In this homework assignment, we will explore, analyze and model a data set containing information on crime for various neighborhoods of a major city. Each record has a response variable indicating whether or not the crime rate is above the median crime rate (1) or not (0).
Our objective is to build a binary logistic regression model on the training data set to predict whether the neighborhood will be at risk for high crime levels. We will provide classifications and probabilities for the evaluation data set using your binary logistic regression model. We can only use the variables given to (or, variables that we derive from the variables provided). Below is a short description of the variables of interest in the data set:
Variable Descriptions
zn: Proportion of residential land zoned for large lots (over 25,000 square feet) (predictor variable)
indus: Proportion of non-retail business acres per suburb (predictor variable)
chas: Dummy variable indicating whether the suburb borders the Charles River (1 = yes, 0 = no) (predictor variable)
nox: Nitrogen oxides concentration (parts per 10 million) (predictor variable)
rm: Average number of rooms per dwelling (predictor variable)
age: Proportion of owner-occupied units built prior to 1940 (predictor variable)
dis: Weighted mean of distances to five Boston employment centers (predictor variable)
rad: Index of accessibility to radial highways (predictor variable)
tax: Full-value property-tax rate per $10,000 (predictor variable)
ptratio: Pupil–teacher ratio by town (predictor variable)
lstat: Percentage of lower-status population (predictor variable)
medv: Median value of owner-occupied homes in $1,000s (predictor variable)
target: Whether the crime rate is above the median (1 = high crime, 0 = low crime) (response variable)
Data
We begin by importing the dataset into the R environment and creating two objects, train and test, which will be used throughout the analysis. The training set is used for exploratory analysis, feature preparation, and model development, while the test set is reserved for evaluating model performance on unseen data. This separation ensures an unbiased assessment of how well our final model generalizes beyond the data it was trained on.
With the dataset fully loaded, we begin our approach to predicting which neighborhoods are at risk of being classified as “unsafe.” Our process starts with exploratory data analysis (EDA) to understand variable distributions and relationships, followed by data preparation through cleaning, centering, and scaling. We then proceed to experiment with multiple logistic regression models to identify the most accurate and interpretable solution for predicting neighborhood safety.
EDA
We use the skimr package to generate a quick statistical overview of the dataset, including the minimum, maximum, mean, and interquartile range for each variable. The summary also provides a compact histogram for every numeric feature, allowing us to quickly visualize each variable’s spread, central tendency, and potential skewness. This approach offers an efficient first look at the dataset’s structure, helps identify possible outliers, and sets the foundation for deeper exploratory analysis and data preprocessing steps.
Code
my_skim <-skim_with(numeric =sfl(median ),append =TRUE)# Now run skimmy_skim(train)|>arrange((complete_rate))
Data summary
Name
train
Number of rows
466
Number of columns
13
_______________________
Column type frequency:
numeric
13
________________________
Group variables
None
Variable type: numeric
skim_variable
n_missing
complete_rate
mean
sd
p0
p25
p50
p75
p100
hist
median
zn
0
1
11.58
23.36
0.00
0.00
0.00
16.25
100.00
▇▁▁▁▁
0.00
indus
0
1
11.11
6.85
0.46
5.15
9.69
18.10
27.74
▇▆▁▇▁
9.69
chas
0
1
0.07
0.26
0.00
0.00
0.00
0.00
1.00
▇▁▁▁▁
0.00
nox
0
1
0.55
0.12
0.39
0.45
0.54
0.62
0.87
▇▇▅▃▁
0.54
rm
0
1
6.29
0.70
3.86
5.89
6.21
6.63
8.78
▁▂▇▂▁
6.21
age
0
1
68.37
28.32
2.90
43.88
77.15
94.10
100.00
▂▂▂▃▇
77.15
dis
0
1
3.80
2.11
1.13
2.10
3.19
5.21
12.13
▇▅▂▁▁
3.19
rad
0
1
9.53
8.69
1.00
4.00
5.00
24.00
24.00
▇▂▁▁▃
5.00
tax
0
1
409.50
167.90
187.00
281.00
334.50
666.00
711.00
▇▇▅▁▇
334.50
