Graduate Admissions
Josephine Wijaya
April 27, 2021
1 Explanation
To get accepted to your favorite University is everyone’s dream, but there are a lot of things that need to be considered. So, we want to analyze what factor that influence the the chance of admissions to Graduate Programs using linear regression. This data was downloaded from kaggle about Graduate Admissions from an Indian perspective.
Some relevant columns in the data :
- GRE.Score : GRE Scores ( out of 340 )
- TOEFL.Score : TOEFL Scores ( out of 120 )
- University.Rating : University Rating ( out of 5 )
- SOP : Statement of Purpose ( out of 5 )
- LOR : Letter of Recommendation Strength ( out of 5 )
- CGPA : Undergraduate GPA ( out of 10 )
- Research : Research Experience ( either 0 or 1 )
- Chance.of.Admit : Chance of Admit ( ranging from 0 to 1 )
1.1 Attaching Packages
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
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## filter, lag## The following objects are masked from 'package:base':
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## intersect, setdiff, setequal, unionlibrary(ggplot2)
library(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectlibrary(leaps)## Warning: package 'leaps' was built under R version 4.0.5library(lmtest)## Warning: package 'lmtest' was built under R version 4.0.5## Loading required package: zoo## Warning: package 'zoo' was built under R version 4.0.5##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
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## as.Date, as.Date.numericlibrary(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
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## recodelibrary(MLmetrics)## Warning: package 'MLmetrics' was built under R version 4.0.5##
## Attaching package: 'MLmetrics'## The following object is masked from 'package:base':
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## Recalllibrary(manipulate)2 Data Explanatory
2.1 Input Data
admission <- read.csv("data/Admission_Predict_Ver1.1.csv")2.2 Data Inspection
head(admission)## Serial.No. GRE.Score TOEFL.Score University.Rating SOP LOR CGPA Research
## 1 1 337 118 4 4.5 4.5 9.65 1
## 2 2 324 107 4 4.0 4.5 8.87 1
## 3 3 316 104 3 3.0 3.5 8.00 1
## 4 4 322 110 3 3.5 2.5 8.67 1
## 5 5 314 103 2 2.0 3.0 8.21 0
## 6 6 330 115 5 4.5 3.0 9.34 1
## Chance.of.Admit
## 1 0.92
## 2 0.76
## 3 0.72
## 4 0.80
## 5 0.65
## 6 0.90tail(admission)## Serial.No. GRE.Score TOEFL.Score University.Rating SOP LOR CGPA Research
## 495 495 301 99 3 2.5 2.0 8.45 1
## 496 496 332 108 5 4.5 4.0 9.02 1
## 497 497 337 117 5 5.0 5.0 9.87 1
## 498 498 330 120 5 4.5 5.0 9.56 1
## 499 499 312 103 4 4.0 5.0 8.43 0
## 500 500 327 113 4 4.5 4.5 9.04 0
## Chance.of.Admit
## 495 0.68
## 496 0.87
## 497 0.96
## 498 0.93
## 499 0.73
## 500 0.842.3 Subsetting Data
We want to use the data that have the chance of admit more than 75%.
adm <- admission %>%
filter(Chance.of.Admit > 0.75) %>%
dplyr::select(-Serial.No.)2.4 Data Cleansing
To check if there is any missing value from the data.
anyNA(adm)## [1] FALSEcolSums(is.na(adm))## GRE.Score TOEFL.Score University.Rating SOP
## 0 0 0 0
## LOR CGPA Research Chance.of.Admit
## 0 0 0 0In this data there is no missing value.
Check data type for each column.
glimpse(adm)## Rows: 210
## Columns: 8
## $ GRE.Score <int> 337, 324, 322, 330, 327, 328, 328, 334, 336, 340, 32~
## $ TOEFL.Score <int> 118, 107, 110, 115, 111, 112, 116, 119, 119, 120, 10~
## $ University.Rating <int> 4, 4, 3, 5, 4, 4, 5, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5~
## $ SOP <dbl> 4.5, 4.0, 3.5, 4.5, 4.0, 4.0, 5.0, 5.0, 4.0, 4.5, 4.~
## $ LOR <dbl> 4.5, 4.5, 2.5, 3.0, 4.5, 4.5, 5.0, 4.5, 3.5, 4.5, 3.~
## $ CGPA <dbl> 9.65, 8.87, 8.67, 9.34, 9.00, 9.10, 9.50, 9.70, 9.80~
## $ Research <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1~
## $ Chance.of.Admit <dbl> 0.92, 0.76, 0.80, 0.90, 0.84, 0.78, 0.94, 0.95, 0.97~ggcorr(adm, label = T, label_size = 3, hjust = 0.9)
All variable have positive correlation with Chance.of.Admit, but the strongest positive correlation with Chance.of.Admit is CGPA, TOEFL.Score, and GRE.Score.
