Model Specific Interpretability - Learning Module

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Linear Model Coefficients

The Original Interpretable Model

In the era of deep neural networks and complex ensemble methods, it's easy to forget that one of the most powerful tools in machine learning remains fundamentally transparent: linear models. When someone asks 'how does your model work?', a linear model provides a complete, mathematically precise answer through its coefficients.

Linear models—including linear regression, logistic regression, and their regularized variants—offer intrinsic interpretability. Unlike black-box models that require post-hoc explanation methods, linear models wear their decision-making logic on their sleeve. Each coefficient directly tells you how much each feature contributes to the prediction.

But this apparent simplicity is deceptive. Correctly interpreting linear model coefficients requires deep understanding of standardization, multicollinearity, regularization effects, causal reasoning, and the distinction between statistical significance and practical importance. Misinterpreting coefficients leads to flawed conclusions, incorrect interventions, and broken trust in machine learning systems.

What You Will Learn

By the end of this page, you will master the complete framework for interpreting linear model coefficients. You'll understand the mathematical foundations, recognize common interpretation pitfalls, correctly handle different feature scales, navigate multicollinearity, interpret regularized coefficients, distinguish correlation from causation, and apply these principles in real-world scenarios.

Linear Model Foundations

Before diving into interpretation, we must establish the mathematical framework. A linear model expresses predictions as a weighted sum of input features:

Linear Regression: $$\hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_p x_p$$

Logistic Regression: $$P(y=1|x) = \sigma(\beta_0 + \beta_1 x_1 + \cdots + \beta_p x_p)$$

where $\sigma(z) = \frac{1}{1 + e^{-z}}$ is the sigmoid function.

Each coefficient $\beta_j$ represents the marginal effect of feature $x_j$ on the outcome, holding all other features constant. This 'holding constant' assumption is crucial—it's the foundation of coefficient interpretation.

Key Components of Linear Models

•Intercept ($\beta_0$) — The predicted value when all features equal zero. Often has no meaningful interpretation (e.g., when features are age or salary).
•Feature Coefficients ($\beta_j$) — The change in predicted outcome for a one-unit increase in feature $x_j$, assuming all other features remain constant.
•Prediction Function — The mechanism combining intercept and weighted features to produce outputs.
•Link Function — In generalized linear models (GLMs), the function connecting the linear predictor to the expected response.

The Ceteris Paribus Assumption

The phrase 'holding all other features constant' is easy to say but hard to achieve. In practice, features are often correlated, and changing one feature while holding others constant may be impossible or create unrealistic scenarios. This is a fundamental limitation of coefficient interpretation that we'll address throughout this page.

Interpreting Coefficient Magnitude and Sign

The most basic interpretation of a coefficient involves its sign and magnitude:

Sign Interpretation:

Positive coefficient ($\beta_j > 0$): Increasing $x_j$ increases the predicted outcome
Negative coefficient ($\beta_j < 0$): Increasing $x_j$ decreases the predicted outcome
Zero coefficient ($\beta_j = 0$): Feature has no linear effect on outcome

Magnitude Interpretation (Linear Regression): For a coefficient $\beta_j = 2.5$ on a feature measured in thousands of dollars, the interpretation is:

"For every additional $1,000 increase in [feature], the predicted [outcome] increases by 2.5 units, holding all other variables constant."

Concrete Example: House Price PredictionUnderstanding coefficient interpretation in a real estate context

Input

Output

Interpreting Logistic Regression Coefficients:

Logistic regression coefficients require special handling because the model predicts log-odds, not probabilities directly:

$$\log\left(\frac{P(y=1)}{P(y=0)}\right) = \beta_0 + \beta_1 x_1 + \cdots$$

The coefficient $\beta_j$ represents the change in log-odds for a one-unit increase in $x_j$. More intuitively, $e^{\beta_j}$ gives the odds ratio:

$e^{\beta_j} > 1$: Increasing $x_j$ increases odds of positive class
$e^{\beta_j} < 1$: Increasing $x_j$ decreases odds of positive class
$e^{\beta_j} = 1$: Feature has no effect on odds

