GRE Score prediction from CGPA

Simple Linear Regression

Sumukha Venkatesha Murthy (s3797866)
Anilkumar Lingaraj Biradar (s3798024)
Amogha Amaresh (s3789160)

Last updated: 27 October, 2019

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Introduction

This investigation aims to understand the relationship between a student’s CGPA and GRE Score.
To apply for Masters Programs a student must give the GRE exam after his/her undergraduate study.
Simple linear regression is used to examine the relationship between CGPA and GRE score.
Thus from the student CGPA we can predict his/her GRE Score.

Problem Statement

CGPA is average of Grade Points obtained for all semesters of an Undergraduate student.The GRE is a standardized test that is an requirement for admission into many graduate schools.
Do they have a significant linear relationship between them?
Can CGPA be used to predict GRE Score of the student?
Simple Linear Regression is the best solution to investigate whether there is a relationship between CGPA (predictor variable) and GRE Score (dependent variable).

Data

The dataset contains a number of parameters that are considered important during the Masters Programs application. The parameters included are:
- GRE Scores
- TOEFL Scores
- University Rating
- Statement of Purpose
- Letter of Recommendation Strength
- Undergraduate GPA
- Research Experience ( ranging from 0 to 1 )
This dataset was collected from the kaggle website https://www.kaggle.com/mohansacharya/graduate-admissions

Data (cont..)

The dataset consists of 9 variables and 400 observations.
List of variables -

Serial No. - This number represents the row number.
GRE Score - Graduate Record Examinations is scored out of 340 .
TOEFL Score - The TOEFL test is scored on a scale of 0 to 120 points.
University Rating - This column has various rating levels of University ranging from 1 to 5.
SOP - The rating of Statement Of Purpose on a scale of 1 to 5.
LOR - The rating of Letter Of Recommendation on a scale of 1 to 5.
CGPA - This column shows the Cumulative Grade Points Average obtained by a undergraduate student. This grading point is out of 10 .
Research - This column represents whether the student has research experience (1) or no research experience (0).
Chance of Admit - This column represents chances of student getting admit ranging from 0 to 1.

Descriptive Statistics

The dataset did not have any missing values.

admit <- read_csv("Admission_Predict.csv")
#Descriptive Statisctics for CGPA
admit %>%summarise(Min = min(CGPA,na.rm = TRUE),
                                           Q1 = quantile(CGPA,probs = .25,na.rm = TRUE),
                                           Median = median(CGPA, na.rm = TRUE),
                                           Q3 = quantile(CGPA,probs = .75,na.rm = TRUE),
                                           Max = max(CGPA,na.rm = TRUE),
                                           Mean = mean(CGPA, na.rm = TRUE),
                                           SD = sd(CGPA, na.rm = TRUE),n = n(),Missing = sum(is.na(CGPA))) -> table1
knitr::kable(table1)

Min	Q1	Median	Q3	Max	Mean	SD	n	Missing
6.8	8.17	8.61	9.0625	9.92	8.598925	0.5963171	400	0

#Descriptive Statisctics for GRE Score
admit %>%summarise(Min = min(`GRE Score`,na.rm = TRUE),
                                           Q1 = quantile(`GRE Score`,probs = .25,na.rm = TRUE),
                                           Median = median(`GRE Score`, na.rm = TRUE),
                                           Q3 = quantile(`GRE Score`,probs = .75,na.rm = TRUE),
                                           Max = max(`GRE Score`,na.rm = TRUE),
                                           Mean = mean(`GRE Score`, na.rm = TRUE),
                                           SD = sd(`GRE Score`, na.rm = TRUE),n = n(),Missing = sum(is.na(`GRE Score`))) -> table2
knitr::kable(table2)

Min	Q1	Median	Q3	Max	Mean	SD	n	Missing
290	308	317	325	340	316.8075	11.47365	400	0

Data Issues

Only CGPA had an outlier in 59th row which was found using Tukey’s method and is removed.

par(mfrow=c(1,2))
Boxplot(admit$CGPA , ylab = "CGPA")

