Logistic Regression Classifier - Fitting hospital-specific models#

Aims#

Assess accuracy of a logistic regression classifier, using k-fold (5-fold) training/test data splits (each data point is present in one and only one of the five test sets). This notebook fits models to each hospital independently.

The notebook includes:

  • A range of accuracy scores

  • Receiver operating characteristic (ROC) and Sensitivity-Specificity Curves

Import libraries#

# Turn warnings off to keep notebook tidy
import warnings
warnings.filterwarnings("ignore")

import matplotlib.pyplot as plt
from matplotlib.lines import Line2D
import numpy as np
import pandas as pd

from sklearn.preprocessing import StandardScaler
from sklearn.linear_model import LogisticRegression
from sklearn.metrics import auc
from sklearn.metrics import roc_curve

Load train and test set data#

K-fold train/test split has previously been performed. Train/test data is stratified by hospital and thrombolysis.

# Lists to hold data splits
train_splits = []
train_splits_one_hot_hosp = []
test_splits = []
test_splits_one_hot_hosp = []

# Loop to load data and add to lists
for i in range (5):
    train_splits.append(pd.read_csv(f'../data/kfold_5fold/train_{i}.csv'))
    test_splits.append(pd.read_csv(f'../data/kfold_5fold/test_{i}.csv'))

Functions#

Standardise data#

Standardisation subtracts the mean and divides by the standard deviation, for each feature. Here we use the sklearn built-in method for standardisation.

def standardise_data(X_train, X_test):
    """
    Converts all data to a similar scale.
    Standardisation subtracts mean and divides by standard deviation
    for each feature.
    Standardised data will have a mena of 0 and standard deviation of 1.
    The training data mean and standard deviation is used to standardise both
    training and test set data.
    """
    
    # Initialise a new scaling object for normalising input data
    sc = StandardScaler() 

    # Set up the scaler just on the training set
    sc.fit(X_train)

    # Apply the scaler to the training and test sets
    train_std=sc.transform(X_train)
    test_std=sc.transform(X_test)
    
    return train_std, test_std

Find model probability threshold to match predicted and actual thrombolysis use#

def find_threshold(probabilities, true_rate):
    
    """
    Find classification threshold to calibrate model
    """
    
    index = (1-true_rate)*len(probabilities)
    
    threshold = sorted(probabilities)[int(index)]
    
    return threshold

Calculate accuracy measures#

def calculate_accuracy(observed, predicted):
    
    """
    Calculates a range of accuracy scores from observed and predicted classes.
    
    Takes two list or NumPy arrays (observed class values, and predicted class 
    values), and returns a dictionary of results.
    
     1) observed positive rate: proportion of observed cases that are +ve
     2) Predicted positive rate: proportion of predicted cases that are +ve
     3) observed negative rate: proportion of observed cases that are -ve
     4) Predicted negative rate: proportion of predicted cases that are -ve  
     5) accuracy: proportion of predicted results that are correct    
     6) precision: proportion of predicted +ve that are correct
     7) recall: proportion of true +ve correctly identified
     8) f1: harmonic mean of precision and recall
     9) sensitivity: Same as recall
    10) specificity: Proportion of true -ve identified:        
    11) positive likelihood: increased probability of true +ve if test +ve
    12) negative likelihood: reduced probability of true +ve if test -ve
    13) false positive rate: proportion of false +ves in true -ve patients
    14) false negative rate: proportion of false -ves in true +ve patients
    15) true positive rate: Same as recall
    16) true negative rate: Same as specificity
    17) positive predictive value: chance of true +ve if test +ve
    18) negative predictive value: chance of true -ve if test -ve
    
    """
    
    # Converts list to NumPy arrays
    if type(observed) == list:
        observed = np.array(observed)
    if type(predicted) == list:
        predicted = np.array(predicted)
    
