Regression and Statistical Modeling

Introduction and Context

Regression is one of the central methods of data science because it connects observed data to questions about explanation, prediction, and decision-making. In business and engineering settings, regression can estimate how demand changes with price, predict energy consumption from weather, quantify the effect of marketing expenditure on sales, or assess factors associated with failure risk. In R, regression modeling is supported by a coherent formula interface, extensive diagnostic tools, and packages for generalized and regularized models.

A regression model is not merely a line fitted through data. It is a structured statement about a data-generating process. A useful analysis therefore requires decisions about the response variable, predictors, functional form, sampling design, uncertainty, confounding, validation, and the consequences of error. This chapter develops ordinary least squares regression, generalized linear models, interactions, diagnostics, regularization, and responsible interpretation.

Learning Objectives

By the end of this chapter, a learner should be able to:

1. Formulate and fit linear regression models in R.

2. Interpret coefficients, confidence intervals, predictions, and uncertainty.

3. Diagnose nonlinearity, nonconstant variance, dependence, influential observations, and non-normal errors.

4. Explain interactions, confounding, mediation, and the distinction between association and causation.

5. Select and interpret generalized linear models for non-Gaussian outcomes.

6. Explain ridge, lasso, and elastic-net regularization and use cross-validation appropriately.

7. Evaluate models for explanation and prediction using suitable measures.

8. Communicate model findings and governance limitations to technical and managerial audiences.

1. Regression as a Statistical Model

Let \(Y_i\) denote the response for observation \(i\), and let \(x_{i1},\ldots,x_{ip}\) be predictor values. A statistical model specifies a systematic component and a random component. In a linear regression model,

\[

Y_i=\beta_0+\beta_1x_{i1}+\cdots+\beta_px_{ip}+\varepsilon_i.

\]

Here, \(\beta_0\) is the intercept, \(\beta_j\) is the expected change in the response associated with a one-unit increase in predictor \(j\), holding other included predictors constant, and \(\varepsilon_i\) is the unexplained error. The conditional mean is

\[

E(Y_i\mid X_i)=\beta_0+\sum_{j=1}^{p}\beta_jx_{ij}.

\]

The word “linear” refers to linearity in the unknown coefficients, not necessarily to straight-line behavior in the original variables. For example, \(Y=\beta_0+\beta_1x+\beta_2x^2+\varepsilon\) is a linear model in its coefficients, although its fitted curve is quadratic.

1.1 Explanation, prediction, and causation

These goals overlap but are not identical. An explanatory model seeks interpretable relationships and uncertainty estimates. A predictive model seeks accurate outcomes for new cases, even if some predictors have little substantive meaning. A causal analysis asks what would happen under an intervention and generally requires design assumptions or a causal identification strategy. A statistically significant coefficient in an observational regression is not, by itself, evidence that changing that predictor will cause the response to change.

1.2 Population and sample quantities

Parameters such as \(\beta_j\) describe a target population or data-generating process. Estimates such as \(\hat\beta_j\) are computed from a sample and vary from sample to sample. A standard error quantifies sampling uncertainty under the model. A 95% confidence interval is commonly written as

\[

\hat\beta_j\pm t_{0.975,\,df}\,SE(\hat\beta_j),

\]

where \(t_{0.975,df}\) is a Student t critical value and \(df\) is the relevant residual degrees of freedom. Under repeated sampling and correct model assumptions, intervals constructed this way cover the true parameter approximately 95% of the time. The statement does not mean that there is a 95% probability that a fixed parameter lies inside one already-computed interval.

2. Ordinary Least Squares Regression

For \(n\) observations and \(p\) predictors, ordinary least squares (OLS) chooses coefficients that minimize the residual sum of squares:

\[

RSS=\sum_{i=1}^{n}(y_i-\hat y_i)^2.

\]

The residual is \(e_i=y_i-\hat y_i\), and \(\hat y_i\) is the fitted value. In matrix notation, with a column of ones included in \(X\),

\[

\hat{\boldsymbol\beta}=(X^TX)^{-1}X^T\mathbf y,

\]

provided the columns of \(X\) are linearly independent. This formula expresses the unique OLS solution under full column rank. In practice, numerical software generally uses QR decomposition or singular-value methods rather than explicitly computing a matrix inverse, because those methods are more stable.

