HW 10
Ayan Elmi
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 ──
## ✔ dplyr 1.1.4 ✔ readr 2.1.5
## ✔ forcats 1.0.1 ✔ stringr 1.5.1
## ✔ ggplot2 4.0.0 ✔ tibble 3.3.0
## ✔ lubridate 1.9.4 ✔ tidyr 1.3.1
## ✔ purrr 1.1.0
## ── 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 Regression Model
## Enter your code here
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: 4Calculating R^2
r_square <- 1 - (logistic$deviance/logistic$null.deviance)
r_square## [1] 0.25626What does it indicate about the model’s explanatory power? This indicates that model explains around 25.6% of the variation in diabetes outcomes.
What does the intercept represent (log-odds of diabetes when predictors are zero)?
The intercept represents the log-odds of having diabetes when BMI, Age and Glucose are = to 0. These values are not realistic because to have a meaningful real-world interpretation because there is no human with BMI of 0, AGE 0 and Glucose 0. Therefore, the intercept does not have meaningful real-world interpretation. It purpose is as the model’s baseline starting point for calculating log-odds.
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)? The three predictors (glucose, BMI and Age), meaning that a one-unit increase in glucose increases the odds of diabetes. All the three p-values is less than 0.05, so it is a statistically significant predictor. As well as, the BMI coefficient is positive, so higher BMI increases the odds of diabetes. The Age coefficient is positive, meaning older age increases the odds of diabetes. so Age is significant, although its effect is smaller than Glucose or BMI.
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(probabilty_diabetes=logistic$fitted.values,
Glucose = data_subset$Glucose,
BMI= data_subset$BMI,
AGE= data_subset$Age)
predicted.data## probabilty_diabetes Glucose BMI AGE
## 1 0.66360006 148 33.6 50
## 2 0.06101402 85 26.6 31
## 3 0.61834186 183 23.3 32
## 4 0.06043396 89 28.1 21
## 5 0.65771328 137 43.1 33
## 6 0.14802668 116 25.6 30
## 7 0.06116212 78 31.0 26
## 8 0.28013239 115 35.3 29
## 9 0.90283168 197 30.5 53
## 11 0.29185049 110 37.6 30
## 12 0.79018904 168 38.0 34
## 13 0.49423685 139 27.1 57
## 14 0.88904285 189 30.1 59
## 15 0.65652969 166 25.8 51
## 16 0.13393352 100 30.0 32
## 17 0.54057167 118 45.8 31
## 18 0.15678020 107 29.6 31
## 19 0.36875552 103 43.3 33
## 20 0.28484895 115 34.6 32
## 21 0.43753713 126 39.3 27
## 22 0.28886248 99 35.4 50
## 23 0.93606735 196 39.8 41
## 24 0.20309566 119 29.0 29
## 25 0.68988838 143 36.6 51
## 26 0.34957706 125 31.1 41
## 27 0.72382298 147 39.4 43
## 28 0.05362751 97 23.2 22
## 29 0.43793280 145 22.2 57
## 30 0.32692619 117 34.1 38
## 31 0.44902956 109 36.0 60
## 32 0.55575994 158 31.6 28
## 33 0.04535146 88 24.8 22
## 34 0.04022137 92 19.9 28
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## 766 0.17967203 121 26.2 30
## 767 0.37685726 126 30.1 47
## 768 0.08803680 93 30.4 23# Predicted classes
predicted.classes <- ifelse(predicted.data$probabilty_diabetes> 0.5, 1, 0)
# Confusion matrix
confusion <- table(
Predicted = factor(predicted.classes, levels = c(0, 1)),
Actual = factor(data_subset$Outcome_num, levels = c(0, 1))
)
confusion## Actual
## Predicted 0 1
## 0 429 114
## 1 59 150Now let’s calculate key performance metrics:
#Extract Values:
TN <- 429
FP <- 59
FN <- 114
TP <- 150
#Metrics
accuracy <- (TP + TN) / (TP + TN + FP + FN)
sensitivity <- TP / (TP + FN) # also called recall or true positive rate
specificity <- TN / (TN + FP) # true negative rate
precision <- TP / (TP + FP) # positive predictive value
f1_score <- 2 * (precision * sensitivity) / (precision + sensitivity)
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 of 77% means it performs moderately well and that it correctly identifies most cases.
The model is better at identifying non-diabetic individuals because it has a high specificity of 87.9% than detecting people with diabetes. The sensitivity is 56.8% which is average meaning that the model misses a lot of people with diabetes. The precision is 71.8 percent showing that the positive predictions are mostly correct, the lower sensitivity suggest that the he model misses a lot of people with diabetes. In conclusion, the model is better at identify people without diabetes than cases of people with diabetes, this is important to consider in medicine because missing cases is detrimental to people’s livelihood and health.
Question 3: ROC Curve, AUC, and Interpretation
Plot the ROC curve, use the “data_subset” from Q2.
Calculate AUC.
#Enter your code here
library(pROC)## Type 'citation("pROC")' for a citation.##
## Attaching package: 'pROC'## The following objects are masked from 'package:stats':
##
## cov, smooth, var# ROC curve & AUC on full data
roc_obj <- roc(response = data_subset$Outcome_num,
predictor = logistic$fitted.values,
levels = c(0, 1),
direction = "<")
# Print AUC value
auc_val <- auc(roc_obj); auc_val## Area under the curve: 0.828# Plot ROC with AUC displayed
plot.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 model helps at distinguishing between two classes. AUC 0.5 means that the model is no different from random guessing and AUC 1.0= perfect classification.
For diabetes diagnosis, prioritize sensitivity (catching cases) or specificity (avoiding false positives)? Suggest a threshold and explain. For diabetes diagnosis, it’s better to focus on sensitivity because catching people who really have diabetes is more important than avoiding a few false alarms. A false negative is worse since the person won’t get the treatment they need, while a false positive just means more testing. To help catch more real cases, we can lower the threshold from 0.5 to something like 0.3 so the model marks more people as possibly having diabetes. This raises sensitivity, even if specificity drops a bit, and that’s usually the safer choice in medical screening.