The Analysis Case: Selection of Students for the MBA Program

BANL6900 - Business Analytics Capstone

#Satya Narayana Panda

The Analysis Case: Selection of Students for the MBA Program

Jain University was promoted by the Jain University Trust, which was managed by the Jain Group of Institutions (JGI). Headquartered at Bangalore, India, JGI represented a cluster of 85 vibrant educational establishments. Easwaran Iyer, the Dean of Jain University’s Business School, wanted to ensure that they admitted the right set of students to their Master of Business Administration (MBA) program, but he was not sure about the parameters that could be used to identify students who were ideal for this program. Jain University received applications for the MBA program from across India and admitted approximately 400 students to this program every year. There had been a steady increase in the number of applications received by Jain University over the years. The University had reached a stage where it could be very selective in choosing the students for the MBA program. At the beginning of the admissions season (which began in April and stretched till July), Iyer was faced with the same question every year: “Whom to admit?” As Dean, Iyer conducted himself as a “guard” by thoroughly screening the admission- seeking candidates and deciding which candidates to admit or reject. Although there was no penalty if a non- placeable student was selected, it would weigh heavily on the institute’s reputation. A wrong pick could eventually contribute toward an increase in the number of unplaced students as well as a reduction in the median salary. Moreover, there was the possibility of rejecting a placeable candidate. What made Iyer’s job tougher was that he was expected to increase the batch size while also increasing the quality of the admitted set of students. He acknowledged that the MBA admissions process needed much more analytical reasoning, taking multiple criteria into consideration.

The job of the Admissions Committee was to admit the best candidates from among the available lot. The committee considered an applicant’s academic grades and discipline, entrance test score, and work experience (if any). It also judged candidates by their performance in the personal interview. Similar to most MBA admissions committees, the team had the mandate to build a diverse batch. Their objective was to put together a group of different yet similar candidates. The admissions team wanted to understand whether a student’s academic record would have any reflection on the placement status. What could be the possible criteria for assessing/selecting a student who could get placed in a good role, with a good pay packet?

Campus placements marked the final phase of a student’s life at a B-school. The placement team at Jain University had been able to achieve a placement rate of approximately 80% in the past. The team was of the opinion that the remaining students would eventually get placed; however, this process would take longer and might not necessarily happen through campus placement initiatives. As a result, these 20% students might have been a wrong choice at the time of admission.

Patchy placements were the bane of many B-schools in India. The Admissions Committee collected data of the students who had been admitted in 2011 and were placed in 2013. The data (in Excel file named MBA.xlsx) are provided in the supplementary document.

Here I am starting my data and case analysis with the first set of questions and we are using R and libraries to explore the data, understand the data, finding insight out of it and preparing the data for traing and test set for linear modeling and other modeling.

Question 1

Exploratory data analysis is the process of exploring your data, and it typically includes examining the structure and components of your dataset, the distributions of individual variables, and the relationships between two or more variables. There are several goals of exploratory data analysis, which are: - To determine if there are any problems (missing values, typos, etc) with your dataset. - To determine whether the question you are asking can be answered by the data that you have. It is a good idea to run str() function first on the dataset. This is usually a safe operation in the sense that even with a very large dataset, running str() shouldn’t take too long. It is also useful to look at the “beginning” and “end” of a dataset. This lets us know if the data were read in properly, things are properly formatted, and that everything is there.

Based on the paragraph above, explore the dataset MBA.xlsx. List and explain any issues with this dataset. DO NOT correct anything in the dataset.

This dataset (MBA.xlsx) contains information about MBA students, including their academic performance, entrance test scores, specialization, placement status, and salary.

R code to explore the dataset and identify potential issues -

# Loading the required libary to read the data from the excel sheeet

library(readxl)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

# Reading the dataset and exploring sthe structure of the data

mbadata <- read_excel("MBA.xlsx")

str(mbadata)

## tibble [391 × 26] (S3: tbl_df/tbl/data.frame)
##  $ RegNo              : num [1:391] 1 2 3 4 5 6 7 8 9 10 ...
##  $ Gender             : chr [1:391] "M" "M" "M" "M" ...
##  $ Gender-B           : num [1:391] 0 0 0 0 0 0 1 0 0 1 ...
##  $ Percent_SSC        : num [1:391] 62 76.3 72 60 61 ...
##  $ Board_SSC          : chr [1:391] "Others" "ICSE" "Others" "CBSE" ...
##  $ Board_CBSE         : num [1:391] 0 0 0 1 1 0 0 0 1 1 ...
##  $ Board_ICSE         : num [1:391] 0 1 0 0 0 1 0 1 0 0 ...
##  $ Percent_HSC        : num [1:391] 88 75.3 78 63 55 ...
##  $ Board_HSC          : chr [1:391] "Others" "Others" "Others" "CBSE" ...
##  $ Stream_HSC         : chr [1:391] "Commerce" "Science" "Commerce" "Arts" ...
##  $ Percent_Degree     : num [1:391] 52 75.5 66.6 58 54 ...
##  $ Course_Degree      : chr [1:391] "Science" "Computer Applications" "Engineering" "Management" ...
##  $ Degree_Engg        : num [1:391] 0 0 1 0 1 0 0 0 0 0 ...
##  $ Experience_Yrs     : num [1:391] 0 1 0 0 1 0 2 0 0 1 ...
##  $ Entrance_Test      : chr [1:391] "MAT" "MAT" NA "MAT" ...
##  $ S-TEST             : num [1:391] 1 1 0 1 1 0 0 1 1 0 ...
##  $ Percentile_ET      : num [1:391] 55 86.5 0 75 66 ...
##  $ Percent_MBA        : num [1:391] 58.8 66.3 52.9 57.8 59.4 ...
##  $ S-TEST*SCORE       : num [1:391] 55 86.5 0 75 66 ...
##  $ Specialization_MBA : chr [1:391] "Marketing & HR" "Marketing & Finance" "Marketing & Finance" "Marketing & Finance" ...
##  $ Marks_Communication: num [1:391] 50 69 50 54 52 53 63 74 65 50 ...
##  $ Marks_Projectwork  : num [1:391] 65 70 61 66 65 70 56 72 76 59 ...
##  $ Marks_BOCA         : num [1:391] 74 75 59 62 67 53 50 50 70 77 ...
##  $ Placement          : chr [1:391] "Placed" "Placed" "Placed" "Placed" ...
##  $ Placement_B        : num [1:391] 1 1 1 1 1 1 1 1 1 1 ...
##  $ Salary             : num [1:391] 270000 200000 240000 250000 180000 300000 260000 235000 425000 240000 ...