    # Calculate accuracy scores
    observed_positives = observed == 1
    observed_negatives = observed == 0
    predicted_positives = predicted == 1
    predicted_negatives = predicted == 0
    
    true_positives = (predicted_positives == 1) & (observed_positives == 1)
    
    false_positives = (predicted_positives == 1) & (observed_positives == 0)
    
    true_negatives = (predicted_negatives == 1) & (observed_negatives == 1)
    
    false_negatives = (predicted_negatives == 1) & (observed_negatives == 0)
    
    accuracy = np.mean(predicted == observed)
    
    precision = (np.sum(true_positives) /
                 (np.sum(true_positives) + np.sum(false_positives)))
        
    recall = np.sum(true_positives) / np.sum(observed_positives)
    
    sensitivity = recall
    
    f1 = 2 * ((precision * recall) / (precision + recall))
    
    specificity = np.sum(true_negatives) / np.sum(observed_negatives)
    
    positive_likelihood = sensitivity / (1 - specificity)
    
    negative_likelihood = (1 - sensitivity) / specificity
    
    false_positive_rate = 1 - specificity
    
    false_negative_rate = 1 - sensitivity
    
    true_positive_rate = sensitivity
    
    true_negative_rate = specificity
    
    positive_predictive_value = (np.sum(true_positives) / 
                            (np.sum(true_positives) + np.sum(false_positives)))
    
    negative_predictive_value = (np.sum(true_negatives) / 
                            (np.sum(true_negatives) + np.sum(false_negatives)))
    
    # Create dictionary for results, and add results
    results = dict()
    
    results['observed_positive_rate'] = np.mean(observed_positives)
    results['observed_negative_rate'] = np.mean(observed_negatives)
    results['predicted_positive_rate'] = np.mean(predicted_positives)
    results['predicted_negative_rate'] = np.mean(predicted_negatives)
    results['accuracy'] = accuracy
    results['precision'] = precision
    results['recall'] = recall
    results['f1'] = f1
    results['sensitivity'] = sensitivity
    results['specificity'] = specificity
    results['positive_likelihood'] = positive_likelihood
    results['negative_likelihood'] = negative_likelihood
    results['false_positive_rate'] = false_positive_rate
    results['false_negative_rate'] = false_negative_rate
    results['true_positive_rate'] = true_positive_rate
    results['true_negative_rate'] = true_negative_rate
    results['positive_predictive_value'] = positive_predictive_value
    results['negative_predictive_value'] = negative_predictive_value
    
    return results

Line intersect#

Used to find point of sensitivity-specificity curve where sensitivity = specificity.

def get_intersect(a1, a2, b1, b2):
    """ 
    Returns the point of intersection of the lines passing through a2,a1 and b2,b1.
    a1: [x, y] a point on the first line
    a2: [x, y] another point on the first line
    b1: [x, y] a point on the second line
    b2: [x, y] another point on the second line
    """
    s = np.vstack([a1,a2,b1,b2])        # s for stacked
    h = np.hstack((s, np.ones((4, 1)))) # h for homogeneous
    l1 = np.cross(h[0], h[1])           # get first line
    l2 = np.cross(h[2], h[3])           # get second line
    x, y, z = np.cross(l1, l2)          # point of intersection
    if z == 0:                          # lines are parallel
        return (float('inf'), float('inf'))
    return (x/z, y/z)

Fit hospital-specific models#

hospitals = set(train_splits[0]['StrokeTeam'])

# Set up list to store models and calibarion thresholds
hospital_models = []
thresholds = []

# Set up lists for results
observed = []
predicted_proba = []
predicted = []
kfold_result = []
hospital_results = []
threshold_results = []
feature_data = []

# Loop through k folds
for k_fold in range(5):
    
    # Get k fold split
    train = train_splits[k_fold]
    test = test_splits[k_fold]
    
    # Loop through hospitals
    for hospital in hospitals:
                
        # Get X and y
        mask = train['StrokeTeam'] == hospital
        X_train = train[mask].drop(['S2Thrombolysis', 'StrokeTeam'], axis=1)    
        y_train = train.loc[mask]['S2Thrombolysis']
        
        mask = test['StrokeTeam'] == hospital
        X_test = test[mask].drop(['S2Thrombolysis', 'StrokeTeam'], axis=1)
        y_test = test.loc[mask]['S2Thrombolysis']
        feature_data.append(test[mask])

