HSCI 410 — Lesson 3

Model-Building Strategies

Exploratory Data Analysis For Epidemiology

Kiffer G. Card, PhD, Faculty of Health Sciences, Simon Fraser University

Learning objectives for this lesson:

  • Develop a full (maximal) model incorporating biological understanding of the system under study
  • Carry out procedures to reduce a large number of predictors to a manageable subset
  • Address issues related to the functional form of continuous predictors and missing values
  • Build regression-type models using both statistical and non-statistical criteria
  • Evaluate the reliability of a regression-type model
  • Present the results from an analysis in a meaningful way

This course was developed by Kiffer G. Card, PhD, as a companion to Dohoo, I. R., Martin, S. W., & Stryhn, H. (2012). Methods in Epidemiologic Research. VER Inc.

Section 1

Introduction & Steps in Model Building

⏱ Estimated time: 20 minutes

Why Model-Building Strategies Matter

When building a regression model, we need to decide on the goals of the analysis, incorporate both statistical considerations and subject matter knowledge, and balance the desire for parsimony (simplicity) with the desire for a model that “best fits” the data. The definition of “best fit” depends on the goal of the analysis, and the principles discussed in this chapter apply to all types of regression models.

Key Concept

Regression models are generally built to meet one of two broad objectives: (1) to build the best model for predicting future observations, or (2) to understand the causal relationship(s) between predictors and the outcome. The approach to model building differs depending on which goal you are pursuing.

Goals of the Analysis

If the goal is prediction, we want to keep any variables whose relationship with the dependent variable is questionable—because excluding them might lead to inaccurate predictions when future observations have extreme values for those variables. The details of specific predictors are of little consequence; we just want overall accuracy.

If the goal is understanding biological relationships, we want precise estimates of coefficients for the variables of interest. Careful attention must be paid to interaction and confounding effects. Factors likely to be confounders should be retained in the model regardless of statistical significance, while factors that are almost certainly not confounders should generally be excluded—especially if they are intervening variables, as their inclusion may bias results.

🎯
Prediction Goal
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🔬
Causal Understanding
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⚖
Parsimony vs Fit
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Steps in Building a Regression Model

The process of building a regression model follows a systematic set of steps. While statistical software handles the computation, the researcher must make many decisions along the way that require both subject matter expertise and statistical reasoning.

Step 1: Specify the Maximum Model

Identify the outcome variable and determine whether it needs transformation (e.g., natural log). Then identify the full set of predictors to consider. The maximum model includes all possible predictors of interest. While a large model prevents overlooking important predictors, adding too many increases the risks of collinearity and spurious associations. Key sub-steps include: drawing a causal diagram, potentially reducing predictors, considering missing values, evaluating effects of continuous predictors, and deciding on interaction terms.

Step 2: Specify the Selection Criteria

Decide how you will determine which variables to retain. Criteria can be non-statistical (e.g., is it a primary predictor of interest? Is it a known confounder?) or statistical (e.g., partial F-tests, likelihood-ratio tests, information criteria like AIC or BIC). Both types of criteria should be considered together.

Step 3: Specify the Selection Strategy

Choose how to apply the criteria. Options include: examining all possible subsets, forward selection (adding variables one at a time), backward elimination (starting with all variables and removing), or stepwise procedures (combining forward and backward). The strategy determines the order in which variables are evaluated.

Steps 4–6: Conduct, Evaluate, and Present

Step 4: Conduct the analyses using your chosen strategy and criteria. Step 5: Evaluate the reliability of the chosen model using diagnostics and sensitivity analyses. Step 6: Present the results in a meaningful way, ensuring they are interpretable to your audience and that the model-building process is transparent.

Building a Causal Model

Before beginning the model-building process, it is imperative to have a causal model in place, usually presented as a causal diagram. The diagram identifies potential causal relationships among the predictors and the outcome of interest.

📋 Example: Cigarette Smoking and Birth Weight

Suppose you want to study the effects of cigarette smoking on birth weight, and you also have data on the mother’s race, education level, total birth order, gestation length, number of babies born, and weight gain during pregnancy.

A causal diagram would show that gestation length and weight gain are intervening variables—they lie on the causal pathway between smoking and birth weight. If the objective is to quantify the total effect of smoking on birth weight, you would not include gestation length or weight gain in the model, because doing so would remove the effect of smoking that is mediated through them.