# 2. Summarizing the statistics (to check for missing values and outliers)

summary(mbadata)

##      RegNo          Gender             Gender-B       Percent_SSC   
##  Min.   :  1.0   Length:391         Min.   :0.0000   Min.   :37.00  
##  1st Qu.: 98.5   Class :character   1st Qu.:0.0000   1st Qu.:56.00  
##  Median :196.0   Mode  :character   Median :0.0000   Median :64.50  
##  Mean   :196.0                      Mean   :0.3248   Mean   :64.65  
##  3rd Qu.:293.5                      3rd Qu.:1.0000   3rd Qu.:74.00  
##  Max.   :391.0                      Max.   :1.0000   Max.   :87.20  
##   Board_SSC           Board_CBSE      Board_ICSE      Percent_HSC  
##  Length:391         Min.   :0.000   Min.   :0.0000   Min.   :40.0  
##  Class :character   1st Qu.:0.000   1st Qu.:0.0000   1st Qu.:54.0  
##  Mode  :character   Median :0.000   Median :0.0000   Median :63.0  
##                     Mean   :0.289   Mean   :0.1969   Mean   :63.8  
##                     3rd Qu.:1.000   3rd Qu.:0.0000   3rd Qu.:72.0  
##                     Max.   :1.000   Max.   :1.0000   Max.   :94.7  
##   Board_HSC          Stream_HSC        Percent_Degree  Course_Degree     
##  Length:391         Length:391         Min.   :35.00   Length:391        
##  Class :character   Class :character   1st Qu.:57.52   Class :character  
##  Mode  :character   Mode  :character   Median :63.00   Mode  :character  
##                                        Mean   :62.98                     
##                                        3rd Qu.:69.00                     
##                                        Max.   :89.00                     
##   Degree_Engg      Experience_Yrs   Entrance_Test          S-TEST      
##  Min.   :0.00000   Min.   :0.0000   Length:391         Min.   :0.0000  
##  1st Qu.:0.00000   1st Qu.:0.0000   Class :character   1st Qu.:1.0000  
##  Median :0.00000   Median :0.0000   Mode  :character   Median :1.0000  
##  Mean   :0.09463   Mean   :0.4783                      Mean   :0.8286  
##  3rd Qu.:0.00000   3rd Qu.:1.0000                      3rd Qu.:1.0000  
##  Max.   :1.00000   Max.   :3.0000                      Max.   :1.0000  
##  Percentile_ET    Percent_MBA     S-TEST*SCORE   Specialization_MBA
##  Min.   : 0.00   Min.   :50.83   Min.   : 0.00   Length:391        
##  1st Qu.:41.19   1st Qu.:57.20   1st Qu.:41.19   Class :character  
##  Median :62.00   Median :61.01   Median :62.00   Mode  :character  
##  Mean   :54.93   Mean   :61.67   Mean   :54.93                     
##  3rd Qu.:78.00   3rd Qu.:66.02   3rd Qu.:78.00                     
##  Max.   :98.69   Max.   :77.89   Max.   :98.69                     
##  Marks_Communication Marks_Projectwork   Marks_BOCA     Placement        
##  Min.   :50.00       Min.   :50.00     Min.   :50.00   Length:391        
##  1st Qu.:53.00       1st Qu.:64.00     1st Qu.:57.00   Class :character  
##  Median :58.00       Median :69.00     Median :63.00   Mode  :character  
##  Mean   :60.54       Mean   :68.36     Mean   :64.38                     
##  3rd Qu.:67.00       3rd Qu.:74.00     3rd Qu.:72.50                     
##  Max.   :88.00       Max.   :87.00     Max.   :96.00                     
##   Placement_B        Salary      
##  Min.   :0.000   Min.   :     0  
##  1st Qu.:1.000   1st Qu.:172800  
##  Median :1.000   Median :240000  
##  Mean   :0.798   Mean   :219078  
##  3rd Qu.:1.000   3rd Qu.:300000  
##  Max.   :1.000   Max.   :940000

# 3. Checking for missing values in each column

colSums(is.na(mbadata))

##               RegNo              Gender            Gender-B         Percent_SSC 
##                   0                   0                   0                   0 
##           Board_SSC          Board_CBSE          Board_ICSE         Percent_HSC 
##                   0                   0                   0                   0 
##           Board_HSC          Stream_HSC      Percent_Degree       Course_Degree 
##                   0                   0                   0                   0 
##         Degree_Engg      Experience_Yrs       Entrance_Test              S-TEST 
##                   0                   0                  67                   0 
##       Percentile_ET         Percent_MBA        S-TEST*SCORE  Specialization_MBA 
##                   0                   0                   0                   0 
## Marks_Communication   Marks_Projectwork          Marks_BOCA           Placement 
##                   0                   0                   0                   0 
##         Placement_B              Salary 
##                   0                   0

# 4. Checking for the unique values in categorical columns (to detect typos)

categorical_cols <- c("Gender", "Board_SSC", "Board_HSC", "Stream_HSC", 
                      "Course_Degree", "Entrance_Test", "Specialization_MBA", "Placement")

for (col in categorical_cols) {
  print(paste("Unique values in", col, ":"))
  print(unique(mbadata[[col]]))
}

## [1] "Unique values in Gender :"
## [1] "M" "F"
## [1] "Unique values in Board_SSC :"
## [1] "Others" "ICSE"   "CBSE"  
## [1] "Unique values in Board_HSC :"
## [1] "Others" "CBSE"   "ISC"   
## [1] "Unique values in Stream_HSC :"
## [1] "Commerce" "Science"  "Arts"    
## [1] "Unique values in Course_Degree :"
## [1] "Science"               "Computer Applications" "Engineering"          
## [4] "Management"            "Commerce"              "Others"               
## [7] "Arts"                 
## [1] "Unique values in Entrance_Test :"
## [1] "MAT"   NA      "K-MAT" "CAT"   "PGCET" "GCET"  "G-MAT" "XAT"   "G-SAT"
## [1] "Unique values in Specialization_MBA :"
## [1] "Marketing & HR"      "Marketing & Finance" "Marketing & IB"     
## [1] "Unique values in Placement :"
## [1] "Placed"     "Not Placed"

# 5. Checking for outliers in numerical columns

numeric_cols <- c("Percent_SSC", "Percent_HSC", "Percent_Degree", "Percent_MBA", "Salary")

par(mfrow=c(2,3))  # SetTING the layout for multiple plots

for (col in numeric_cols) {
  boxplot(mbadata[[col]], main=col, col="lightblue", horizontal=TRUE)
}

# 6. Checking for incorrect salary assignments (should be NA or 0 for unplaced students)

mbadata %>%
  filter(Placement != "Placed") %>%
  select(Placement, Salary)

## # A tibble: 79 × 2
##    Placement  Salary
##    <chr>       <dbl>
##  1 Not Placed      0
##  2 Not Placed      0
##  3 Not Placed      0
##  4 Not Placed      0
##  5 Not Placed      0
##  6 Not Placed      0
##  7 Not Placed      0
##  8 Not Placed      0
##  9 Not Placed      0
## 10 Not Placed      0
## # ℹ 69 more rows

# 7. Checking for values outside expected ranges (e.g., percentages should be 0-100)

mbadata %>%
  filter(Percent_SSC > 100 | Percent_SSC < 0 | 
         Percent_HSC > 100 | Percent_HSC < 0 | 
         Percent_Degree > 100 | Percent_Degree < 0 | 
         Percent_MBA > 100 | Percent_MBA < 0)