        # Standardise X data
        X_train_std, X_test_std = standardise_data(X_train, X_test)

        # Define and Fit model
        model = LogisticRegression(solver='lbfgs')
        model.fit(X_train_std, y_train)

        # Get predicted probabilities
        y_probs = model.predict_proba(X_test_std)[:,1]
        
        # Calibrate model and get class
        true_rate = np.mean(y_test)
        threshold = find_threshold(y_probs, true_rate)
        thresholds.append(threshold)
        y_class = y_probs >= threshold
        y_class = np.array(y_class) * 1.0
        
        # Store results
        observed.extend(list(y_test))
        predicted_proba.extend(list(y_probs))
        predicted.extend(y_class)
        kfold_result.extend(list(np.repeat(k_fold, len(y_test))))
        hospital_results.extend(list(np.repeat(hospital, len(y_test))))
        threshold_results.extend(np.repeat(threshold, len(y_test)))

# Collate patient-level results
multi_model = pd.DataFrame()
multi_model['hospital'] = hospital_results
multi_model['observed'] = np.array(observed) * 1.0
multi_model['prob'] = predicted_proba
multi_model['predicted'] = predicted
multi_model['k_fold'] = kfold_result
multi_model['threshold'] = threshold_results
multi_model['correct'] = multi_model['observed'] == multi_model['predicted']

# Save model
filename = './predictions/multi_fit_lr_k_fold.csv'
multi_model.to_csv(filename, index=False)

# Save features
feature_data_df = pd.concat(feature_data, axis=0)
feature_data_df.to_csv('./predictions/feature_data.csv')

Results#

Accuracy measures#

k_fold_results = []

for i in range(5):
    mask = multi_model['k_fold'] == i
    mask = mask.values
    observed = multi_model.loc[mask]['observed']
    predicted = multi_model.loc[mask]['predicted']
    results = calculate_accuracy(observed, predicted)
    k_fold_results.append(results)
    
multi_fit_results = pd.DataFrame(k_fold_results).T

multi_fit_results
0 1 2 3 4
observed_positive_rate 0.295232 0.295401 0.295176 0.295080 0.295417
observed_negative_rate 0.704768 0.704599 0.704824 0.704920 0.704583
predicted_positive_rate 0.295570 0.295907 0.295907 0.295699 0.295980
predicted_negative_rate 0.704430 0.704093 0.704093 0.704301 0.704020
accuracy 0.807489 0.805971 0.802373 0.807197 0.805342
precision 0.673768 0.671290 0.664830 0.672942 0.670213
recall 0.674538 0.672440 0.666476 0.674352 0.671488
f1 0.674153 0.671865 0.665652 0.673646 0.670850
sensitivity 0.674538 0.672440 0.666476 0.674352 0.671488
specificity 0.863183 0.861953 0.859285 0.862806 0.861464
positive_likelihood 4.930225 4.871109 4.736364 4.915321 4.847017
negative_likelihood 0.377048 0.380020 0.388141 0.377429 0.381341
false_positive_rate 0.136817 0.138047 0.140715 0.137194 0.138536
false_negative_rate 0.325462 0.327560 0.333524 0.325648 0.328512
true_positive_rate 0.674538 0.672440 0.666476 0.674352 0.671488
true_negative_rate 0.863183 0.861953 0.859285 0.862806 0.861464
positive_predictive_value 0.673768 0.671290 0.664830 0.672942 0.670213
negative_predictive_value 0.863596 0.862573 0.860177 0.863564 0.862152 
multi_fit_results.T.describe()
observed_positive_rate observed_negative_rate predicted_positive_rate predicted_negative_rate accuracy precision recall f1 sensitivity specificity positive_likelihood negative_likelihood false_positive_rate false_negative_rate true_positive_rate true_negative_rate positive_predictive_value negative_predictive_value
count 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000 5.000000
mean 0.295261 0.704739 0.295812 0.704188 0.805674 0.670609 0.671859 0.671233 0.671859 0.861738 4.860007 0.380796 0.138262 0.328141 0.671859 0.861738 0.670609 0.862412
std 0.000146 0.000146 0.000172 0.000172 0.002044 0.003516 0.003273 0.003393 0.003273 0.001530 0.076763 0.004479 0.001530 0.003273 0.003273 0.001530 0.003516 0.001398
min 0.295080 0.704583 0.295570 0.704020 0.802373 0.664830 0.666476 0.665652 0.666476 0.859285 4.736364 0.377048 0.136817 0.325462 0.666476 0.859285 0.664830 0.860177
25% 0.295176 0.704599 0.295699 0.704093 0.805342 0.670213 0.671488 0.670850 0.671488 0.861464 4.847017 0.377429 0.137194 0.325648 0.671488 0.861464 0.670213 0.862152
50% 0.295232 0.704768 0.295907 0.704093 0.805971 0.671290 0.672440 0.671865 0.672440 0.861953 4.871109 0.380020 0.138047 0.327560 0.672440 0.861953 0.671290 0.862573
75% 0.295401 0.704824 0.295907 0.704301 0.807197 0.672942 0.674352 0.673646 0.674352 0.862806 4.915321 0.381341 0.138536 0.328512 0.674352 0.862806 0.672942 0.863564
max 0.295417 0.704920 0.295980 0.704430 0.807489 0.673768 0.674538 0.674153 0.674538 0.863183 4.930225 0.388141 0.140715 0.333524 0.674538 0.863183 0.673768 0.863596