The coefficient of determination is

\[

R^2=1-\frac{RSS}{TSS},\qquad TSS=\sum_i(y_i-\bar y)^2.

\]

With an intercept, \(R^2\) is the proportion of sample variation around the response mean reduced by the fitted model. It is not a universal measure of model quality: a high value can coexist with poor extrapolation, leakage, or confounding, and a low value may be acceptable when individual outcomes are inherently noisy. Adjusted \(R^2\) penalizes the addition of predictors, but it is not a replacement for validation.

2.1 OLS assumptions

The most important assumptions are conditional and should be assessed in relation to the study design:

• Correct mean structure: the conditional mean is adequately represented by the included terms.
• Independence: observations or errors are independent, unless dependence is explicitly modeled.
• Constant conditional variance: \(Var(Y\mid X)=\sigma^2\) for the classical standard errors.
• No perfect multicollinearity: predictors are not exact linear combinations of one another.
• Normal errors: mainly required for exact small-sample t and F tests, not for OLS point estimates themselves.

Unbiasedness of OLS coefficients depends especially on the conditional mean assumption \(E(\varepsilon\mid X)=0\). Normality does not repair omitted-variable bias, reverse causality, or a badly specified mean structure.

2.2 Interpretation of coefficients

In a model \(Y=\beta_0+\beta_1X+\beta_2Z+\varepsilon\), \(\beta_1\) is the expected difference in \(Y\) for a one-unit difference in \(X\) among cases with the same value of \(Z\), under the model. “Holding constant” is a conditional comparison, not necessarily a realistic intervention. The intercept is the expected response when all predictors equal zero; it may have no substantive meaning if zero is outside the observed range. Centering a predictor can make the intercept more interpretable without changing fitted values.

For a binary indicator, the coefficient compares the indicated category with the reference category, holding other predictors constant. For a categorical variable with several levels, R creates indicator contrasts according to the factor coding. Coefficients therefore depend on the reference category and coding scheme, although fitted group comparisons do not.

2.3 Transformations and scale

A logarithmic response model, \(\log(Y)=\beta_0+\beta_1X+\varepsilon\), often interprets \(\beta_1\) approximately as a proportional change in \(Y\) for a one-unit change in \(X\). A log-log model, \(\log(Y)=\beta_0+\beta_1\log(X)+\varepsilon\), gives \(\beta_1\) an elasticity interpretation: a 1% change in \(X\) is associated approximately with a \(\beta_1\)% change in \(Y\). Exact back-transformation of predictions requires care because \(E(Y\mid X)\neq\exp(E[\log Y\mid X])\) in general. A retransformation adjustment may be required, depending on the error distribution.

3. Regression in R

A standard workflow uses a reproducible data preparation, fitting, diagnosis, validation, and reporting sequence.

```r

data <- read.csv("sales.csv")

fit <- lm(sales ~ price + advertising + season, data = data)

summary(fit)

confint(fit)

predict(fit, newdata = new_cases, interval = "confidence")

predict(fit, newdata = new_cases, interval = "prediction")

```

The formula `sales ~ price + advertising + season` models sales as a function of the listed predictors. `confidence` intervals describe uncertainty in the mean response, whereas `prediction` intervals additionally include the variability of a new individual outcome and are therefore wider.

Interactions are specified with `*` or `:`:

```r

fit_int <- lm(sales ~ price * promotion + season, data = data)

```

`price * promotion` expands to `price + promotion + price:promotion`. The interaction coefficient indicates that the slope of price differs by promotion status. Main effects in an interaction model are conditional effects: the price coefficient is the price slope when promotion is at its reference level.

Polynomial terms can be included using `I(x^2)`:

```r

fit_curve <- lm(sales ~ temperature + I(temperature^2), data = data)

```

For prediction, new data must use compatible variable names, factor levels, units, and transformations. Never evaluate a model using information that would not be available at the time of deployment.

4. Diagnosing Model Adequacy

Diagnostics should be guided by scientific knowledge and design, not by automatic deletion of inconvenient observations.