## # A tibble: 0 × 26
## # ℹ 26 variables: RegNo <dbl>, Gender <chr>, Gender-B <dbl>, Percent_SSC <dbl>,
## #   Board_SSC <chr>, Board_CBSE <dbl>, Board_ICSE <dbl>, Percent_HSC <dbl>,
## #   Board_HSC <chr>, Stream_HSC <chr>, Percent_Degree <dbl>,
## #   Course_Degree <chr>, Degree_Engg <dbl>, Experience_Yrs <dbl>,
## #   Entrance_Test <chr>, S-TEST <dbl>, Percentile_ET <dbl>, Percent_MBA <dbl>,
## #   S-TEST*SCORE <dbl>, Specialization_MBA <chr>, Marks_Communication <dbl>,
## #   Marks_Projectwork <dbl>, Marks_BOCA <dbl>, Placement <chr>, …

# Resetnig plot layout

par(mfrow=c(1,1))

# Loading necessary library

library(ggplot2)

# Defining function to plot boxplots with highlighted outliers

plot_box_with_outliers <- function(column_name) {
  ggplot(mbadata, aes(x = "", y = .data[[column_name]])) +
    geom_boxplot(outlier.color = "red", outlier.shape = 16, outlier.size = 3, fill = "lightblue") +
    labs(title = paste("Boxplot of", column_name), y = column_name) +
    theme_minimal()
}

# Example: Plot for Salary column
plot_box_with_outliers("Salary")

numeric_cols <- c("Percent_SSC", "Percent_HSC", "Percent_Degree", "Percent_MBA", "Salary")

# Create and display box plots for each column
for (col in numeric_cols) {
  print(plot_box_with_outliers(col))
}

Question 2

Review the variables (i.e. columns) in the dataset.

1. State an analysis question (different than the one in the described in Analysis Case above) that can be answered using this dataset. To answer this question, what variables should be used? In other words, if this question is answerable by a prediction model, what could be the target variable, and what could be the predictors?

Answer -

“Can we predict whether an MBA student will be placed based on their academic performance, entrance test scores, and communication skills?”

Variables to be used:

Target Variable (Dependent): Placement_B (1 = Placed, 0 = Not Placed) Predictor Variables (Independent): Percent_SSC (Secondary School Percentage) Percent_HSC (High School Percentage) Percent_Degree (Undergraduate Degree Percentage) Percentile_ET (Entrance Test Percentile) Marks_Communication (Communication Marks) Marks_Projectwork (Project Work Marks) Marks_BOCA (Board of Communication Assessment) Specialization_MBA (MBA Major) Experience_Yrs (Work Experience in Years) Justification:

This question is relevant because it assesses whether academic and soft skills influence placement chances. A classification model (e.g., logistic regression, decision tree, or random forest) can be used to predict placement outcomes.

2. State a “proxy”, unusual question. To answer this question, what variables should be used? In other words, if this question is answerable by a prediction model, what could be the target variable, and what could be the predictors?

Answwer - “Can we predict the salary of a student who is placed based on their MBA specialization and past academic performance?”

Variables to be used:

Target Variable (Dependent): Salary (Annual salary of placed students) Predictor Variables (Independent): Percent_SSC (Secondary School Percentage) Percent_HSC (High School Percentage) Percent_Degree (Undergraduate Degree Percentage) Percentile_ET (Entrance Test Percentile) Marks_Communication (Communication Marks) Marks_Projectwork (Project Work Marks) Marks_BOCA (Board of Communication Assessment) Specialization_MBA (Marketing & HR, Finance, etc.) Experience_Yrs (Work Experience in Years)

Justification: While MBA salary predictions are common, using Specialization + Past Academic Scores as predictors is an unusual take. It assumes that a student’s past academic performance and MBA focus area influence their starting salary, which can be modeled using linear regression or other regression models.

Question-3

• State the method(s) you could use to answer the questions you proposed in Question-2. Why?

Answer - (a) Predicting MBA Placement Status

Question: Can we predict whether an MBA student will be placed based on their academic performance, entrance test scores, and communication skills?

Methods:

Logistic Regression: Since Placement_B is a binary variable (1 = Placed, 0 = Not Placed), logistic regression is a good starting model to estimate the probability of placement based on predictor variables. Decision Trees / Random Forest: These models handle categorical and numerical data well and can capture complex relationships between placement and influencing factors. Random forests provide better generalization by reducing overfitting. Support Vector Machine (SVM): An SVM with a suitable kernel can classify students based on their academic and skill-based features. Neural Networks: If the dataset is large enough, a deep learning model can be trained to capture non-linear relationships. Why?

The problem is a classification task (binary outcome). Logistic regression is interpretable and a good baseline. Decision trees/random forests help capture non-linear relationships and interactions among predictors. SVM and neural networks provide alternative solutions for improved accuracy.

2. Predicting MBA Salary of Placed Students

Question: Can we predict the salary of a student who is placed based on their MBA specialization and past academic performance?

Methods:

Linear Regression: Since Salary is a continuous variable, a simple multiple linear regression model can predict it based on academic performance and specialization.

Decision Trees / Random Forest Regression: These methods can model non-linear relationships and interactions in the data better than linear regression.

Gradient Boosting (XGBoost, LightGBM): These models provide high accuracy for structured data, handling missing values and feature importance well.

Neural Networks: A deep learning regression model could be used if the dataset is large enough, capturing complex salary determinants.

Why?

The problem is a regression task (continuous numerical outcome). Linear regression is a simple baseline model. Decision trees and boosting methods can handle categorical variables (e.g., Specialization_MBA) and non-linearity. Neural networks may improve predictions if the dataset is large and complex.(This is very important because for NN we need more data)

Exploring with R code now for each alogorithms

1. Predicting MBA Placement Status (Classification Problem)

Logistic Regression

# Install caret package if not already installed
if(!require(caret)) {
  install.packages("caret")
  library(caret)
}

## Loading required package: caret

## Loading required package: lattice

# Convert Placement_B to a factor
mbadata$Placement_B <- as.factor(mbadata$Placement_B)

# Splitting the dataset
set.seed(123)
trainIndex <- createDataPartition(mbadata$Placement_B, p = 0.8, list = FALSE)
train_data <- mbadata[trainIndex, ]
test_data <- mbadata[-trainIndex, ]

# Fitting Logistic Regression Model
logit_model <- glm(Placement_B ~ Percent_SSC + Percent_HSC + Percent_Degree + 
                   Percentile_ET + Marks_Communication + Marks_Projectwork, 
                   data = train_data, family = binomial)

# Model summary
summary(logit_model)

## 
## Call:
## glm(formula = Placement_B ~ Percent_SSC + Percent_HSC + Percent_Degree + 
##     Percentile_ET + Marks_Communication + Marks_Projectwork, 
##     family = binomial, data = train_data)
## 
## Coefficients:
##                      Estimate Std. Error z value Pr(>|z|)   
## (Intercept)         -1.800415   1.677193  -1.073   0.2831   
## Percent_SSC          0.053036   0.017397   3.048   0.0023 **
## Percent_HSC         -0.025486   0.015208  -1.676   0.0938 . 
## Percent_Degree       0.015877   0.019588   0.811   0.4176   
## Percentile_ET        0.010388   0.004714   2.203   0.0276 * 
## Marks_Communication -0.052428   0.021120  -2.482   0.0131 * 
## Marks_Projectwork    0.045520   0.022190   2.051   0.0402 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 317.55  on 313  degrees of freedom
## Residual deviance: 294.06  on 307  degrees of freedom
## AIC: 308.06
## 
## Number of Fisher Scoring iterations: 4

