Çoklu Doğrusal Regresyonda Varsayım Testleri

Why do the assumptions matter?

Multiple linear regression relates an outcome to several explanatory variables. Once the model is fitted, Python or R will always hand you a coefficient table — whether or not the model is appropriate. Assumption testing is how you work out how much to trust the numbers in that table.

There is a distinction that often gets skipped: some assumptions affect the coefficient estimates themselves, others only affect standard errors, p-values and confidence intervals. Confusing the two leads either to needless panic or to false comfort.

Another caution: diagnostic tests are not a stamp of "model correct / model wrong". In large samples even tiny deviations make tests significant; in small samples serious problems are hard to detect. Read the test alongside the plot, and the plot alongside domain knowledge.

The assumptions at a glance

Note what is not on the list: explanatory variables do not need to be normally distributed. This is one of the most widespread misconceptions. A regressor can be categorical, skewed, binary or entirely irregular — OLS does not mind.

Setting up the example model

The example below generates synthetic sales data and fits an OLS model. In a real project, adapt the variables, missing-value handling and category encoding to your own data.

import numpy as np
import pandas as pd
import statsmodels.formula.api as smf

rng = np.random.default_rng(42)
n = 250

adspend = rng.normal(50, 12, n)
price = rng.normal(20, 4, n)
store = rng.normal(100, 18, n)
error = rng.normal(0, 12, n)

df = pd.DataFrame({
    "sales": 30 + 1.8*adspend - 2.1*price + 0.7*store + error,
    "adspend": adspend,
    "price": price,
    "store": store,
})

model = smf.ols("sales ~ adspend + price + store", data=df).fit()
print(model.summary())

The formula interface adds the intercept automatically. Coefficients give the association between a one-unit increase in the variable and the conditional mean outcome, holding the others constant — and "holding constant" is exactly the phrase that starts losing meaning when collinearity is present.

1. Linearity and model form

"Linearity" in regression does not mean the outcome must follow a straight line against every raw variable. The model must be linear in the coefficients; a squared term, a logarithm or an interaction can all enter, and it is still linear regression.

The check is the residuals-versus-fitted plot. We want a flat cloud. A curved pattern means the model form is incomplete; a funnel-shaped spread is the subject of the next section.

import matplotlib.pyplot as plt
import seaborn as sns

sns.residplot(
    x=model.fittedvalues, y=model.resid,
    lowess=True, line_kws={"color": "red"}
)
plt.axhline(0, color="gray", linestyle="--")
plt.xlabel("Fitted value")
plt.ylabel("Residual")
plt.show()

If the red lowess line does not run flat around zero, the model is missing something systematic. A transformation, a polynomial term or an interaction can help — but do not add a term merely to flatten the plot. Every extra term needs a justification; otherwise you are fitting the noise.

2. Constant variance (homoskedasticity)

Homoskedasticity means the spread of residuals stays roughly constant across fitted levels. With variables such as sales, income or spending this assumption is often violated: deviation grows with the magnitude, and the plot takes on a funnel shape.

The Breusch–Pagan test examines the null hypothesis that residual variance is not related to the explanatory variables. A small p-value is evidence for heteroskedasticity.

from statsmodels.stats.diagnostic import het_breuschpagan

lm, lm_p, f_stat, f_p = het_breuschpagan(
    model.resid, model.model.exog
)
print({"LM p": round(lm_p, 4), "F p": round(f_p, 4)})

An important point here: heteroskedasticity does not bias the coefficients. The estimates are still centred correctly. What breaks is the standard errors — which makes your p-values and confidence intervals unreliable.

So the fix usually is not to change the model. The most practical route is robust standard errors:

# HC3 is a common choice for small to medium samples
robust = model.get_robustcov_results(cov_type="HC3")
print(robust.summary())

The coefficients stay the same; the standard errors are corrected. Alternatives: taking the logarithm of the dependent variable (sensible if the relationship is multiplicative), or weighted least squares.

3. Independence of errors

If observations are ordered in time, consecutive errors can carry into one another. The model then computes uncertainty too small: your p-values come out lower than they should, and you take a relationship to be significant when it is not.