ptratio
0
1
18.40
2.20
12.60
16.90
18.90
20.20
22.00
▁▃▅▅▇
18.90
lstat
0
1
12.63
7.10
1.73
7.04
11.35
16.93
37.97
▇▇▅▂▁
11.35
medv
0
1
22.59
9.24
5.00
17.02
21.20
25.00
50.00
▂▇▅▁▁
21.20
target
0
1
0.49
0.50
0.00
0.00
0.00
1.00
1.00
▇▁▁▁▇
0.00
Code
my_skim(test)|>arrange((complete_rate))
Data summary
Name
test
Number of rows
40
Number of columns
12
_______________________
Column type frequency:
numeric
12
________________________
Group variables
None
Variable type: numeric
skim_variable
n_missing
complete_rate
mean
sd
p0
p25
p50
p75
p100
hist
median
zn
0
1
8.88
22.97
0.00
0.00
0.00
0.00
90.00
▇▁▁▁▁
0.00
indus
0
1
11.51
7.11
1.76
5.69
8.91
18.10
25.65
▇▅▁▇▂
8.91
chas
0
1
0.05
0.22
0.00
0.00
0.00
0.00
1.00
▇▁▁▁▁
0.00
nox
0
1
0.56
0.11
0.38
0.47
0.54
0.63
0.74
▇▇▇▅▇
0.54
rm
0
1
6.21
0.68
3.56
5.87
6.14
6.53
8.25
▁▁▇▅▁
6.14
age
0
1
70.99
26.27
6.80
56.62
83.25
93.10
100.00
▁▂▂▂▇
83.25
dis
0
1
3.79
2.12
1.20
2.04
3.37
4.53
9.09
▇▅▃▂▂
3.37
rad
0
1
9.78
9.06
1.00
4.00
5.00
24.00
24.00
▇▂▁▁▃
5.00
tax
0
1
393.50
177.33
188.00
276.75
307.00
666.00
666.00
▇▇▂▁▇
307.00
ptratio
0
1
19.12
1.63
14.70
18.40
19.60
20.20
21.20
▂▁▅▅▇
19.60
lstat
0
1
12.90
7.67
2.96
6.44
11.68
17.36
34.02
▇▆▂▃▁
11.68
medv
0
1
21.88
8.77
8.40
16.98
20.55
25.00
50.00
▃▇▂▁▁
20.55
There is a categorical variables in our data set lets look at its distribution
Code
cat("F T Charles River \n no yes \n==========\n", table(train$chas))
F T Charles River
no yes
==========
433 33
We see that our neighborhoods mostly do not border the Charles River.
Now lets look at our target variable to see if we have a balanced data set.
Code
cat("F T TARGET \n no yes \n==========\n", table(train$target))
F T TARGET
no yes
==========
237 229
Lets convert our char predictor to a factor so that R can better handle it.
library(tidyr)# Select numeric variables onlynum_vars <- train %>%dplyr::select(where(is.numeric))# Convert to long format for facet plottingnum_long <- num_vars %>%pivot_longer(cols =everything(), names_to ="variable", values_to ="value")# Density plots onlyggplot(num_long, aes(x = value)) +geom_density(fill ="grey", alpha =0.6, color ="black") +facet_wrap(~ variable, scales ="free", ncol =3) +labs(title ="Density Plots of Numeric Predictors",x ="Value", y ="Density" ) +theme_minimal(base_size =12)
.The density plots reveal that several predictors are not normally distributed, showing varying degrees of skewness. Variables such as zn, rad, tax, dis, and lstat display strong right skew, indicating that most neighborhoods have small values with a few much larger ones. In contrast, age shows a slight left skew, while rm, ptratio, and nox appear approximately symmetric. The variable indus is bimodal, suggesting the presence of distinct neighborhood types or zoning patterns. These distribution shapes highlight the diversity of Boston neighborhoods and suggest that some features may benefit from transformations or standardization prior to modeling. Overall, the density plots provide valuable insight into the range, shape, and variability of each predictor before further statistical analysis.
The variables zn, medv, and rm exhibit the highest number of outliers, with approximately 48, 38, and 28 observations outside the interquartile range, respectively. In comparison, dis and lstat contain relatively few outliers (around 6 and 5, respectively). Lets take a look and ponder how this relates to our data is noise or is data with meaningful information for us.
The variables rm (average number of rooms per dwelling) and medv (median value of owner-occupied homes) both show clear high-value outliers in the boxplots. These likely correspond to neighborhoods with unusually large homes or significantly higher property values. Since these observations appear to represent genuine, high-end housing markets rather than data entry errors, they will be retained for modeling but may benefit from transformation to reduce skewness.
The variable lstat, representing the percentage of the population with lower socioeconomic status, also exhibits a wide spread with several extreme values, suggesting substantial socioeconomic diversity across neighborhoods.
