title: “Homework 4”
Chances of Graduate Admissions for Applicants with Research Experience.
For my research question, I use dataset from Kaggle which provides the data related to chances of graduate admissions to universities based on whether the applicants have research experience and what are their academic scores (GRE, TOEFT, etc). I am including the following variables in my analysis: GRE Scores, TOEFL scores, University Rating (out of 5 ), Research Experience ( either 0 or 1 ), Chance of Admit ( ranging from 0 to 1 ). I am mostly intersted to find out if having a research experience for an applicant could drastically increases the chances of his/her graduate admission. My working hypothesis is that it does correlate. The students with a research experience are much more likely to be admitted to the graduate programs in the than students without the research experience. Planing to construct three models, I will include academic scores, like GRE and TOEFL in one of the models to further examine the relational effect of research experience on the chances of graduate admissions.
First, I downloaded the data and imported it in R.
Grad_ad <- read.csv ("C:/Users/Marcy/Documents/soc 712/Grad_ad.csv")
head (Grad_ad)## 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 0.90Model 1: This model is to view the correlation between having a research experince and the chance of graduate admission.
mod1 <- glm(Research ~ Chance.of.Admit, family = "binomial", data = Grad_ad)
summary (mod1)##
## Call:
## glm(formula = Research ~ Chance.of.Admit, family = "binomial",
## data = Grad_ad)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.1063 -0.9103 0.3819 0.8151 2.6727
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -7.3783 0.7363 -10.02 <2e-16 ***
## Chance.of.Admit 10.6533 1.0232 10.41 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 685.93 on 499 degrees of freedom
## Residual deviance: 514.33 on 498 degrees of freedom
## AIC: 518.33
##
## Number of Fisher Scoring iterations: 4As shown in the coefficients, there is a significant correlation between having a research experience and chances of graduate admissions. (Three stars indicate significance). Model 2 will include another independant variable, like University Ratings to see if there is any difference in an established correlation of two previously examined variables.
mod2 <- glm(Research ~ Chance.of.Admit + University.Rating, family = "binomial", data = Grad_ad)
summary (mod2)##
## Call:
## glm(formula = Research ~ Chance.of.Admit + University.Rating,
## family = "binomial", data = Grad_ad)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.2111 -0.9053 0.3557 0.8091 2.6759
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -7.1627 0.7390 -9.693 < 2e-16 ***
## Chance.of.Admit 9.3817 1.2207 7.686 1.52e-14 ***
## University.Rating 0.2333 0.1309 1.783 0.0746 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 685.93 on 499 degrees of freedom
## Residual deviance: 511.16 on 497 degrees of freedom
## AIC: 517.16
##
## Number of Fisher Scoring iterations: 4Result: As shown, there is no much difference in the university rates and chances of admissions based on research experience. However, the chances of admissions are strongly correlated with research experience as I stated in my hypothesis.
What would be the outcome if I am to include GRE Score and Toefl scores, as well as University ratings? How it would impact the chances of admission in relation to having research experience? In model 3 it would be shown. Here, the chances of admission will also intreact with GRE score to examine how different range of GRE score numbers and chances of admissions are related within the model 3 of all these additional variables (University rating, TOEFL score).
mod3 <- glm(Research ~ Chance.of.Admit * GRE.Score + University.Rating +TOEFL.Score, family = "binomial", data = Grad_ad)
summary (mod3)##
## Call:
## glm(formula = Research ~ Chance.of.Admit * GRE.Score + University.Rating +
## TOEFL.Score, family = "binomial", data = Grad_ad)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -3.0833 -0.7739 0.1469 0.6814 2.0407
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 46.07474 21.78825 2.115 0.034459 *
## Chance.of.Admit -106.46727 31.78835 -3.349 0.000810 ***
## GRE.Score -0.13985 0.07009 -1.995 0.046025 *
## University.Rating 0.10218 0.14166 0.721 0.470714
## TOEFL.Score -0.06269 0.03591 -1.746 0.080851 .
## Chance.of.Admit:GRE.Score 0.35724 0.10188 3.506 0.000454 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 685.93 on 499 degrees of freedom
## Residual deviance: 469.51 on 494 degrees of freedom
## AIC: 481.51
##
## Number of Fisher Scoring iterations: 5Evidently, there are several insignificant correlations, however there is a strong positive correlation between chances of admission / GRE score and having a research experience.