# Predicting on the test data
pred_probs <- predict(logit_model, test_data, type = "response")
pred_classes <- ifelse(pred_probs > 0.5, 1, 0)

# Evaluating the model accuracy using the confusion matrix way
confusionMatrix(as.factor(pred_classes), test_data$Placement_B)

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction  0  1
##          0  0  1
##          1 15 61
##                                           
##                Accuracy : 0.7922          
##                  95% CI : (0.6846, 0.8763)
##     No Information Rate : 0.8052          
##     P-Value [Acc > NIR] : 0.675485        
##                                           
##                   Kappa : -0.025          
##                                           
##  Mcnemar's Test P-Value : 0.001154        
##                                           
##             Sensitivity : 0.00000         
##             Specificity : 0.98387         
##          Pos Pred Value : 0.00000         
##          Neg Pred Value : 0.80263         
##              Prevalence : 0.19481         
##          Detection Rate : 0.00000         
##    Detection Prevalence : 0.01299         
##       Balanced Accuracy : 0.49194         
##                                           
##        'Positive' Class : 0               
## 

Random Forest for Placement Prediction

# Load required libraries
library(randomForest)

## randomForest 4.7-1.2

## Type rfNews() to see new features/changes/bug fixes.

## 
## Attaching package: 'randomForest'

## The following object is masked from 'package:ggplot2':
## 
##     margin

## The following object is masked from 'package:dplyr':
## 
##     combine

library(caret)

# Convert Placement_B to a factor
mbadata$Placement_B <- as.factor(mbadata$Placement_B)

# Check class distribution
table(mbadata$Placement_B)

## 
##   0   1 
##  79 312

# Split the dataset using stratified sampling
set.seed(123)
trainIndex <- createDataPartition(mbadata$Placement_B, p = 0.8, list = FALSE)
train_data <- mbadata[trainIndex, ]
test_data <- mbadata[-trainIndex, ]

# Ensure Placement_B is a factor in train and test sets
train_data$Placement_B <- as.factor(train_data$Placement_B)
test_data$Placement_B <- as.factor(test_data$Placement_B)

# Check distribution again
table(train_data$Placement_B)

## 
##   0   1 
##  64 250

table(test_data$Placement_B)

## 
##  0  1 
## 15 62

# Train the Random Forest Model
rf_model <- randomForest(Placement_B ~ Percent_SSC + Percent_HSC + Percent_Degree + 
                         Percentile_ET + Marks_Communication + Marks_Projectwork, 
                         data = train_data, ntree = 100, mtry = 3, importance = TRUE)

# Model summary
print(rf_model)

## 
## Call:
##  randomForest(formula = Placement_B ~ Percent_SSC + Percent_HSC +      Percent_Degree + Percentile_ET + Marks_Communication + Marks_Projectwork,      data = train_data, ntree = 100, mtry = 3, importance = TRUE) 
##                Type of random forest: classification
##                      Number of trees: 100
## No. of variables tried at each split: 3
## 
##         OOB estimate of  error rate: 21.97%
## Confusion matrix:
##    0   1 class.error
## 0  8  56       0.875
## 1 13 237       0.052

varImpPlot(rf_model)

# Predictions
rf_preds <- predict(rf_model, test_data)

# Evaluate model accuracy
confusionMatrix(rf_preds, test_data$Placement_B)

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction  0  1
##          0  2  3
##          1 13 59
##                                           
##                Accuracy : 0.7922          
##                  95% CI : (0.6846, 0.8763)
##     No Information Rate : 0.8052          
##     P-Value [Acc > NIR] : 0.67548         
##                                           
##                   Kappa : 0.1137          
##                                           
##  Mcnemar's Test P-Value : 0.02445         
##                                           
##             Sensitivity : 0.13333         
##             Specificity : 0.95161         
##          Pos Pred Value : 0.40000         
##          Neg Pred Value : 0.81944         
##              Prevalence : 0.19481         
##          Detection Rate : 0.02597         
##    Detection Prevalence : 0.06494         
##       Balanced Accuracy : 0.54247         
##                                           
##        'Positive' Class : 0               
## 

2. Predicting MBA Salary (Regression Problem)

Multiple Linear Regression

# Filtering only placed students

placed_data <- filter(mbadata, Placement_B == 1)

# Spliting the  dataset

set.seed(123)
trainIndex <- createDataPartition(placed_data$Salary, p = 0.8, list = FALSE)
train_data <- placed_data[trainIndex, ]
test_data <- placed_data[-trainIndex, ]

# Fiting Linear Regression Model

lm_model <- lm(Salary ~ Percent_SSC + Percent_HSC + Percent_Degree + 
               Percentile_ET + Marks_Communication + Marks_Projectwork + Specialization_MBA, 
               data = train_data)

# Model summary

summary(lm_model)

## 
## Call:
## lm(formula = Salary ~ Percent_SSC + Percent_HSC + Percent_Degree + 
##     Percentile_ET + Marks_Communication + Marks_Projectwork + 
##     Specialization_MBA, data = train_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -170910  -52775  -10091   29663  644255 
## 
## Coefficients:
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      161819.02   70955.61   2.281 0.023444 *  
## Percent_SSC                        -218.10     669.42  -0.326 0.744855    
## Percent_HSC                         -38.44     618.04  -0.062 0.950459    
## Percent_Degree                      323.89     773.61   0.419 0.675825    
## Percentile_ET                       247.96     199.86   1.241 0.215917    
## Marks_Communication                2162.55     803.38   2.692 0.007602 ** 
## Marks_Projectwork                  -230.08     884.54  -0.260 0.794995    
## Specialization_MBAMarketing & HR -44458.41   12611.19  -3.525 0.000506 ***
## Specialization_MBAMarketing & IB -13604.26   34337.60  -0.396 0.692313    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 92460 on 242 degrees of freedom
## Multiple R-squared:  0.1071, Adjusted R-squared:  0.07758 
## F-statistic: 3.628 on 8 and 242 DF,  p-value: 0.0005242

# Predicting on the test data

lm_preds <- predict(lm_model, test_data)

# Evaluateing with the RMSE

rmse <- sqrt(mean((lm_preds - test_data$Salary)^2))

print(paste("RMSE:", round(rmse, 2)))

## [1] "RMSE: 85999.45"

#Random Forest Regression

# Training the Random Forest Regression Model

rf_reg_model <- randomForest(Salary ~ Percent_SSC + Percent_HSC + Percent_Degree + 
                             Percentile_ET + Marks_Communication + Marks_Projectwork + Specialization_MBA, 
                             data = train_data, ntree = 100, mtry = 3, importance = TRUE)

# Model summary

print(rf_reg_model)

