Breast Cancer Detection

Breast cancer detection using Artificial intelligence

analysis
R
Author

Aditya Ranade

Published

April 11, 2025

I found this dataset on UCI machine learning repository which gives the dataset for breast cancer detection. It has 10 basic variables which indicates different aspects of measurements in a medical examination. For the 10 variables, the dataset provided mean, sd and the worst measurement. We will focus on the mean measurement.

library(reshape2)
library(ggplot2)
library(dplyr)
library(ggh4x)
library(GGally)
library(pROC)
library(glmnet)

# Get data from github repo
path <- "https://raw.githubusercontent.com/adityaranade/portfolio/refs/heads/main/breast_cancer/wdbc.data"
data00 <- read.table(path, sep = ",", header = FALSE)
data0 <- data00[,2:12] # Use only first 11 columns

# Data processing
head(data0)
  V2    V3    V4     V5     V6      V7      V8     V9     V10    V11     V12
1  M 17.99 10.38 122.80 1001.0 0.11840 0.27760 0.3001 0.14710 0.2419 0.07871
2  M 20.57 17.77 132.90 1326.0 0.08474 0.07864 0.0869 0.07017 0.1812 0.05667
3  M 19.69 21.25 130.00 1203.0 0.10960 0.15990 0.1974 0.12790 0.2069 0.05999
4  M 11.42 20.38  77.58  386.1 0.14250 0.28390 0.2414 0.10520 0.2597 0.09744
5  M 20.29 14.34 135.10 1297.0 0.10030 0.13280 0.1980 0.10430 0.1809 0.05883
6  M 12.45 15.70  82.57  477.1 0.12780 0.17000 0.1578 0.08089 0.2087 0.07613
# change column names
colnames(data0) <- c("diagnosis", "radius",
                     "texture", "perimeter",
                     "area","smoothness",
                     "compactness","concavity",
                     "concave_points","symmetry",
                     "fractal_dimension")

Since the measurements are taken on the same object, there is bound to be some correlation between the variables. For example, there will be strong correlation between radius, perimeter and area. So using all the three variables in a model will not be a good idea as it might not give a good reliable model.

# data (select all the explanatory variables)
data <- data0 |> dplyr::select(radius, texture, perimeter,
                               area, smoothness, compactness, 
                               concavity, concave_points,
                               symmetry, fractal_dimension)

# convert the response to factor and categorical variable
data$response <- ifelse(data0$diagnosis=="B",0,
                        ifelse(data0$diagnosis=="M",1,99))

# Check the first 6 rows of the dataset
data |> head()
  radius texture perimeter   area smoothness compactness concavity
1  17.99   10.38    122.80 1001.0    0.11840     0.27760    0.3001
2  20.57   17.77    132.90 1326.0    0.08474     0.07864    0.0869
3  19.69   21.25    130.00 1203.0    0.10960     0.15990    0.1974
4  11.42   20.38     77.58  386.1    0.14250     0.28390    0.2414
5  20.29   14.34    135.10 1297.0    0.10030     0.13280    0.1980
6  12.45   15.70     82.57  477.1    0.12780     0.17000    0.1578
  concave_points symmetry fractal_dimension response
1        0.14710   0.2419           0.07871        1
2        0.07017   0.1812           0.05667        1
3        0.12790   0.2069           0.05999        1
4        0.10520   0.2597           0.09744        1
5        0.10430   0.1809           0.05883        1
6        0.08089   0.2087           0.07613        1
# Check the type of data
data |> str()
'data.frame':   569 obs. of  11 variables:
 $ radius           : num  18 20.6 19.7 11.4 20.3 ...
 $ texture          : num  10.4 17.8 21.2 20.4 14.3 ...
 $ perimeter        : num  122.8 132.9 130 77.6 135.1 ...
 $ area             : num  1001 1326 1203 386 1297 ...
 $ smoothness       : num  0.1184 0.0847 0.1096 0.1425 0.1003 ...
 $ compactness      : num  0.2776 0.0786 0.1599 0.2839 0.1328 ...
 $ concavity        : num  0.3001 0.0869 0.1974 0.2414 0.198 ...
 $ concave_points   : num  0.1471 0.0702 0.1279 0.1052 0.1043 ...
 $ symmetry         : num  0.242 0.181 0.207 0.26 0.181 ...
 $ fractal_dimension: num  0.0787 0.0567 0.06 0.0974 0.0588 ...
 $ response         : num  1 1 1 1 1 1 1 1 1 1 ...
# Check the rows which do not have any entries
sum(is.na(data)) # no NA values
[1] 0

We look at the distribution of the continuous data variables based on if response variable is 1 which indicates the tumor is malignant (cancerous) or response variable is 0 which indicates the tumor is benign (non cancerous).

