Introduction to Regression Analysis

Regression analysis is presented as a tool that boosts scientific productivity by encouraging researchers to “step backwards” and view the bigger picture rather than focusing on individual data points. This backward step mirrors the literal meaning of the word “regression” and helps uncover overall patterns that would be missed when counting each element separately.

Forward or Backward?

A series of pop‑quiz examples—counting flower petals, tallying fruit in a basket, viewing the Earth from space, and observing a forest—illustrate the choice between moving forward (counting each item) and moving backward (understanding the larger phenomenon). The decision depends on the research objective: detailed enumeration versus holistic insight.

The Elephant Analogy

When observers stand too close to an elephant, they see only a fragment and draw incomplete conclusions. Stepping back provides a full view of the animal, just as stepping back in regression reveals the overall relationship among variables.

Core Components of Regression

Regression models consist of two essential parts: the outcome (dependent variable, left‑hand side) and the predictor (independent variable, right‑hand side). These components are the building blocks for any regression analysis.

Exploratory vs. Confirmatory Analysis

Two research approaches are distinguished.
Exploratory analysis is likened to a “fishing expedition,” where researchers probe data without a pre‑specified hypothesis.
Confirmatory analysis resembles aiming a telescope at a known star to test a specific hypothesis about its brightness.

A second pop‑quiz classifies research questions—such as factors influencing customer satisfaction or drug dosage effects—into exploratory or confirmatory categories based on their specificity.

Models as Simplified Representations

Regression produces models that simplify reality. Like a picture of an apple that cannot be eaten but still conveys essential features, a regression model helps explain and predict phenomena when direct data are unavailable. The classic maxim “All models are wrong, but some are useful” underscores this point.

Foundational Statistical Concepts

Key concepts supporting regression include hypothesis testing, P‑values, Type I (false positive) and Type II (false negative) errors, bias, validity, statistical power, and sample size. Understanding these ideas is crucial for correctly interpreting regression results.

Truth, Chance, and Bias

Observed outcomes can be attributed to three forces:
Truth (validity) – the actual effect researchers aim to uncover.
Chance – random error that introduces variability.
Bias* – systematic error that consistently skews results.

Common biases include confounding, selection, and measurement bias. Regression analysis seeks to isolate truth by controlling for chance (through P‑values) and bias (through proper design and adjustment).

Distinguishing Univariate, Bivariate, Multivariable, and Multivariate Analyses

• Univariate – analysis of a single variable.
• Bivariate – analysis of two variables and their relationship.
• Multivariable – one dependent variable with two or more independent predictors.
• Multivariate – two or more dependent variables.

The brief notes frequent misuse of “multivariate” when “multivariable” is intended.

Adjusting for Variables

An analogy of passengers on an airplane illustrates adjustment: to study one passenger’s movement, the others are strapped in (held constant). In regression, this translates to holding other predictors constant—often using reference groups—to isolate the effect of the variable of interest.

Sample Size, Power, and Precision

Adequate sample size is essential for statistical power—the ability to detect true differences. Small samples increase the risk of Type II errors. Sample‑size calculations differ for exploratory versus confirmatory studies.

The K‑Quest platform is highlighted as a tool for calculating sample sizes, requiring inputs such as outcome prevalence, effect size, confidence level, desired power, control‑to‑case ratio, and anticipated response rate. Example calculations include:
Confirmatory study: 393 participants per group (total 786).
Case‑control study: 56 cases and 221 controls for a 1:4 ratio.

Cohort studies depend on outcome prevalence, competing risks, and follow‑up duration. Structural Equation Modeling (SEM) follows standard sample‑size methods but places extra emphasis on measurement error and bias.

Types of Bias in Research

Bias is explored in depth:
Information bias – misclassification (differential or non‑differential).
Confounding bias – involving colliders and causal pathways.

Measurement bias arises from how variables are measured, including questionnaire design and participant‑investigator interactions. Social desirability bias reflects participants tailoring responses to please the researcher.

Research Designs and Advanced Techniques

Various designs are discussed: surveys, clinical trials, time‑series data, and joint‑point regression for detecting trend changes. Repeated surveys collect independent samples over time, while longitudinal surveys follow the same individuals. Joint‑point regression is suited for population‑level time‑varying data.

Key Distinctions

• Precision vs. Power – precision concerns the width of confidence intervals; power concerns the ability to detect true effects.
• Precision vs. Validity – a precise estimate can still be invalid if it lacks external generalizability.
• Exploratory vs. Confirmatory – exploratory work generates hypotheses and emphasizes precise estimates; confirmatory work tests predefined hypotheses and emphasizes power.
• Repeated vs. Longitudinal Surveys – repeated surveys use new samples each wave; longitudinal surveys track the same participants.