Foundations of Regression Modelling in Clinical Research

Mayta
Jun 7
4 min read

Introduction

In clinical research, understanding how health outcomes are associated with predictors such as age, treatment, or biomarker levels is essential. Regression modelling provides a systematic approach to quantify and interpret these relationships. Unlike simple correlation or group comparisons, regression allows for adjustment, prediction, and hypothesis testing within a unified framework. This article introduces the fundamental concepts underpinning regression analysis, explaining how regression models are constructed, assessed, and interpreted in the context of clinical data.

Understanding Correlation Versus Association

Before delving into regression, it is important to distinguish correlation from association. Correlation quantifies the linear relationship between two continuous variables using a correlation coefficient (commonly Pearson’s r), which ranges from -1 to +1. A value of zero implies no linear correlation, but does not necessarily mean no relationship.

Importantly, correlation does not imply causation or control for other variables. For instance, the systolic blood pressure and age of individuals may be positively correlated, but this observation alone does not account for medications, comorbidities, or lifestyle factors that also influence blood pressure.

Concept of Regression

Regression modelling extends correlation by capturing both the strength and nature of the relationship between an outcome (dependent variable) and one or more predictors (independent variables). It allows us to:

Estimate the direction and magnitude of associations.
Predict outcomes for individuals given known predictor values.
Adjust for confounding variables to isolate effects of interest.

At its core, a regression model fits a mathematical line (or surface in multivariable contexts) through observed data, aiming to minimize the overall prediction error.

The Simple Linear Regression Equation

The most basic regression model is a simple linear regression:

Y = a + bX

Where:

Y is the outcome variable (e.g., systolic blood pressure),
X is the predictor (e.g., age),
a is the intercept (expected Y when X=0),
b is the regression coefficient (change in Y per unit change in X).

This equation forms a straight line that best fits the observed data, assuming a linear relationship between the variables.

Defining the “Best Fit” Line

Determining the “best” line involves minimizing the discrepancy between the actual data points and the predicted values on the line. This is achieved using the Ordinary Least Squares (OLS) method, which minimizes the Residual Sum of Squares (RSS):

RSS = Σ(Ŷᵢ − Yᵢ)²

Where Ŷᵢ is the predicted outcome for observation i and Yᵢ is the observed outcome.

A smaller RSS indicates that the model predictions closely align with observed outcomes. However, RSS is sensitive to the number of data points, so additional metrics are often used.

Quantifying Prediction Error

Two widely used measures of prediction error are:

Mean Squared Error (MSE): RSS divided by the number of observations. It estimates average squared error across all predictions.
Root Mean Squared Error (RMSE): The square root of MSE. This re-expresses the error in the same units as the outcome, improving interpretability.

A lower RMSE implies better model fit, but an excessively low value may indicate overfitting, especially in small samples.

Evaluating Model Fit: Explained Variance

While RMSE quantifies absolute error, R-squared (R²) expresses the proportion of total variance in the outcome that is explained by the model:

R² = 1 − (RSS / TSS)

Where:

TSS (Total Sum of Squares) represents total variance around the mean,
RSS is the residual (unexplained) variance.

An R² of 1.0 indicates a perfect fit; 0.0 means the model explains none of the variability beyond simply predicting the mean.

Key Assumptions of Linear Regression

For valid inference, regression models rely on several assumptions:

Linearity: The relationship between continuous predictors and the outcome must be linear. Non-linear relationships require transformations or flexible models (e.g., splines, polynomials).
Normality of Residuals: Residuals (prediction errors) should be normally distributed, particularly for Gaussian (linear) regression. Diagnostic tools include histograms, Q-Q plots, and statistical tests (e.g., Shapiro-Wilk).
Homoskedasticity: Variance of residuals should be constant across all levels of the predictor.
Independence: Observations (and residuals) must be independent. This is especially critical in longitudinal or clustered data.
Outlier Sensitivity: Extreme values can disproportionately influence regression coefficients. Influence diagnostics (e.g., leverage plots, Cook's distance) help identify problematic observations.

Types of Regression Models

Regression models are categorized by the nature of the outcome variable:

Gaussian Regression: For continuous, normally distributed outcomes. Coefficients estimate mean differences.
Logistic Regression: For binary outcomes (e.g., yes/no events). Coefficients represent log-odds; exponentiated values yield odds ratios.
Additional types (not discussed in depth here) include Poisson, ordinal, and Cox regression, suited for counts, ordered categories, and time-to-event data, respectively.

Strategies for Model Building

The purpose of analysis determines the modelling strategy:

1. Explanatory Modelling

Seeks to test causal hypotheses.
Requires careful confounding control, guided by theory or directed acyclic graphs (DAGs).
All important covariates should be included, regardless of statistical significance.

2. Exploratory Modelling

Used when identifying potential associations.
Less emphasis on confounding; focus is on uncovering relationships.
Multiple predictors are examined without strict theoretical constraints.

3. Predictive Modelling

Aims to forecast outcomes as accurately as possible.
Prioritizes parsimony and performance.
Often uses automated selection methods (e.g., stepwise regression, regularization).

Each strategy has distinct goals and implications for interpretation, variable selection, and validation.

Conclusion

Regression modelling is a foundational technique in clinical research for evaluating, quantifying, and predicting the relationship between variables. From estimating blood pressure trends across age groups to identifying risk factors for disease, regression offers a versatile toolkit. However, responsible use requires a firm grasp of assumptions, appropriate model selection, and clear alignment with research objectives—be it explanation, exploration, or prediction. As clinical data grow in complexity, so too must our analytical approaches, making regression an indispensable skill for evidence-based inquiry.