In order to see which model fits best to examine my hypothesis, I conduct ANOVA and lmtest.
anova(mod1, mod2, mod3, test= "Chisq")## Analysis of Deviance Table
##
## Model 1: Research ~ Chance.of.Admit
## Model 2: Research ~ Chance.of.Admit + University.Rating
## Model 3: Research ~ Chance.of.Admit * GRE.Score + University.Rating +
## TOEFL.Score
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 498 514.33
## 2 497 511.16 1 3.177 0.07468 .
## 3 494 469.51 3 41.651 4.758e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Likelihood Ratio Test
library (texreg)## Version: 1.36.23
## Date: 2017-03-03
## Author: Philip Leifeld (University of Glasgow)
##
## Please cite the JSS article in your publications -- see citation("texreg").lmtest::lrtest(mod1, mod2, mod3)## Likelihood ratio test
##
## Model 1: Research ~ Chance.of.Admit
## Model 2: Research ~ Chance.of.Admit + University.Rating
## Model 3: Research ~ Chance.of.Admit * GRE.Score + University.Rating +
## TOEFL.Score
## #Df LogLik Df Chisq Pr(>Chisq)
## 1 2 -257.17
## 2 3 -255.58 1 3.1771 0.07468 .
## 3 6 -234.75 3 41.6509 4.758e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1As the resutls above show, model 3 has significant correlations to examine. To confirm that and choose it as the best fit model, I conduct the AIC/BIC test.
library(texreg)
# htmlreg(list(mod1,mod2,mod3), doctype=FALSE)| Model 1 | Model 2 | Model 3 | ||
|---|---|---|---|---|
| (Intercept) | -7.38*** | -6.36*** | 49.53* | |
| (0.74) | (0.83) | (22.81) | ||
| Chance.of.Admit | 10.65*** | 9.50*** | -110.12*** | |
| (1.02) | (1.24) | (33.39) | ||
| University.Rating2 | -0.66 | -0.31 | ||
| (0.47) | (0.47) | |||
| University.Rating3 | -0.18 | 0.11 | ||
| (0.47) | (0.48) | |||
| University.Rating4 | 0.28 | 0.22 | ||
| (0.53) | (0.55) | |||
| University.Rating5 | 0.08 | -0.22 | ||
| (0.64) | (0.69) | |||
| GRE.Score | -0.15* | |||
| (0.07) | ||||
| TOEFL.Score | -0.06 | |||
| (0.04) | ||||
| Chance.of.Admit:GRE.Score | 0.37*** | |||
| (0.11) | ||||
| AIC | 518.33 | 518.40 | 484.93 | |
| BIC | 526.76 | 543.69 | 522.87 | |
| Log Likelihood | -257.17 | -253.20 | -233.47 | |
| Deviance | 514.33 | 506.40 | 466.93 | |
| Num. obs. | 500 | 500 | 500 | |
| p < 0.001, p < 0.01, p < 0.05 | ||||
The Lowerst numbers of AIC/BIC indicate better fit for a model, thus, we are going to plot model 3 which has also provides more variables to examine.
As seen below, having a research experience increases probability of higher chances of admission.
library(visreg)
visreg(mod3, "Chance.of.Admit",scale="response")## Conditions used in construction of plot
## GRE.Score: 317
## University.Rating: 3
## TOEFL.Score: 107
visreg(mod3, "Chance.of.Admit", by ='GRE.Score', scale="response")
There is an interesting finding to observe while analyzing the plots and making a conclusion on the hypothesis: Overall, a research experience does increase the chances of admission to graduate programs. However, in a range of the lowest 33% of the GRE scores, the research experience does not increase the chances of admissions by any significant number. Only, reaching the GRE score of 317 and up a research experience creates a significant positive effect on probability of admissions. And, even more interestingly, in the highest range of the top GRE scores (top 5-8%), having a research experience does not impact chances of admissions significantly (if at all) because the chances of admissions are high with, or without a research experience. This particular finding of irrelevance of having a research experience when your GRE scores are very high is surprising to discover.
Thus, accordning to this constructed model, for any prospective applicant with above average but not very high GRE, it is true that a research experience will significantly increase his or her chances of gradute admissions.