D-error is a way of summarizing how good or bad a design is at extracting information from respondents in a choice experiment. A design with a low D-error is better than a design with a high D-error, provided that both designs are for the same experiment; comparing D-error between designs for different experiments is meaningless. Many other related measures exist that also serve this purpose, such as D-optimality.

This article gives an overview of D-error and demonstrates how to compute D-error by working through an example. Concepts in this article are covered in more (mathematical) detail here.

Prior parameter assumptions

When computing D-error, a prior assumption about the respondent parameters needs to be made. D0-error assumes that all parameters are zero โ€” i.e., respondents have no preference for any of the attribute levels. DP-error assumes that all respondent parameters are equal to a parameter vector. On the other hand, 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.

DP-error example

A small choice experiment design is shown below:

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

The first step is to encode the design, with either dummy coding or effects coding. I use dummy coding in this example, and split the encoded design by its four questions:

    \[\textbf{X}_1=\left[\begin{matrix}0&1&1\\1&0&0\end{matrix}\right],\textbf{X}_2=\left[\begin{matrix}1&0&1\\1&1&0\end{matrix}\right]\]

    \[ \textbf{X}_3=\left[\begin{matrix}1&0&1\\0&1&0\end{matrix}\right],\textbf{X}_4=\left[\begin{matrix}1&1&1\\0&0&0\end{matrix}\right] \]

For DP-error, I assume that the respondent parameters are given by \mathbit{\beta}=[0.5, -0.8, 1.0]. The next step is to compute the multinomial logit probabilities, using the formula

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

where \mathbittextbf{p}_{q,i} refers to the probability of selecting alternative i out of J alternatives in question q. The probabilities are

    \[\mathbittextbf{p}_1=[0.43,0.57],\mathbittextbf{p}_2=[0.86,0.14],\mathbittextbf{p}_3=[0.91,0.09],\mathbittextbf{p}_4=[0.67,0.33]\]

These probabilities are then used to construct the Fisher information matrix, using the formula

    \[\textbf{M}=\sum_{q=1}^{Q}{{\textbf{X}^\prime}_q(\textrm{diag}(\mathbittextbf{p}_q)-p_q{p_q}\prime)}\textbf{X}_q\]

Plugging the values for \textbf{X} and \mathbittextbf{p} into the formula, the information matrix is

    \[\textbf{M}=\left[\begin{matrix}0.55&-0.11&0.06\\-0.11&0.67&0.26\\0.06&0.26&0.67\end{matrix}\right]\]

The DP-error is {|\textbf{M}|}^{-1/K}=1.72, where K=3 is the number of parameters.

D0-error example

Computing D0-error is just a special case of DP-error where \mathbit{\beta} is assumed to be a vector of zeros. In this case, the probabilities \mathbittextbf{p}_{q,i}=1/J (=1/2 in this example) and the information matrix is

    \[\textbf{M}=\left[\begin{matrix}0.75&-0.25&0.25\\-0.25&1&0\\0.25&0&1\end{matrix}\right]\]

The D0-error is {|\mathbittextbf{M}|}^{-1/K}=1.17.

DB-error (Bayesian) Example

DB-error is defined as the integral of DP-error over an assumed prior distribution of the respondent parameters. One way to compute this numerically is known as Monte Carlo estimation. It involves calculating the average DP-error of many sets of parameters randomly drawn from the prior distribution. To illustrate this, assume that the parameter distribution is multivariate normal with mean \mu=[0.5, -0.8, 1.0] and a covariance matrix that is diagonal with standard deviations \sigma=[0.4, 0.4, 0.4]. I draw 1000 samples from this distribution, as partially shown in the table below:

DrawParameter 1Parameter 2Parameter 3Dp-error
10.25-0.730.671.45
21.14-0.670.671.90
30.69-0.501.231.92
...............
10001.15-1.670.572.82

DB-error is estimated as the mean DP-error, which is 1.90 in this example. Another more computationally efficient but complicated way of computing the integral using quadrature exists1 but is beyond the scope of this article.

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References

1 Christopher M. Gotwalt, Bradley A. Jones & David M. Steinberg, โ€œFast Computation of Designs Robust to Parameter Uncertainty for Nonlinear Settings,โ€ Technometrics (2012) 51:1, 88-95, DOI: 10.1198/TECH.2009.0009