For small probabilities, odds ratios approximately equal relative risk. For larger probabilities, odds ratios can substantially overstate relative risk.

coefficient_interpretation.py
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import numpy as np
from sklearn.linear_model import LinearRegression, LogisticRegression
from sklearn.preprocessing import StandardScaler
 
# Example: Linear Regression Interpretation
X = np.array([[1500, 3, 20], [2000, 4, 15], [1200, 2, 30], [1800, 3, 10]])
y = np.array([300000, 450000, 200000, 400000])
feature_names = ['sqft', 'bedrooms', 'age']
 
lr = LinearRegression()
lr.fit(X, y)
 
print("Linear Regression Coefficients:")
print(f"Intercept: ${lr.intercept_:, .2f}")
for name, coef in zip(feature_names, lr.coef_):
    print(f"  {name}: ${coef:,.2f} per unit increase")
 
# Example: Logistic Regression - Odds Ratio Interpretation
    X_binary = np.array([[25, 50000], [35, 80000], [45, 60000], [30, 100000]])
    y_binary = np.array([0, 1, 0, 1])  # Purchased premium product
 
    log_reg = LogisticRegression()
    log_reg.fit(X_binary, y_binary)
 
    print("\nLogistic Regression Coefficients (Log-Odds):")
    for name, coef in zip(['age', 'income'], log_reg.coef_[0]):
        print(f"  {name}: {coef:.4f}")
    print(f"    Odds Ratio: {np.exp(coef):.4f}")
    
# Interpretation for income:
# If odds ratio = 1.00005, then each $1 increase in income
# multiplies the odds of purchase by 1.00005
# For $10,000 increase: 1.00005 ^ 10000 ≈ 1.65(65 % increase in odds)

The Scale Problem: Why Standardization Matters

A critical insight that many practitioners miss: raw coefficients cannot be compared across features with different scales.

Consider a model predicting loan default with features:

Income (range: $20,000 - $500,000)
Credit Score (range: 300 - 850)
Debt-to-Income Ratio (range: 0 - 2)

If the coefficients are $\beta_{income} = -0.00001$, $\beta_{credit} = -0.01$, and $\beta_{dti} = 2.5$, which feature is most important? You cannot answer this question from raw coefficients because the scales differ by orders of magnitude.

The Solution: Standardized Coefficients

By standardizing features to have mean 0 and standard deviation 1, coefficients become comparable: $$x_j^{\text{std}} = \frac{x_j - \mu_j}{\sigma_j}$$

Standardized coefficients answer: "How many standard deviations does the outcome change for a one standard deviation change in the feature?"

Raw Coefficients

•Depend on original units of measurement
•Cannot compare importance across features
•Large coefficient may indicate small effect (if feature has tiny scale)
•Useful for prediction formula
•Interpretation requires knowing units

Standardized Coefficients

•Unit-free, scale-independent
•Directly comparable across features
•Magnitude indicates relative importance
•Useful for interpretation
•All features on same scale

standardized_coefficients.py
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import numpy as np
from sklearn.linear_model import LinearRegression
        from sklearn.preprocessing import StandardScaler
 
# Simulate loan default data
    np.random.seed(42)
    n = 1000
 
    income = np.random.uniform(20000, 500000, n)
    credit_score = np.random.uniform(300, 850, n)
    dti_ratio = np.random.uniform(0, 2, n)
 
# True relationship(with different scales)
    default_prob = -income * 0.00001 - credit_score * 0.008 + dti_ratio * 1.5
    y = (default_prob > np.random.normal(0, 1, n)).astype(int)
 
    X = np.column_stack([income, credit_score, dti_ratio])
    feature_names = ['income', 'credit_score', 'dti_ratio']
 
# Fit on raw data
    lr_raw = LinearRegression()
    lr_raw.fit(X, y)
 
    print("Raw Coefficients:")
    for name, coef in zip(feature_names, lr_raw.coef_):
        print(f"  {name}: {coef:.8f}")
 
# Fit on standardized data
    scaler = StandardScaler()
    X_std = scaler.fit_transform(X)
 
    lr_std = LinearRegression()
    lr_std.fit(X_std, y)
 
    print("\nStandardized Coefficients:")
    for name, coef in zip(feature_names, lr_std.coef_):
        print(f"  {name}: {coef:.4f}")
 
