Alright, let’s get down to the brass tacks of what these numbers in your regression output actually mean. You’ve run your model, you’ve got a neat table of coefficients, p-values, and other assorted stats. It’s tempting to just glance at the p-values, circle the ones below 0.05, and declare victory. Resist that urge. That’s how bad science—and frankly, bad data science—happens. Let’s learn to read the whole story.

What a Coefficient Actually Represents

Think of a coefficient as the model’s way of telling you the leverage or influence of a feature. In a linear regression, it’s beautifully straightforward. For a continuous predictor, the coefficient is the amount you’d expect the target variable to change for a one-unit increase in the predictor, holding all other variables constant.

That last bit is the statistical version of “all other things being equal,” and it’s the secret sauce that makes regression so powerful. It isolates the relationship between one feature and the target.

Let’s make this concrete. Imagine we’re predicting house prices (price) based on square footage (sqft) and number of bedrooms (bedrooms).

import statsmodels.api as sm
import pandas as pd
import numpy as np

# Let's generate some plausible fake data
np.random.seed(42) # for reproducibility
n_houses = 100
sqft = np.random.normal(2000, 500, n_houses)
bedrooms = np.random.poisson(3, n_houses) # Most houses have around 3 beds

# Let's say each sqft adds $200, each bedroom adds $10,000, and there's a base price
price = 50000 + 200 * sqft + 10000 * bedrooms + np.random.normal(0, 25000, n_houses)

df = pd.DataFrame({'price': price, 'sqft': sqft, 'bedrooms': bedrooms})

# Fit the model using Ordinary Least Squares (OLS)
# Note: statsmodels doesn't add an intercept by default for you, so you have to do it.
X = df[['sqft', 'bedrooms']]
X = sm.add_constant(X)  # Adds a constant term (y-intercept) to the model
y = df['price']

model = sm.OLS(y, X).fit()
print(model.summary())

Looking at the output, you’d hopefully find a coefficient for sqft somewhere around 200. The interpretation: “Holding the number of bedrooms constant, for every additional square foot, the house price increases by approximately $200.” The coefficient for bedrooms would be around 10,000: “Holding square footage constant, each additional bedroom is associated with an increase in price of about $10,000.”

The P-Value: A Measure of Surprise

The p-value is the most used and abused statistic in the book. Here’s the correct way to think about it, minus the textbook jargon:

The p-value for a coefficient asks: “Assuming there was absolutely no real relationship between this feature and the target (i.e., the true coefficient is zero), what is the probability that we would have gotten a coefficient as extreme as the one we did, purely by random chance?”

A very low p-value (conventionally below 0.05) is you saying, “Wow, that’s a really weird coincidence. The coefficient I observed is so extreme that it would be very unlikely to happen if there was no real effect. Therefore, I don’t believe the ’no effect’ hypothesis. I believe this feature probably has a real relationship with the target.”

It is NOT the probability that the feature is unimportant. It is NOT a measure of the relationship’s strength. A tiny p-value can accompany a minuscule, practically useless coefficient if you have a huge dataset.

The Confidence Interval: The Co-efficient’s Bodyguard

This is where p-values get some much-needed context. A 95% confidence interval gives you a range of values that you can be 95% confident contains the true value of the coefficient.

If your model spits out a coefficient of 150 for sqft with a 95% CI of [100, 200], you’re saying: “Hey, based on my data, I believe the true value is 150, but I acknowledge there’s uncertainty. I’d bet $5 that the real value is somewhere between 100 and 200.” It’s a much richer piece of information than a p-value alone.

A p-value below 0.05 will always correspond to a 95% confidence interval that does not include zero. Look at the CI first. Is the range tight and far from zero? Great, you have a precise and significant effect. Is the range wide and barely misses zero? Your effect is probably real but you have low precision on its size. Is the range huge and includes zero? Your data is telling you it has no idea what’s going on with this feature.

Common Pitfalls and How to Avoid Them

  1. The P-Value Fishing Expedition: Don’t just throw 100 features into a model and blindly trust the ones with p < 0.05. By pure probability, you’d expect about 5 of them to be “significant” even if they’re all garbage. This is a classic case of p-hacking. Have a hypothesis first.

  2. Ignoring Scale: The size of a coefficient is meaningless without knowing the scale of the feature. A coefficient of 1000 for annual_income (in dollars) is massive. A coefficient of 1000 for national_debt (in dollars) is negligible. Standardizing your features (subtracting the mean, dividing by the standard deviation) can make coefficients comparable to each other.

  3. Correlated Predictors (Multicollinearity): This is a big one. If two of your features are highly correlated (e.g., sqft and number_of_rooms), the model has a hard time untangling their individual effects. It might assign importance to one and give the other a high p-value, or it might make both of their coefficients unstable and their standard errors explode, leading to wide confidence intervals. The model’s overall prediction might still be good, but your interpretation of the individual drivers goes out the window. Check a correlation matrix or calculate Variance Inflation Factors (VIFs).

from statsmodels.stats.outliers_influence import variance_inflation_factor

# Calculate VIF for each feature
vif_data = pd.DataFrame()
vif_data["feature"] = X.columns
vif_data["VIF"] = [variance_inflation_factor(X.values, i) for i in range(len(X.columns))]
print(vif_data)

A VIF above 5 or 10 is a major red flag for problematic multicollinearity.

In the end, interpreting a model is an art informed by science. The coefficients, p-values, and confidence intervals are your tools. Use them together, understand their limitations, and never, ever let a single number like a p-value tell the whole story. Your model is a conversation with your data, not a dictate from it.