Concept Architecture
Concept
Theoretically, Regression Analysis is a statistical methodology used to quantify and model the relationship between one or more explanatory variables and an outcome variable. It provides a mathematical framework for estimating associations, testing hypotheses, explaining variation, and predicting outcomes. Regression analysis forms one of the foundations of statistical modelling and is extensively applied throughout health economics to evaluate treatment effects, identify determinants of costs and outcomes, estimate resource utilisation, and support economic decision-making.
Mathematically, regression analysis represents the expected value of an outcome variable as a function of one or more predictors. Depending on the nature of the outcome and underlying assumptions, the framework encompasses linear, logistic, Poisson, Cox, and other regression models. Parameters are commonly estimated using ordinary least squares or maximum likelihood estimation, with statistical inference based on estimated coefficients, standard errors, confidence intervals, and hypothesis tests.
In practice, regression analysis is implemented using statistical software to estimate adjusted relationships while controlling for confounding variables. In health economics it is routinely used to model healthcare costs, quality-adjusted life years, resource utilisation, treatment effectiveness, survival outcomes, and disease progression, providing adjusted effect estimates for economic evaluation, observational studies, and clinical trials.
Purpose
Used to estimate relationships between variables, adjust for confounding, explain variation in outcomes, generate predictions, quantify uncertainty, and support evidence generation and decision-making in health economics.
Mathematical Formulae
Primary Formula
General multiple regression model:
Y = ?? + ??X? + ??X? + ? + ??X? + �
where:
- Y = outcome variable
- ?? = intercept
- ?? ? ?? = regression coefficients
- X? ? X? = explanatory variables
- � = random error term
Supporting Formulae
Ordinary Least Squares estimator:
?? = (X?X)??X?Y
Residual:
e? = y? ? ??
Coefficient of Determination:
R� = 1 ? RSS � TSS
Residual Sum of Squares:
RSS = ?(y? ? ??)�
Mean Squared Error:
MSE = RSS � (n ? p ? 1)
Related Mathematical Methods
- Linear Regression
- Logistic Regression
- Poisson Regression
- Cox Regression
- Generalised Linear Models
- Ordinary Least Squares
- Maximum Likelihood Estimation
- Weighted Least Squares
- Mixed Effects Model
- Residual Analysis
Example
A health economist models annual healthcare costs using age, disease severity, and treatment status.
The fitted model is:
Cost = 1,850 + 145(Age) + 980(Severity) ? 1,200(Treatment)
For a 65-year-old patient with a severity score of 3 receiving treatment:
Cost = 1,850 + (145 ? 65) + (980 ? 3) ? (1,200 ? 1)
= �13,015
The model predicts annual healthcare costs of �13,015 after adjusting for all explanatory variables.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(YRange,XRange,TRUE,TRUE) | Estimate regression coefficients for healthcare cost or outcome models. |
| TREND | =TREND(YRange,XRange,NewX) | Predict future healthcare outcomes from fitted regression models. |
| FORECAST.LINEAR | =FORECAST.LINEAR(A2,YRange,XRange) | Predict continuous health economic outcomes. |
| RSQ | =RSQ(YRange,PredictedRange) | Calculate the coefficient of determination (R�). |
| SUMXMY2 | =SUMXMY2(ObservedRange,PredictedRange) | Calculate the residual sum of squares for model evaluation. |
VBA (Optional)
Automate regression model estimation, prediction generation, coefficient reporting, and diagnostic summaries for health economic analyses.
Sources
- Draper NR, Smith H. Applied Regression Analysis.
- Kutner MH, Nachtsheim CJ, Neter J, Li W. Applied Linear Statistical Models.
- Montgomery DC, Peck EA, Vining GG. Introduction to Linear Regression Analysis.
- Weisberg S. Applied Linear Regression.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
- NICE Health Technology Evaluation Manual.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is regression analysis?
Statistical techniques modelling the relationship between one or more predictor variables and an outcome variable, from simple linear to survival regression.
Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.
What relationship does regression analysis quantify?
Regression analysis quantifies how an outcome relates to one or more predictor variables, expressing the connection in a model whose coefficients say how the outcome is expected to change as each predictor changes. It serves both to describe associations, adjusting for other factors, and to predict outcomes from known characteristics. The family runs from simple linear regression through logistic regression for binary outcomes to survival regression for time-to-event data. Modelling how predictors relate to an outcome is its purpose. Kirkwood and Sterne (2003) describe these techniques.
Source: Kirkwood & Sterne 2003
What are the uses of regression analysis?
The uses of regression analysis include prediction, estimating the outcome from the predictors; description and inference, quantifying and testing the relationships between the outcome and predictors; and adjustment, controlling for confounding variables when estimating an effect. So regression analysis is used to predict, to describe and test associations, and to adjust for other variables, which makes it versatile across research, from building prediction models to estimating adjusted exposure effects, and this range of uses, together with the variety of regression methods for different outcome types, is why regression analysis is central to quantitative research.
Source: Casella & Berger 2002
What are the main types of regression analysis?
The main types of regression analysis include linear regression for continuous outcomes; logistic regression for binary outcomes, giving odds ratios; Poisson and negative binomial regression for counts; and survival regression, such as the Cox model, for time-to-event outcomes. So regression analysis encompasses several methods suited to different outcome types, all sharing the aim of relating an outcome to predictors, with the choice determined by the nature of the outcome, which is why understanding the type of outcome is the first step in selecting an appropriate regression method for an analysis.
Source: Casella & Berger 2002
What assumptions underlie regression analysis?
The assumptions underlying regression analysis depend on the type but commonly include a specified form for the relationship, such as linearity in the appropriate scale; independence of observations; and, for some methods, assumptions about the distribution of the outcome or errors. Violations can bias estimates or invalidate inference. So regression analysis rests on assumptions that vary by method, and its validity depends on these holding, which is why they are checked through diagnostics, since departures such as non-linearity, correlated observations, or incorrect distributional assumptions can affect the estimates or standard errors and may call for a different model or remedial measures.
Source: Casella & Berger 2002
How is regression analysis interpreted?
Regression analysis is interpreted through its estimated coefficients, which give the effect of each predictor on the outcome, adjusted for the others, expressed according to the method, such as changes in the outcome for linear regression, odds ratios for logistic regression, or hazard ratios for Cox regression, together with their confidence intervals. So regression analysis is interpreted as showing how the outcome relates to each predictor while accounting for the others, with the coefficients and their uncertainty conveying the direction, size, and precision of the associations, though causal interpretation requires that confounding be addressed, since a coefficient reflects association within the model rather than necessarily a causal effect.
Source: Casella & Berger 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 24 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/regression-analysis
- Term code
- HE-ES-SA-175
Stable URI · Machine-readable · Resolvable · CC BY 4.0