on 8 and 94 DF, p-value: 7.574e-05Insights from the Regression Analysis: - p-value of 7.574e-05 (<0.05) suggests that this is overall a good model. - R-squared value of 0.2241 suggests that the explanatory variables explain only 22.4% of the covariance which means there must be quite a few other explanatory variables not taken into account here. - Only the beta-coefficients of first language and work years have statistical significance (i.e. p-value < 0.05). For every 1 year increase in work experience, the starting salary increases by 2331 units.
Logistic regression
Here we apply logit regression to check for the categorical variable got_jobs (1 = Job, 0 = No Job) that we had added to the all_sal dataframe:
model <- glm(got_job ~ gmat_tpc + gmat_tot + s_avg + f_avg + work_yrs + sex + frstlang + satis, family=binomial(link='logit'), data=all_sal)
summary(model)##
## Call:
## glm(formula = got_job ~ gmat_tpc + gmat_tot + s_avg + f_avg +
## work_yrs + sex + frstlang + satis, family = binomial(link = "logit"),
## data = all_sal)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7969 -1.1750 0.7812 1.0610 1.7901
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.243768 3.154526 -0.077 0.9384
## gmat_tpc 0.073538 0.044501 1.652 0.0984 .
## gmat_tot -0.016302 0.009382 -1.738 0.0823 .
## s_avg 0.652323 0.495360 1.317 0.1879
## f_avg -0.081864 0.347219 -0.236 0.8136
## work_yrs -0.087415 0.046405 -1.884 0.0596 .
## sex 0.210544 0.337300 0.624 0.5325
## frstlang 0.062156 0.576303 0.108 0.9141
## satis 0.437869 0.204119 2.145 0.0319 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 266.68 on 192 degrees of freedom
## Residual deviance: 251.21 on 184 degrees of freedom
## AIC: 269.21
##
## Number of Fisher Scoring iterations: 4The results suggest that the beta-coefficients of satis (satisfaction level) is the only statistically significant coefficient while others are statistically insignificant