## [1] 59

Boxplot(admit$`GRE Score` , ylab = "GRE Score")

Figure 1

#Removing outlier 
admit <- admit[-59,]

Visualisation

Scatter Plot (Figure 2) is used to visualize the relationship between CGPA and GRE Score.
From the plot we can infer that as CGPA increases, so too does GRE Score (linear trend). This is a positive relationship. Thus we can proceed with fitting the linear regression model.

ggplot(admit, aes(x=CGPA, y=`GRE Score`))+
  geom_point(color="deepskyblue3" )

Figure 2

Hypothesis Testing

Hypotheses for the overall linear regression model:
\(H_0:\) The data does not fit the linear regression model.
\(H_A:\) The data fits the linear regression model.
Assumptions

Independence
Linearity
Normality of residuals
Homoscedasticity
Residual vs Leverage(Influential cases)

Decision Rules
- Reject \(H_0:\) If p-value < 0.05 ( significance level)
- Otherwise, fail to reject \(H_0\).
Conclusion:
- Test will be statistically significant if we reject \(H_0\).
- Otherwise, the test is not statistically significant.

Testing the Overall Model

The p-value for the F-test is really small, F(1,397) = 905.4, p < .001.The CGPA explained 69.52% of the variability in GRE Score readings. Thus the linear model is statistically significant. Now lets interpret the model coefficients.

cgpaGREmodel <- lm(`GRE Score` ~ CGPA, data = admit)
cgpaGREmodel %>% summary()

## 
## Call:
## lm(formula = `GRE Score` ~ CGPA, data = admit)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -18.1178  -4.2131   0.4833   3.7378  26.9171 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 177.6010     4.6387   38.29   <2e-16 ***
## CGPA         16.1852     0.5379   30.09   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.334 on 397 degrees of freedom
## Multiple R-squared:  0.6952, Adjusted R-squared:  0.6944 
## F-statistic: 905.4 on 1 and 397 DF,  p-value: < 2.2e-16

Linear Regression - Testing Model Parameters

Testing the statistical significance of the constant. The following is statistical hypotheses: \(H_0:α=0\) \(H_A:α≠0\)
Decision Rules:
- Reject \(H_0:\) If p-value < 0.05 (significance level) and 95% CI of the parameter does not capture \(H_0:α=0\)
- Otherwise, fail to reject \(H_0\).
Conclusion:
- Test will be statistically significant if we reject \(H_0\).
- Otherwise, the test is not statistically significant.
From the t statistic, reported as t = 38.28697, p < .001. The constant is statistically significant at the 0.05 level. This means that there is statistically significant evidence that the constant is not 0.
From the 95% CI for α to be [168.48154 , 186.72043]. \(H_0:α=0\) is clearly not captured by this interval, so it is rejected.

cgpaGREmodel %>% summary() %>% coef()

##              Estimate Std. Error  t value      Pr(>|t|)
## (Intercept) 177.60099   4.638680 38.28697 2.408109e-135
## CGPA         16.18524   0.537905 30.08940 1.843967e-104

cgpaGREmodel %>% confint()

##                 2.5 %    97.5 %
## (Intercept) 168.48154 186.72043
## CGPA         15.12774  17.24274

Linear Regression - Testing Model Parameters (cont..)

Testing the statistical significance of the slope. The following is statistical hypotheses: \(H_0:β=0\) \(H_A:β≠0\)
Decision Rules:
- Reject \(H_0:\) If p-value < 0.05 (significance level) and 95% CI of the parameter does not capture \(H_0:β=0\)
- Otherwise, fail to reject \(H_0\).
Conclusion:
- Test will be statistically significant if we reject \(H_0\).
- Otherwise, the test is not statistically significant.
From the t statistic, reported as t = 30.08940, p < .001. The slope is statistically significant at the 0.05 level. This means that there is statistically significant evidence that the slope is not 0.
From the 95% CI for β to be [15.12774 , 17.24274]. \(H_0:β=0\) is clearly not captured by this interval, so it is rejected.

cgpaGREmodel %>% summary() %>% coef()