# Alternative: compute standardized coefficients from raw
    print("\nStandardized from Raw (verification):")
    for name, coef, scale in zip(feature_names, lr_raw.coef_, scaler.scale_):
        std_coef = coef * scale  # Convert to standardized
    print(f"  {name}: {std_coef:.4f}")

When to Use Which

Use raw coefficients when you need exact predictions or care about the effect in original units (e.g., 'each $10,000 income increase reduces default probability by X'). Use standardized coefficients when comparing the relative importance of features or communicating which features matter most. Always report both for complete transparency.

The Multicollinearity Challenge

One of the most dangerous pitfalls in coefficient interpretation is multicollinearity—when predictor variables are correlated with each other. Multicollinearity doesn't prevent you from making good predictions, but it severely undermines coefficient interpretation.

Why Multicollinearity Breaks Interpretation:

The interpretation 'effect of $x_j$ holding other variables constant' becomes problematic when $x_j$ is correlated with other features. If increasing $x_j$ naturally co-occurs with changes in $x_k$, what does 'holding $x_k$ constant' even mean?

Signs of Multicollinearity:

Coefficients change dramatically when adding/removing correlated features
Standard errors are inflated (wide confidence intervals)
Coefficients have unexpected signs
Model performance is good but individual coefficients seem wrong
Variance Inflation Factor (VIF) is high (> 5 or > 10)

Impact of Multicollinearity on Coefficient Interpretation
Correlation Level	VIF Approximation	Interpretation Reliability	Recommended Action
0 - 0.3	1.0 - 1.1	High - Coefficients reliable	Proceed with standard interpretation
0.3 - 0.5	1.1 - 1.3	Moderate - Minor inflation	Check confidence intervals
0.5 - 0.7	1.3 - 2.0	Low - Notable uncertainty	Consider removing/combining features
0.7 - 0.9	2.0 - 5.0	Very Low - High variance	Must address before interpretation
0.9 - 1.0	5.0	None - Coefficients meaningless	Remove features or use regularization

multicollinearity_diagnosis.py
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import numpy as np
    import pandas as pd
from sklearn.linear_model import LinearRegression
        from statsmodels.stats.outliers_influence import variance_inflation_factor
 
# Create data with multicollinearity
np.random.seed(42)
    n = 500
 
# sqft and num_rooms are highly correlated
    sqft = np.random.uniform(1000, 3000, n)
    num_rooms = sqft / 300 + np.random.normal(0, 0.5, n)  # Correlated!
    age = np.random.uniform(5, 50, n)
 
    y = 50 * sqft + 10000 * num_rooms - 1000 * age + np.random.normal(0, 20000, n)
 
    X = pd.DataFrame({ 'sqft': sqft, 'num_rooms': num_rooms, 'age': age })
 
# Check correlation matrix
    print("Correlation Matrix:")
    print(X.corr().round(3))
 
# Calculate VIF for each feature
print("\nVariance Inflation Factors:")
for i, col in enumerate(X.columns):
            vif = variance_inflation_factor(X.values, i)
    print(f"  {col}: {vif:.2f}")
 
# Fit model and observe unstable coefficients
    lr = LinearRegression()
    lr.fit(X, y)
 
    print("\nCoefficients with multicollinearity:")
    for name, coef in zip(X.columns, lr.coef_):
        print(f"  {name}: {coef:.2f}")
 
# Notice: sqft coefficient is much lower than true 50
# because num_rooms absorbs some of the effect
 
# Demonstration: Remove correlated feature
    X_no_rooms = X[['sqft', 'age']]
    lr2 = LinearRegression()
    lr2.fit(X_no_rooms, y)
 
    print("\nCoefficients without num_rooms:")
    for name, coef in zip(X_no_rooms.columns, lr2.coef_):
        print(f"  {name}: {coef:.2f}")
# Now sqft coefficient closer to true value

Critical Warning

In the presence of high multicollinearity, individual coefficient values become essentially arbitrary. Two equivalent models might have completely different coefficient values for correlated features. Never interpret individual coefficients when VIF > 5 without addressing multicollinearity first.