On the other hand, race and college education might be confounders and should be retained regardless of statistical significance. Building the causal diagram first helps ensure you do not accidentally adjust for intervening variables.

⚠ Important Distinction

Confounders should be retained in the model to avoid bias. Intervening variables should generally be excluded when estimating total effects, because including them removes the indirect effect that passes through them. A causal diagram drawn before model building helps you distinguish between the two.

✔ Check Your Understanding

1. When the goal of a regression model is to understand biological relationships, which of the following is true?

When the goal is causal understanding, confounders should be retained regardless of significance to avoid biased estimates. Intervening variables should generally be excluded when estimating total effects.

2. What is the first step in building a regression model?

The first step is to specify the maximum model—identify the outcome variable and the full set of predictors to consider. This includes drawing a causal diagram and determining whether transformations are needed.

3. In a study of cigarette smoking’s effect on birth weight, why should gestation length generally NOT be included in the model?

Gestation length is an intervening variable—smoking affects gestation length, which in turn affects birth weight. Including it would remove the indirect effect of smoking that is mediated through gestation length, underestimating the total effect.

✎ Reflection

Think about a research question in your own field. What would the causal diagram look like? How would you distinguish confounders from intervening variables?

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Section 2

Reducing Predictors & Missing Values

⏱ Estimated time: 25 minutes

Reducing the Number of Predictors

It is sometimes necessary to reduce the number of predictors in the model-building process. Before undertaking any reduction, it is essential to identify the primary variables of interest and any variables that might be confounders or interacting variables—these should always be retained for consideration.

Practical Tip

The most appropriate procedure for managing a large number of predictors is often to design a more focused study that collects high-quality data on fewer predictors. This greatly reduces the risk of identifying spurious associations.

Screening Predictors Based on Descriptive Statistics

Before starting any model building, become thoroughly familiar with your data using descriptive statistics (means, variances, percentiles for continuous variables; frequency tabulations for categorical variables). This helps identify variables of little value. Guidelines include:

  • Avoid variables with large numbers of missing observations
  • Select only variables with substantial variability (e.g., if almost all subjects are male, sex will not be a useful predictor)
  • If a categorical variable has many categories with small counts, consider combining categories or eliminating the variable

Correlation Analysis

Examining pairwise correlations among predictor variables identifies pairs that contain essentially the same information. Highly correlated predictors (typically r > 0.9) produce multicollinearity, leading to unstable coefficient estimates and incorrect standard errors.

If highly correlated pairs are found, select one based on criteria such as biological plausibility, ease of measurement, and fewer missing values. Note that pairwise screening will not detect multicollinearity arising from linear combinations of multiple predictors.

Creation of Indices & Cronbach’s Alpha

Related predictors can sometimes be combined into a single index. For example, the Hamilton Rating Scale for Depression combines 22 characteristics into an overall depression score.

Cronbach’s alpha evaluates the internal consistency of such a scale—how well each predictor correlates with the overall scale. Interpretation guidelines:

  • < 0.60: Unacceptable
  • 0.60–0.65: Undesirable
  • 0.66–0.70: Minimally acceptable
  • 0.71–0.80: Respectable
  • 0.81–0.90: Very good
  • > 0.90: Consider shortening the scale

One drawback of indices is that they preclude evaluating the effects of the individual factors that were combined.

Screening Variables Based on Unconditional Associations

A common approach is to select only predictors with unconditional associations significant at a liberal P-value (e.g., 0.15 or 0.20). Simple univariable regression models are used for this screening.

One drawback: an important predictor might be excluded if its effect is masked by another variable (i.e., confounding is present). Using a liberal P-value helps prevent this. Another approach is to build the model with significant predictors, then add back excluded predictors one at a time to check if any become significant after adjusting for other variables.

PCA, Factor Analysis & Correspondence Analysis

Principal Components Analysis (PCA) converts a set of k predictor variables into k orthogonal (uncorrelated) principal components, each containing a decreasing proportion of total variation. A small subset of components is then used as predictors, eliminating multicollinearity. Coefficients can be back-transformed to the original predictors, though interpretation is less direct.