3 Linear Regression Model
3.1 Strongest Correlation
Firstly, we will make a linear regression model that use CGPA, TOEFL.Score, and GRE.Score as the predictor.
model_one <- lm(Chance.of.Admit ~ CGPA + TOEFL.Score + GRE.Score, adm)
summary(model_one)##
## Call:
## lm(formula = Chance.of.Admit ~ CGPA + TOEFL.Score + GRE.Score,
## data = adm)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.091157 -0.016801 0.002086 0.020893 0.077820
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.8480784 0.0929162 -9.127 < 2e-16 ***
## CGPA 0.1115444 0.0083104 13.422 < 2e-16 ***
## TOEFL.Score 0.0030727 0.0006916 4.442 1.45e-05 ***
## GRE.Score 0.0010480 0.0004403 2.380 0.0182 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02977 on 206 degrees of freedom
## Multiple R-squared: 0.7917, Adjusted R-squared: 0.7887
## F-statistic: 261 on 3 and 206 DF, p-value: < 2.2e-16Using the first model we got the Adjusted R-squared = 0.7887
3.2 Step-wise Regression
Next, we will choose predictor using Step-wise Regression with backward elimination to make the second model.
model_step <- lm(Chance.of.Admit ~ ., adm)
step(model_step, direction = "backward")## Start: AIC=-1506.92
## Chance.of.Admit ~ GRE.Score + TOEFL.Score + University.Rating +
## SOP + LOR + CGPA + Research
##
## Df Sum of Sq RSS AIC
## - LOR 1 0.000786 0.14962 -1507.8
## - SOP 1 0.001166 0.15000 -1507.3
## - GRE.Score 1 0.001300 0.15013 -1507.1
## <none> 0.14883 -1506.9
## - Research 1 0.007299 0.15613 -1498.9
## - University.Rating 1 0.009019 0.15785 -1496.6
## - TOEFL.Score 1 0.017296 0.16613 -1485.8
## - CGPA 1 0.090177 0.23901 -1409.5
##
## Step: AIC=-1507.82
## Chance.of.Admit ~ GRE.Score + TOEFL.Score + University.Rating +
## SOP + CGPA + Research
##
## Df Sum of Sq RSS AIC
## - GRE.Score 1 0.001103 0.15072 -1508.3
## <none> 0.14962 -1507.8
## - SOP 1 0.001936 0.15156 -1507.1
## - Research 1 0.007283 0.15690 -1499.8
## - University.Rating 1 0.011135 0.16076 -1494.7
## - TOEFL.Score 1 0.016821 0.16644 -1487.4
## - CGPA 1 0.097553 0.24718 -1404.4
##
## Step: AIC=-1508.28
## Chance.of.Admit ~ TOEFL.Score + University.Rating + SOP + CGPA +
## Research
##
## Df Sum of Sq RSS AIC
## <none> 0.15072 -1508.3
## - SOP 1 0.001751 0.15248 -1507.8
## - Research 1 0.009631 0.16036 -1497.3
## - University.Rating 1 0.011987 0.16271 -1494.2
## - TOEFL.Score 1 0.027459 0.17818 -1475.1
## - CGPA 1 0.119510 0.27024 -1387.7##
## Call:
## lm(formula = Chance.of.Admit ~ TOEFL.Score + University.Rating +
## SOP + CGPA + Research, data = adm)
##
## Coefficients:
## (Intercept) TOEFL.Score University.Rating SOP
## -0.505701 0.003387 0.011247 0.005708
## CGPA Research
## 0.098242 0.020271model_two <- lm(formula = Chance.of.Admit ~ TOEFL.Score + University.Rating +
SOP + CGPA + Research, data = adm)
summary(model_two)##
## Call:
## lm(formula = Chance.of.Admit ~ TOEFL.Score + University.Rating +
## SOP + CGPA + Research, data = adm)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.09466 -0.01305 0.00267 0.01702 0.09020
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.5057014 0.0558446 -9.056 < 2e-16 ***
## TOEFL.Score 0.0033872 0.0005556 6.096 5.35e-09 ***
## University.Rating 0.0112466 0.0027921 4.028 7.94e-05 ***
## SOP 0.0057084 0.0037084 1.539 0.125281
## CGPA 0.0982415 0.0077245 12.718 < 2e-16 ***
## Research 0.0202706 0.0056145 3.610 0.000385 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02718 on 204 degrees of freedom
## Multiple R-squared: 0.828, Adjusted R-squared: 0.8238
## F-statistic: 196.4 on 5 and 204 DF, p-value: < 2.2e-16Step-wise Regression will make a model using the most appropriate predictor based on the AIC value where the lower the AIC value the less information loss that happened on the model. Using the second model we got the Adjusted R-squared = 0.8238.
