Concept Architecture
Concept
Theoretically, Prediction Interval is a statistical interval estimate that defines the range within which a future individual observation is expected to fall with a specified probability, given a fitted statistical model. Unlike a confidence interval, which quantifies uncertainty in an estimated population parameter such as the mean, a prediction interval incorporates both uncertainty in parameter estimation and the inherent variability of individual observations. Prediction intervals are fundamental to regression analysis, forecasting, and statistical prediction.
Mathematically, prediction intervals are derived from the estimated regression model by combining the variance of the estimated mean response with the residual variance representing random variation around the regression line. Under the assumptions of linear regression, prediction intervals are constructed using the Student's t-distribution and become wider as the uncertainty of prediction increases, particularly for observations distant from the centre of the predictor distribution.
In practice, prediction intervals are estimated following regression model fitting and are widely used in health economics to forecast future healthcare costs, disease incidence, resource utilisation, quality-adjusted life years, and other economic outcomes. They provide decision-makers with an estimate of the uncertainty surrounding individual future predictions rather than average population effects.
Purpose
Used to quantify the uncertainty surrounding future individual observations, supporting forecasting, risk assessment, economic modelling, model validation, and decision-making in health economics.
Mathematical Formulae
Primary Formula
PI = ?? � t?�/?,n?p?1? ? s ? �[1 + 1/n + (x? ? x?)� / ?(x? ? x?)�]
where:
- ?? = predicted value
- s = residual standard error
- n = sample size
- p = number of predictors
- x? = predictor value for the new observation
- x? = mean predictor value
Supporting Formulae
Residual Standard Error:
s = �[RSS / (n ? p ? 1)]
Residual Sum of Squares:
RSS = ?(y? ? ??)�
Prediction Error:
e? = y? ? ??
Related Mathematical Methods
- Linear Regression
- Confidence Interval
- Residual Analysis
- Mean Squared Error
- Forecasting
- Regression Diagnostics
- Model Validation
Example
A regression model predicts annual healthcare costs for patients with chronic disease.
Predicted annual cost:
? = �8,500
Residual standard error:
s = �1,200
Sample size:
n = 150
For a new patient, the calculated standard error of prediction is �1,260. Using a 95% prediction interval with t � 1.98:
Prediction Interval = �8,500 � (1.98 ? �1,260)
= �8,500 � �2,495
= (�6,005, �10,995)
The model predicts that the patient's annual healthcare cost is likely to fall between approximately �6,005 and �10,995 with 95% confidence.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| FORECAST.LINEAR | =FORECAST.LINEAR(A2,KnownY,KnownX) | Predict future healthcare costs or utilisation. |
| STEYX | =STEYX(KnownY,KnownX) | Calculate the residual standard error required for prediction intervals. |
| T.INV.2T | =T.INV.2T(0.05,n-p-1) | Obtain the critical t-value for interval construction. |
| SQRT | =SQRT(1+1/n+((A2-XBar)^2/SSX)) | Calculate the prediction standard error. |
| CONFIDENCE.T | =CONFIDENCE.T(0.05,s,n) | Calculate the margin of error for comparison with confidence intervals. |
VBA (Optional)
Automate calculation of prediction intervals for regression forecasts and generate uncertainty bands for projected health economic outcomes.
Sources
- Draper NR, Smith H. Applied Regression Analysis.
- Montgomery DC, Peck EA, Vining GG. Introduction to Linear 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.
- 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 a prediction interval?
A range calculated to contain a future individual observation with a specified confidence, unlike a confidence interval describing a population parameter.
Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.
What does a prediction interval aim to contain?
A prediction interval is a range calculated to contain a single future observation with a stated probability, rather than a population average. This distinguishes it from a confidence interval, which brackets a population parameter such as a mean; the prediction interval must also allow for the natural variation of individuals around that mean. That extra source of variability is why a prediction interval is always wider than the corresponding confidence interval. Bracketing a future individual value is its aim. Kirkwood and Sterne (2003) describe this.
Source: Kirkwood & Sterne 2003
How does a prediction interval differ from a confidence interval?
A prediction interval is designed to contain a future individual observation, while a confidence interval is designed to contain a population parameter, such as a mean. The prediction interval is wider because it accounts for both the uncertainty in estimating the parameter and the variability of individual observations around it, whereas the confidence interval reflects only the former. So the two differ in what they cover and in their width, with the prediction interval concerning where a new single value will fall and the confidence interval where the true parameter lies, which is why prediction intervals are wider and are used when the interest is in individual predictions rather than in a summary parameter.
Source: Casella & Berger 2002
How is a prediction interval calculated?
A prediction interval is calculated by combining the uncertainty in the predicted value from the model with the variability of individual observations around the prediction, so its width reflects both sources; for a regression prediction, this adds the residual variance to the variance of the estimated mean prediction. So a prediction interval is calculated to encompass both the uncertainty in the estimated relationship and the individual observation variability, which makes it wider than the corresponding confidence interval, and the calculation depends on the model and its assumptions, with the interval indicating the range within which a new individual value is expected to fall at the stated confidence.
Source: Casella & Berger 2002
When is a prediction interval used?
A prediction interval is used when the interest is in the likely value of a future individual observation rather than in a population parameter, such as predicting a new patient's outcome or a future measurement, where the range of plausible individual values matters. So a prediction interval is used for individual predictions, providing a range expected to contain a new observation, which is relevant when decisions concern specific future cases rather than averages, and it is preferred over a confidence interval in such situations because it accounts for the variability of individual values, giving a realistic range for a single new outcome.
Source: Casella & Berger 2002
Why is a prediction interval wider than a confidence interval?
A prediction interval is wider than a confidence interval because it must account for two sources of uncertainty: the uncertainty in estimating the underlying relationship or parameter, which the confidence interval also includes, and the additional natural variability of individual observations around the predicted value, which the confidence interval does not. So a prediction interval is wider because it captures individual observation variability on top of parameter uncertainty, reflecting that predicting a single new value is inherently more uncertain than estimating an average, which is why a prediction interval, aimed at a future individual observation, exceeds the width of a confidence interval aimed at a population parameter.
Source: Casella & Berger 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 23 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/prediction-interval
- Term code
- HE-ES-SA-163
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