HW 10
Tejas Shrestha
Introduction:
In this homework, you will apply logistic regression to a real-world dataset: the Pima Indians Diabetes Database. This dataset contains medical records from 768 women of Pima Indian heritage, aged 21 or older, and is used to predict the onset of diabetes (binary outcome: 0 = no diabetes, 1 = diabetes) based on physiological measurements.
The data is publicly available from the UCI Machine Learning Repository and can be imported directly.
Dataset URL: https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv
Columns (no header in the CSV, so we need to assign them manually):
- Pregnancies: Number of times pregnant
- Glucose: Plasma glucose concentration (2-hour test)
- BloodPressure: Diastolic blood pressure (mm Hg)
- SkinThickness: Triceps skin fold thickness (mm)
- Insulin: 2-hour serum insulin (mu U/ml)
- BMI: Body mass index (weight in kg/(height in m)^2)
- DiabetesPedigreeFunction: Diabetes pedigree function (a function scoring genetic risk)
- Age: Age in years
- Outcome: Class variable (0 = no diabetes, 1 = diabetes)
Task Overview: You will load the data, build a logistic regression model to predict diabetes onset using a subset of predictors (Glucose, BMI, Age), interpret the model, evaluate it with a confusion matrix and metrics, and analyze the ROC curve and AUC.
Cleaning the dataset Don’t change the following code
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ✔ ggplot2 4.0.2 ✔ tibble 3.3.0
## ✔ lubridate 1.9.4 ✔ tidyr 1.3.1
## ✔ purrr 1.2.0
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorsurl <- "https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv"
data <- read.csv(url, header = FALSE)
colnames(data) <- c("Pregnancies", "Glucose", "BloodPressure", "SkinThickness", "Insulin", "BMI", "DiabetesPedigreeFunction", "Age", "Outcome")
data$Outcome <- as.factor(data$Outcome)
# Handle missing values (replace 0s with NA because 0 makes no sense here)
data$Glucose[data$Glucose == 0] <- NA
data$BloodPressure[data$BloodPressure == 0] <- NA
data$BMI[data$BMI == 0] <- NA
colSums(is.na(data))## Pregnancies Glucose BloodPressure
## 0 5 35
## SkinThickness Insulin BMI
## 0 0 11
## DiabetesPedigreeFunction Age Outcome
## 0 0 0Question 1: Create and Interpret a Logistic Regression Model - Fit a logistic regression model to predict Outcome using Glucose, BMI, and Age.
Provide the model summary.
Calculate and interpret R²: 1 - (model\(deviance / model\)null.deviance). What does it indicate about the model’s explanatory power?
logistic <- glm(Outcome ~ Glucose + BMI + Age, data=data, family="binomial")
summary(logistic)##
## Call:
## glm(formula = Outcome ~ Glucose + BMI + Age, family = "binomial",
## data = data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -9.032377 0.711037 -12.703 < 2e-16 ***
## Glucose 0.035548 0.003481 10.212 < 2e-16 ***
## BMI 0.089753 0.014377 6.243 4.3e-10 ***
## Age 0.028699 0.007809 3.675 0.000238 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 974.75 on 751 degrees of freedom
## Residual deviance: 724.96 on 748 degrees of freedom
## (16 observations deleted due to missingness)
## AIC: 732.96
##
## Number of Fisher Scoring iterations: 4r_square <- 1 - (logistic$deviance/logistic$null.deviance)
r_square## [1] 0.25626The R-squared value is 0.25626 which translates to 25.6% of the variation in the outcome can be explained by Glucose, BMI, and Age.
What does the intercept represent (log-odds of diabetes when predictors are zero)?
The intercept (-9.032377) represents the log-odds of diabetes when Glucose, BMI, and Age are zero.
For each predictor (Glucose, BMI, Age), does a one-unit increase raise or lower the odds of diabetes? Are they significant (p-value < 0.05)?
For each predictor, a one-unit increase raises the odds of diabetes. All three predictors have positive coefficients and all p-values are statistically significant.