The variable dis (weighted mean distance to five Boston employment centers) shows two notable high-value outliers. These observations likely represent neighborhoods that are geographically distant from the city’s core employment hubs. Such areas may be more isolated and less accessible compared to the more central and densely connected neighborhoods within the Boston region.
Finally, zn (proportion of residential land zoned for large lots) contains high-value outliers, indicating a small number of neighborhoods dominated by large-lot zoning. These reflect real structural zoning differences rather than noise and will be preserved.
Transformations
Code
library(caret)library(dplyr)# Select numeric predictors (exclude categorical and target)num_features <- train %>% dplyr::select(where(is.numeric)) # Create preprocessing model (center + scale only)preproc_model <-preProcess(num_features, method =c("center", "scale"))# Apply transformation to training datatrain_scaled <-predict(preproc_model, num_features)# Combine back with categorical and target variablestrain_final <-bind_cols(train_scaled, train %>% dplyr::select(chas, target))# Quick checksummary(train_final)
zn indus nox rm
Min. :-0.4955 Min. :-1.5550 Min. :-1.4169 Min. :-3.4442
1st Qu.:-0.4955 1st Qu.:-0.8706 1st Qu.:-0.9112 1st Qu.:-0.5724
Median :-0.4955 Median :-0.2067 Median :-0.1398 Median :-0.1145
Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
3rd Qu.: 0.2000 3rd Qu.: 1.0218 3rd Qu.: 0.5973 3rd Qu.: 0.4811
Max. : 3.7845 Max. : 2.4299 Max. : 2.7145 Max. : 3.5317
age dis rad tax
Min. :-2.3116 Min. :-1.2654 Min. :-0.9821 Min. :-1.3252
1st Qu.:-0.8648 1st Qu.:-0.8041 1st Qu.:-0.6367 1st Qu.:-0.7653
Median : 0.3101 Median :-0.2870 Median :-0.5215 Median :-0.4467
Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
3rd Qu.: 0.9086 3rd Qu.: 0.6734 3rd Qu.: 1.6659 3rd Qu.: 1.5277
Max. : 1.1169 Max. : 3.9540 Max. : 1.6659 Max. : 1.7957
ptratio lstat medv chas target
Min. :-2.6395 Min. :-1.5350 Min. :-1.9037 no border:433 0:237
1st Qu.:-0.6821 1st Qu.:-0.7870 1st Qu.:-0.6022 borders : 33 1:229
Median : 0.2283 Median :-0.1804 Median :-0.1504
Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
3rd Qu.: 0.8200 3rd Qu.: 0.6053 3rd Qu.: 0.2609
Max. : 1.6394 Max. : 3.5679 Max. : 2.9666
Code
test_scaled <-predict(preproc_model, test)
We perform a simple centering and scaling of the numeric predictors to standardize their ranges. This step ensures that all variables contribute equally to the model and helps with numerical stability during optimization. Logistic regression does not require linearity or normal distributions of predictors, so no additional transformations are necessary at this stage. With the data standardized, we can now proceed to train our logistic regression models.Partitioning Data
Now will begin modeling before we do we will split our training data to be able to test our model since our test set does not have labeled data to calculate evaluation metrics.
Code
library(caret)set.seed(123) # for reproducibility# Create index for 70% trainingtrain_index <-createDataPartition(train_final$target, p =0.7, list =FALSE)# Split datadev_set <- train_final[train_index, ]test_set <- train_final[-train_index, ]# Check sizesnrow(dev_set)
[1] 327
Code
nrow(test_set)
[1] 139
Printing the distribution of target.
Code
prop.table(table(dev_set$target))
0 1
0.5076453 0.4923547
Code
prop.table(table(test_set$target))
0 1
0.5107914 0.4892086
Perfect now the models!
Models
We train a model with all predictor to begin and a model with what I perceive to be easy to interpret predictors, and we will also include a step-wise model that iterates to find the optimal model using AIC .
Akaike Information Criterion (AIC)
The Akaike Information Criterion (AIC) is a measure used to evaluate the quality of a statistical model by balancing goodness of fit with model simplicity. It penalizes models that use too many predictors while rewarding those that explain the data well.
\(AIC = 2k -2 \times ln(\hat L)\)
where:
\((\hat L)\) is the max log-likelihood measures how well the model fits the observed data
\(k\) is the number of parameters in the model (including the intercept).
Interpretation:
A lower AIC value indicates a better model, meaning it achieves a strong fit with fewer parameters.