# Data for histogram
melted_data <- melt(na.omit(data0), id="diagnosis")
melted_data$diagnosis <- ifelse(melted_data$diagnosis == "M","Malignant",
                                ifelse(melted_data$diagnosis == "B","Benign","NA"))


# Plot the histogram of all the variables
ggplot(melted_data,aes(value))+
  geom_histogram(aes(),bins = 30)+
  facet_grid2(diagnosis~variable, scales="free")+theme_bw()

There is a noticeable difference between the distribution of the variables for the two categories. Let us look at pairs plot which will help us understand the correlation between each pair of explanatory variables.

# Pairs plot between the explanatory variables to 
# check correlation between each pair of the variables
ggpairs(data[,-ncol(data)])

As expected, there is multicollinearity in the data. One way to mitigate the effect of multicollinearity is to use \(L2\) regularization (often called as ridge regression). Another way is to transform the data using principal component analysis (PCA) and use that data for regression. In this instance, we will first look at logistic regression and then logistic regression using L2 regularization.

We first split the data into training and testing set.

# split the data into training and testing data
seed <- 23
set.seed(seed)

ind <- sample(floor(0.8*nrow(data)),
              replace = FALSE)

# Training dataset
data_train <- data[ind,]

# Testing dataset
data_test <- data[-ind,]

Let us look at the results of logistic regression.

# Fit an binary logistic regression model
model <- glm(response ~ ., data = data_train, family = binomial)

# Check the summary of the model
model |> summary()

Call:
glm(formula = response ~ ., family = binomial, data = data_train)

Coefficients:
                   Estimate Std. Error z value Pr(>|z|)    
(Intercept)       -10.90839   14.88757  -0.733  0.46373    
radius             -2.64293    4.05814  -0.651  0.51487    
texture             0.47784    0.08329   5.737 9.63e-09 ***
perimeter           0.02482    0.57887   0.043  0.96580    
area                0.04017    0.01981   2.027  0.04263 *  
smoothness        107.08151   39.67654   2.699  0.00696 ** 
compactness        -2.66325   22.90510  -0.116  0.90744    
concavity           6.57732    9.57425   0.687  0.49210    
concave_points     56.61819   32.77129   1.728  0.08405 .  
symmetry            2.79133   12.47816   0.224  0.82299    
fractal_dimension -40.94988   92.09458  -0.445  0.65657    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 615.54  on 454  degrees of freedom
Residual deviance: 112.51  on 444  degrees of freedom
AIC: 134.51

Number of Fisher Scoring iterations: 9
# Prediction on the testing dataset
y_pred_prob <- predict(model, data_test,
                  type = "response")

y_pred <- ifelse(y_pred_prob>0.5,1,0)

# Generate the confusion matrix
conf_matrix <- caret::confusionMatrix(as.factor(y_pred),
                                      as.factor(data_test$response),
                                      positive = "1")
conf_matrix
Confusion Matrix and Statistics

          Reference
Prediction  0  1
         0 77  2
         1 11 24
                                          
               Accuracy : 0.886           
                 95% CI : (0.8129, 0.9379)
    No Information Rate : 0.7719          
    P-Value [Acc > NIR] : 0.001462        
                                          
                  Kappa : 0.7113          
                                          
 Mcnemar's Test P-Value : 0.026500        
                                          
            Sensitivity : 0.9231          
            Specificity : 0.8750          
         Pos Pred Value : 0.6857          
         Neg Pred Value : 0.9747          
             Prevalence : 0.2281          
         Detection Rate : 0.2105          
   Detection Prevalence : 0.3070          
      Balanced Accuracy : 0.8990          
                                          
       'Positive' Class : 1

Our logistic regression model has accuracy around 88.6% on the testing dataset. The misclassification rate on the testing data is (2+11)/114 = 0.1140. We will look at the ROC curve and AUC.

# Compute ROC curve
roc_curve <- roc(data_test$response,as.vector(y_pred_prob))
# Calculate AUC
auc_value <- auc(roc_curve)

# Plot the ROC curve
plot(roc_curve, col = "blue", lwd = 3, main = "ROC Curve")
# Add AUC to the plot
legend("bottomright", legend = paste("AUC =", round(auc_value, 3)), col = "blue", lwd = 3)

We will now try the logistic regression with \(L2\) regularization.

