Concept Architecture
Concept
Theoretically, Standardised Residual is a regression diagnostic statistic obtained by scaling an ordinary residual according to its estimated standard deviation. Standardisation removes the influence of differing residual variances across observations, allowing residuals to be compared on a common scale. Standardised residuals are fundamental to regression diagnostics because they facilitate the identification of outliers, model misspecification, heteroscedasticity, and departures from distributional assumptions.
Mathematically, a standardised residual is calculated by dividing the ordinary residual by an estimate of its standard deviation, which incorporates both the residual variance and the leverage of the corresponding observation. Under the assumptions of linear regression, standardised residuals are approximately Normally distributed with mean zero and variance close to one. Large absolute values indicate observations that deviate substantially from the fitted model.
In practice, standardised residuals are calculated following regression model estimation and are routinely examined using residual plots, Normal probability plots, and influence diagnostics. In health economics they are used when modelling healthcare costs, quality-adjusted life years, healthcare utilisation, and other continuous outcomes to evaluate model adequacy and identify observations requiring further investigation.
Purpose
Used to evaluate regression model fit, identify unusual observations, assess model assumptions, detect potential outliers, and support diagnostic evaluation of health economic regression models.
Mathematical Formulae
Primary Formula
r? = e? � (s�(1 ? h?))
where:
- r? = standardised residual
- e? = ordinary residual
- s = residual standard error
- h? = leverage for observation i
Supporting Formulae
Residual:
e? = y? ? ??
Residual standard error:
s = �[RSS � (n ? p ? 1)]
Residual Sum of Squares:
RSS = ?(y? ? ??)�
Related Mathematical Methods
- Residual Analysis
- Studentised Residual
- Cook's Distance
- Leverage
- Regression Diagnostics
- Ordinary Least Squares
- Mean Squared Error
- Influence Analysis
Example
A health economist fits a regression model predicting annual healthcare costs.
For one patient:
Observed cost:
�12,500
Predicted cost:
�11,700
Residual:
e = �800
Residual standard error:
s = �600
Leverage:
h = 0.11
The standardised residual is:
r = 800 � (600 ? �(1 ? 0.11))
= 1.41
Since the standardised residual is well below commonly used thresholds of �2 or �3, the observation would not normally be considered an unusual outlier.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SQRT | =SQRT(1-H2) | Calculate the leverage adjustment. |
| POWER | =POWER(B2-C2,2) | Calculate squared residuals used in residual variance estimation. |
| SUMXMY2 | =SUMXMY2(ObservedRange,PredictedRange) | Calculate the residual sum of squares. |
| SQRT | =SQRT(RSS/(n-p-1)) | Calculate the residual standard error. |
| Formula | =(B2-C2)/(ResidualSE*SQRT(1-H2)) | Calculate the standardised residual for each observation. |
VBA (Optional)
Automate calculation of standardised residuals and generate regression diagnostic reports highlighting potential outliers and influential observations.
Sources
- Belsley DA, Kuh E, Welsch RE. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity.
- Draper NR, Smith H. Applied Regression Analysis.
- Kutner MH, Nachtsheim CJ, Neter J, Li W. Applied Linear Statistical Models.
- 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.
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 a standardised residual?
A regression diagnostic dividing an observation's raw residual by its estimated standard deviation, allowing comparison on a common scale.
Source: Cook & Weisberg 1982
What does a standardised residual put on a common scale?
A standardised residual divides an observation's raw residual by an estimate of its standard deviation, placing residuals of differing natural sizes on a common scale. This lets them be compared directly and judged against familiar benchmarks, so a standardised residual beyond about two or three flags a point the model fits poorly, a possible outlier. Putting residuals in comparable units is what makes them useful for spotting cases the model struggles with. Rescaling residuals for comparison is its purpose. Kirkwood and Sterne (2003) describe this diagnostic.
Source: Kirkwood & Sterne 2003
How is a standardised residual calculated?
A standardised residual is calculated by dividing the raw residual of an observation by an estimate of its standard deviation, which accounts for the variability of the residuals. This yields a value indicating how many standard deviations the residual is from zero. So a standardised residual is calculated by scaling the raw residual by its estimated standard deviation, which puts residuals on a common scale where their magnitude can be interpreted consistently, and this is why standardised residuals beyond a rough threshold, such as two or three in absolute value, are flagged as unusually large, indicating observations the model fits poorly.
Source: Cook & Weisberg 1982
Why are standardised residuals used?
Standardised residuals are used because raw residuals depend on the scale of the data and can have differing variances, making them hard to compare, whereas standardising puts them on a common scale so that unusually large ones, indicating poorly fitted observations or outliers, can be identified consistently. So standardised residuals are used to detect outliers and assess model fit more reliably than raw residuals, since expressing residuals in standard deviation units allows a consistent criterion for what counts as large, which is why they are a standard diagnostic for flagging observations that depart markedly from the model.
Source: Cook & Weisberg 1982
How are standardised residuals interpreted?
Standardised residuals are interpreted by their magnitude, with values far from zero, such as beyond about two or three in absolute value, indicating observations that the model fits poorly and that may be outliers warranting investigation. So standardised residuals are interpreted as showing how unusual each observation is relative to the model on a common scale, with large absolute values flagging potential outliers, though such thresholds are rough guides rather than strict rules, and a flagged observation is examined to understand why it fits poorly rather than automatically removed, since it may be a genuine but unusual case or reveal a model shortcoming.
Source: Cook & Weisberg 1982
How do standardised residuals differ from studentized residuals?
Standardised residuals divide the raw residual by an estimate of its standard deviation that uses all the data, while studentized residuals use a standard deviation estimated with the observation in question excluded, making them more sensitive for detecting influential points, since an outlier does not inflate its own standard deviation estimate. So standardised and studentized residuals differ in whether the observation contributes to its own standard deviation estimate, with studentized residuals being more robust for identifying outliers and influential observations, which is why studentized residuals are often preferred for detecting points that the model fits poorly, though both put residuals on a comparable scale.
Source: Cook & Weisberg 1982
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 25 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-202
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