Code
# Model 1 — Full modelmodel_full <-glm(target ~ ., data = dev_set, family = binomial)# Model 2 — Reduced model (focus on key interpretable predictors)model_reduced <-glm(target ~ rm + lstat + medv + rad + tax, data = dev_set, family = binomial)
Call:
glm(formula = target ~ ., family = binomial, data = dev_set)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.1355 0.8036 2.657 0.007876 **
zn -1.1131 0.8140 -1.367 0.171504
indus -0.1941 0.3732 -0.520 0.602936
nox 5.1768 0.9868 5.246 1.56e-07 ***
rm -0.3765 0.5618 -0.670 0.502757
age 0.8864 0.4139 2.141 0.032237 *
dis 1.4039 0.5320 2.639 0.008312 **
rad 5.6441 1.5579 3.623 0.000291 ***
tax -1.3969 0.5758 -2.426 0.015263 *
ptratio 0.9166 0.3192 2.872 0.004079 **
lstat 0.3001 0.4192 0.716 0.474076
medv 1.3905 0.6792 2.047 0.040641 *
chasborders 1.5809 0.9002 1.756 0.079082 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 453.24 on 326 degrees of freedom
Residual deviance: 146.40 on 314 degrees of freedom
AIC: 172.4
Number of Fisher Scoring iterations: 9
The logistic regression model explains a large portion of the variance in crime classification.
Significant predictors include nox, rad, ptratio, age, dis, tax, and medv, while chas is borderline significant.
The direction and magnitude of the coefficients align with known relationships from the Boston housing data higher pollution, highway accessibility, and older housing correlate with elevated crime likelihoods.
The model’s AIC (172.4) and substantial deviance reduction confirm a strong fit to the
Code
summary(model_reduced)
Call:
glm(formula = target ~ rm + lstat + medv + rad + tax, family = binomial,
data = dev_set)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.2934 0.4598 2.813 0.004906 **
rm 0.2428 0.3402 0.714 0.475354
lstat 1.2395 0.2762 4.489 7.17e-06 ***
medv 0.3457 0.3708 0.932 0.351170
rad 3.1268 0.8153 3.835 0.000126 ***
tax 0.1307 0.3090 0.423 0.672396
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 453.24 on 326 degrees of freedom
Residual deviance: 266.51 on 321 degrees of freedom
AIC: 278.51
Number of Fisher Scoring iterations: 8
The reduced logistic regression model, which included rm, lstat, medv, rad, and tax, explains a moderate amount of variation in crime classification. Among these predictors, rad, tax, and medv were statistically significant, while rm and lstat were not. The results suggest that neighborhoods with greater highway accessibility (rad) and higher median home values (medv) are more likely to fall into the high-crime category, whereas areas with higher property taxes (tax) tend to have lower crime rates, likely reflecting wealthier or better-maintained neighborhoods. Overall, the reduced model fits the data less effectively than the full model, showing a higher AIC and smaller deviance reduction. While it is easier to interpret, it loses some predictive strength by omitting key predictors such as nox, ptratio, and dis.
Code
# Stepwise model based on AICmodel_step <-step(model_full, direction ="both", trace =FALSE)# View selected variablessummary(model_step)
Call:
glm(formula = target ~ zn + nox + age + dis + rad + tax + ptratio +
medv + chas, family = binomial, data = dev_set)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.1512 0.7804 2.756 0.00584 **
zn -1.1742 0.7905 -1.485 0.13747
nox 4.9028 0.8871 5.527 3.26e-08 ***
age 0.8685 0.3407 2.549 0.01081 *
dis 1.3495 0.5102 2.645 0.00817 **
rad 5.7486 1.4760 3.895 9.83e-05 ***
tax -1.4925 0.5107 -2.923 0.00347 **
ptratio 0.7981 0.2836 2.814 0.00489 **
medv 0.8124 0.3452 2.353 0.01860 *
chasborders 1.7220 0.8865 1.942 0.05208 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 453.24 on 326 degrees of freedom
Residual deviance: 148.07 on 317 degrees of freedom
AIC: 168.07
Number of Fisher Scoring iterations: 9
The stepwise logistic regression model retained nine predictors: zn, nox, age, dis, rad, tax, ptratio, medv, and chas—and achieved a strong overall fit, with a substantial reduction in deviance (from 453.24 to 148.07) and an AIC of 168.1, indicating improved model efficiency compared to the full model. Several predictors were statistically significant, including nox, age, dis, rad, tax, ptratio, and medv, while chas was marginally significant. The direction of the coefficients aligns with expectations: higher pollution (nox), greater highway accessibility (rad), and higher pupil–teacher ratios (ptratio) are associated with increased odds of a neighborhood being classified as high-crime, whereas higher tax rates (tax) are linked to lower crime likelihood. Overall, the model provides strong explanatory power while remaining more parsimonious than the full version.
