Multiple Linear Regression, the game

More than one feature

Multiple linear regression predicts one dependent variable from several independent variables at once.

The key phrase is “holding the other features constant”. Each coefficient tells you what happens when only that one feature changes.

Build your model

Every feature you add makes R² (the share of variance explained) go up or stay the same, even if the feature is useless. Adjusted R² fixes that by penalizing extra features:

Below are the real values for every possible model predicting a car’s fuel efficiency (mpg) on R’s built-in mtcars data (32 cars, 10 features). The original article uses a fish-weight dataset; the idea is identical.

Pick features like a pro

With hundreds of features you need a strategy:

• Backward elimination: start with every feature and repeatedly remove the least significant one (highest p-value above 0.05). Like an entrance exam where the weakest candidates leave.
• Forward selection: start with no features and add the most useful one at a time. Like picking a cricket team one player at a time.
• Stepwise selection: add features like forward selection, but re-check after each step and remove any that are no longer significant.

Code it in Python

The article’s workflow on the Kaggle Fish Market dataset (159 fish, 7 columns), with two fixes: species is one-hot encoded (category codes would wrongly imply an order), and the metric functions are imported.

import numpy as np
import pandas as pd
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score, mean_squared_error

data = pd.read_csv("Fish.csv")
print(data.shape, data.isnull().sum().sum())   # (159, 7) and no missing values

# One-hot encode the species column (no fake ordering)
data = pd.get_dummies(data, columns=["Species"], drop_first=True)

X = data.drop(columns="Weight")
y = data["Weight"]
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0)

model = LinearRegression()
model.fit(X_train, y_train)
y_pred = model.predict(X_test)

print("R²:", r2_score(y_test, y_pred))
print("RMSE:", np.sqrt(mean_squared_error(y_test, y_pred)))

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Keep levelling up

You learned the multiple regression equation, how to read its coefficients, why R² always rises with more features while adjusted R² doesn’t, three feature selection methods, and the full Python workflow.