Factor analysis is similar but assumes factors with inherent meaning can be created from the original variables. Unlike PCA, the composition of factors varies as the number selected changes. Predictors with high “factor loadings” are identified as important determinants.

Correspondence analysis is designed for categorical variables. It produces a visual summary (2D scatterplot) of complex relationships, showing which clusters of predictors are associated with which clusters of outcome values.

The Problem of Missing Values

Missing data are common in observational studies. Statistical programs use complete case analysis by default—only observations with no missing values for any variable are included. Even a relatively low overall percentage of missing values can result in a substantial reduction of the sample if missing data are spread across observations.

🎲
MCAR
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🔍
MAR
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⚠
MNAR
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Dealing with Missing Data: Imputation

The two main alternatives to complete case analysis are imputation and analysis methods where missing data are ignorable. Imputation involves replacing missing data points with values predicted from available data.

Single vs Multiple Imputation

Single imputation derives one estimate for each missing value. However, analysis based on single imputed data does not account for the uncertainty of the estimated values. Multiple imputation generates multiple imputed datasets and combines results, properly accounting for this uncertainty. Multiple imputation is generally preferred over single imputation.

Maximum Likelihood & Bayesian Estimation

Maximum likelihood (ML) and Bayesian estimation are procedures that make missing values ignorable under the MAR assumption. ML requires specification of the distribution of missing values for predictors, but this is unnecessary for outcome missing values. These methods are closely linked to multiple imputation conceptually.

✔ Check Your Understanding

1. What does Cronbach’s alpha measure?

Cronbach’s alpha evaluates the internal consistency of a scale—how well each individual predictor (item) correlates with the overall scale. It is used to assess the reliability of combining related predictors into a single index.

2. Under which missing data mechanism is complete case analysis most likely to produce biased results?

Under MCAR, complete case analysis produces unbiased estimates (though with reduced power). Under MAR and MNAR, complete cases may not be representative of the full sample, leading to biased estimates. MNAR is the most problematic because the missingness depends on the unobserved data itself.

3. Why is multiple imputation generally preferred over single imputation?

Multiple imputation generates multiple completed datasets and combines results, which properly reflects the uncertainty associated with the imputed values. Single imputation treats imputed values as if they were known, underestimating the true variability in the data.

✎ Reflection

Consider a dataset you have worked with (or imagine one). Which type of missing data mechanism (MCAR, MAR, MNAR) do you think was most likely present, and why? What approach would you take to handle it?

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Section 3

Effects of Continuous Predictors

⏱ Estimated time: 25 minutes

Evaluating Continuous Predictor–Outcome Relationships

It is important to evaluate the structure of the relationship between a continuous predictor and the outcome before starting model building. The underlying assumption of linearity can be evaluated through diagnostics after fitting the model, but it is useful to explore the nature of the relationship beforehand.

Key Approaches

Four main approaches to evaluating the effect of continuous predictors are: (1) scatterplots and smoothed line plots, (2) categorising the continuous variable, (3) exploring polynomial models, and (4) using splines.

Scatterplots & Smoothed Lines

Scatterplots are 2-way plots of the outcome (Y-axis) versus the continuous predictor (X-axis). They are most useful for continuous outcomes; scatterplots of dichotomous outcomes present as two lines of dots and are rarely informative by themselves.

Scatterplots can be greatly improved by adding a smoothed line through the centre of the data. All smoothed lines have a local-influence property: the position of the line at any value of x is influenced by nearby points but not by distant points.

Types of Smoothed Lines

There are several types of smoothed line functions:

  • Running mean smoother: Computes a simple average of y values in the neighbourhood
  • Running line smoother: Fits a simple linear regression through observations in the neighbourhood
  • Lowess smoother: Fits a weighted linear regression where points closer to xi receive larger weight (using tricube weighting)
  • Local polynomial smoother: Fits a weighted polynomial regression in the neighbourhood

The bandwidth controls the size of the neighbourhood. A bandwidth of 0.8 means 80% of the data is used for each point. Larger bandwidths produce smoother lines but may miss important features.

Caution with Extreme Values

All smoothed line functions can have problems at the extreme values of the predictor distribution. This is because the neighbourhood at the tails is not symmetrical and contains relatively few data points. It is important not to pay much attention to the extremes of the fitted line. Vertical dashed lines marking the 2.5th and 97.5th percentiles can help delineate where most of the data falls.