The Durbin–Watson statistic is a common diagnostic for first-order serial correlation. A value near 2 means a weak signal; approaching 0 is consistent with positive, approaching 4 with negative autocorrelation.

from statsmodels.stats.stattools import durbin_watson

print("DW:", round(durbin_watson(model.resid), 3))

# Plotting residuals in observation order says more than the test
plt.plot(model.resid, marker=".", linestyle="-", alpha=0.6)
plt.axhline(0, color="gray", linestyle="--")
plt.xlabel("Observation order"); plt.ylabel("Residual")
plt.show()

4. Normality of residuals

This diagnostic applies to the residuals, not to the explanatory variables. If the points on a Q–Q plot follow the reference line, the distribution is close to normal; deviations in the tails point to outlying observations.

import statsmodels.api as sm
from statsmodels.stats.stattools import jarque_bera

sm.qqplot(model.resid, line="45", fit=True)
plt.title("Q–Q plot of residuals")
plt.show()

# The statsmodels version returns 4 values
jb, jb_p, skew, kurt = jarque_bera(model.resid)
print({"JB p": round(jb_p, 4),
       "skew": round(skew, 3),
       "kurtosis": round(kurt, 3)})

Avoid two traps when interpreting. A large p-value does not prove normality; it only says no evidence against it was found. And in large samples even practically irrelevant deviations come out significant — with 5,000 observations Jarque–Bera rejects almost every time. In that situation, look at the shape of the Q–Q plot rather than the test.

5. Multicollinearity

When explanatory variables behave like linear combinations of each other, coefficient standard errors grow and the coefficients move with small changes in the data. The variance inflation factor (VIF) summarises this per variable with one number: VIF = 1/(1 − R²_j).

from statsmodels.stats.outliers_influence import variance_inflation_factor

X = model.model.exog
names = model.model.exog_names

vif = pd.DataFrame({
    "variable": names,
    "VIF": [variance_inflation_factor(X, i)
            for i in range(X.shape[1])],
})
print(vif[vif["variable"] != "Intercept"].round(2))

Do not interpret the intercept's VIF. Thresholds such as 5 or 10 are signals to investigate, not mechanical decision rules. When you see a high VIF, look for a structural error first: was the same variable added twice, or is a total sitting alongside its components?

6. Influential observations

This rarely appears on assumption lists yet causes trouble most often in practice. A single extreme observation can drag the coefficients on its own. Cook's distance measures how much removing an observation would change the fitted values.

influence = model.get_influence()
cooks = influence.cooks_distance[0]

print("Five most influential observations:")
print(pd.Series(cooks).nlargest(5).round(4))

Running the diagnostics together

1. Verify the data first. Faulty measurement, wrong coding, unexpected missing values? Half of all diagnostics resolve here.
2. Look at the plots. Residuals-versus-fitted and Q–Q carry more information than numerical tests. Name the pattern: curvature, funnel, clustering?
3. Run the tests, but remember that applying all of them at once raises the chance of a false alarm. Choose the diagnostic that matches your model's question.
4. Examine influential points. Cook's distance and leverage show whether a single row is driving the result.
5. Report the limitations. Which assumption, on what evidence; what you did and what you did not.

Checklist

• The model question and variable selection are justified.
• The residuals-versus-fitted plot was examined and the pattern named.
• A Q–Q plot was drawn; the normality test was read with sample size in mind.
• Breusch–Pagan was run; robust standard errors were used where needed.
• With a time ordering, residuals were plotted in sequence and DW computed.
• VIF was computed with the intercept excluded.
• Influential observations were sought with Cook's distance.
• The before/after effect of any correction on the coefficients was reported.

Conclusion

Regression assumptions are not a checklist of exams to pass. Each diagnostic asks you a different question: are the coefficients biased, are the standard errors trustworthy, does the result rest on a single observation?

The practical approach: look at the plots first, use the tests to interpret the plots, meet any problem you find with a justified correction, and state the remaining limitations plainly. "All assumptions were satisfied" is rarely an honest sentence; "this assumption deviates in this way, and the effect is this large" is always stronger.