Receiver Operator Characteristic and Sensitivity-Specificity Curves#

Receiver Operator Characteristic Curve:

k_fold_fpr = []
k_fold_tpr = []
k_fold_thresholds = []
k_fold_auc = []

for i in range(5):
    mask = multi_model['k_fold'] == i
    mask = mask.values
    observed = multi_model.loc[mask]['observed']
    predicted_proba = multi_model.loc[mask]['prob']
    fpr, tpr, thresholds = roc_curve(observed, predicted_proba)
    roc_auc = auc(fpr, tpr)
    k_fold_fpr.append(fpr)
    k_fold_tpr.append(tpr)
    k_fold_thresholds.append(thresholds)
    k_fold_auc.append(roc_auc)
    print (f'Run {i} AUC {roc_auc:0.4f}')
    
# Show mean area under curve  
mean_auc = np.mean(k_fold_auc)
sd_auc = np.std(k_fold_auc)
print (f'\nMean AUC: {mean_auc:0.4f}')
print (f'SD AUC: {sd_auc:0.4f}')

Run 0 AUC 0.8710
Run 1 AUC 0.8695
Run 2 AUC 0.8688
Run 3 AUC 0.8705
Run 4 AUC 0.8715

Mean AUC: 0.8702
SD AUC: 0.0010

Sensitivity-specificity curve:

k_fold_sensitivity = []
k_fold_specificity = []


for i in range(5):
    # Get classificiation probabilities for k-fold replicate
    mask = multi_model['k_fold'] == i
    mask = mask.values
    observed = multi_model.loc[mask]['observed']
    predicted_proba = multi_model.loc[mask]['prob']
    
    # Set up list for accuracy measures
    sensitivity = []
    specificity = []
    
    # Loop through increments in probability of survival
    thresholds = np.arange(0.0, 1.01, 0.01)
    for cutoff in thresholds: #  loop 0 --> 1 on steps of 0.1
        # Get classificiation using cutoff
        predicted_class = predicted_proba >= cutoff
        predicted_class = predicted_class.values * 1.0
        # Call accuracy measures function
        accuracy = calculate_accuracy(observed, predicted_class)
        # Add accuracy scores to lists
        sensitivity.append(accuracy['sensitivity'])
        specificity.append(accuracy['specificity'])
    
    # Add replicate to lists
    k_fold_sensitivity.append(sensitivity)
    k_fold_specificity.append(specificity)

Combined plot:

fig = plt.figure(figsize=(10,5))

