Concept Architecture
Concept
Theoretically, Studentized Residual is a regression diagnostic statistic obtained by scaling an ordinary residual using an estimate of its own standard deviation. Unlike a standardised residual, a studentized residual accounts for the influence of each observation on the estimated residual variance, making it more suitable for identifying unusual observations and assessing model adequacy. Studentized residuals are fundamental tools in regression diagnostics because they are approximately distributed according to Student's t-distribution under the assumptions of the linear regression model.
Mathematically, studentized residuals divide each residual by an estimate of its standard error that incorporates leverage. External studentized residuals further improve the calculation by estimating the residual variance after temporarily removing the observation under investigation. This prevents unusually large residuals from artificially inflating the estimated residual variance and masking their own influence. Large absolute studentized residuals indicate observations that deviate substantially from the fitted regression model.
In practice, studentized residuals are calculated following regression model estimation and are routinely examined alongside leverage, Cook's Distance, and DFBETAS to identify influential observations, assess model assumptions, and evaluate goodness of fit. In health economics they are used when modelling healthcare costs, quality-adjusted life years, healthcare utilisation, and clinical outcomes to identify observations requiring further investigation.
Purpose
Used to identify outlying observations, assess regression model assumptions, evaluate model fit, and support regression diagnostics by providing residuals with approximately known sampling distributions.
Mathematical Formulae
Primary Formula
Internal studentized residual:
t? = e? � (s�(1 ? h?))
where:
- t? = studentized residual
- e? = ordinary residual
- s = residual standard error
- h? = leverage
Supporting Formulae
External studentized residual:
t?* = e? � (s????�(1 ? h?))
where:
- s???? = residual standard error estimated after removing observation i
Residual:
e? = y? ? ??
Residual standard error:
s = �[RSS � (n ? p ? 1)]
Related Mathematical Methods
- Residual Analysis
- Standardised Residual
- Leverage
- Cook's Distance
- DFBETAS
- Regression Diagnostics
- Ordinary Least Squares
- Influence Analysis
Example
A regression model predicts annual healthcare costs.
For one patient:
Observed cost:
�13,400
Predicted cost:
�12,200
Residual:
e = �1,200
Residual standard error after removing the observation:
s???? = �550
Leverage:
h = 0.16
The external studentized residual is:
t* = 1,200 � (550 ? �(1 ? 0.16))
= 2.38
Since the absolute value exceeds 2, the observation would typically be investigated as a potential outlier or influential case.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SQRT | =SQRT(1-H2) | Calculate the leverage adjustment for each observation. |
| SUMXMY2 | =SUMXMY2(ObservedRange,PredictedRange) | Calculate the residual sum of squares. |
| POWER | =POWER(B2-C2,2) | Calculate squared residuals used in estimating residual variance. |
| SQRT | =SQRT(RSS/(n-p-1)) | Calculate the residual standard error. |
| Formula | =(B2-C2)/(ResidualSE*SQRT(1-H2)) | Calculate internal studentized residuals for regression diagnostics. |
VBA (Optional)
Automate calculation of internal and external studentized residuals and generate regression diagnostic reports highlighting influential observations.
Sources
- Belsley DA, Kuh E, Welsch RE. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity.
- Cook RD, Weisberg S. Residuals and Influence in Regression.
- Draper NR, Smith H. Applied Regression Analysis.
- Kutner MH, Nachtsheim CJ, Neter J, Li W. Applied Linear Statistical Models.
- 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)
Frequently Asked Questions (6)
What is a studentized residual?
A standardised residual excluding an observation's own contribution to the model fit, a more robust diagnostic for identifying influential data points.
Source: Cook & Weisberg 1982
How does a studentized residual improve on a standardised one for spotting outliers?
A studentized residual is a scaled residual that leaves out an observation's own influence on the model when estimating the spread used to scale it. This matters because a genuine outlier can inflate the ordinary estimate of variability and so disguise itself; by excluding the point's own contribution, the studentized version keeps its scale honest and makes such a point stand out more clearly. It is therefore a sharper tool than the standardised residual for detecting outliers. Scaling each residual by a fit that excludes it is its refinement. Kirkwood and Sterne (2003) describe this diagnostic.
Source: Kirkwood & Sterne 2003
How is a studentized residual calculated?
A studentized residual is calculated by dividing an observation's residual by an estimate of its standard deviation that is obtained with the observation excluded from the estimation, so the observation does not influence the scale against which its own residual is judged. So a studentized residual is calculated using a leave-one-out standard deviation estimate, which distinguishes it from the standardised residual that uses all the data, and this exclusion makes the studentized residual more sensitive to outliers, since an extreme observation cannot inflate the standard deviation used to assess it, allowing such points to stand out more clearly.
Source: Cook & Weisberg 1982
Why are studentized residuals more robust for detecting outliers?
Studentized residuals are more robust for detecting outliers because they use a standard deviation estimated without the observation being assessed, so a genuine outlier does not inflate the scale against which its own residual is judged and therefore cannot mask itself, as it might with a standardised residual that includes it. So studentized residuals better reveal outliers because excluding the observation from its own standard deviation estimate keeps an extreme point from reducing its own apparent unusualness, which is why studentized residuals are preferred over standardised residuals for identifying observations that the model fits poorly and that may unduly influence the fit.
Source: Cook & Weisberg 1982
How is a studentized residual interpreted?
A studentized residual is interpreted by its magnitude, with large absolute values, such as beyond about two or three, indicating observations that the model fits poorly and that may be outliers or influential points warranting investigation, and it can be compared with a reference distribution for a formal test. So a studentized residual is interpreted like a standardised residual but with greater sensitivity, flagging unusual observations more reliably, and large values prompt examination of why the observation departs from the model, since it may be an error, a genuine but unusual case, or a sign of model inadequacy, rather than being removed automatically.
Source: Cook & Weisberg 1982
How does a studentized residual differ from a standardised residual?
A studentized residual differs from a standardised residual in the standard deviation used to scale the residual: the studentized residual uses an estimate obtained with the observation excluded, while the standardised residual uses one based on all the data. This makes the studentized residual more sensitive to outliers, since the observation does not affect its own standard deviation estimate. So the two differ in whether the observation contributes to its own scaling, with the studentized residual being more robust for detecting outliers and influential points, which is why it is often preferred as a diagnostic, though both express residuals on a comparable scale for assessing model fit.
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-208
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