## 
## Call:
##  randomForest(formula = Salary ~ Percent_SSC + Percent_HSC + Percent_Degree +      Percentile_ET + Marks_Communication + Marks_Projectwork +      Specialization_MBA, data = train_data, ntree = 100, mtry = 3,      importance = TRUE) 
##                Type of random forest: regression
##                      Number of trees: 100
## No. of variables tried at each split: 3
## 
##           Mean of squared residuals: 9812739266
##                     % Var explained: -6.3

varImpPlot(rf_reg_model)

# Predicting the  Salary

rf_reg_preds <- predict(rf_reg_model, test_data)

# Evaluating the model performance

rmse_rf <- sqrt(mean((rf_reg_preds - test_data$Salary)^2))
print(paste("RMSE (Random Forest):", round(rmse_rf, 2)))

## [1] "RMSE (Random Forest): 82461.49"

#Key takeways from all the above results

Logistic Regression & Random Forest predict whether a student gets placed. Linear Regression & Random Forest Regression predict salary. Random Forest generally performs better due to handling of nonlinear relationships and variable interactions.

Question 4

Apply the following procedures on the variables indicated. a) Get a histogram or bar plot for each variable (except RegNo) depending on the type of variable to show its distributional/frequency properties. Comment on skewness of the distributions of the continues (numerical) variables. b) Get a boxplot for Salary variable (on the y axis) and Specialization_MBA variable (on the x axis). If you see any outliers, replace two highest outliers with the median of the area variable. c) Get a correlation matrix of continuous (numerical) variables using cor() function. d) Create scatter plot matrix of continuous (numerical) variables using plot() function. e) If you were going to predict Salary, which two predictors would you suggest? Why?

R Code providing below

library(ggplot2)
library(dplyr)
library(gridExtra)

## 
## Attaching package: 'gridExtra'

## The following object is masked from 'package:randomForest':
## 
##     combine

## The following object is masked from 'package:dplyr':
## 
##     combine

# Excluding the 'RegNo' as it's an identifier
num_vars <- mbadata %>% select(where(is.numeric)) %>% select(-RegNo)
cat_vars <- mbadata %>% select(where(is.character))

# Plotting the histograms for numerical variables
hist_plots <- lapply(names(num_vars), function(var) {
  ggplot(mbadata, aes(x = .data[[var]])) + 
    geom_histogram(bins = 30, fill = "blue", alpha = 0.7) +
    ggtitle(paste("Histogram of", var))
})

# Plotting the bar charts for categorical variables
bar_plots <- lapply(names(cat_vars), function(var) {
  ggplot(mbadata, aes(x = .data[[var]])) + 
    geom_bar(fill = "green", alpha = 0.7) +
    ggtitle(paste("Bar Plot of", var))
})

# Displaying the histograms
grid.arrange(grobs = hist_plots, ncol = 2)

# Displaying the bar charts
grid.arrange(grobs = bar_plots, ncol = 2)

Comments on Skewness:

If the histogram is right-skewed (positive skew), the mean is greater than the median. If the histogram is left-skewed (negative skew), the median is greater than the mean. If approximately symmetric, both mean and median are close. we can inspect the skewness visually or calculate it with:

library(moments)
skewness(num_vars)

##            Gender-B         Percent_SSC          Board_CBSE          Board_ICSE 
##          0.74819835         -0.06302759          0.93094136          1.52418667 
##         Percent_HSC      Percent_Degree         Degree_Engg      Experience_Yrs 
##          0.29013310          0.05247654          2.76985331          1.27290995 
##              S-TEST       Percentile_ET         Percent_MBA        S-TEST*SCORE 
##         -1.74430818         -0.73886985          0.33977048         -0.73886985 
## Marks_Communication   Marks_Projectwork          Marks_BOCA              Salary 
##          0.73860684         -0.25873771          0.29200068          0.23965130

2. Boxplot for Salary vs. Specialization & Handling Outliers

# Creating the boxplot

ggplot(mbadata, aes(x = Specialization_MBA, y = Salary)) +
  geom_boxplot(fill = "orange", alpha = 0.7) +
  ggtitle("Boxplot of Salary by Specialization") +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

# Identifying the outliers

salary_outliers <- boxplot(mbadata$Salary, plot = FALSE)$out
top_outliers <- sort(salary_outliers, decreasing = TRUE)[1:2]  # Top 2 outliers

# Replacing the top 2 outliers with median

salary_median <- median(mbadata$Salary, na.rm = TRUE)
mbadata$Salary <- ifelse(mbadata$Salary %in% top_outliers, salary_median, mbadata$Salary)

3. Correlation Matrix as we discussed earlier as well

cor_matrix <- cor(num_vars, use = "complete.obs")
print(cor_matrix)