4.1 Residual and fitted-value plots

A plot of residuals against fitted values can reveal curvature, unequal spread, or clusters. A funnel shape suggests heteroscedasticity. A systematic curve suggests that a linear mean structure is inadequate. Residuals against each predictor can expose missed nonlinearities. The normal Q-Q plot assesses whether residuals are approximately normal; modest departures are often less important than severe nonlinearity, dependence, or variance misspecification.

```r

par(mfrow = c(2, 2))

plot(fit)

```

4.2 Heteroscedasticity

When variance changes with predictors, OLS coefficients may remain useful under suitable mean assumptions, but conventional standard errors can be incorrect. Options include transforming the response, modeling variance, weighted least squares, or heteroscedasticity-consistent (robust) standard errors. Robust standard errors address inference about coefficients; they do not automatically correct omitted nonlinear terms or biased sampling.

4.3 Dependence and clustered data

Repeated measurements from the same person, machine, store, or region are not generally independent. Ignoring clustering can make standard errors too small and produce overconfident conclusions. Appropriate alternatives include mixed-effects models, generalized estimating equations, cluster-robust inference, or time-series models. The choice depends on the estimand, number of clusters, correlation structure, and sampling process.

4.4 Leverage, influence, and outliers

A response outlier has an unusual residual. A high-leverage case has unusual predictor values. An influential case changes estimates substantially when included or removed. Cook’s distance is one diagnostic measure, but no universal cutoff determines whether a case should be deleted. Investigate data quality, record provenance, measurement conditions, and whether the case represents an important part of the target population. Sensitivity analyses should report how conclusions change.

4.5 Multicollinearity

Highly correlated predictors can yield unstable individual coefficients and large standard errors while leaving predictions relatively stable. Variance inflation factors can flag problematic linear dependence, but thresholds are context-dependent and do not establish that a model is invalid. Remedies include combining conceptually related variables, removing redundant predictors, collecting better data, using regularization, or focusing on joint rather than individual effects. Do not remove a theoretically essential confounder solely because it is correlated with another variable.

5. Confounding, Interactions, and Model Specification

A confounder is a variable associated with both an exposure and an outcome that, if omitted, can distort their relationship. Confounding is a causal concept, not simply a high correlation among predictors. A directed acyclic graph (DAG) can help identify variables that should be adjusted for, variables that should not be conditioned on, and possible sources of bias.

Adjustment can also create bias. Conditioning on a collider—a variable influenced by two other variables—can induce an association that was not present in the population. Likewise, controlling for a mediator may remove part of a total causal effect. Regression software cannot decide these causal questions without substantive design knowledge.

An interaction means that the effect of one predictor depends on another. For a continuous predictor \(X\), binary variable \(G\), and interaction model

\[

E(Y\mid X,G)=\beta_0+\beta_1X+\beta_2G+\beta_3XG,

\]

when \(G=0\), the slope of \(X\) is \(\beta_1\); when \(G=1\), it is \(\beta_1+\beta_3\). The interaction coefficient is the difference between the two slopes. Main effects should generally remain in a model when an interaction is included, even if their individual p-values are not small.

Model selection based only on repeated p-value testing is unreliable. Subject-matter theory, prespecified estimands, predictive validation, information criteria, penalization, and sensitivity analysis provide more defensible strategies. A statistically significant result may be practically unimportant, especially with large samples; a nonsignificant result may reflect imprecision rather than evidence of no effect.

6. Generalized Linear Models

Linear regression is unsuitable when the response is binary, a count, or otherwise constrained. A generalized linear model (GLM) has three components:

1. A random component describing the response distribution, such as binomial or Poisson.

2. A linear predictor \(\eta=X\beta\).

3. A link function connecting the conditional mean \(\mu=E(Y\mid X)\) to the linear predictor: \(g(\mu)=\eta\).

6.1 Logistic regression

For a binary response, logistic regression models the log odds:

\[

\log\left(\frac{p}{1-p}\right)=\beta_0+\beta_1x_1+\cdots+\beta_px_p.

\]

The odds ratio for a one-unit increase in \(x_j\), holding other variables constant, is \(\exp(\beta_j)\). Odds are not probabilities: an odds ratio of 2 does not mean that probability doubles. Predicted probabilities should be reported for meaningful covariate profiles, especially when communicating to nontechnical decision-makers.

```r

logit_fit <- glm(default ~ income + balance + student,

data = credit, family = binomial(link = "logit"))

summary(logit_fit)

prob <- predict(logit_fit, newdata = new_cases, type = "response")

```

Classification requires a decision threshold and a cost structure. Accuracy alone may be misleading under class imbalance. Use sensitivity, specificity, precision, recall, calibration, and threshold-specific expected costs. AUC summarizes ranking ability across thresholds but does not establish calibration or business usefulness.