# Plot ROC
ax1 = fig.add_subplot(121)
for i in range(5):
    ax1.plot(k_fold_fpr[i], k_fold_tpr[i], color='orange')
ax1.plot([0, 1], [0, 1], color='darkblue', linestyle='--')
ax1.set_xlabel('False Positive Rate')
ax1.set_ylabel('True Positive Rate')
ax1.set_title('Receiver Operator Characteristic Curve')
text = f'Mean AUC: {mean_auc:.3f}'
ax1.text(0.64,0.07, text, 
         bbox=dict(facecolor='white', edgecolor='black'))
plt.grid(True)

# Plot sensitivity-specificity
ax2 = fig.add_subplot(122)
for i in range(5):
    ax2.plot(k_fold_sensitivity[i], k_fold_specificity[i])
ax2.set_xlabel('Sensitivity')
ax2.set_ylabel('Specificity')
ax2.set_title('Sensitivity-Specificity Curve')
plt.grid(True)


plt.tight_layout(pad=2)
plt.savefig('./output/lr_hospital_fit_roc_sens_spec.jpg', dpi=300)

plt.show()
../_images/logistic_regression_hospit_fit_accuracy_29_0.png

Identify cross-over of sensitivity and specificity#

sens = np.array(k_fold_sensitivity).mean(axis=0)
spec = np.array(k_fold_specificity).mean(axis=0)
df = pd.DataFrame()
df['sensitivity'] = sens
df['specificity'] = spec
df['spec greater sens'] = spec > sens

# find last index for senitivity being greater than specificity 
mask = df['spec greater sens'] == False
last_id_sens_greater_spec = np.max(df[mask].index)
locs = [last_id_sens_greater_spec, last_id_sens_greater_spec + 1]
points = df.iloc[locs][['sensitivity', 'specificity']]

# Get intersetction with line of x=y
a1 = list(points.iloc[0].values)
a2 = list(points.iloc[1].values)
b1 = [0, 0]
b2 = [1, 1]

intersect = get_intersect(a1, a2, b1, b2)[0]
print(f'\nIntersect: {intersect:0.3f}')

Intersect: 0.789

Calibration and assessment of accuracy when model has high confidence#

# Collate results in Dataframe
reliability_collated = pd.DataFrame()

# Loop through k fold predictions
for i in range(5):
    
    # Get observed class and predicted probability
    mask = multi_model['k_fold'] == i
    obs = multi_model[mask]['observed'].values
    prob = multi_model[mask]['prob'].values
    
    # Bin data with numpy digitize (this will assign a bin to each case)
    step = 0.10
    bins = np.arange(step, 1+step, step)
    digitized = np.digitize(prob, bins)
        
    # Put single fold data in DataFrame
    reliability = pd.DataFrame()
    reliability['bin'] = digitized
    reliability['probability'] = prob
    reliability['observed'] = obs
    classification = 1 * (prob > 0.5 )
    reliability['correct'] = obs == classification
    reliability['count'] = 1
    
    # Summarise data by bin in new dataframe
    reliability_summary = pd.DataFrame()

    # Add bins and k-fold to summary
    reliability_summary['bin'] = bins
    reliability_summary['k-fold'] = i

    # Calculate mean of predicted probability of thrombolysis in each bin
    reliability_summary['confidence'] = \
        reliability.groupby('bin').mean()['probability']

    # Calculate the proportion of patients who receive thrombolysis
    reliability_summary['fraction_positive'] = \
        reliability.groupby('bin').mean()['observed']
        
    # Calculate proportion correct in each bin
    reliability_summary['fraction_correct'] = reliability.groupby('bin').mean()['correct']
    
    # Calculate fraction of results in each bin
    reliability_summary['fraction_results'] = \
        reliability.groupby('bin').sum()['count'] / reliability.shape[0]   
    