Code
# Predictions from FULL modeltest_set$prob_full <-predict(model_full, newdata = test_set, type ="response")test_set$pred_full <-ifelse(test_set$prob_full >0.5, 1, 0)# Predictions from REDUCED modeltest_set$prob_reduced <-predict(model_reduced, newdata = test_set, type ="response")test_set$pred_reduced <-ifelse(test_set$prob_reduced >0.5, 1, 0)# Predicitons for the Steotest_set$prob_step <-predict(model_step, newdata = test_set, type ="response")test_set$pred_step <-ifelse(test_set$prob_step >0.5, 1, 0)# Confusion matrixcm_step <-confusionMatrix(as.factor(test_set$pred_step), test_set$target,positive ="1")cm_full <-confusionMatrix(as.factor(test_set$pred_full), test_set$target,positive ="1")cm_reduced <-confusionMatrix(as.factor(test_set$pred_reduced), test_set$target,positive ="1")
Code
print(cm_full)
Confusion Matrix and Statistics
Reference
Prediction 0 1
0 65 6
1 6 62
Accuracy : 0.9137
95% CI : (0.8541, 0.9546)
No Information Rate : 0.5108
P-Value [Acc > NIR] : <2e-16
Kappa : 0.8273
Mcnemar's Test P-Value : 1
Sensitivity : 0.9118
Specificity : 0.9155
Pos Pred Value : 0.9118
Neg Pred Value : 0.9155
Prevalence : 0.4892
Detection Rate : 0.4460
Detection Prevalence : 0.4892
Balanced Accuracy : 0.9136
'Positive' Class : 1
Code
print(cm_reduced)
Confusion Matrix and Statistics
Reference
Prediction 0 1
0 66 16
1 5 52
Accuracy : 0.8489
95% CI : (0.7784, 0.904)
No Information Rate : 0.5108
P-Value [Acc > NIR] : <2e-16
Kappa : 0.6967
Mcnemar's Test P-Value : 0.0291
Sensitivity : 0.7647
Specificity : 0.9296
Pos Pred Value : 0.9123
Neg Pred Value : 0.8049
Prevalence : 0.4892
Detection Rate : 0.3741
Detection Prevalence : 0.4101
Balanced Accuracy : 0.8471
'Positive' Class : 1
Code
print(cm_step)
Confusion Matrix and Statistics
Reference
Prediction 0 1
0 64 4
1 7 64
Accuracy : 0.9209
95% CI : (0.8628, 0.9598)
No Information Rate : 0.5108
P-Value [Acc > NIR] : <2e-16
Kappa : 0.8418
Mcnemar's Test P-Value : 0.5465
Sensitivity : 0.9412
Specificity : 0.9014
Pos Pred Value : 0.9014
Neg Pred Value : 0.9412
Prevalence : 0.4892
Detection Rate : 0.4604
Detection Prevalence : 0.5108
Balanced Accuracy : 0.9213
'Positive' Class : 1
The full logistic regression model performed very well on the test set, achieving an overall accuracy of 91.4% with nearly balanced sensitivity (91.2%) and specificity (91.6%). This indicates that the model correctly classified both high- and low-crime neighborhoods with similar success rates. The Kappa statistic (0.83) reflects strong agreement beyond chance, confirming that the model generalizes effectively to unseen data. In addition, the McNemar’s test p-value (1.00) suggests no significant difference between false positives and false negatives, indicating balanced performance across both classes.
The reduced logistic regression model, by comparison, achieved a lower accuracy of 84.9%, with sensitivity dropping to 76.5% but slightly higher specificity (92.9%). This pattern shows that the reduced model is more conservative—it better identifies low-crime areas but misses more high-crime neighborhoods. The Kappa value (0.70) indicates moderate agreement, and the McNemar’s test p-value (0.029) reveals an imbalance in misclassification, suggesting the model’s errors are not evenly distributed.
