Concept Architecture
Concept
Theoretically, Residual Analysis is the examination of residuals, defined as the differences between observed and predicted values, to evaluate whether a statistical model adequately satisfies its underlying assumptions. Residual analysis is a fundamental component of regression diagnostics because systematic patterns in residuals indicate potential model misspecification, omitted variables, non-linearity, heteroscedasticity or violations of distributional assumptions. In health economics, residual analysis is routinely used to assess regression models, survival models and other statistical models that provide inputs for economic evaluations.
Mathematically, residual analysis is based on the calculation of residuals for each observation and the evaluation of their statistical properties. For an appropriately specified model, residuals should have an expected value of zero, exhibit constant variance and display no systematic relationship with fitted values or explanatory variables. Standardised, studentised and deviance residuals are commonly examined to identify influential observations and departures from modelling assumptions.
In practice, analysts inspect residual plots, normal probability plots and summary statistics after estimating statistical models. Health economists use residual analysis to evaluate cost models, utility mapping equations, risk prediction models and resource utilisation models before incorporating estimated parameters into decision-analytic models. Residual analysis is interpreted alongside other diagnostic measures such as goodness of fit, influence diagnostics and validation procedures.
Purpose
Used to assess whether statistical model assumptions are satisfied, identify model misspecification and improve the reliability of parameter estimates used in health economic evaluation.
Mathematical Formulae
Primary Formula
The residual for observation i is
e? = y? ? ??
where:
- y? = observed value
- ?? = predicted value
Supporting Formulae
Standardised residual:
r? = e? / [�?�(1 ? h??)]
where:
- �? = estimated residual standard deviation
- h?? = leverage of observation i
Residual Sum of Squares:
RSS = �???� e?�
Related Mathematical Methods
- Regression diagnostics
- Goodness of fit assessment
- Standardised residuals
- Studentised residuals
- Cook's Distance
- DFBETA
- Leverage analysis
- Variance Inflation Factor
Example
A health economist develops a regression model predicting annual healthcare costs for 600 patients.
For one patient:
Observed annual cost:
�3,200
Predicted annual cost:
�3,050
The residual is
e = 3200 ? 3050 = 150
Residual plots for all observations show no systematic trend and approximately constant variance, suggesting that the regression model satisfies the principal modelling assumptions and provides an adequate representation of the data.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
- | =B2-C2 | Calculates the residual for each observation. |
| ABS | =ABS(B2-C2) | Calculates absolute residuals for diagnostic review. |
| SUMSQ | =SUMSQ(D2:D601) | Calculates the residual sum of squares. |
| AVERAGE | =AVERAGE(D2:D601) | Verifies that residuals are centred near zero. |
| STDEV.S | =STDEV.S(D2:D601) | Calculates the residual standard deviation. |
VBA (Optional)
Automate calculation of residual diagnostics and generate residual plots and summary statistics for regression model validation.
Sources
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Harrell FE. Regression Modeling Strategies. 2nd ed.
- Cook RD, Weisberg S. Residuals and Influence in Regression.
- Draper NR, Smith H. Applied Regression Analysis. 3rd ed.
- ISPOR Good Practice Reports on statistical modelling and model validation.
Related Concepts (2)
Library
Publications
1
Modeling Good Research Practices — Overview: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-1 — Caro, Briggs, Siebert & Kuntz, Task Force Report 1 ed., 2012 (Value in Health / Medical Decision Making)
The overview paper of the seven-part ISPOR-SMDM modelling good-practice series, setting out best-practice recommendations across model design, technique selection, implementation, validation, parameterisation, uncertainty and use in decision making.
Journal ArticleView source →
Frequently Asked Questions (6)
What is residual analysis?
A statistical technique examining the differences between a model's predicted values and observed data, checking for systematic patterns left unexplained.
Source: Cook & Weisberg 1982
What is a residual in a fitted model?
A residual is the gap between a value the model predicts and the value actually observed, one for each data point. Examining these gaps checks whether what the model failed to predict is mere random scatter or shows a systematic pattern. Scatter with no structure suggests the model has captured the signal, whereas a trend or curve in the residuals reveals something the model is missing. The residuals therefore diagnose where and how a model departs from the data. Draper and Smith (1998) describe this analysis.
Source: Draper & Smith 1998
How is residual analysis performed?
Residual analysis is performed by computing the residuals for each observation, the observed value minus the model's prediction, and examining them, often by plotting them against predicted values, against predictors, or over time, and checking their distribution. Random scatter without pattern suggests the model fits well, while systematic patterns, trends, or unusual spread indicate problems. Formal tests and diagnostic plots help detect departures from the model's assumptions. The examination reveals where and how the model fails to capture the data.
Source: Cook & Weisberg 1982
What do patterns in residuals indicate?
Patterns in residuals indicate that the model has not captured some structure in the data, signalling problems such as a missing predictor, an incorrect functional form, non-linearity not represented, or violated assumptions like non-constant variance. For example, a trend in residuals against a predictor suggests the relationship is misspecified, and a funnel shape suggests changing variance. Because a correct model should leave random residuals, systematic patterns point to specific inadequacies, so residual analysis diagnoses how the model should be improved.
Source: Cook & Weisberg 1982
Why is residual analysis useful?
Residual analysis is useful because it reveals ways a model fails to fit the data that summary measures of overall fit may hide, showing not just whether but how and where the model departs from the observations. By exposing systematic patterns, it identifies specific problems, such as missing terms or violated assumptions, guiding improvement of the model. It also helps detect outliers and influential points. As a diagnostic examining what the model leaves unexplained, residual analysis is central to checking and refining a fitted model.
Source: Belsley, Kuh & Welsch 1980
How does residual analysis relate to model diagnostics?
Residual analysis is a core part of regression and model diagnostics, the checks that assess whether a fitted model is adequate and its assumptions hold. Examining residuals reveals misspecification, violated assumptions, and influential observations, complementing other diagnostics such as influence measures. Together these diagnostics establish whether the model fits well and its assumptions are met, so residual analysis contributes to the broader assessment of a model's adequacy, providing detailed evidence of how the model relates to the data.
Source: Cook & Weisberg 1982
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/residual-analysis
- Term code
- HE-EM-MV-069
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