A Monte Carlo study of design-generating algorithms for the latent class mixed logit model

Abstract

We compare different procedures which generate $D_{B}$-efficient designs for choice-based conjoint analysis using the latent class mixed logit model which captures latent consumer heterogeneity in a flexible way. These procedures are the Coordinate-Exchange algorithm, the Relabel-Swap-Cycle algorithm, and the remaining six combinations of the individual components of the latter. Halton draws and a minimum potential design for prior draws both of which reduce computation times serve to determine $D_{B}$-errors of designs. We simulate choices for each set of generated designs and constellations which differ with respect to number of choice sets, number of clusters, within cluster heterogeneity, amount of stochastic error, relative cluster size and cluster similarity. Using these artificial choices we estimate parameters of the latent class mixed logit model in the next step. Designs are evaluated by TOPSIS scores which combine estimation accuracy and run time. ANOVA with TOPSIS scores as dependent variable shows that Relabel alone yields the best results of all procedures investigated. Coordinate-Exchange, Swap alone and the combination of Relabel and Swap turn out to be second best. Relabel also leads to much lower run times than the other procedures. We recommend to use Relabel and to avoid Cycle altogether because it performs worst.

Appendices

Appendix A: Model estimation

We set initial cluster shares $g_{c}^{0}$ to equal values, initial cluster means $b_{c}^{0}$ to a random value in the interval [$\beta - 2 \cdot \sigma , \beta + 2 \cdot \sigma$] with $\beta$ being the true value of the cluster means and $\sigma =0.1$, which corresponds to the higher value of factor 3. The initial covariance matrix is diagonal with all diagonal elements equaling 0.5. We decide to initialize parameters this way, because we focus on the comparison of design-generating procedures and do not intend to evaluate estimation algorithms.

The proper estimation algorithm is taken from Train (2008):

• 1. For each subject R random values are drawn from the cluster-specific normal distribution $N(b_{c}^{0},V_{c}^{0})$. The r-th draw for subject n in cluster c is labeled as ${\hat{\beta }}_{ncr}^{0}$. A preliminary study has shown that four draws ($R=4$) are sufficient.

• 2. For each subject in each cluster and each draw a weight is calculated as

  $$\begin{aligned} h_{ncr}=\frac{g_{c}^{0} K_{n}\left( {\hat{\beta }}_{ncr}^{0}\right) }{\sum _{c} g_{c}^{0} \sum _{r}K_{n}({\hat{\beta }}_{ncr}^{0}/R} \end{aligned}$$

  with $K_n(\hat{\beta }_{ncr}^0)$ as probability of the choices $y_n$ conditional on $\hat{\beta }_{ncr}^0$ ($y_{nt}$ is the index of the alternative chosen from choice set t)

  $$\begin{aligned} K_{n}({\hat{\beta }}_{ncr}^{0})=\prod _{t}{\frac{e^{x'_{ny_{nt}t} {\hat{\beta }}_{ncr}^{0}}}{\sum _j{e^{x'_{njt}{\hat{\beta }}_{ncr}^{0}}}}} \end{aligned}$$

• 3. Cluster means, covariances and shares are updated as follows:

  $$\begin{aligned} b_{c}^{1}= & {} \frac{\sum _{n}\sum _{r} h_{ncr}{\hat{\beta }}_{ncr}^{0}}{\sum _{n}\sum _{r}h_{ncr}}\\ V_{c}^{1}= & {} \frac{\sum _{n}\sum _{r} h_{ncr} [({\hat{\beta }}^{0}_{ncr} -b_{c}^{1})({\hat{\beta }}_{ncr}^{0}-b_{c}^{1})']}{\sum _{n}\sum _{r}h_{ncr}},\\ g_{c}^{1}= & {} \frac{\sum _{n}\sum _{r}h_{ncr}}{\sum _{n}\sum _{c'} \sum _{r}h_{nc'r}} \end{aligned}$$

• 4. If the absolute difference between all previous and updated parameter values is less than $10^{-6}$ or 10000 iterations have been made, the algorithm stops; otherwise, it sets the old parameter values to their updated values and goes back to step 1.

We need individual coefficients ${\hat{\beta }}_{n}$ to compute the estimation accuracy defined in expression (14). Individual coefficients are a linear combination of cluster specific mean coefficients weighted by cluster membership probabilities of the respective subject ${\hat{\beta }}_{n} = \sum _{c} u_{nc} \, b_{c}$, where $u_{nc}$ is the cluster membership probability of subject n to cluster c:

$$\begin{aligned} u_{nc} =\frac{\sum _{r}h_{ncr}}{\sum _{c'} \sum _{r}h_{nc'r}} \end{aligned}$$

Appendix B: TOPSIS

Let Z be an $(m \times n)$ matrix where m is the number of values, n the number of criteria and let $z_{ij}$ denote an element of this matrix.

• 1. Normalize to get

  $$\begin{aligned} r_{ij} =\frac{z_{ij}}{\sqrt{\sum _{i=1}^{m} z_{ij}^{2}}} \end{aligned}$$

• 2. Calculate weighted normalized values (we set the weight for estimation accuracy to 0.67 and that for run time to 0.33):

  $$\begin{aligned} t_{ij} = w_{j}r_{ij} \end{aligned}$$

• 3. Determine the best and worst value $t_{bj}$ and $t_{wj}$ for every criterion j.

• 4. Calculate the distance of every value i to the best and worst values:

  $$\begin{aligned} d_{ib} = \sqrt{\sum _{j=1}^{n}(t_{ij}-t_{bj})^{2}}, \quad d_{iw} = \sqrt{\sum _{j=1}^{n}(t_{ij}-t_{wj})^{2}} \end{aligned}$$

• 5. Calculate for each value i its similarity to the best values:

  $$\begin{aligned} s_{i}=\frac{d_{iw}}{d_{ib}+d_{iw}} \end{aligned}$$

$s_{i}$ lies in the interval [0, 1] with 0 marking the worst value and 1 marking the best value.