The stepwise logistic regression model, which selected only the most informative predictors based on AIC, achieved the best overall performance, with an accuracy of 92.1%, sensitivity of 94.1%, and specificity of 90.1%. The Kappa statistic (0.84) indicates excellent agreement beyond chance, while the McNemar’s test p-value (0.55) suggests balanced classification errors. This model strikes the best balance between predictive accuracy and simplicity, maintaining nearly identical performance to the full model while using fewer predictors.
Overall, while the full model provides a strong and comprehensive fit, the stepwise model offers a more efficient alternative without sacrificing performance. The reduced model remains the most interpretable but at a clear cost in predictive power. The results suggest that a carefully selected subset of predictors like nox, rad, ptratio, and tax capture most of the explanatory power needed to accurately classify neighborhoods by crime level.
Setting levels: control = 0, case = 1
Setting direction: controls < cases
Code
# Plot all three ROC curvesplot(roc_full, col ="blue", lwd =2, main ="ROC Curves - Logistic Regression Models")plot(roc_reduced, col ="red", lwd =2, add =TRUE)plot(roc_step, col ="darkgreen", lwd =2, add =TRUE)# Add legendlegend("bottomright",legend =c(paste0("Full Model (AUC = ", round(auc(roc_full), 3), ")"),paste0("Reduced Model (AUC = ", round(auc(roc_reduced), 3), ")"),paste0("Stepwise Model (AUC = ", round(auc(roc_step), 3), ")") ),col =c("blue", "red", "darkgreen"),lwd =2,bty ="n")
TheThe ROC comparison shows that all three models perform well, with AUC values above 0.90, indicating strong overall classification ability. The full logistic regression model achieves the highest AUC (0.982), followed very closely by the stepwise model (0.98). Both demonstrate excellent discrimination between high- and low-crime neighborhoods. The reduced model performs noticeably worse (AUC = 0.93), suggesting some predictive information was lost when limiting the number of predictors. Overall, the stepwise model offers nearly the same predictive performance as the full model but with fewer variables, making it the most efficient and practical option for classification
Conclusion
Code
library(dplyr)library(knitr)library(kableExtra)
Attaching package: 'kableExtra'
The following object is masked from 'package:dplyr':
group_rows
Code
# Build a summary table of all model performance metricsmodel_comparison <-data.frame(Model =c("Full Model", "Reduced Model", "Stepwise Model"),Accuracy =c( cm_full$overall["Accuracy"], cm_reduced$overall["Accuracy"], cm_step$overall["Accuracy"] ),Precision =c( cm_full$byClass["Pos Pred Value"], cm_reduced$byClass["Pos Pred Value"], cm_step$byClass["Pos Pred Value"] ),Recall =c( cm_full$byClass["Sensitivity"], cm_reduced$byClass["Sensitivity"], cm_step$byClass["Sensitivity"] ),F1_Score =c( cm_full$byClass["F1"], cm_reduced$byClass["F1"], cm_step$byClass["F1"] ),AUC =c(auc(roc_full),auc(roc_reduced),auc(roc_step) ))# Format the table neatly for Quarto outputmodel_comparison %>%mutate(across(-Model, ~round(.x, 3))) %>%arrange(desc(F1_Score)) %>%kbl(caption ="Performance Comparison of Logistic Regression Models") %>%kable_styling(full_width =FALSE, position ="center", bootstrap_options =c("striped", "hover", "condensed"))
Performance Comparison of Logistic Regression Models
Model
Accuracy
Precision
Recall
F1_Score
AUC
Stepwise Model
0.921
0.901
0.941
0.921
0.980
Full Model
0.914
0.912
0.912
0.912
0.982
Reduced Model
0.849
0.912
0.765
0.832
0.930
Based on the results of all three logistic regression models, the stepwise model provides the best balance between predictive performance and simplicity. It achieved an accuracy of 92% and an AUC of 0.98, nearly identical to the full model (AUC = 0.982) but with fewer predictors, making it more efficient and easier to interpret. The analysis identified several statistically significant predictors of neighborhood crime rate, including nitric oxide concentration (nox), highway accessibility (rad), pupil–teacher ratio (ptratio), distance to employment centers (dis), age of housing (age), and property tax rate (tax). Higher values of nox, rad, ptratio, and age are associated with a greater likelihood of a neighborhood being classified as high-crime, while higher tax rates are linked to lower crime probability, reflecting more affluent areas.
Overall, the stepwise logistic regression model is recommended as the final model due to its strong predictive accuracy, interpretability, and parsimony. It effectively captures the key environmental and socioeconomic factors influencing crime while avoiding unnecessary model complexity.
Eval Predictions
This section provides predictions for the eval set provided.
