Interpreting Coefficients in Linear and Logistic Regression

Interpreting coefficients in linear and logistic regression is essential for understanding the relationship between variables in statistical and machine learning models. In linear regression, coefficients quantify how much the dependent variable changes for a one-unit increase in an independent variable, assuming all other variables remain constant. Logistic regression, used for binary classification, provides coefficients that explain the impact of predictors on the log-odds of an event occurring, which can be further converted into odds ratios for easier interpretation. By understanding these coefficients, practitioners can gain insights into the significance, magnitude, and direction of predictors, enabling informed decision-making and better model explanations.

Linear Regression

Linear regression models the relationship between a dependent variable and one or more independent variables using a straight line. The coefficients represent the change in the dependent variable for a one-unit increase in the independent variable.

Example: Predicting house prices based on square footage.

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression

# Sample data
data = {'Square_Feet': [1500, 1800, 2400, 3000, 3500],
        'Price': [300000, 350000, 400000, 500000, 600000]}
df = pd.DataFrame(data)

# Model
X = df[['Square_Feet']]
y = df['Price']
model = LinearRegression()
model.fit(X, y)

# Coefficients
print("Coefficient (Slope):", model.coef_[0])
print("Intercept:", model.intercept_)

# Interpretation
# For every additional square foot, the house price increases by model.coef_[0] units.
        

The previous code block consist of the following code lines:

• The code imports necessary libraries: numpy, pandas, and LinearRegression from sklearn.linear_model.
• A dictionary named data is created with two keys: Square_Feet (independent variable) and Price (dependent variable), representing house sizes and their corresponding prices.
• The dictionary is converted into a pandas DataFrame called df for easier manipulation.
• The independent variable (Square_Feet) is assigned to X, and the dependent variable (Price) is assigned to y.
• An instance of LinearRegression is created and stored in the variable model.
• The model is trained on the data using model.fit(X, y), where the algorithm learns the relationship between square footage and price.
• The slope (coefficient) of the regression line is retrieved using model.coef_[0], which indicates how much the price increases for each additional square foot.
• The y-intercept of the regression line is retrieved using model.intercept_, representing the price of a house when the square footage is 0.
• The code prints the slope and intercept values to interpret the linear relationship between the variables.
• Interpretation: The coefficient (model.coef_[0]) indicates that for every additional square foot of house size, the price increases by the given amount (in the same units as Price).

When the code is executed the following result is obtained.

Coefficient (Slope): 144.21669106881407
Intercept: 78111.27379209368

The interpretation of the output is as follows:

• Coefficient (Slope): 144.21669106881407 For every additional square foot of house size, the house price increases by approximately 144.22 units. In this context, if the price is in dollars, then for every extra square foot, the price increases by $144.22.
• Intercept: 78111.27379209368 When the house size is 0 square feet (which is theoretical and may not have practical meaning), the predicted house price is approximately $78,111.27. The intercept represents the baseline value of the dependent variable (price) when all predictors (square footage) are zero.
• Practical Interpretation: The model suggests that larger houses cost more, with an increase of $144.22 for each additional square foot.

Key Point: The coefficient for Square_Feet shows how much the price changes per square foot.

Logistic Regression

Logistic regression is used for classification problems, predicting the probability of a binary outcome. The coefficients represent the change in the log-odds of the outcome for a one-unit increase in the predictor variable.

Example: Predicting whether a customer will buy a product based on income.

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

# Sample data
data = {'Income': [30000, 45000, 60000, 80000, 100000],
        'Purchased': [0, 0, 1, 1, 1]}
df = pd.DataFrame(data)

# Model
X = df[['Income']]
y = df['Purchased']
model = LogisticRegression()
model.fit(X, y)

# Coefficients
print("Coefficient (Log-Odds):", model.coef_[0][0])
print("Intercept:", model.intercept_[0])

# Probability Interpretation
import math
odds_ratio = math.exp(model.coef_[0][0])
print("Odds Ratio:", odds_ratio)

# For every additional dollar in income, the odds of purchase increase by odds_ratio times.
        

The previous code block example consist of the following code lines:

• Imports: The code imports necessary libraries for the task:
• NumPy - a library for numerical operations in Python, although it is not directly used in the code.
• Pandas - a library used for data manipulation and analysis. It is used to create the DataFrame `df` containing the sample data.
• LogisticRegression from sklearn.linear_model - a machine learning model used for binary classification tasks, in this case, predicting whether a purchase will be made based on income.
• Sample Data: The dictionary `data` contains two key-value pairs:
• 'Income': The income values of 5 individuals, used as the independent variable for prediction.
• 'Purchased': A binary target variable (0 or 1) representing whether the individual made a purchase (1) or not (0).