Regularization Effects on Coefficient Interpretation

Regularization (L1/Lasso, L2/Ridge, Elastic Net) fundamentally changes how we should interpret coefficients. Regularization introduces intentional bias into coefficient estimates to reduce variance and improve generalization—but this bias affects interpretation.

L2 Regularization (Ridge):

Ridge shrinks all coefficients toward zero but rarely to exactly zero: $$\hat{\beta} = \arg\min \left( \sum(y_i - X_i\beta)^2 + \lambda\sum\beta_j^2 \right)$$

Interpretation Impact: Ridge coefficients underestimate true effect sizes. The stronger the regularization (higher $\lambda$), the more coefficients are attenuated toward zero. Comparisons between features remain somewhat valid, but absolute magnitudes are biased.

L1 Regularization (Lasso):

Lasso creates sparse models by driving some coefficients exactly to zero: $$\hat{\beta} = \arg\min \left( \sum(y_i - X_i\beta)^2 + \lambda\sum|\beta_j| \right)$$

Interpretation Impact: Non-zero coefficients in Lasso are biased toward zero, often substantially. Zero coefficients indicate 'not selected' rather than 'no effect'. Feature selection depends on $\lambda$ choice.

Interpreting Regularized Coefficients

•Don't interpret magnitudes literally — Regularized coefficients are intentionally biased; their magnitudes don't represent true effects.
•Relative comparisons are approximate — Larger coefficients generally indicate more important features, but differences are compressed.
•Zero in Lasso ≠ no effect — A feature set to zero may have effects that are captured by correlated features or deemed unnecessary given regularization strength.
•Stability across λ values matters — Robust features remain non-zero across a range of regularization strengths.
•Consider debiasing techniques — For unbiased estimates, fit regularized model for selection, then unregularized on selected features.

regularization_interpretation.py
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import numpy as np
    import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression, Ridge, Lasso
 
# Create data with true known coefficients
    np.random.seed(42)
    n, p = 200, 10
 
# True coefficients: first 5 matter, last 5 are noise
    true_coefs = np.array([5, 3, 2, 1, 0.5, 0, 0, 0, 0, 0])
 
    X = np.random.randn(n, p)
    y = X @true_coefs + np.random.randn(n) * 2
 
# Fit OLS, Ridge, and Lasso
    ols = LinearRegression().fit(X, y)
    ridge = Ridge(alpha = 10).fit(X, y)
    lasso = Lasso(alpha = 0.5).fit(X, y)
 
# Compare coefficients
    print("True vs Estimated Coefficients:")
    print(f"{'Feature':<10} {'True':<10} {'OLS':<10} {'Ridge':<10} {'Lasso':<10}")
    print("-" * 50)
    for i in range(p):
        print(f"x{i:<9} {true_coefs[i]:<10.2f} {ols.coef_[i]:<10.2f} "
          f"{ridge.coef_[i]:<10.2f} {lasso.coef_[i]:<10.2f}")
 
# Key observations:
# 1. OLS coefficients are closest to true values
# 2. Ridge shrinks all coefficients(even true zeros)
# 3. Lasso zeroes out noise features but shrinks others
 
# Stability analysis: How coefficients change with regularization
alphas = np.logspace(-2, 2, 50)
    ridge_coefs = []
    lasso_coefs = []
 
    for alpha in alphas:
        ridge_coefs.append(Ridge(alpha = alpha).fit(X, y).coef_)
    lasso_coefs.append(Lasso(alpha = alpha, max_iter = 10000).fit(X, y).coef_)
 
    ridge_coefs = np.array(ridge_coefs)
    lasso_coefs = np.array(lasso_coefs)
 
# Features that remain stable across alpha values are more reliable

Post-Selection Inference

For valid statistical inference after Lasso selection, use specialized techniques like selective inference, data splitting, or stability selection. Standard confidence intervals are invalid when the same data is used for both selection and inference.