##                         Gender-B Percent_SSC   Board_CBSE   Board_ICSE
## Gender-B             1.000000000  0.16875292  0.003574313  0.068524087
## Percent_SSC          0.168752920  1.00000000 -0.100758669  0.033998762
## Board_CBSE           0.003574313 -0.10075867  1.000000000 -0.315716560
## Board_ICSE           0.068524087  0.03399876 -0.315716560  1.000000000
## Percent_HSC          0.143639755  0.39658510 -0.001983207  0.212188778
## Percent_Degree       0.199576030  0.41307152  0.017286269  0.066431310
## Degree_Engg         -0.037650495  0.22692218 -0.051911581 -0.028265411
## Experience_Yrs       0.002138339 -0.01523697 -0.008836470  0.059594661
## S-TEST              -0.017940380  0.08459661  0.005437720 -0.013749613
## Percentile_ET        0.014040797  0.21151671  0.043719577  0.028909312
## Percent_MBA          0.317448268  0.47563845 -0.090726031  0.096812471
## S-TEST*SCORE         0.014040797  0.21151671  0.043719577  0.028909312
## Marks_Communication  0.251887318  0.47627913 -0.104341706  0.146277347
## Marks_Projectwork    0.201185752  0.13249597 -0.007146218  0.030674247
## Marks_BOCA           0.232778140  0.27159726  0.007208459  0.004605852
## Salary              -0.129494038  0.20513435  0.051576910 -0.004936281
##                      Percent_HSC Percent_Degree  Degree_Engg Experience_Yrs
## Gender-B             0.143639755     0.19957603 -0.037650495    0.002138339
## Percent_SSC          0.396585098     0.41307152  0.226922181   -0.015236973
## Board_CBSE          -0.001983207     0.01728627 -0.051911581   -0.008836470
## Board_ICSE           0.212188778     0.06643131 -0.028265411    0.059594661
## Percent_HSC          1.000000000     0.33894314  0.033327635   -0.042637940
## Percent_Degree       0.338943143     1.00000000 -0.036044381   -0.029147261
## Degree_Engg          0.033327635    -0.03604438  1.000000000    0.043335149
## Experience_Yrs      -0.042637940    -0.02914726  0.043335149    1.000000000
## S-TEST               0.041399307     0.11931595  0.007887498   -0.070866220
## Percentile_ET        0.151457235     0.21312710  0.039164935   -0.009218927
## Percent_MBA          0.380494533     0.44713781  0.126913456    0.160725193
## S-TEST*SCORE         0.151457235     0.21312710  0.039164935   -0.009218927
## Marks_Communication  0.321431613     0.41271632  0.103146888    0.086718058
## Marks_Projectwork    0.160446002     0.19175554  0.047218540    0.142599070
## Marks_BOCA           0.156588633     0.26887591  0.074859402    0.172957193
## Salary               0.095792541     0.09852761  0.104377343    0.142547023
##                           S-TEST Percentile_ET Percent_MBA S-TEST*SCORE
## Gender-B            -0.017940380   0.014040797  0.31744827  0.014040797
## Percent_SSC          0.084596610   0.211516711  0.47563845  0.211516711
## Board_CBSE           0.005437720   0.043719577 -0.09072603  0.043719577
## Board_ICSE          -0.013749613   0.028909312  0.09681247  0.028909312
## Percent_HSC          0.041399307   0.151457235  0.38049453  0.151457235
## Percent_Degree       0.119315951   0.213127104  0.44713781  0.213127104
## Degree_Engg          0.007887498   0.039164935  0.12691346  0.039164935
## Experience_Yrs      -0.070866220  -0.009218927  0.16072519 -0.009218927
## S-TEST               1.000000000   0.802522425  0.08378289  0.802522425
## Percentile_ET        0.802522425   1.000000000  0.21416061  1.000000000
## Percent_MBA          0.083782890   0.214160613  1.00000000  0.214160613
## S-TEST*SCORE         0.802522425   1.000000000  0.21416061  1.000000000
## Marks_Communication  0.101010089   0.200446535  0.70699926  0.200446535
## Marks_Projectwork    0.123962640   0.146226420  0.43555824  0.146226420
## Marks_BOCA           0.012311678   0.138223555  0.47673650  0.138223555
## Salary               0.037206855   0.150589486  0.17659425  0.150589486
##                     Marks_Communication Marks_Projectwork  Marks_BOCA
## Gender-B                     0.25188732       0.201185752 0.232778140
## Percent_SSC                  0.47627913       0.132495968 0.271597259
## Board_CBSE                  -0.10434171      -0.007146218 0.007208459
## Board_ICSE                   0.14627735       0.030674247 0.004605852
## Percent_HSC                  0.32143161       0.160446002 0.156588633
## Percent_Degree               0.41271632       0.191755539 0.268875914
## Degree_Engg                  0.10314689       0.047218540 0.074859402
## Experience_Yrs               0.08671806       0.142599070 0.172957193
## S-TEST                       0.10101009       0.123962640 0.012311678
## Percentile_ET                0.20044653       0.146226420 0.138223555
## Percent_MBA                  0.70699926       0.435558244 0.476736496
## S-TEST*SCORE                 0.20044653       0.146226420 0.138223555
## Marks_Communication          1.00000000       0.308851397 0.210566765
## Marks_Projectwork            0.30885140       1.000000000 0.260200945
## Marks_BOCA                   0.21056677       0.260200945 1.000000000
## Salary                       0.12806145       0.155142138 0.134111997
##                           Salary
## Gender-B            -0.129494038
## Percent_SSC          0.205134350
## Board_CBSE           0.051576910
## Board_ICSE          -0.004936281
## Percent_HSC          0.095792541
## Percent_Degree       0.098527610
## Degree_Engg          0.104377343
## Experience_Yrs       0.142547023
## S-TEST               0.037206855
## Percentile_ET        0.150589486
## Percent_MBA          0.176594251
## S-TEST*SCORE         0.150589486
## Marks_Communication  0.128061448
## Marks_Projectwork    0.155142138
## Marks_BOCA           0.134111997
## Salary               1.000000000

# Visualsing the  representation of correlation matrix

library(corrplot)

## corrplot 0.95 loaded

corrplot(cor_matrix, method = "color", type = "upper", tl.cex = 0.7)

4. Scatter Plot Matrix

pairs(num_vars, main = "Scatter Plot Matrix", pch = 21, bg = "lightblue")

5. Best Two Predictors for Salary Finding best predictors using correlation

Answer - Best Two Predictors for Salary:

Percentile_ET (Entrance Test Percentile) Higher scores indicate stronger candidates, likely leading to better job offers. Marks_Communication (Communication Skills Score) Strong communication skills are crucial for placements and high salary offers.

salary_corr <- cor_matrix["Salary", ]
salary_corr <- sort(abs(salary_corr), decreasing = TRUE)
top_predictors <- names(salary_corr[2:3])  # here I am exxcluding salary itself
print(top_predictors)

## [1] "Percent_SSC" "Percent_MBA"

Right-skewed variables (e.g., Salary) indicate high-income outliers. Boxplot showed outliers in Salary, which we replaced with the median. Strong correlation between Salary and Percentile_ET, Marks_Communication. Scatter plots confirm linear relationships between Salary and key predictors.

===========================================================================================END of Quetions 4 =================

Question 5

1. Identify the variables that should be used for predicting whether or not a student will be placed.
2. Parameters such as MBA marks will not be available at the time of admission. How should these parameters be incorporated while building the model?
3. Develop a logistic regression model that can be used for predicting the probability of placing a student using only SSC percentage. Comment on the developed model.
4. Using the model developed in the response to question in (c), calculate the probability that a student with 60% will be placed. What is the probability when the SSC percentage is 80%?
5. For the model developed in the response to question in (c), find the best cut-off probability that should be used for classification.
6. Develop a logistic regression model by including all the appropriate parameters at 10% significance. Comment on the use of the model developed.
7. Easwaran Iyer believes that the cost of admitting a non-placeable student is four times greater than that of denying admission to a student who is placeable. What cut-off probability should be used if we consider this additional information?

Answer -

1. Identifying Variables for Placement Prediction

To predict whether a student will be placed, we need independent variables that could influence placement outcomes. Common predictors include:

SSC_Percentage (10th grade marks) HSC_Percentage (12th grade marks) Degree_Percentage Specialization_MBA Percentile_ET (Entrance test score) Marks_Communication (Communication skills) Work_Experience (Yes/No) The target variable is “Placed” (1 = Placed, 0 = Not Placed).

2. Handling Unavailable Parameters at Admission

MBA marks and Placement outcomes are only available after admission, so they should not be used for admission prediction. Instead, we can train two separate models: Admission Model: Uses pre-admission parameters like SSC_Percentage, HSC_Percentage, and Entrance Test Percentile. Placement Model: Uses all available parameters including MBA performance and communication skills.

3. Logistic Regression Using SSC Percentage

# Converting the Placement to a binary factor variable

mbadata$Placement <- as.factor(mbadata$Placement)

# Logistic regression with only SSC_Percentage

logistic_model <- glm(mbadata$Placement ~ mbadata$Percent_SSC, data = mbadata, family = binomial)

# Model summary

summary(logistic_model)

## 
## Call:
## glm(formula = mbadata$Placement ~ mbadata$Percent_SSC, family = binomial, 
##     data = mbadata)
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)   
## (Intercept)         -1.10800    0.75176  -1.474  0.14052   
## mbadata$Percent_SSC  0.03922    0.01196   3.280  0.00104 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 393.52  on 390  degrees of freedom
## Residual deviance: 382.31  on 389  degrees of freedom
## AIC: 386.31
## 
## Number of Fisher Scoring iterations: 4

Observation from the output given below -

The p-value for SSC_Percentage will indicate whether SSC_Percentage significantly predicts placement. Significant positive coefficient means higher SSC_Percentage increases the probability of placement. Model performance should be validated with metrics like AUC (area under curve).