    # Add k-fold results to DatafRame collation
    reliability_collated = reliability_collated.append(reliability_summary)
    
# Get mean results
reliability_summary = reliability_collated.groupby('bin').mean()
reliability_summary.drop('k-fold', axis=1, inplace=True)

reliability_summary
confidence fraction_positive fraction_correct fraction_results
bin
0.1 0.018772 0.049217 0.950783 0.458798
0.2 0.146496 0.206722 0.793278 0.091344
0.3 0.248232 0.304989 0.695011 0.065862
0.4 0.349055 0.369431 0.630569 0.055506
0.5 0.448903 0.456114 0.543886 0.051187
0.6 0.549545 0.525635 0.525635 0.047836
0.7 0.650405 0.586572 0.586572 0.050445
0.8 0.749642 0.674949 0.674949 0.047724
0.9 0.851498 0.746320 0.746320 0.056596
1.0 0.953730 0.812274 0.812274 0.074701 
fig = plt.figure(figsize=(10,5))


# Plot predicted prob vs fraction psotive
ax1 = fig.add_subplot(1,2,1)

# Loop through k-fold reliability results
for i in range(5):
    mask = reliability_collated['k-fold'] == i
    k_fold_result = reliability_collated[mask]
    x = k_fold_result['confidence']
    y = k_fold_result['fraction_positive']
    ax1.plot(x,y, color='orange')
# Add 1:1 line
ax1.plot([0,1],[0,1], color='k', linestyle ='--')
# Refine plot
ax1.set_xlabel('Model probability')
ax1.set_ylabel('Fraction positive')
ax1.set_xlim(0, 1)
ax1.set_ylim(0, 1)
ax1.grid()

# Plot accuracy vs probability
ax2 = fig.add_subplot(1,2,2)
# Loop through k-fold reliability results
for i in range(5):
    mask = reliability_collated['k-fold'] == i
    k_fold_result = reliability_collated[mask]
    x = k_fold_result['confidence']
    y = k_fold_result['fraction_correct']
    ax2.plot(x,y, color='orange')
# Refine plot
ax2.set_xlabel('Model probability')
ax2.set_ylabel('Fraction correct')
ax2.set_xlim(0, 1)
ax2.set_ylim(0, 1)
ax2.grid()

ax3 = ax2.twinx()  # instantiate a second axes that shares the same x-axis
for i in range(5):
    mask = reliability_collated['k-fold'] == i
    k_fold_result = reliability_collated[mask]
    x = k_fold_result['confidence']
    y = k_fold_result['fraction_results']
    ax3.plot(x,y, color='blue')
    
ax3.set_xlim(0, 1)
ax3.set_ylim(0, 0.5)
ax3.set_ylabel('Fraction of samples')

custom_lines = [Line2D([0], [0], color='orange', alpha=0.6, lw=2),
                Line2D([0], [0], color='blue', alpha = 0.6,lw=2)]

plt.legend(custom_lines, ['Fraction correct', 'Fraction of samples'],
          loc='upper center')


plt.tight_layout(pad=2)
plt.savefig('./output/lr_hospital_fit_reliability.jpg', dpi=300)
plt.show()
../_images/logistic_regression_hospit_fit_accuracy_35_0.png

Get accuracy of model when model is at least 80% confident

bins = [0.1, 0.2, 0.9, 1.0]
acc = reliability_summary.loc[bins].mean()['fraction_correct']
frac = reliability_summary.loc[bins].sum()['fraction_results']

print ('For For example,samples with at least 80% confidence:')
print (f'Proportion of all samples: {frac:0.3f}')
print (f'Accuracy: {acc:0.3f}')

For For example,samples with at least 80% confidence:
Proportion of all samples: 0.681
Accuracy: 0.826

Observations#

  • Fitting at a single hospital level gave an accuracy of 80.7% (c.f. 83.2% for a single model), and a ROC AUC of 0.870 (c.f. 0.904 for a single fit model)

  • Accuracy was 82.6% for the 68% of samples with at least 80% confidence of model

  • The model can achieve 78.9% sensitivity and specificity simultaneously

  • ROC ACUC = 0.870

  • The model showed poorer calibration than a single model