The Causation Problem: Coefficients Are Not Causal Effects

Perhaps the most common and dangerous misinterpretation of linear model coefficients is treating them as causal effects. A coefficient tells you how the prediction changes with a feature—not what happens if you intervene to change that feature.

The Fundamental Distinction:

Observational Association: When people with higher education have higher incomes, and your model has a positive education coefficient.
Causal Effect: What happens to income if you send a random person to college.

These are different questions with potentially different answers. The observational association might be driven by confounders (people who go to college might have greater motivation, better opportunities, smarter parents, etc.).

Simpson's Paradox in Linear Models:

Consider the Berkeley admissions example. A model might show negative coefficients for being female when predicting admission. But controlling for department (women applied to more competitive departments), the coefficient becomes positive. The 'effect' entirely depends on what you control for.

Coefficient Interpretation: Predictive vs Causal
Aspect	Predictive Interpretation	Causal Interpretation
Question Answered	How does prediction change?	What happens if we intervene?
Assumption Required	Model approximates E[Y\|X]	No confounding, correct model
Validity Context	Often valid for forecasting	Rarely valid without experiments
Actionability	For prediction only	For decision-making
Example	'Smokers have higher lung cancer risk'	'Quitting smoking reduces risk by X%'

The Intervention Fallacy

Never say 'increasing X by 1 unit WILL increase Y by β units' unless you have causal identification (randomized experiment, natural experiment, or rigorous causal design). Instead say 'the model prediction increases by β for each unit increase in X' or 'X is associated with β higher Y'.

When Can Coefficients Be Causal?

Coefficients approximate causal effects under specific conditions:

Randomized Controlled Trial: Feature was randomly assigned (rare in observational ML)
Conditional Ignorability: You've measured and controlled for all confounders
Natural Experiment: Some external factor created as-if random variation
Instrumental Variables: Valid instruments create exogenous variation
Regression Discontinuity: Sharp threshold creates local randomization

In typical ML applications, none of these hold. Treat coefficients as associations, not effects.

causal_warning_example.py
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import numpy as np
from sklearn.linear_model import LinearRegression
 
# Simulation: Confounding makes coefficient misleading
    np.random.seed(42)
    n = 1000
 
# True causal structure:
# Motivation -> Education AND Income(confounder)
# Education -> Income(small true effect)
 
    motivation = np.random.randn(n)  # Unobserved confounder
 
# Both education and income driven by motivation
    education_years = 12 + 2 * motivation + np.random.randn(n) * 1
    income = 30000 + 5000 * motivation + 500 * education_years + np.random.randn(n) * 5000
 
# True causal effect of education on income: $500 / year
# But if we don't control for motivation...
 
# Naive regression(confounded)
    lr_naive = LinearRegression()
    lr_naive.fit(education_years.reshape(-1, 1), income)
    print(f"Naive coefficient: ${lr_naive.coef_[0]:.0f} per year of education")
# Output: ~$1, 500 - much higher than true $500!
 
# If we could control for motivation(usually unobserved)
X_controlled = np.column_stack([education_years, motivation])
lr_controlled = LinearRegression()
lr_controlled.fit(X_controlled, income)
    print(f"Controlled coefficient: ${lr_controlled.coef_[0]:.0f} per year of education")
# Output: ~$500 - close to true effect
 
# Lesson: Without controlling for motivation,
# coefficient overstates causal effect 3x
# In practice, motivation is unobserved
# The naive model PREDICTS well but misleads for intervention

Categorical Variables and Interaction Terms

Interpreting coefficients becomes more nuanced with categorical variables and interaction terms. These require understanding reference categories and conditional effects.

Categorical Variable Encoding:

For a categorical feature with K categories, we typically create K-1 dummy variables. The omitted category becomes the reference category, and all coefficients are interpreted relative to it.

Example: Region {North, South, East, West} with North as reference:

$β_{South} = 5,000$ means South houses cost $5,000 more than North, on average
$β_{East} = -3,000$ means East houses cost $3,000 less than North
$β_{West} = 2,000$ means West costs $2,000 more than North

The choice of reference category affects coefficient signs but not predictions!

Dummy Variable Trap

Including all K dummy variables (without dropping one) causes perfect multicollinearity. Always omit one category or use regularization. Be explicit about which category is the reference when reporting results.