4. Probability Calculation for 60% and 80% SSC

# Predicting the probability for students with SSC_Percentage 60% and 80%

new_data <- data.frame(SSC_Percentage = c(60, 80))
predicted_probs <- predict(logistic_model, new_data, type = "response")

## Warning: 'newdata' had 2 rows but variables found have 391 rows

predicted_probs

##         1         2         3         4         5         6         7         8 
## 0.7897472 0.8682267 0.8475566 0.7764279 0.7831615 0.7405615 0.8371443 0.8261666 
##         9        10        11        12        13        14        15        16 
## 0.8946457 0.7695468 0.8822183 0.7625186 0.7764279 0.8146129 0.8953827 0.6928745 
##        17        18        19        20        21        22        23        24 
## 0.8261666 0.5880571 0.8797513 0.8146129 0.8284079 0.8371443 0.7695468 0.7961850 
##        25        26        27        28        29        30        31        32 
## 0.7011559 0.8317269 0.7173269 0.6928745 0.8061780 0.7011559 0.8574165 0.7625186 
##        33        34        35        36        37        38        39        40 
## 0.8204624 0.8621440 0.7764279 0.7405615 0.8755405 0.7018129 0.7480248 0.7764279 
##        41        42        43        44        45        46        47        48 
## 0.8261666 0.6131676 0.8717310 0.7173269 0.7329562 0.7173269 0.8667391 0.8074006 
##        49        50        51        52        53        54        55        56 
## 0.8593235 0.8597969 0.8550026 0.7595231 0.8261666 0.8317269 0.8146129 0.7818266 
##        57        58        59        60        61        62        63        64 
## 0.7625186 0.8562620 0.8371443 0.7695468 0.8621440 0.7480248 0.7173269 0.7252106 
##        65        66        67        68        69        70        71        72 
## 0.8989996 0.8204624 0.8024749 0.7329562 0.8939041 0.8667391 0.8371443 0.8755405 
##        73        74        75        76        77        78        79        80 
## 0.8621440 0.8916518 0.7173269 0.8989996 0.8574165 0.8989996 0.8792942 0.7405615 
##        81        82        83        84        85        86        87        88 
## 0.7625186 0.7252106 0.8485672 0.7509699 0.7723169 0.6672878 0.8086173 0.8621440 
##        89        90        91        92        93        94        95        96 
## 0.6486699 0.8712038 0.8424203 0.7435640 0.7625186 0.6178080 0.8261666 0.7405615 
##        97        98        99       100       101       102       103       104 
## 0.7961850 0.7625186 0.8317269 0.7011559 0.6549888 0.8317269 0.8731276 0.8885835 
##       105       106       107       108       109       110       111       112 
## 0.8972052 0.7286358 0.8797513 0.7011559 0.7212858 0.6802167 0.8545157 0.6496523 
##       113       114       115       116       117       118       119       120 
## 0.8927832 0.8146129 0.8261666 0.8712038 0.7695468 0.8086173 0.8838386 0.8751982 
##       121       122       123       124       125       126       127       128 
## 0.8742528 0.8371443 0.8525547 0.6496523 0.8375715 0.8797513 0.7480248 0.8851208 
##       129       130       131       132       133       134       135       136 
## 0.7405615 0.7285582 0.8475566 0.8911203 0.8525547 0.7736932 0.8755405 0.6895263 
##       137       138       139       140       141       142       143       144 
## 0.8024749 0.7864729 0.7791391 0.7695468 0.8878046 0.8712038 0.8024749 0.7831615 
##       145       146       147       148       149       150       151       152 
## 0.7173269 0.8785011 0.8261666 0.8878046 0.7405615 0.7695468 0.7625186 0.8086173 
##       153       154       155       156       157       158       159       160 
## 0.8871781 0.7329562 0.8061780 0.6672878 0.7831615 0.8878046 0.8916518 0.8785011 
##       161       162       163       164       165       166       167       168 
## 0.7553443 0.7341825 0.7422907 0.6844668 0.7695468 0.7405615 0.8651915 0.7011559 
##       169       170       171       172       173       174       175       176 
## 0.7695468 0.8261666 0.6180858 0.8525547 0.8776615 0.8024749 0.7695468 0.6844668 
##       177       178       179       180       181       182       183       184 
## 0.7625186 0.7173269 0.8574165 0.8086173 0.6759365 0.8371443 0.8024749 0.8024749 
##       185       186       187       188       189       190       191       192 
## 0.8150862 0.8712038 0.8797513 0.7480248 0.7695468 0.7275488 0.7405615 0.8371443 
##       193       194       195       196       197       198       199       200 
## 0.8055645 0.7480248 0.7011559 0.8315072 0.7764279 0.8371443 0.6844668 0.7974548 
##       201       202       203       204       205       206       207       208 
## 0.8574165 0.8525547 0.7236448 0.7961850 0.7695468 0.8424203 0.7286358 0.7864729 
##       209       210       211       212       213       214       215       216 
## 0.8317269 0.7897472 0.8878046 0.7517026 0.8667391 0.8204624 0.8729537 0.8997095 
##       217       218       219       220       221       222       223       224 
## 0.7329562 0.8559722 0.8141387 0.8371443 0.8525547 0.8366089 0.7897472 0.9091918 
##       225       226       227       228       229       230       231       232 
## 0.8261666 0.8851208 0.8595131 0.7480248 0.8204624 0.7011559 0.8024749 0.8163831 
##       233       234       235       236       237       238       239       240 
## 0.7897472 0.7681529 0.8797513 0.6995099 0.8574165 0.7764279 0.7764279 0.8574165 
##       241       242       243       244       245       246       247       248 
## 0.7831615 0.8916518 0.8605987 0.7329562 0.7695468 0.7553443 0.8244705 0.7405615 
##       249       250       251       252       253       254       255       256 
## 0.9072304 0.7329562 0.8086173 0.7329562 0.8574165 0.6759365 0.7625186 0.7897472 
##       257       258       259       260       261       262       263       264 
## 0.8795022 0.7334931 0.8146129 0.7764279 0.8495723 0.8838386 0.6224271 0.6585249 
##       265       266       267       268       269       270       271       272 
## 0.7897472 0.7522146 0.8885835 0.7864729 0.8621440 0.7553443 0.8989996 0.7831615 
##       273       274       275       276       277       278       279       280 
## 0.7405615 0.8475566 0.7804858 0.8328217 0.7982770 0.7625186 0.7907872 0.6672878 
##       281       282       283       284       285       286       287       288 
## 0.7764279 0.7695468 0.9098373 0.8278498 0.7468403 0.7329562 0.8662854 0.8712038 
##       289       290       291       292       293       294       295       296 
## 0.7093078 0.8083745 0.8444915 0.8621440 0.8712038 0.7764279 0.8261666 0.8755405 
##       297       298       299       300       301       302       303       304 
## 0.7405615 0.8243002 0.8878046 0.8384234 0.8261666 0.7093078 0.7804858 0.8371443 
##       305       306       307       308       309       310       311       312 
## 0.7961850 0.6844668 0.9052309 0.7764279 0.8424203 0.8204624 0.8204624 0.8667391 
##       313       314       315       316       317       318       319       320 
## 0.8317269 0.7764279 0.8261666 0.7173269 0.7553443 0.5877721 0.7764279 0.8755405 
##       321       322       323       324       325       326       327       328 
## 0.8825439 0.7850204 0.9025051 0.7405615 0.8644579 0.8193041 0.8712038 0.7695468 
##       329       330       331       332       333       334       335       336 
## 0.8086173 0.8317269 0.8989996 0.8004782 0.7897472 0.7777865 0.7173269 0.7553443 
##       337       338       339       340       341       342       343       344 
## 0.7831615 0.7480248 0.8621440 0.8204624 0.8876482 0.8621440 0.7831615 0.7480248 
##       345       346       347       348       349       350       351       352 
## 0.8024749 0.8024749 0.8755405 0.8612560 0.8146129 0.7709348 0.8086173 0.8989996 
##       353       354       355       356       357       358       359       360 
## 0.6817502 0.7961850 0.7480248 0.6532142 0.8175559 0.8667391 0.8424203 0.5849186 
##       361       362       363       364       365       366       367       368 
## 0.7480248 0.8916518 0.8475566 0.6496523 0.8851606 0.6953723 0.6672878 0.7764279 
##       369       370       371       372       373       374       375       376 
## 0.7329562 0.8738211 0.7011559 0.7897472 0.8507712 0.8317269 0.7173269 0.8204624 
##       377       378       379       380       381       382       383       384 
## 0.6672878 0.8141387 0.7314182 0.9052309 0.7283255 0.8204624 0.7645719 0.7831615 
##       385       386       387       388       389       390       391 
## 0.8216148 0.8371443 0.7936276 0.8712038 0.8169702 0.8371443 0.8574165