##              Estimate Std. Error  t value      Pr(>|t|)
## (Intercept) 177.60099   4.638680 38.28697 2.408109e-135
## CGPA         16.18524   0.537905 30.08940 1.843967e-104

cgpaGREmodel %>% confint()

##                 2.5 %    97.5 %
## (Intercept) 168.48154 186.72043
## CGPA         15.12774  17.24274

Summarisation of the linear relationship

There was a statistically significant positive linear relationship between CGPA and GRE Score.
Plotting line of best fit on the scatter plot (Figure 3).

ggplot(admit, aes(x=CGPA, y=`GRE Score`))+
  geom_point(color="deepskyblue3" )+
  geom_smooth(method=lm,se=FALSE)

Figure 3

Validation of all the assumptions for linear regression

Independence: Independence was assumed as each CGPA and GRE Score measurement came from different people.
Linearity: The scatter plot (Figure 4) suggested a linear relationship. Other non-linear relationships are ruled out. There is no non-linear trends in the Residual vs. fitted plot.
Normality of residuals: Normal Q-Q plot (Figure 5) didn’t show any obvious departures from normality. Most of the residuals fell close to the line.

par(mfrow=c(1,2))
cgpaGREmodel %>% plot(which=1)
cgpaGREmodel %>% plot(which=2)

Figure 4 Figure 5

Validation of all the assumptions for linear regression (cont..)

Homoscedasticity: Homoscedasticity looked fine according to the scale-location plot (Figure 6). The variance in residuals appeared constant across predicted values.
Residual vs Leverage: In the plot (Figure 7), there are no values that fall outside the bands, therefore, there appeared to be no influential cases.

par(mfrow=c(1,2))
cgpaGREmodel %>% plot(which=3)
cgpaGREmodel %>% plot(which=5)

Figure 6 Figure 7

Discussion

Major findings of the investigation:
A linear regression model was fitted to predict the dependent variable, GRE Score, using CGPA as a single predictor. Prior to fitting the regression, a scatter plot assessing the bivariate relationship between GRE Score and CGPA was inspected. The scatter plot demonstrated evidence of a positive linear relationship. The overall regression model was statistically significant, F(1,397) = 905.4, p < .001, and explained 69.52% of the variability in GRE Score, \(R^2\) = 0.6952. The estimated regression equation was GRE Score = 177.6010+16.1852 ∗ CGPA The positive slope for CGPA was statistically significant, b = 16.18524, t(397) = 30.08940, p < .001, 95% CI [15.12774 , 17.24274]. Final inspection of the residuals supported normality and homoscedasticity. There were no influential cases too. Thus we can conclude that there was a statistically significant positive linear relationship between a CGPA and GRE Score.
Strengths:

The dataset was easy to observe and understand. It also includes essential metadata.
The variables were informative and performed the required hypothesis testing without any hindrance.

Limitations:

This investigation is constrained to two quantitative variables (CGPA and GRE Score).
There are several other parameters which can be considered for prediction of GRE Score.

Discussion (cont..)

Future investigation:

Besides, an investigation in the future should have an expansion in the dataset where it includes other parameters such as IQ Level, strength in solving maths problems, mastery in english, to predict the GRE Score of the student.
In the future, Multiple Linear Regression can be carried out by considering multiple predictor variables.

Thus from the findings of the above investigation conducted, Simple Linear Regression found that there is a statistically significant positive linear relationship between a CGPA and GRE Score. Thus, the investigation concludes as the CGPA score increases, the GRE Score also rises.

References

Linearity, photograph, viewed on 24 October 2019
https://linearity.be/wp-content/uploads/2017/09/linearity8.jpg
Graduate Admission, Admission_Predict CSV , Graduate Admission Dataset , Viewed on 22nd October 2019 https://www.kaggle.com/mohansacharya/graduate-admissions.
Mohan S Acharya, Asfia Armaan, Aneeta S Antony : A Comparison of Regression Models for Prediction of Graduate Admissions, IEEE International Conference on Computational Intelligence in Data Science 2019