Interaction Terms:

Interaction terms allow the effect of one variable to depend on another:

$$y = β_0 + β_1 x_1 + β_2 x_2 + β_3 (x_1 \times x_2) + ε$$

Interpretation of Main Effects Changes:

With an interaction, $β_1$ is no longer 'the effect of $x_1$'. Instead:

$β_1$ = effect of $x_1$ when $x_2 = 0$
Actual effect of $x_1$ = $β_1 + β_3 x_2$ (depends on value of $x_2$)

Similarly, $β_2$ = effect of $x_2$ when $x_1 = 0$.

categorical_interactions.py
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import numpy as np
    import pandas as pd
from sklearn.linear_model import LinearRegression
        from sklearn.preprocessing import OneHotEncoder
 
# Example: Salary prediction with department and experience
    np.random.seed(42)
    n = 500
 
    departments = np.random.choice(['Engineering', 'Sales', 'HR'], n)
    experience = np.random.uniform(0, 20, n)
 
# True model: Engineering pays most, experience matters more in Engineering
    base_salary = 50000
    dept_effects = { 'Engineering': 30000, 'Sales': 15000, 'HR': 0 }
    exp_effects = { 'Engineering': 4000, 'Sales': 2000, 'HR': 1500 }
 
    salary = np.array([
        base_salary + dept_effects[d] + exp_effects[d] * e + np.random.randn() * 5000
    for d, e in zip(departments, experience)
])
 
# Create DataFrame
    df = pd.DataFrame({
        'department': departments,
        'experience': experience,
        'salary': salary
    })
 
# One - hot encode departments(drop_first: HR as reference)
    df_encoded = pd.get_dummies(df, columns = ['department'], drop_first = True)
 
# Model without interaction
    X_main = df_encoded[['experience', 'department_Engineering', 'department_Sales']]
    lr_main = LinearRegression().fit(X_main, df_encoded['salary'])
 
    print("Model WITHOUT interaction:")
    print(f"  Intercept (HR, 0 years): ${lr_main.intercept_: .0f
                                    }")
    for name, coef in zip(X_main.columns, lr_main.coef_):
    print(f"  {name}: ${coef:.0f}")
 
# Model WITH interaction
    df_encoded['exp_x_eng'] = df_encoded['experience'] * df_encoded['department_Engineering']
    df_encoded['exp_x_sales'] = df_encoded['experience'] * df_encoded['department_Sales']
 
    X_inter = df_encoded[['experience', 'department_Engineering', 'department_Sales',
        'exp_x_eng', 'exp_x_sales']]
    lr_inter = LinearRegression().fit(X_inter, df_encoded['salary'])
 
    print("\nModel WITH interaction:")
    print(f"  Intercept (HR, 0 years): ${lr_inter.intercept_:.0f}")
    for name, coef in zip(X_inter.columns, lr_inter.coef_):
        print(f"  {name}: ${coef:.0f}")
 
# Interpretation:
# 'experience' coefficient now means experience effect for HR(reference)
# 'exp_x_eng' is ADDITIONAL experience effect for Engineering
# Total experience effect in Engineering = experience + exp_x_eng

Confidence Intervals and Statistical Significance

A coefficient estimate is useless without understanding its uncertainty. Confidence intervals and p-values quantify this uncertainty, but they're frequently misinterpreted.

Confidence Interval Interpretation:

A 95% confidence interval [a, b] for $β_j$ does NOT mean:

'95% probability that $β_j$ is between a and b' (Bayesian credible interval meaning)

It DOES mean:

'If we repeated this experiment many times, 95% of the calculated intervals would contain the true $β_j$'

Statistical Significance ≠ Practical Importance:

With enough data, tiny effects become statistically significant. A coefficient of $0.0001 with p < 0.001 is highly significant but practically meaningless. Conversely, with small samples, large effects might not achieve significance.

Always report:

Point estimate (coefficient value)
Confidence interval (uncertainty range)
Practical significance (is the effect size meaningful?)