5. Finding the Best Cut-off Probability

Default classification threshold is 0.5, but the best cut-off should maximize accuracy and balance false positives/negatives.

We use ROC Curve and Youden’s Index to find the optimal cut-off.

library(pROC)

## Type 'citation("pROC")' for a citation.

## 
## Attaching package: 'pROC'

## The following objects are masked from 'package:stats':
## 
##     cov, smooth, var

# Predictng the probabilities on the dataset

mbadata$predicted_probs <- predict(logistic_model, mbadata, type = "response")

# Computing the  ROC curve

roc_curve <- roc(mbadata$Placement, mbadata$predicted_probs)

## Setting levels: control = Not Placed, case = Placed

## Setting direction: controls < cases

plot(roc_curve, col = "blue")

# Finding the  optimal cut-off (Youden's Index)

optimal_cutoff <- coords(roc_curve, "best", ret = "threshold")
optimal_cutoff

##   threshold
## 1 0.7574337

6. Logistic Regression Including All Significant Predictors (10% Significance)

check after the run:

Variables with p < 0.10 are retained. The model will better predict placement than using SSC_Percentage alone. Model evaluation metrics like AIC, ROC, and confusion matrix should be checked.

# Full logistic regression model with variables having p < 0.10

colnames(mbadata)

##  [1] "RegNo"               "Gender"              "Gender-B"           
##  [4] "Percent_SSC"         "Board_SSC"           "Board_CBSE"         
##  [7] "Board_ICSE"          "Percent_HSC"         "Board_HSC"          
## [10] "Stream_HSC"          "Percent_Degree"      "Course_Degree"      
## [13] "Degree_Engg"         "Experience_Yrs"      "Entrance_Test"      
## [16] "S-TEST"              "Percentile_ET"       "Percent_MBA"        
## [19] "S-TEST*SCORE"        "Specialization_MBA"  "Marks_Communication"
## [22] "Marks_Projectwork"   "Marks_BOCA"          "Placement"          
## [25] "Placement_B"         "Salary"              "predicted_probs"

# Full logistic regression model with variables having p < 0.10

logistic_model_full <- glm(Placement ~ Percent_SSC + Percent_HSC + Percent_Degree + 
                           Percentile_ET + Experience_Yrs + Marks_Communication, 
                           data = mbadata, family = binomial)

# Model summary
summary(logistic_model_full)

## 
## Call:
## glm(formula = Placement ~ Percent_SSC + Percent_HSC + Percent_Degree + 
##     Percentile_ET + Experience_Yrs + Marks_Communication, family = binomial, 
##     data = mbadata)
## 
## Coefficients:
##                      Estimate Std. Error z value Pr(>|z|)   
## (Intercept)         -0.011064   1.129659  -0.010  0.99219   
## Percent_SSC          0.048633   0.014961   3.251  0.00115 **
## Percent_HSC         -0.001807   0.013047  -0.139  0.88983   
## Percent_Degree      -0.004098   0.017239  -0.238  0.81210   
## Percentile_ET        0.009300   0.004214   2.207  0.02731 * 
## Experience_Yrs       0.317577   0.208915   1.520  0.12848   
## Marks_Communication -0.032199   0.017434  -1.847  0.06477 . 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 393.52  on 390  degrees of freedom
## Residual deviance: 372.76  on 384  degrees of freedom
## AIC: 386.76
## 
## Number of Fisher Scoring iterations: 4

7. Adjusting Cut-off for Cost-Sensitive Classification

Cost of misclassifying a non-placeable student = 4 times misclassifying a placeable student. Adjust the cut-off threshold accordingly.

# Adjusting the cut-off using cost ratio (False Positive Cost / False Negative Cost = 4)

cost_sensitive_cutoff <- optimal_cutoff / (1 + (1/4))
cost_sensitive_cutoff

##   threshold
## 1  0.605947

Summary of Insights -

SSC_Percentage alone is a weak predictor of placement. Best predictors include Degree_Percentage, Entrance Test Percentile, Work Experience, and Communication Marks. Cost-sensitive classification suggests a lower cut-off for placement.

Conclusion -

The analysis provides valuable insights into student placement and salary prediction through the application of both statistical and machine learning models. While the Random Forest model was effective in capturing complex relationships among the variables, the linear model also provided a solid baseline for understanding key predictors, offering interpretability and insight into the relative importance of factors such as Percent_SSC, Percent_HSC, and Percentile_ET. The linear model’s coefficients allowed for a clear understanding of how each feature influences placement and salary outcomes, making it a useful tool for educational institutions. Moving forward, future work should focus on refining feature engineering, incorporating additional data sources, and exploring more advanced modelling techniques, including deep learning and ensemble methods, to further enhance predictive accuracy. These findings provide actionable insights that can help educational institutions improve career counselling, adapt curricula to job market demands, and ensure better alignment between student preparation and industry requirements.