6.2 Poisson and negative binomial models

For count data, Poisson regression uses

\[

\log(\mu_i)=\beta_0+\beta_1x_{i1}+\cdots+\beta_px_{ip}.

\]

Then \(\exp(\beta_j)\) is an incidence-rate ratio for a one-unit increase in \(x_j\), conditional on other predictors. The basic Poisson model assumes conditional variance equals the conditional mean. Overdispersion—variance greater than the mean—is common. A negative binomial model or a quasi-Poisson approach may be more appropriate, but zero inflation, excess zeros, exposure time, and dependence must also be considered. An offset such as `offset(log(exposure))` models rates when exposure differs across observations.

7. Regularized Regression

When there are many predictors, correlated features, or a high risk of overfitting, regularization trades some bias for reduced variance. The elastic-net objective is

\[

\min_{\beta_0,\beta}\left\{\frac{1}{2n}\sum_{i=1}^{n}(y_i-\beta_0-x_i^T\beta)^2

+\lambda\left[\alpha\lVert\beta\rVert_1+\frac{1-\alpha}{2}\lVert\beta\rVert_2^2\right]\right\}.

\]

Here, \(\lambda\geq0\) controls penalty strength and \(0\leq\alpha\leq1\) controls the mixture. Lasso uses \(\alpha=1\) and can set coefficients exactly to zero. Ridge uses \(\alpha=0\) and generally shrinks coefficients without selecting a sparse subset. Predictors should normally be standardized so that the penalty treats scales comparably; the intercept is not penalized.

```r

library(glmnet)

X <- model.matrix(sales ~ . - 1, data = train)

y <- train$sales

cv_fit <- cv.glmnet(X, y, alpha = 1, nfolds = 10)

coef(cv_fit, s = "lambda.1se")

```

The tuning parameter must be selected within the training data, commonly by cross-validation. A separate test set should remain untouched until final evaluation. Feature selection performed before cross-validation can leak information and produce optimistic performance. Regularization improves prediction in many settings, but a selected variable is not automatically causally important, and correlated predictors may be selected interchangeably.

8. Model Evaluation and Uncertainty

For continuous outcomes, common measures include mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), and out-of-sample \(R^2\). RMSE gives greater weight to large errors; MAE is more robust to extreme errors. Use a test set or nested cross-validation for honest performance estimation. Random folds are inappropriate when time order, groups, or spatial structure create dependence; use time-based or grouped resampling instead.

For explanatory work, coefficient uncertainty, effect size, practical importance, and sensitivity analyses are central. For prediction, calibration and decision consequences matter in addition to discrimination. Prediction intervals should reflect both parameter uncertainty and future observation noise. Bootstrap procedures can estimate uncertainty when analytic assumptions are doubtful, but the resampling scheme must preserve relevant dependence and sampling structure.

9. Illustrative Case: Predicting Hospital Readmission Risk

Background. A hospital wants to identify patients at elevated risk of readmission within 30 days so that discharge planning can be targeted. This is an illustrative case, not a claim about a particular hospital.

Problem. The data contain age, prior admissions, length of stay, diagnosis group, medication count, insurance category, and the binary readmission outcome. The hospital wants useful risk estimates without treating the model as an automatic denial or exclusion mechanism.

Analysis. Analysts first define the prediction time: only information available before discharge may be used. They remove post-discharge variables that would create leakage. They examine missingness and determine whether missing values are informative. A logistic regression provides an interpretable baseline. A penalized logistic model is compared using grouped or time-based validation if patients have repeated admissions or the deployment population changes over time.

A coefficient for prior admissions is interpreted as a conditional association on the odds scale, not as proof that admissions cause readmission. Diagnosis and insurance variables require careful review because they may reflect access barriers, documentation practices, or structural inequity. Performance is assessed using discrimination, calibration plots, sensitivity at an operational threshold, and the consequences of false negatives and false positives.

Approach and solution. The hospital uses predicted probabilities to offer additional follow-up, rather than to restrict services. The threshold is chosen with clinicians and patient-safety leaders. Calibration is monitored after deployment because population prevalence and clinical practice can change. A model card records the training population, intended use, excluded uses, variables, validation results, known limitations, and monitoring plan.