The Adjusted R-squared from the first model is 0.7887 and the Adjusted R-squared from the second model is 0.8238. So we can conclude that the second model got higher Adjusted R-squared than the first model. The higher the Adjusted R-squared the better the model is.
Models: \[Chance.of.Admit = 0.111544(CGPA) + 0.003073(TOEFL.Score) + 0.001048(GRE.Score) - 0.848078 \] \[Chance.of.Admit = 0.003387(TOEFL.Score) + 0.011247(University.Rating) + 0.005708(SOP) + 0.098242(CGPA) + 0.020271(Research) - 0.505701\]
4 Prediction Model and Error
predict(model_one, data.frame(Chance.of.Admit = 0.92,
CGPA = 9.65,
TOEFL.Score = 118,
GRE.Score = 337),
interval = "confidence", level = 0.95)## fit lwr upr
## 1 0.9440649 0.936391 0.9517387predict(model_two, data.frame(Chance.of.Admit = 0.92,
CGPA = 9.65,
TOEFL.Score = 118,
University.Rating = 4,
SOP = 4.5,
Research = 1),
interval = "confidence", level = 0.95)## fit lwr upr
## 1 0.9329629 0.9255154 0.9404104sqrt((0.92-0.9440649)^2)## [1] 0.0240649sqrt((0.92-0.9329629)^2)## [1] 0.0129629The smaller the RSE value the better the model is so we can conclude that the second model is better than the first model.
5 Model Evaluation
5.1 Normality of Residual
hist(model_one$residuals, breaks = 20)
shapiro.test(model_one$residuals)##
## Shapiro-Wilk normality test
##
## data: model_one$residuals
## W = 0.9897, p-value = 0.138hist(model_two$residuals, breaks = 20)
shapiro.test(model_two$residuals)##
## Shapiro-Wilk normality test
##
## data: model_two$residuals
## W = 0.97511, p-value = 0.0008914P-value of the first model is more than 0.05, while the p-value of the second model is less than 0.05. So, the residual of the second model isn’t normally distributed. To make the residual of the second model to normally distributed we could transform the data or we could increase the data.
5.2 Homoscedasticity of Residual
plot(adm$Chance.of.Admit, model_one$residuals)
abline(h = 0, col = "red")
bptest(model_one)##
## studentized Breusch-Pagan test
##
## data: model_one
## BP = 7.6526, df = 3, p-value = 0.05376plot(adm$Chance.of.Admit, model_two$residuals)
abline(h = 0, col = "red")
bptest(model_two)##
## studentized Breusch-Pagan test
##
## data: model_two
## BP = 10.754, df = 5, p-value = 0.05648The p-value of first model and the p-value of second model is more than 0.05. So, the residual is constant and didn’t make a pattern (Heteroscedasticity).
5.3 No Multicolinearity
vif(model_one)## CGPA TOEFL.Score GRE.Score
## 2.185335 2.407613 2.557778vif(model_two)## TOEFL.Score University.Rating SOP CGPA
## 1.863324 1.863578 1.910092 2.264315
## Research
## 1.127349There are no values more than 10 in the both of the model. So, there is no predictor that depend with each other (Multicolinearity).
6 Conclusion
If we only look at the Adjusted R-squared and the RSE value, the second model is better than the first model. The Adjusted R-squared from the first model is 0.7887 and the Adjusted R-squared from the second model is 0.8238. But, the second model didn’t pass the Normality of Residual while the first model pass all of the model evaluation.
So, I think the first model still better than the second model. It means the factor that most influence the chance of admissions to Graduate Programs are Undergraduate GPA, GRE Scores, and TOEFL Scores.