Concept Architecture
Concept
Theoretically, Linear Regression is a statistical modelling method that quantifies the linear relationship between a dependent variable and one or more independent variables. It is founded on the principle that changes in the outcome variable can be explained, wholly or partly, by systematic variation in explanatory variables together with random error. Linear regression exists to estimate effect sizes, explain variation, make predictions and evaluate associations between variables. It is one of the most widely used analytical methods in health economics for analysing costs, resource utilisation, health outcomes and determinants of healthcare expenditure.
Mathematically, linear regression represents the expected value of the dependent variable as a linear combination of regression coefficients and explanatory variables. Model parameters are estimated using Ordinary Least Squares (OLS), which minimises the residual sum of squares between observed and predicted values. Statistical inference is based on the estimated coefficients, their standard errors, hypothesis tests and confidence intervals, while overall model fit is commonly assessed using measures such as R� and adjusted R�.
In practice, linear regression is implemented by specifying an outcome variable and one or more predictors, estimating regression coefficients and evaluating model assumptions including linearity, independence, homoscedasticity and normality of residuals. In health economics, linear regression is used to estimate treatment effects, identify determinants of healthcare costs, predict quality-adjusted life years, analyse resource utilisation and adjust for confounding in observational studies.
Purpose
Used to quantify linear relationships between variables, estimate treatment effects, identify determinants of health outcomes and costs, generate predictions and support statistical inference in health economic research.
Mathematical Formulae
Primary Formula
Multiple linear regression model:
Y = ?? + ??X? + ??X? + ? + ??X? + �
where:
- Y = dependent variable
- ?? = intercept
- ????? = regression coefficients
- X??X? = explanatory variables
- � = random error
Ordinary Least Squares estimator:
?? = (X?X)??X?Y
Supporting Formulae
Residual:
e? = y? ? ??
Residual Sum of Squares:
RSS = ?(y? ? ??)�
Coefficient of Determination:
R� = 1 ? RSS / TSS
Adjusted R�:
Adjusted R� = 1 ? [(RSS / (n ? p ? 1)) / (TSS / (n ? 1))]
Related Mathematical Methods
- Ordinary Least Squares
- Multiple Linear Regression
- Generalised Linear Models
- Weighted Least Squares
- Ridge Regression
- Lasso Regression
- Robust Regression
Example
A health economist models annual healthcare costs using age, number of chronic conditions and treatment group as predictors.
The estimated regression model is:
Annual Cost = 2,150 + 95(Age) + 1,480(Chronic Conditions) ? 820(Treatment)
For a patient aged 65 years with three chronic conditions receiving the treatment:
Predicted Cost
= 2,150 + (95 ? 65) + (1,480 ? 3) ? 820
= �11,945
The model explains 72% of the variation in annual healthcare costs (R� = 0.72), indicating good predictive performance.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LINEST | =LINEST(B2:B501,C2:E501,TRUE,TRUE) | Estimate regression coefficients and model statistics. |
| TREND | =TREND(B2:B501,C2:E501,C502:E502) | Predict healthcare costs or outcomes for new patients. |
| FORECAST.LINEAR | =FORECAST.LINEAR(F2,B2:B501,C2:C501) | Predict outcomes using a single predictor. |
| RSQ | =RSQ(B2:B501,F2:F501) | Calculate the coefficient of determination (R�). |
| STEYX | =STEYX(B2:B501,F2:F501) | Estimate the standard error of prediction. |
VBA (Optional)
A VBA routine can automate regression model estimation, diagnostic testing, prediction generation and reporting across multiple health economic datasets.
Sources
- Draper NR, Smith H. Applied Regression Analysis. Wiley.
- Montgomery DC, Peck EA, Vining GG. Introduction to Linear Regression Analysis. Wiley.
- Kutner MH, Nachtsheim CJ, Neter J, Li W. Applied Linear Statistical Models. McGraw-Hill.
- Wooldridge JM. Introductory Econometrics: A Modern Approach. Cengage Learning.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- 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 linear regression?
A modelling technique estimating the linear relationship between a continuous outcome and one or more predictors, giving expected change per unit predictor.
Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.
What relationship does linear regression estimate?
Linear regression estimates the straight-line relationship between a continuous outcome and one or more predictor variables, fitting the line that best captures how the outcome changes as the predictors change. Each coefficient gives the expected change in the outcome for a one-unit rise in its predictor, holding the others fixed, so the model both describes associations and predicts values. It is the foundation on which much of statistical modelling is built. Fitting a line through the data is its essence. Kirkwood and Sterne (2003) describe this technique.
Source: Kirkwood & Sterne 2003
How does linear regression work?
Linear regression works by fitting a line, or plane, that relates the outcome to the predictors, choosing the coefficients that minimise the sum of the squared differences between the observed and predicted outcomes, a method called least squares. The coefficients estimate the expected change in the outcome per unit change in each predictor. So linear regression works by estimating the linear relationship that best fits the data in the least-squares sense, yielding coefficients that quantify each predictor's association with the outcome, which can be used to predict the outcome and to interpret how it depends on the predictors, under the model's assumptions.
Source: Casella & Berger 2002
What are the assumptions of linear regression?
The assumptions of linear regression include a linear relationship between the predictors and the outcome; independence of the observations; constant variance of the errors, or homoscedasticity; and, for inference, approximately normally distributed errors. Violations can bias estimates or invalidate inference. So linear regression rests on assumptions of linearity, independence, constant error variance, and normality of errors, and its validity depends on these holding reasonably, which is why they are checked through residual diagnostics, since departures such as non-linearity, correlated errors, or non-constant variance can affect the estimates or the standard errors and may call for remedies or alternative models.
Source: Casella & Berger 2002
How are the coefficients of linear regression interpreted?
The coefficients of linear regression are interpreted as the expected change in the outcome for a one-unit increase in the corresponding predictor, holding the other predictors constant. The intercept is the expected outcome when all predictors are zero. So each coefficient in linear regression is interpreted as the estimated effect of its predictor on the outcome, adjusted for the other predictors, which allows the independent association of each variable to be assessed, though interpretation assumes the model is correct and, for causal claims, that confounding is addressed, since a regression coefficient reflects association within the model rather than necessarily a causal effect.
Source: Casella & Berger 2002
What is linear regression used for?
Linear regression is used for prediction, estimating the outcome from the predictors; for description and inference, quantifying the relationship between the outcome and predictors and testing whether associations differ from zero; and for adjustment, accounting for confounders when estimating an effect. So linear regression is used widely to model continuous outcomes, serving to predict, to describe relationships, and to adjust for other variables, which makes it a foundational tool in statistics and health research, applied whenever the relationship between a continuous outcome and one or more predictors is of interest, subject to its assumptions being reasonable.
Source: Casella & Berger 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 17 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/linear-regression
- Term code
- HE-ES-SA-103
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