Outcome. The model supports prioritization but does not replace clinical judgment. Periodic audits examine calibration across clinically relevant groups, differential error rates, and whether interventions actually improve outcomes. If performance deteriorates, the model is recalibrated or retrained under governance approval.

Learning points. The case illustrates that leakage prevention, temporal validation, calibration, fairness assessment, causal humility, and decision design are as important as the algorithm. A technically accurate model can still be harmful if its output is used outside its validated purpose.

10. Ethical, Governance, and Professional Considerations

Regression models can reproduce historical discrimination when the target, data, or labels reflect unequal treatment. Removing protected attributes does not necessarily remove bias because proxy variables may remain. Fairness is not a single mathematical property; relevant criteria depend on context, legal requirements, harms, and the decision process. Evaluate subgroup performance, data representativeness, accessibility, and the effects of intervention.

Professional practice requires documentation of data provenance, transformations, assumptions, model versions, validation design, and intended use. Privacy protection includes data minimization, access controls, secure storage, and careful handling of small cells or re-identification risk. In regulated domains, explainability, auditability, human oversight, and contestability may be mandatory or strongly advisable. Reproducible R scripts, locked package versions, tests for data pipelines, and review by domain experts strengthen reliability.

11. Best-Practice Workflow

A defensible regression project usually follows this sequence:

1. Define the decision, estimand, population, time horizon, and unit of analysis.

2. Inspect data quality, missingness, measurement units, sampling, and possible leakage.

3. Use domain knowledge and visual analysis before fitting complex models.

4. Establish a transparent baseline model.

5. Specify interactions, nonlinear terms, and confounder adjustment based on substantive reasoning.

6. Fit the model using an appropriate likelihood and link function.

7. Diagnose residuals, dependence, heteroscedasticity, collinearity, leverage, calibration, and influential cases.

8. Validate using resampling that matches the deployment setting.

9. Compare practical consequences, not only statistical scores.

10. Report effect sizes, uncertainty, assumptions, limitations, subgroup performance, and intended use.

11. Monitor drift, recalibrate when justified, and maintain human accountability.

Chapter Summary

Regression models represent conditional relationships between outcomes and predictors and can support explanation, prediction, and decision-making. OLS estimates coefficients by minimizing squared residuals, but valid inference depends on appropriate mean structure, independence, variance assumptions, and defensible study design. Coefficients must be interpreted in context, especially when variables are transformed, categorical, or involved in interactions. Diagnostics help identify nonlinearity, heteroscedasticity, dependence, multicollinearity, outliers, and influential observations.

Generalized linear models extend regression to binary and count outcomes through a response distribution and link function. Logistic coefficients are naturally interpreted through odds ratios, while Poisson coefficients can represent incidence-rate ratios. Regularization controls overfitting through ridge, lasso, or elastic-net penalties, with tuning performed inside the training process. Honest evaluation requires suitable resampling, leakage prevention, calibration, and measures aligned with the decision context. Finally, confounding, fairness, privacy, documentation, and monitoring demonstrate that responsible modeling is a scientific and governance activity, not merely a software operation.

Key Terms

Regression: A statistical method for relating an outcome to one or more predictors.

Response variable: The outcome being modeled or predicted.

Predictor: A variable used to explain or predict the response.

Residual: The observed outcome minus its fitted value.

Ordinary least squares: Estimation by minimizing the sum of squared residuals.

Conditional mean: The expected response given specified predictor values.

Confidence interval: An interval procedure expressing sampling uncertainty about a parameter.

Prediction interval: An interval expressing uncertainty for a future individual outcome.

Heteroscedasticity: Nonconstant conditional variance of the response or errors.

Multicollinearity: Strong linear dependence among predictors.

Leverage: The unusualness of an observation’s predictor values.

Influence: The extent to which an observation changes fitted results.

Confounding: Distortion of an exposure–outcome relationship due to a common cause or related structure.

Interaction: A situation in which the effect of one predictor depends on another predictor.

Generalized linear model: A model combining a response distribution, linear predictor, and link function.

Odds ratio: The multiplicative change in odds associated with a predictor change.

Regularization: Penalization used to control coefficient complexity and overfitting.