The dictionary is converted into a DataFrame `df` using pd.DataFrame(data).

• Model Training: The logistic regression model is trained using the data:
• X: The independent variable, which is the 'Income' column from the DataFrame, selected using df[['Income']].
• y: The target variable, which is the 'Purchased' column from the DataFrame, selected using df['Purchased'].
• Logistic Regression Model: An instance of LogisticRegression() is created and trained using the fit method with the input data X and the target variable y.
• Model Coefficients: After the model is trained, the coefficients are displayed:
• Coefficient (Log-Odds): The model’s coefficient is extracted using model.coef_[0][0], which represents the log-odds for a one-unit increase in income. This is printed out.
• Intercept: The model’s intercept is extracted using model.intercept_[0], which represents the log-odds of the baseline (when income = 0). This is printed out as well.
• Probability Interpretation: The odds ratio is calculated to interpret the model’s prediction:
• Odds Ratio: The odds ratio is calculated using the formula math.exp(model.coef_[0][0]), which converts the log-odds to the actual odds ratio. This shows how much the odds of purchasing increase for every additional dollar of income.
• Conclusion: The print statement "For every additional dollar in income, the odds of purchase increase by odds_ratio times." concludes the interpretation of the odds ratio, giving insight into the model’s behavior.

When the code is executed the following output is obtained.

Coefficient (Log-Odds): 1.652730135568006e-05
Intercept: -6.136333210253191e-10
Odds Ratio: 1.0000165274379322
        

Here is the explanation of the obtained results:

• Coefficient (Log-Odds): 1.652730135568006e-05
• This is the coefficient (log-odds) obtained for the "Income" variable in the logistic regression model. It represents the change in the log-odds of purchasing a product for a one-unit increase in income.
• The value of 1.652730135568006e-05 (which is a very small number) suggests that for every 1-dollar increase in income, the log-odds of purchasing the product increase by approximately 0.0000165. This is a very small effect.
• Intercept: -6.136333210253191e-10
• The intercept (log-odds) represents the baseline log-odds when income is 0 (i.e., no income). The value -6.136333210253191e-10 is a very small negative number, suggesting that with an income of 0, the log-odds of purchasing the product are extremely close to zero, which makes sense because it would be highly unlikely that someone with no income would make a purchase.
• Odds Ratio: 1.0000165274379322
• The odds ratio is calculated by exponentiating the coefficient (log-odds). In this case, exp(1.652730135568006e-05) gives an odds ratio of 1.0000165274379322.
• An odds ratio of approximately 1 means that the increase in income has a very small effect on the odds of making a purchase. Specifically, for every additional dollar in income, the odds of making a purchase increase by a factor of 1.0000165, which is a very slight increase. The odds ratio close to 1 indicates that income has only a minimal effect on the probability of purchasing in this model.

Key Point: Convert the coefficient to an odds ratio using the exponential function to interpret it in terms of probability.

Conclusion

In both linear and logistic regression, the coefficients are essential for understanding the relationship between the independent variables (predictors) and the dependent variable (outcome). In linear regression, the coefficient represents the change in the dependent variable for each one-unit change in the independent variable. A positive coefficient indicates a direct relationship, while a negative coefficient suggests an inverse relationship between the two variables. On the other hand, in logistic regression, the coefficient represents the change in the log-odds of the outcome occurring for a one-unit change in the independent variable. Although interpreting log-odds is not as straightforward as interpreting linear regression coefficients, the results can be converted into an odds ratio by exponentiating the coefficient, which is easier to interpret.

The odds ratio in logistic regression helps to understand how the odds of the event change with each one-unit increase in the independent variable. An odds ratio of 1 means no effect on the odds, while values greater than 1 or less than 1 indicate an increase or decrease in the odds, respectively. In our example, the odds ratio of approximately 1 suggests that income has a minimal effect on the likelihood of making a purchase. This indicates that other factors beyond income may have a greater influence on purchasing behavior.

In summary, understanding how to interpret coefficients in both linear and logistic regression models is crucial for making informed decisions based on model predictions. The coefficients provide insights into how each independent variable contributes to the outcome, and the odds ratio in logistic regression offers a more intuitive way to interpret the relationship between the predictors and the event being studied.

Thank you for reading the tutorial! Try running the Python code and let me know in the comments if you got the same results. If you have any questions or need further clarification, feel free to leave a comment. Thanks again!