Heteroskedasticity Explained: Meaning, Detection and Fixes, with Real Data

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Below, we explain heteroskedasticity in plain language and show it clearly in real data from ScholarsColab.

Homoskedasticity vs heteroskedasticity at a glance

Worked example: does the h-index vary more over a research career?

We detect and fix heteroskedasticity in a real regression, notebook-style. Every result is calculated live, and you can run the same code yourself.

The data: one row per ScholarsColab mentor with at least five dated papers, giving their discipline, career length, number of papers, h-index and total citations. The file is anonymised: no names or identifiers, rows are shuffled and citation totals are rounded.

Step 1: Load the data

import numpy as np
import pandas as pd
from datetime import date

# The ScholarsColab Research Methods Index (mentors level): anonymised, updated daily
DATA = "https://scholarscolab.in/wp-json/scholarscolab/v1/research-methods-data?level=mentors"
mentors = pd.read_csv(DATA, comment="#")   # lines starting with # hold the citation

def p_text(p):
    return "< 0.001" if p < 0.001 else f"= {p:.3f}"

print(mentors.shape)
mentors.head()

(891, 6)

Step 2: Fit a regression and plot its residuals

We regress h-index on career length (years since a researcher’s first paper), then plot each residual (actual minus predicted) against the predicted value.

import statsmodels.formula.api as smf
import matplotlib.pyplot as plt

model = smf.ols("h_index ~ career_years", data=mentors).fit()
print(f"h_index = {model.params['Intercept']:.2f} + {model.params['career_years']:.3f} × career_years")

fig, ax = plt.subplots(figsize=(8, 4.5))
ax.scatter(model.fittedvalues, model.resid, s=12, alpha=0.4)
ax.axhline(0, color="grey")
ax.set_xlabel("Fitted h-index")
ax.set_ylabel("Residual")
plt.show()

h_index = 3.80 + 0.425 × career_years

Step 3: Confirm it with a table

bands = pd.cut(mentors["career_years"], [0, 5, 10, 15, 20, 30, 70],
               labels=["1–5", "6–10", "11–15", "16–20", "21–30", "31+"])
mentors.groupby(bands, observed=True)["h_index"].agg(["count", "median", "std"]).round(1)

Step 4: Test it formally

from statsmodels.stats.diagnostic import het_breuschpagan
lm, lm_p, f, f_p = het_breuschpagan(model.resid, model.model.exog)
print(f"Breusch-Pagan test: LM = {lm:.1f}, p {p_text(lm_p)}")

Breusch-Pagan test: LM = 35.8, p < 0.001

The Breusch-Pagan test checks whether the size of the residuals depends on the predictors. The p-value is < 0.001, so the test confirms heteroskedasticity: the size of the residuals depends on career length.

Step 5: Fix the standard errors

robust = smf.ols("h_index ~ career_years", data=mentors).fit(cov_type="HC3")
print(f"Ordinary standard error of the slope: {model.bse['career_years']:.4f}")
print(f"Robust (HC3) standard error:          {robust.bse['career_years']:.4f}")

Ordinary standard error of the slope: 0.0316
Robust (HC3) standard error:          0.0434

How to detect and fix heteroskedasticity

1. Plot the residuals of your regression against the predicted values; a funnel shape is the classic sign.
2. Use a formal test if needed: Breusch-Pagan or White’s test.
3. Transform the outcome (for example, log or square root) when the spread grows with the size of values.
4. Use heteroskedasticity-robust standard errors (also called White or HC standard errors).
5. Consider a better-suited model, such as weighted least squares or a count model for count data.

Common mistakes reviewers point out

• Never checking the residuals of a regression.
• Reporting ordinary standard errors when heteroskedasticity is obvious.
• Removing high values as “outliers” when they are a genuine part of the pattern.
• Confusing heteroskedasticity with non-linearity; both can appear in residual plots and need different fixes.

Frequently asked questions

Use this data in your own work

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