# Now try the logistic regression ridge (L2) regularization
model_l2 <- cv.glmnet(as.matrix(data_train[,-ncol(data_train)]), 
                      data_train[,ncol(data_train)], 
                      family = "binomial",
                      alpha=0)

# Plot cross-validation results 
plot(model_l2) 

# View the best lambda 
best_lambda_l2 <- model_l2$lambda.min 
print(best_lambda_l2) 
[1] 0.03789937
# Fit the final model with the best lambda 
final_model_l2 <- glmnet(as.matrix(data_train[,-ncol(data_train)]), 
                      data_train[,ncol(data_train)],
                      family = "binomial", 
                      alpha = 0, 
                      lambda = best_lambda_l2)

coef(final_model_l2,s = best_lambda_l2)
11 x 1 sparse Matrix of class "dgCMatrix"
                             s1
(Intercept)       -14.254070800
radius              0.174447577
texture             0.189604875
perimeter           0.025035861
area                0.001661682
smoothness         34.526223296
compactness         5.508633703
concavity           5.821321243
concave_points     17.322503544
symmetry            5.139945976
fractal_dimension -29.003877807
# Check the summary of the model
final_model_l2 |> summary()
           Length Class     Mode     
a0          1     -none-    numeric  
beta       10     dgCMatrix S4       
df          1     -none-    numeric  
dim         2     -none-    numeric  
lambda      1     -none-    numeric  
dev.ratio   1     -none-    numeric  
nulldev     1     -none-    numeric  
npasses     1     -none-    numeric  
jerr        1     -none-    numeric  
offset      1     -none-    logical  
classnames  2     -none-    character
call        6     -none-    call     
nobs        1     -none-    numeric  
# Prediction on the testing dataset
y_pred_prob_l2 <- predict(final_model_l2, 
                          as.matrix(data_test[,-ncol(data_test)]),
                          type = "response")

y_pred_l2 <- ifelse(y_pred_prob_l2>0.5,1,0)

# Generate the confusion matrix
conf_matrix_l2 <- caret::confusionMatrix(as.factor(y_pred_l2),as.factor(data_test$response))
conf_matrix_l2
Confusion Matrix and Statistics

          Reference
Prediction  0  1
         0 84  2
         1  4 24
                                         
               Accuracy : 0.9474         
                 95% CI : (0.889, 0.9804)
    No Information Rate : 0.7719         
    P-Value [Acc > NIR] : 3.302e-07      
                                         
                  Kappa : 0.8545         
                                         
 Mcnemar's Test P-Value : 0.6831         
                                         
            Sensitivity : 0.9545         
            Specificity : 0.9231         
         Pos Pred Value : 0.9767         
         Neg Pred Value : 0.8571         
             Prevalence : 0.7719         
         Detection Rate : 0.7368         
   Detection Prevalence : 0.7544         
      Balanced Accuracy : 0.9388         
                                         
       'Positive' Class : 0

Now the misclassification rate on the testing data is (2+4)/114 = 0.0526. We will now try the logistic regression with \(L2\) regularization, which should improve the performance.

# Compute ROC curve
roc_curve_l2 <- roc(data_test$response,as.vector(y_pred_prob_l2))

# Plot the ROC curve
plot(roc_curve_l2, col = "blue", lwd = 3, main = "ROC Curve")

# Add AUC to the plot
auc_value_l2 <- auc(roc_curve_l2)
legend("bottomright", legend = paste("AUC =", round(auc_value_l2, 3)), col = "blue", lwd = 3)

Our model has an accuracy of around 94% which indicates the model is correctly identifying the positive and negative cases in around 94% of the cases. Next, we look at the Receiver Operating Characteristic (ROC) curve. It plots the True Positive Rate (TPR) against the False Positive Rate (FPR). It is the visualization of trade-off between correctly identifying positive cases and incorrectly identifying negative cases as positive. A good model has ROC curve which goes from bottom left to top left which means the model is perfectly identifying positive cases and does not identify negatives as positive. On the other hand, a ROC curve which is a straight line from bottom left to to right with slope 1 indicates the model is randomly assigning the positive and negative cases. Our curve is somewhere in between these 2 extreme cases and is decent. The area under the curve (AUC) is around 99% which is also good. Let us compare the coefficients of the model parameters for the two models.

coef_model <- coef(model)
coef_model_l2 <- matrix(coef(final_model_l2,s = best_lambda_l2))

coef_combined <- data.frame(coef_model,coef_model_l2)
colnames(coef_combined) <- c("Logistic regression", "Logistic regression with L2 penalty")
coef_combined
                  Logistic regression Logistic regression with L2 penalty
(Intercept)              -10.90838572                       -14.254070800
radius                    -2.64293183                         0.174447577
texture                    0.47783651                         0.189604875
perimeter                  0.02482002                         0.025035861
area                       0.04017044                         0.001661682
smoothness               107.08150915                        34.526223296
compactness               -2.66325384                         5.508633703
concavity                  6.57731768                         5.821321243
concave_points            56.61818911                        17.322503544
symmetry                   2.79132574                         5.139945976
fractal_dimension        -40.94988370                       -29.003877807

This demonstrates that using L2 regularization in logistic regression where there is multicollinearity improves the model without any transformation. We can also try to build a model using the principal component analysis but that is for another day / dataset.