Concept Architecture
Concept
Theoretically, an Orthogonal Design is an experimental design in which the levels of each attribute vary independently of the levels of every other attribute. Based on the principles of experimental design and linear independence, orthogonal designs ensure that parameter estimates are statistically uncorrelated, allowing the independent effects of individual attributes to be estimated efficiently. In health economics, orthogonal designs have traditionally been used in conjoint analysis and discrete choice experiments, although they have largely been superseded by D-efficient and Bayesian efficient designs.
Mathematically, an Orthogonal Design is characterised by a design matrix whose columns are mutually orthogonal, meaning the inner product (or correlation) between attribute columns is zero. This minimises multicollinearity and permits unbiased estimation of the main effects of individual attributes under the assumed statistical model.
In practice, Orthogonal Designs are generated before data collection to construct combinations of attribute levels for stated preference surveys. They are implemented using experimental design software and analysed using regression-based methods to estimate preferences, although modern health economics studies generally favour statistically efficient designs that incorporate prior parameter information.
Purpose
Used to construct experimental designs in which attribute effects can be estimated independently while minimising correlation between explanatory variables.
Mathematical Formulae
Primary Formula
For a design matrix X:
X?X =
?k? 0 ? 0?
?0 k? ? 0?
?? ? ? ??
?0 0 ? k??
where the off-diagonal elements are zero, indicating orthogonality among the design columns.
Supporting Formulae
For any two columns x? and x?:
x??x? = 0??(i ? j)
Related Mathematical Methods
- Experimental Design
- Fractional Factorial Design
- Conjoint Analysis
- Discrete Choice Experiment
- D-Efficient Design
- Linear Regression
Example
A discrete choice experiment is developed to evaluate preferences for asthma treatments using four attributes with two levels each. An orthogonal fractional factorial design is generated so that each attribute varies independently of the others. The resulting survey allows unbiased estimation of the main effect of each attribute, such as treatment effectiveness, dosing frequency, adverse effects and monthly cost.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| MMULT | =MMULT(TRANSPOSE(B2:E17),B2:E17) | Computes X?X to assess orthogonality of the design matrix |
| TRANSPOSE | =TRANSPOSE(B2:E17) | Creates the transpose of the design matrix |
| CORREL | =CORREL(B2:B17,C2:C17) | Verifies that attribute columns are uncorrelated |
| MUNIT | =MUNIT(4) | Compares the cross-product matrix with an ideal diagonal structure |
VBA (Optional)
Automate the generation and validation of orthogonal experimental designs for discrete choice experiments before survey administration.
Sources
- Louviere JJ, Hensher DA, Swait JD. Stated Choice Methods: Analysis and Applications. Cambridge University Press.
- Huber J, Zwerina K. The Importance of Utility Balance in Efficient Choice Designs. Journal of Marketing Research.
- Bridges JFP, Hauber AB, Marshall D, et al. Conjoint Analysis Applications in Health: A Checklist. Value in Health.
- ISPOR Conjoint Analysis Good Research Practices Task Force Reports.
- Kuhfeld WF. Marketing Research Methods in SAS: Experimental Design, Choice, Conjoint and Graphical Techniques.
Related Concepts (2)
Library
Publications
1
Conjoint Analysis Applications in Health — A Checklist: A Report of the ISPOR Good Research Practices for Conjoint Analysis Task Force — Bridges, Hauber, Marshall, Lloyd, Prosser, Regier, Johnson & Mauskopf, Vol. 14, No. 4 ed., 2011 (Value in Health)
The ISPOR good-practice checklist for conjoint analysis and discrete-choice experiments in health — the stated-preference methods used to elicit patient and public preferences over treatment attributes for value assessment and priority-setting.
Journal ArticleView source →
Frequently Asked Questions (6)
What is an orthogonal design?
An experimental design for a choice experiment in which attribute levels are varied independently, so each attribute's effect on preference can be separated statistically.
Source: Louviere, Hensher & Swait 2000
What does an orthogonal design achieve?
It varies attribute levels independently of one another, so that the estimated effect of each attribute is not confounded with any other. That independence is what allows the contribution of each to be separated statistically from a limited number of choice tasks. It requires no assumptions about the size of the effects before the data are collected, which is its principal advantage over designs optimised using prior estimates.
Source: Louviere, Hensher & Swait 2000
How does an orthogonal design differ from an efficient design?
An orthogonal design treats every parameter as equally worth estimating and needs no prior information. An efficient design uses assumed parameter values to minimise the variance of the estimates, concentrating precision where it is wanted. Where those priors are roughly correct the efficient design performs better; where they are badly wrong it can perform worse, which is why orthogonal designs remain a defensible fallback when no pilot data exist.
Source: Huber & Zwerina 1996
What are the limitations of an orthogonal design?
Orthogonality concerns the attributes shown rather than the choices respondents make, so an orthogonal design can still produce choice sets in which one alternative dominates another, and such sets yield no information about trade-offs. It also takes no account of plausibility, so implausible combinations must be excluded separately, and doing so breaks the orthogonality it was constructed to achieve. Constrained designs are therefore common in practice, and a design described simply as orthogonal may in fact be near-orthogonal after the necessary exclusions have been applied.
Source: Lancsar & Louviere 2008
When is an orthogonal design appropriate?
Where no prior information about the parameters exists, so an efficient design cannot be constructed reliably. Where the study is exploratory and the attributes have not been examined before. And where the design must be defensible to an audience unfamiliar with efficiency criteria, since orthogonality is easier to explain and to verify than a statistical efficiency measure. It also remains appropriate where the analysis will estimate interactions rather than main effects alone, since efficiency criteria are normally specified for main-effects models.
Source: Louviere, Hensher & Swait 2000
What should be reported about an orthogonal design?
The number of attributes and levels, the size of the design, how choice sets were formed from the profiles, and how sets were divided among respondents. Any constraints applied to exclude implausible or dominated combinations should be stated, since these compromise orthogonality and a design described as orthogonal after such adjustment is only approximately so. Reporting the design in full, or making it available, allows another analyst to verify the properties claimed for it, which published choice experiments rarely permit.
Source: Bridges et al. 2011
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Verified by Dr Darrin Baines
British health economist
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Verification date: 1 Aug 2025
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