Question 2: Confusion Matrix and Important Metric
Predict probabilities using the fitted model.
Create predicted classes with a 0.5 threshold (1 if probability > 0.5, else 0).
Build a confusion matrix (Predicted vs. Actual Outcome).
Calculate and report the metrics:
Accuracy: (TP + TN) / Total Sensitivity (Recall): TP / (TP + FN) Specificity: TN / (TN + FP) Precision: TP / (TP + FP)
Use the following starter code
# Keep only rows with no missing values in Glucose, BMI, or Age
data_subset <- data[complete.cases(data[, c("Glucose", "BMI", "Age")]), ]
#Create a numeric version of the outcome (0 = no diabetes, 1 = diabetes).This is required for calculating confusion matrices.
data_subset$Outcome_num <- ifelse(data_subset$Outcome == "1", 1, 0)
# Predicted probabilities
predicted.data <- data.frame(
probability.of.outcomes=logistic$fitted.values,
outcome=data_subset$Outcome_num)
predicted.data <- predicted.data[
order(predicted.data$probability.of.outcomes, decreasing=FALSE),]
predicted.data$rank <- 1:nrow(predicted.data)
head(predicted.data)## probability.of.outcomes outcome rank
## 681 0.01422697 0 1
## 63 0.01490161 0 2
## 618 0.01550448 0 3
## 98 0.01718930 0 4
## 91 0.02040067 0 5
## 590 0.02132939 0 6# Predicted classes
predicted.probs <- logistic$fitted.values
predicted.classes <- ifelse(predicted.probs > 0.5, 1, 0)
# Confusion matrix
confusion <- table(
Predicted = factor(predicted.classes, levels = c(0, 1)),
Actual = factor(data_subset$Outcome_num, levels = c(0, 1)))
confusion## Actual
## Predicted 0 1
## 0 429 114
## 1 59 150#Extract Values:
TN <- 429
FP <- 59
FN <- 114
TP <- 150
#Metrics
accuracy <- (TP + TN) / (TN + FP + FN + TP)
sensitivity <- TP / (TP + FN)
specificity <- TN / (TN + FP)
precision <- TP / (TP + FP)
cat("Accuracy:", round(accuracy, 3), "\nSensitivity:", round(sensitivity, 3), "\nSpecificity:", round(specificity, 3), "\nPrecision:", round(precision, 3))## Accuracy: 0.77
## Sensitivity: 0.568
## Specificity: 0.879
## Precision: 0.718Interpret: How well does the model perform? Is it better at detecting diabetes (sensitivity) or non-diabetes (specificity)? Why might this matter for medical diagnosis?
The model has an accuracy rating of 77%, but its performance is scattered. It has a high specificity at 87.9%, meaning it is effective at correctly interpreting non-diabetic individuals. However, its sensitivity is low at 56.8% meaning that it misses a lot of actual diabetes cases. This suggests that the model lacks at observing the disease. This can be bad for medical diagnosis as it is too unreliable for it to be used professional.
Question 3: ROC Curve, AUC, and Interpretation
Plot the ROC curve, use the “data_subset” from Q2.
Calculate AUC.
library(pROC)
roc_obj <- roc(response = data_subset$Outcome_num,
predictor = logistic$fitted.values,
levels = c(0, 1),
direction = "<")
auc_val <- auc(roc_obj); auc_val## Area under the curve: 0.828plot.roc(roc_obj, print.auc = TRUE, legacy.axes = TRUE,
xlab = "False Positive Rate (1 - Specificity)",
ylab = "True Positive Rate (Sensitivity)")
What does AUC indicate (0.5 = random, 1.0 = perfect)?
The AUC of this graph is 0.828 meaning that the model is not completely random, but not perfect.
For diabetes diagnosis, prioritize sensitivity (catching cases) or specificity (avoiding false positives)? Suggest a threshold and explain.
For diabetes diagnosis, sensitive should be prioritized over specificity because failing to figure out true cases of diabetes can lead to complications and severe problems later on. To improve sensitivity, the threshold should be lowered to increase the ability to detect individuals with diabetes