Statistical vs Practical Significance Matrix
Scenario	Statistically Significant?	Practically Important?	Interpretation
Large effect, small sample	No	Yes	Inconclusive - need more data
Small effect, large sample	Yes	No	Real but ignorable effect
Large effect, large sample	Yes	Yes	Confirmed important effect
Small effect, small sample	No	No	No evidence of effect

confidence_intervals.py
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import numpy as np
    import statsmodels.api as sm
 
# Example: Coefficient uncertainty
    np.random.seed(42)
    n_small, n_large = 50, 5000
 
# True coefficients
    true_beta = np.array([10, 0.5, 0])  # Strong, weak, zero effect
 
def analyze_sample(n_samples):
    X = np.random.randn(n_samples, 3)
    y = X @true_beta + np.random.randn(n_samples) * 5
    
    # Add constant for intercept
    X_with_const = sm.add_constant(X)
    
    model = sm.OLS(y, X_with_const).fit()
    
    print(f"\nSample size: n = {n_samples}")
    print(f"{'Feature':<8} {'Coef':>10} {'95% CI':>25} {'p-value':>12}")
    print("-" * 60)
 
    names = ['const', 'x1', 'x2', 'x3']
    true_vals = [0, 10, 0.5, 0]
 
    for i, name in enumerate(names):
        coef = model.params[i]
    ci_low, ci_high = model.conf_int().iloc[i]
    pval = model.pvalues[i]
    sig = '***' if pval < 0.001 else '**' if pval < 0.01 else '*' if pval < 0.05 else ''
    print(f"{name:<8} {coef:>10.3f} [{ci_low:>9.3f}, {ci_high:>9.3f}] {pval:>10.4f} {sig}")
 
# Small sample: uncertainty is large
    analyze_sample(50)
 
# Large sample: even tiny effects become significant
    analyze_sample(5000)
 
# Notice: 
# - x1(true = 10) is significant in both
# - x2(true = 0.5) may not be significant at n = 50, highly significant at n = 5000
# - x3(true = 0) stays insignificant regardless of sample size

Reporting Best Practice

Never report just 'p < 0.05'. Always include: (1) coefficient value, (2) confidence interval, (3) effect size in meaningful units, and (4) domain context for practical significance. A $1/year salary increase might be statistically significant but is practically worthless.

Summary: Best Practices for Coefficient Interpretation

We've covered the complete framework for interpreting linear model coefficients. Let's consolidate these principles into actionable best practices:

Core Interpretation Principles

•State the Ceteris Paribus Condition — Always emphasize 'holding other variables constant' and acknowledge when this is realistic or not.
•Use Standardized Coefficients for Comparison — Never compare raw coefficient magnitudes across different-scale features.
•Check for Multicollinearity — Calculate VIF before interpreting. Address high-VIF features before drawing conclusions.
•Understand Regularization Bias — Regularized coefficients are intentionally shrunk; don't interpret magnitudes literally.
•Never Claim Causation Without Evidence — Unless you have experimental or quasi-experimental design, coefficients are associations only.
•Report Uncertainty — Include confidence intervals and discuss both statistical and practical significance.
•Be Explicit About Reference Categories — For categorical variables, clearly state what category coefficients are relative to.
•Handle Interactions Carefully — Main effects change meaning in the presence of interactions.

Coefficient Interpretation Checklist

Before interpreting any coefficient, verify: (1) Features are appropriately scaled or standardized, (2) VIF is acceptable (< 5), (3) Model assumptions are reasonable, (4) Confidence intervals are available, (5) You're not overclaiming causation, (6) Regularization effects are understood, (7) Reference categories are clear.

What's Next:

Linear models offer the clearest form of interpretability, but their simplicity limits their predictive power. In the next page, we'll explore tree visualization—how to interpret decision trees and ensemble methods through visual and feature importance analysis. Tree-based methods offer a different paradigm for interpretability that complements linear coefficient analysis.

Page Complete

You now have a comprehensive framework for interpreting linear model coefficients. You understand scaling, multicollinearity, regularization effects, causal limitations, and proper statistical reporting. These skills apply whether you're building interpretable models for high-stakes decisions or simply understanding what a linear baseline reveals about your data.