Concept Architecture
Concept
Theoretically, a Leverage Point is an observation whose values on one or more predictor variables are substantially different from those of the remaining observations. Within regression analysis, leverage quantifies the potential influence that an observation has on the fitted regression model due to its location in the predictor space. High-leverage observations are not necessarily influential, but they have the capacity to exert considerable influence on parameter estimates if they are also associated with large residuals.
Mathematically, leverage is defined by the diagonal elements of the projection (hat) matrix in ordinary least squares regression. The leverage value measures the extent to which an observation contributes to determining its own fitted value. Observations with leverage substantially greater than the average leverage are commonly examined as potential influential observations, particularly when accompanied by large residuals or high values of Cook's Distance.
In practice, leverage is calculated during regression diagnostics after model estimation. In health economics, leverage diagnostics are used when modelling healthcare costs, resource utilisation, quality-adjusted life years and other outcomes to identify observations that may disproportionately affect estimated treatment effects, cost functions or prediction models. High-leverage observations should be investigated for data quality issues, unusual patient characteristics or legitimate population heterogeneity before deciding whether any remedial action is appropriate.
Purpose
Used to identify observations with unusual predictor values that have the potential to exert substantial influence on regression estimates, assess model robustness and support regression diagnostic evaluation in health economic analyses.
Mathematical Formulae
Primary Formula
For observation i:
h?? = x??(X?X)??x?
where:
- h?? = leverage of observation i
- x? = predictor vector for observation i
- X = design matrix
Supporting Formulae
Hat matrix:
H = X(X?X)??X?
Average leverage:
h? = p / n
where:
- p = number of estimated regression parameters (including the intercept)
- n = sample size
Common diagnostic threshold:
h?? > 2p / n
or, more conservatively,
h?? > 3p / n
Related Mathematical Methods
- Hat Matrix
- Cook's Distance
- Studentised Residuals
- DFBETAS
- Covariance Ratio (COVRATIO)
- Ordinary Least Squares Regression
- Regression Diagnostics
Example
A health economist fits a multiple regression model predicting annual healthcare costs using five explanatory variables and an intercept from a sample of 500 patients.
The average leverage is:
h? = 6 / 500 = 0.012
A patient has a calculated leverage value of:
h?? = 0.048
Since:
2p / n = 2 ? 6 / 500 = 0.024
the observation exceeds the commonly used threshold and is investigated further. Subsequent examination shows that the patient has an exceptionally high number of chronic conditions. Cook's Distance is also evaluated to determine whether the observation materially influences the estimated cost model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(MMULT(A2:F2,MINVERSE(MMULT(TRANSPOSE(A2:F501),A2:F501))),TRANSPOSE(A2:F2)) | Calculate the leverage value for an observation. |
| MINVERSE | =MINVERSE(MMULT(TRANSPOSE(A2:F501),A2:F501)) | Compute (X?X)?? used in leverage calculations. |
| TRANSPOSE | =TRANSPOSE(A2:F501) | Construct matrix operations required for the hat matrix. |
| COUNT | =COUNT(A2:A501) | Determine the sample size for leverage thresholds. |
| IF | =IF(G2>2*6/500,"High leverage","Typical leverage") | Flag observations requiring further diagnostic investigation. |
VBA (Optional)
A VBA routine can automatically calculate leverage values for all observations, identify high-leverage cases and generate a regression diagnostics report incorporating additional influence measures.
Sources
- Belsley DA, Kuh E, Welsch RE. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. Wiley.
- Cook RD, Weisberg S. Residuals and Influence in Regression. Chapman & Hall.
- Draper NR, Smith H. Applied Regression Analysis. Wiley.
- Montgomery DC, Peck EA, Vining GG. Introduction to Linear Regression Analysis. Wiley.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is a leverage point?
An observation with an unusual combination of predictor values, distinct from the bulk of the data, capable of disproportionately influencing a regression model.
Source: Belsley, Kuh & Welsch 1980
What makes a leverage point unusual in a regression?
A leverage point is an observation whose combination of predictor values is far from that of the rest of the data, sitting out on its own in the space of the predictors. This unusual position gives it the potential to pull a regression line strongly toward itself, so it can shape the fitted model out of proportion to its numbers. A leverage point differs from an outlier, which is unusual in its outcome; a point can have high leverage yet sit neatly on the trend, or be both at once. An extreme position among the predictors is what defines it. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
How is leverage measured?
Leverage is measured by a quantity, often called the hat value, that reflects how far an observation's predictor values lie from the mean of the predictors, taking values between zero and one, with higher values indicating greater leverage. Observations with hat values well above the average are considered high-leverage. So leverage is measured through the hat values from the regression, which quantify each observation's distance from the centre of the predictor space, and points with notably high values are flagged as leverage points, since their unusual predictor values give them the potential to influence the fitted model more than typical observations.
Source: Belsley, Kuh & Welsch 1980
How does a leverage point differ from an outlier?
A leverage point is unusual in its predictor values, lying far from the others in the predictor space, while an outlier is unusual in its outcome, having a large residual from the model. A point can have high leverage without being an outlier, and vice versa, and a point that is both high-leverage and an outlier is especially influential. So a leverage point and an outlier differ in whether the unusualness is in the predictors or the outcome, and although either can affect a regression, it is often the combination, a high-leverage point with a large residual, that most strongly influences the estimates, which is why both are examined.
Source: Cook & Weisberg 1982
Why do leverage points matter?
Leverage points matter because their unusual predictor values give them the potential to influence the regression strongly, so that a single such point, especially if it also has a large residual, can pull the fitted line or surface and distort the estimated coefficients. So leverage points matter for the robustness of a regression, since results driven by a few extreme observations may not reflect the general pattern, which is why leverage is examined alongside residuals and influence measures to identify observations that could unduly affect the model, prompting investigation of whether they are errors, valid but unusual cases, or signs that the model is inadequate.
Source: Belsley, Kuh & Welsch 1980
What should be done about a leverage point?
A leverage point should be investigated rather than automatically removed: the analyst examines whether its unusual predictor values result from an error, represent a legitimate but rare case, and how much the results depend on it, considering influence measures that combine leverage with the residual. So a high-leverage point is scrutinised to understand why it is unusual and whether it distorts the analysis, with the response depending on the cause, since discarding a valid observation merely for being extreme in the predictors can bias the analysis or lose information, whereas an erroneous high-leverage point should be corrected or removed, and this judgement requires examining the point in context.
Source: Belsley, Kuh & Welsch 1980
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/leverage-point
- Term code
- HE-ES-SA-102
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