Categorising Continuous Predictors

The assumption of linearity can be avoided by converting the continuous predictor into categories. However, this is generally not advisable for three reasons:

  1. Categorisation involves the loss of information
  2. It is unlikely that biological processes have a step-function relationship (i.e., sudden changes at specific cutpoints)
  3. The choice of cutpoints is arbitrary and, if data-driven, may lead to biased results

That said, about 5 categories will usually suffice to control for confounding effects. A model with a categorised variable can be compared to one with a continuous (linear) variable using AIC or BIC.

Polynomial Models

Polynomials allow the regression line to follow a curve rather than a straight line. Power terms (e.g., x² or x³) are added to the model. Unlike smoothed lines, polynomial models have a global-influence property—the shape of the entire line is influenced by all the data.

Quadratic Model
Y = β0 + β1x + β2x2 + ε
⚠ Centring to Avoid Collinearity

The original variable (x) is often highly correlated with its squared term (x²), creating collinearity. The solution is to centre the variable by subtracting the mean before squaring. If a quadratic model is insufficient (i.e., the quadratic term is significant but the fit is still poor), a cubic term (x³) can be added.

Fractional Polynomials

Fractional polynomials (FPs) extend the idea of polynomial models by allowing power terms that are not restricted to positive integers. The most common set of powers to consider is: −3, −2, −1, −0.5, 0 (= ln), 0.5, 1, 2, 3. A 2-degree FP can fit a wide range of non-linear shapes and may be the most parsimonious way to model non-linearity.

📊 Example: Birth Weight vs Gestation Length

A quadratic model regressing birth weight on centred gestation length showed R² = 0.29. When fractional polynomials were explored, the best-fitting 2-degree FP used powers of 3 and 3×ln(gest), yielding R² = 0.30 and fitting significantly better than the linear, quadratic, or cubic models. The FP coefficients are not directly interpretable—the only way to make sense of such a model is to display the function graphically.

Splines

An alternative to polynomial models is to fit a piecewise linear function. Points where the slope changes are called knot points. In the absence of prior evidence, knots may be chosen at percentiles of the predictor (e.g., 25th, 50th, 75th). Cubic splines allow for smoother transitions across knots compared to linear splines, producing more biologically plausible curves.

ApproachInfluenceStrengthsLimitations
Smoothed linesLocalFlexible; reveals non-linearityCannot be used in model itself; issues at extremes
CategorisationN/AAvoids linearity assumptionLoses information; arbitrary cutpoints
PolynomialsGlobalSimple to implement; formal testsMay over-fit at extremes; collinearity
Fractional polynomialsGlobalVery flexible with few termsCoefficients not directly interpretable
SplinesLocalFlexible; smooth transitionsSudden shifts at knots (linear splines)

✔ Check Your Understanding

1. Why is categorising a continuous predictor generally not advisable?

Categorisation loses information about the continuous variable, assumes an unlikely step-function relationship between the predictor and outcome, and involves arbitrary cutpoint choices. However, about 5 categories can suffice to control for confounding.

2. What is a key difference between smoothed lines and polynomial models?

Smoothed lines have a local-influence property—the line at any point is influenced primarily by nearby data. Polynomial models have a global-influence property—the entire curve is influenced by all data points. This means polynomials can be heavily influenced by extreme values.

3. Why should you centre a continuous variable before adding its squared term to a regression model?

The original variable (x) is often highly correlated with its squared term (x²), leading to collinearity. Centring (subtracting the mean) before squaring reduces this correlation and produces more stable coefficient estimates.

✎ Reflection

Think about a continuous predictor in your field. Would you expect the relationship with the outcome to be linear? If not, which approach (categorisation, polynomials, fractional polynomials, or splines) would you choose and why?

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Section 4

Interactions & Building the Model

⏱ Estimated time: 25 minutes

Identifying Interaction Terms

It is important to consider including interaction terms when specifying the maximum model. There are five general strategies for creating and evaluating 2-way interactions:

Strategy 1: Evaluate All Possible 2-Way Interactions

This is feasible only when the total number of predictors is small (e.g., ≤ 8). You create and test every possible pair of interaction terms.

