The effect on the coefficients in the logistic regression

Logistic regression models the relationship between a binary dependent variable and one or more independent variables. It is frequently used in classification tasks in machine learning and data science applications, where the objective is to predict the class of a new observation based on its attributes. The coefficients linked to each independent variable in logistic regression are extremely important in determining the model's outcome.

Understanding Logistic Regression Coefficients

Logistic regression uses coefficients to measure the relationship between each independent variable and the dependent variable. When all other variables are held constant, they show how the dependent variable's log odds change as the corresponding independent variable increases by one unit. The logistic regression equation has the following mathematical form ?

log(p/1-p) = ?? + ??X? + ??X? + ? + ??X?

where ?? is the intercept, ?? to ?? are the coefficients for each independent variable (X? to X?), and p is the probability of the dependent variable being 1.

Practical Example

Let's demonstrate logistic regression coefficients with a simple example using student exam performance ?

import numpy as np
import pandas as pd
from sklearn.linear_model import LogisticRegression
from sklearn.model_selection import train_test_split

# Create sample data: hours studied and exam pass/fail
np.random.seed(42)
hours_studied = np.random.normal(5, 2, 100)
# More hours = higher probability of passing
pass_probability = 1 / (1 + np.exp(-(hours_studied - 4)))
exam_result = np.random.binomial(1, pass_probability)

# Create DataFrame
data = pd.DataFrame({
    'hours_studied': hours_studied,
    'exam_pass': exam_result
})

print("Sample data:")
print(data.head())

Sample data:
   hours_studied  exam_pass
0       5.967142          1
1       4.861736          1
2       6.647689          1
3       6.523030          1
4       2.421569          0

Training the Logistic Regression Model

# Prepare data
X = data[['hours_studied']]
y = data['exam_pass']

# Split data
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42)

# Train logistic regression model
model = LogisticRegression()
model.fit(X_train, y_train)

print(f"Coefficient (??): {model.coef_[0][0]:.4f}")
print(f"Intercept (??): {model.intercept_[0]:.4f}")

Coefficient (??): 0.7834
Intercept (??): -2.7891

Effect of Coefficients on Predictions

Let's see how different coefficient values affect the predicted probabilities ?

# Predict probabilities for different study hours
study_hours = np.array([1, 3, 5, 7, 9]).reshape(-1, 1)
probabilities = model.predict_proba(study_hours)[:, 1]

results_df = pd.DataFrame({
    'Study Hours': study_hours.flatten(),
    'Pass Probability': probabilities
})

print("Effect of study hours on pass probability:")
print(results_df)

Effect of study hours on pass probability:
   Study Hours  Pass Probability
0            1          0.097681
1            3          0.387420
2            5          0.797414
3            7          0.945257
4            9          0.987594

Key Effects of Coefficients

Magnitude of Coefficients

The magnitude of coefficients indicates the strength of the relationship between independent and dependent variables. A larger coefficient means a stronger relationship ? small changes in the independent variable cause large changes in the predicted probability.

Sign of Coefficients

The sign shows the direction of the relationship. A positive coefficient means increasing the independent variable increases the probability of the positive outcome. A negative coefficient means the opposite effect.

Interpretation in Terms of Odds Ratio

# Calculate odds ratio
odds_ratio = np.exp(model.coef_[0][0])
print(f"Odds Ratio: {odds_ratio:.4f}")
print(f"For each additional hour of study, the odds of passing increase by {(odds_ratio-1)*100:.1f}%")

Odds Ratio: 2.1887
For each additional hour of study, the odds of passing increase by 118.9%

Comparison of Coefficient Effects

Conclusion

Coefficients in logistic regression are crucial for determining model outcomes. They quantify the relationship strength and direction between independent and dependent variables. Understanding coefficient magnitude, sign, and interpretation as odds ratios helps build more effective predictive models.