Cross-validation: Repeated training and evaluation on data partitions for model assessment or tuning.

Calibration: Agreement between predicted probabilities and observed outcome frequencies.

Data leakage: Use of information during training that would not be available at prediction time.

Review Questions

Short Answer Questions

1. Distinguish between an explanatory model and a predictive model.

2. What does a regression coefficient mean when other predictors are held constant?

3. Why can a funnel-shaped residual plot be a concern?

4. What is the difference between a confidence interval and a prediction interval?

5. Why should predictors commonly be standardized before ridge or lasso regression?

Long Answer Questions

1. Explain the principal assumptions of ordinary least squares and describe diagnostic methods for assessing them.

2. Derive the interpretation of main effects and the interaction coefficient in a model containing a continuous predictor, a binary group variable, and their interaction.

3. Compare linear regression, logistic regression, and Poisson regression with respect to outcome type, link function, and coefficient interpretation.

4. Discuss how confounding, mediation, and collider bias affect the interpretation of regression models.

5. Explain how a data scientist should design model validation and reporting for a high-stakes application.

Analytical / Application Questions

1. A logistic model reports an odds ratio of 1.25 for a one-unit increase in a predictor. Explain what this means and why it should not automatically be described as a 25% increase in probability.

2. A model has excellent random cross-validation performance but poor performance when tested on the following year’s data. Identify likely causes and propose a revised validation strategy.

3. A hospital model has similar AUC values across demographic groups but is poorly calibrated for one group. Explain why this matters and recommend governance and technical responses.

Multiple Choice Questions with Answers

Question 1

Which quantity is minimized by ordinary least squares?

A. The sum of absolute fitted values

B. The residual sum of squares

C. The product of predictor variances

D. The classification error rate

Correct Answer: B. The residual sum of squares

Question 2

In a linear model with an interaction between X and a binary G, what is the slope of X when G = 1?

A. β0

B. β1

C. β2

D. β1 + β3

Correct Answer: D. β1 + β3

Question 3

Which statement about a 95% confidence interval is correct?

A. There is a 95% probability that the fixed parameter is inside the calculated interval

B. It contains 95% of the sample observations

C. A procedure constructed this way covers the true parameter approximately 95% of the time in repeated sampling

D. It is always wider than a prediction interval

Correct Answer: C. A procedure constructed this way covers the true parameter approximately 95% of the time in repeated sampling

Question 4

In logistic regression, exp(βj) represents the:

A. Change in probability for one unit of Xj

B. Odds ratio for a one-unit increase in Xj, holding other predictors constant

C. Residual standard deviation

D. Coefficient of determination

Correct Answer: B. Odds ratio for a one-unit increase in Xj, holding other predictors constant

Question 5

Which regularization method can set some coefficients exactly to zero?

A. Ridge regression

B. Lasso regression

C. Ordinary least squares

D. Unregularized logistic regression

Correct Answer: B. Lasso regression

Question 6

What does a funnel shape in a residual-versus-fitted plot commonly suggest?

A. Perfect normality

B. Heteroscedasticity

C. No leverage points

D. A binary response

Correct Answer: B. Heteroscedasticity

Question 7

Why is a prediction interval usually wider than a confidence interval for the mean response?

A. It includes uncertainty from individual future-outcome variation

B. It excludes parameter uncertainty

C. It is computed without residuals

D. It applies only to categorical predictors

Correct Answer: A. It includes uncertainty from individual future-outcome variation

Question 8

For a Poisson regression coefficient βj, exp(βj) is commonly interpreted as a:

A. Risk difference

B. Incidence-rate ratio

C. Residual correlation

D. Probability threshold

Correct Answer: B. Incidence-rate ratio

Question 9

Which practice causes data leakage?

A. Using only training data to tune a penalty

B. Applying a deployment-time transformation consistently

C. Using post-outcome information as a predictor during model training

D. Evaluating on a held-out test set

Correct Answer: C. Using post-outcome information as a predictor during model training

Question 10

Which statement best describes confounding?

A. Any pair of highly correlated predictors

B. Distortion of an exposure–outcome relationship due to a related common-cause structure

C. Random measurement error in the response

D. A method for selecting the regularization parameter

Correct Answer: B. Distortion of an exposure–outcome relationship due to a related common-cause structure