Strategy 2: Interactions Among Significant Main Effects

After building the final main-effects model, create interactions among all predictors that are statistically significant. This reduces the number of interactions to evaluate but may miss interactions with non-significant main effects.

Strategy 3: Interactions Among Unconditionally Associated Predictors

Create interactions among all predictors that have a significant unconditional association with the outcome. This casts a wider net than Strategy 2.

Strategy 4: Theory-Driven Interactions

Only create interactions among pairs of variables you suspect (based on evidence from the literature or biological reasoning) might interact. This usually focuses on interactions involving the primary predictor(s) of interest and important confounders.

Strategy 5: Exposure-Only Interactions

Only create interactions that involve the exposure variable (primary predictor of interest). This is the most conservative approach but may miss important interactions among covariates.

⚠ Important Rules for Interactions

If an interaction term is included in the model, the main effects that make it up must also be included. Evaluating many interactions increases the risk of identifying spurious associations, so a Bonferroni adjustment or similar correction may be warranted. Three-way interactions are usually very difficult to interpret and should be included only if there is strong a priori reason.

Building the Model: Selection Criteria

Once the maximum model has been specified, you need to decide how to determine which predictors to retain. Both non-statistical and statistical criteria should be considered.

Non-Statistical Considerations

Variables should be retained in the model if they:

  • Are a primary predictor of interest
  • Are thought a priori to be confounders for the primary predictor
  • Show evidence of being a confounder (their removal causes a substantial change in the coefficient of interest)
  • Are a component of an interaction term included in the model

Statistical Criteria for Nested Models

Models where one model’s predictors are a subset of another’s are called nested models. Tests for nested models include:

  • Partial F-test (for linear regression)
  • Wald test (most commonly used; can be unreliable if P-values are near 0.05 or SEs appear suspect)
  • Likelihood-ratio test (LRT) (has the best statistical properties but requires fitting both models)

For categorical variables with multiple indicator terms, evaluate the overall significance of all indicators together, not individual terms.

Information Criteria (AIC & BIC)

For non-nested models, information criteria are used. The general formula is:

Information Criterion (Eq 15.1)
IC = −2 lnL + a × s

Where s is the number of parameters, lnL is the log-likelihood, and a is a penalty constant. For AIC, a = 2. For BIC, a = ln(n). Smaller values indicate a better model. BIC tends to favour more parsimonious models.

Guidelines for interpreting BIC differences between models:

  • 0–<2: Weak evidence
  • 2–<6: Positive evidence
  • 6–<10: Strong evidence
  • ≥10: Very strong evidence

Adjusted R² & Mallow’s Cp

Adjusted R² maximises the variance explained while penalising unnecessary complexity. The model that maximises adjusted R² is preferred.

Mallow’s Cp (Eq 15.2)
Cp = Σ (Y − Ŷ)² / σ² − n + 2k

Where k is the number of predictors in the candidate model, σ² is the MSE from the full model, and n is the sample size. Mallow’s Cp is a special case of the AIC. The model with the lowest Cp is generally considered the best.

Specifying the Selection Strategy

Once criteria are established, there are several strategies for selecting which variables to include in the final model.

📚
All Possible Subsets
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➡
Forward Selection
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⬅
Backward Elimination
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Best Practice Summary

Backward elimination is generally preferred over forward selection because each predictor is evaluated in the context of all others. However, the most important point is to combine statistical procedures with subject matter knowledge: retain known confounders and primary predictors regardless of statistical criteria, and always build a causal model first.

✔ Check Your Understanding

1. If an interaction term between variables A and B is included in a regression model, which of the following must also be true?

If an interaction term (A × B) is included in the model, the main effects for both A and B must also be included. This is required for the interaction term to be properly interpretable and is a standard rule in regression modelling.

2. What is the key difference between AIC and BIC?

Both AIC and BIC use the formula IC = −2 lnL + a × s, but the penalty constant a differs. AIC uses a = 2 while BIC uses a = ln(n), which is larger for any n > 7. BIC therefore imposes a heavier penalty for adding parameters, tending to favour simpler models.

3. Why is backward elimination generally preferred over forward selection?

Backward elimination starts with all predictors in the model, so each variable is evaluated in the presence of all others. This is better at identifying important predictors whose individual effect may be masked or suppressed by confounding. Forward selection examines each predictor in isolation first, potentially missing such effects.

