D-error is a measure that quantifies how good or bad a design is at extracting information from respondents in an experiment. A lower D-error indicates a better design. A design that minimizes D-error is considered D-optimal. Many other related measures exist that also serve this purpose, such as D-optimality. This article describes computing D-error, Bayesian D-error, and D-efficiency.

Computing D-error

I'll show you how to compute D-error to assess the quality of your experiment. There are many different types of experiments. One type is a choice experiment, which represents a design with a table where each row corresponds to a choice alternative and the columns correspond to the version,  task, question, alternative and attributes. I've shown you an example of a design below:

VersionTaskQuestionAlternativeAttribute 1Attribute 2Attribute 3
1111121
1112212
1221122
1222211
1331221
1332112
2411222
2412111
2521222
2522111
2631121
2632212

To compute D-error, first encode the levels of the attributes in the design using dummy coding (or alternatively effects-type coding) and split the encoded design by questions (suppose there are Q of them) to create a set of matrices

    \[ {\textbf{X}}={\mathbit{\textbf{X}}_1,\cdots,\mathbit{\textbf{X}}_Q}. \]

In addition, you need to make a prior assumption regarding the respondent parameters. The different possible assumptions lead to three variants of the D-error. The first is the Dp-error, which assumes that all respondent parameters are given by a parameter vector β . The multinomial logit probability that a respondent chooses alternative 𝑖 in question 𝑞 is

    \[ \textbf{p}_{q,i}=\frac{\exp\funcapply(\textbf{X}_{q,i}\mathbit{\beta})}{\sum_{j=1}^{J}{\exp\funcapply(\textbf{X}_{q,j}\mathbit{\beta})}} \]

where J is the number of alternatives per question and \textbf{p}_q is a vector of length J.

We first construct the Fisher information matrix M using the formula below

    \[ {\textbf{M}}({\textbf{X}},{\beta})=\sum_{q=1}^{Q}{{\textbf{X}}^\prime}_q({\textbf{P}}_q-{\textbf{p}}_q{\textbf{p}}_q\prime)}{\textbf{X}}_q \]

where

    \[ {\textbf{P}}_q=\textrm{diag}({\textbf{p}}_\mathbit{q}).\]

Dp-error is a function of the determinant of the information matrix

    \[ D_P(\textbf{X})=\left|\textbf{M}(\textbf{X},{\beta})\right|^{-1/K} \]

where K is the number of parameters in β.

D0-error

The simpler D0-error assumes that respondents have no preference for any of the attribute levels, which is equivalent to assuming that β=0. The logit probabilities \textbf{p}_{q,j} become 1/J for all questions and alternatives. The information matrix formula in this case reduces to

    \[ \textbf{M}(\textbf{X})=\sum_{q=1}^{Q}{{\textbf{X}^\prime}_q\textbf{X}_q-J^{-1}({\textbf{X}^\prime}_q\textbf{1}_J)({\textbf{1}^\prime}_J\textbf{X}_q)} \]

where 1J is a vector of J ones. Apart from this difference, the D0-error is computed using the same function of the determinant as for DP-error. A design created by optimizing D0-error is known as a utility-neutral design.

DB-error

Lastly, DB-error assumes that respondent parameters are distributed according to a probability distribution, which is often a multivariate normal distribution with a diagonal covariance matrix.

DB-error is simply the integral of DP-error over the prior distribution of respondent parameters

    \[ D_B(\textbf{X})=\int{\log\funcapply\left|\textbf{M}\left(\textbf{X},\mathbit{\beta}\right)\right|\pi(\mathbit{\beta})d\mathbit{\beta}} \]

where π(β) denotes the density of the prior distribution at β. A design created by optimizing DB-error is also known as a Bayesian design.

D-efficiency

D-efficiency is a relative measure based on comparing one design X against a benchmark design X*. You can compute D-efficiency for any of the three D-errors

    \[ \textrm{D\mathrm{-}eff}(\textbf{X},\ \textbf{X}^\ast)=\frac{D(\textbf{X}^\ast)}{D(\textbf{X})} \]

If the benchmark design has a lower D-error, the D-efficiency has a range from 0 to 1, and it will be close to 1 if the design is close to the benchmark.

Ready to find out more about D-Errors and assessing the quality of your design? Check out the Displayr blog