✎ Reflection

Reflect on the tension between statistical model selection (AIC, BIC, stepwise methods) and subject matter knowledge. Why might a model selected purely by statistical criteria fail to answer your research question?

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Final Assessment

Lesson 3 — Final Assessment

15 questions • 100% required to pass

This assessment covers all sections of Lesson 3. You must answer all 15 questions correctly to complete the lesson. Read each question carefully and review the feedback for any incorrect answers before retrying.

✎ Final Reflection

Now that you have completed all four sections, summarise the key steps you would follow when building a regression model for a real-world epidemiological study. What role does subject matter knowledge play at each step?

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✔ Final Assessment

1. What are the two broad objectives of building a regression model?

Regression models are generally built either to predict future observations or to understand the (potentially causal) relationships between predictors and the outcome.

2. Why should parsimony guide model building?

Simple models are more robust, less influenced by idiosyncrasies of the specific dataset, and will generally perform better if applied to new data. However, parsimony should not override biological reasoning for including variables.

3. What is the purpose of drawing a causal diagram before model building?

A causal diagram maps potential causal relationships among predictors and the outcome. It helps distinguish confounders (which should be retained) from intervening variables (which should generally be excluded when estimating total effects).

4. Which technique is used to evaluate whether related predictors can be combined into a single index?

Cronbach’s alpha evaluates the internal consistency of a scale formed from related predictors. It measures how well each item correlates with the overall scale, helping determine whether combining the variables into an index is justified.

5. What is the main difference between PCA and factor analysis?

A key difference is that in PCA, the composition of each principal component does not change regardless of how many components are retained. In factor analysis, the composition of factors varies as the number of factors selected changes.

6. Under the MCAR assumption, complete case analysis:

Under MCAR, the complete cases are a random subset of the full data, so estimates are unbiased. However, because fewer observations are used, statistical power is reduced. Multiple imputation can improve efficiency even under MCAR.

7. What does MNAR mean in the context of missing data?

MNAR (Missing Not at Random) means the probability of a value being missing depends on the unobserved value itself. For example, sicker patients may be less likely to attend follow-up, so the missing health data would be systematically different from the observed data.

8. What is the local-influence property of smoothed lines?

The local-influence property means that the position of the smoothed line at any value xi is determined primarily by data points close to xi, not by points far away. This allows the line to capture local features of the data.

9. Why are fractional polynomials useful?

Fractional polynomials use power terms that can take non-integer values (e.g., −2, −0.5, 0.5), allowing them to fit diverse non-linear shapes. A 2-degree FP may be the most parsimonious way to model complex non-linear relationships.

10. What are knot points in the context of splines?

In spline models, knot points are the values of the predictor where the slope of the fitted line is allowed to change. Between knots, the relationship is assumed to be linear (for linear splines) or follow a polynomial curve (for cubic splines).

11. Which of the following is NOT a non-statistical reason to retain a variable in the model?

Having P < 0.05 in univariable analysis is a statistical criterion, not a non-statistical one. Non-statistical reasons include being a primary predictor, a suspected confounder, or part of an interaction term already in the model.

12. In the formula IC = −2 lnL + a × s, what does ‘a’ equal for BIC?

For BIC (Bayesian Information Criterion), the penalty constant a = ln(n), where n is the sample size. For AIC, a = 2. Because ln(n) > 2 for any sample size greater than 7, BIC imposes a heavier penalty and favours more parsimonious models.

13. Why is backward elimination generally preferred over forward selection?

Backward elimination starts with all predictors, so each one is evaluated while controlling for all others. This is better at identifying important predictors whose individual effect might be suppressed or masked by confounding when examined in isolation (as occurs in forward selection).

14. A BIC difference of 8 between two non-nested models suggests:

According to Raftery’s guidelines: 0–<2 = weak, 2–<6 = positive, 6–<10 = strong, ≥10 = very strong. A BIC difference of 8 falls in the “strong” evidence range (6–<10).

15. When should three-way interaction terms be included in a model?

Three-way interactions are very difficult to interpret and should only be included if there is strong prior reasoning to suspect the effect exists, or if the component variables have significant two-way interactions. They also require all component main effects and two-way interactions to be included.

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