A-optimal designs under a linearized model for discrete choice experiments

Abstract

Discrete choice experiments have proven useful in areas such as marketing, government planning, medical studies and psychological research, to help understand consumer preferences. To aid in these experiments, several groups of authors have contributed to the theoretical development of D-optimal and A-optimal discrete choice designs under the multinomial logit (MNL) model. In the setting in which the class of feasible designs is too large for complete search, Sun and Dean (J Stat Plann Inference 170:144–157, 2016) proposed a construction method for A-optimal designs for estimating a set of orthonormal contrasts in the option utilities via a linearization of the MNL model. In this paper, we show that the set of A-optimal designs that result from this linearization may or may not include the optimal design under the MNL model itself. We provide an alternative linearization that leads to an information matrix which coincides with that under the MNL model and, consequently, selects the same set of designs as being A-optimal. We obtain a bound for the average variance of a set of contrasts of interest under the MNL model, and show that the construction method of Sun and Dean (2016) can be used to identify A-optimal and A-efficient designs under the MNL model for both equal and unequal utilities.

Appendix

Proof of Lemma 1

For the ith choice set, let $\varDelta _i$ be an $L\times m$ index matrix with 1 in column j, row w, if the jth option in $C_i$ is option w, i.e. if $t_{i_j}\in C_i$ is $t_{w}\in T$. Then using the definition of $F_{i}$, $F_{i_j}$ and $f_{i_j,i_k}$ in Sect. 3.1 and $Z_i$ and $D_i$ in (9), it follows that

$$\begin{aligned} F_{i_j}= p_{i_j} \sum _{k=1 (k\ne j)}^{m} -p_{i_k} f_{i_j,i_k} = \left[ \begin{array}{c} 0\\ \vdots \\ 0\\ p_{i_j}\\ 0\\ \vdots \\ 0 \end{array}\right] + p_{i_j} \varDelta _i\left[ \begin{array}{c} -p_{i_1}\\ \vdots \\ -p_{i_2}\\ \vdots \\ -p_{i_j}\\ \vdots \\ -p_{i_m} \end{array}\right] = \left[ \begin{array}{c} 0\\ \vdots \\ 0\\ p_{i_j}\\ 0\\ \vdots \\ 0 \end{array}\right] -p_{i_j}\varDelta _iD_i{\varvec{1}}_m . \end{aligned}$$

Therefore,

$$\begin{aligned} F_i= & {} \left[ F_{i_1}, F_{i_2}, \ldots , F_{i_m}\right] = ~\varDelta _iD_i - \varDelta _iD_i {\varvec{1}}\left[ p_{i_1}, p_{i_2}, \ldots , p_{i_m}\right] \nonumber \\= & {} ~\varDelta _i (D_i - D_i {\varvec{1}}{\varvec{1}}^\top D_i) . \end{aligned}$$

(23)

From (9),

$$\begin{aligned} D_iZ_{i}= & {} \left[ \begin{array}{c} p_{i_1}h^\top _{i_1}\\ \vdots \\ p_{i_j}h^\top _{i_j}\\ \vdots \\ p_{i_m}h^\top _{i_m} \end{array}\right] - \left[ \begin{array}{c} p_{i_1}\\ \vdots \\ p_{i_j}\\ \vdots \\ p_{i_m} \end{array}\right] \left( \sum _{k=1}^mp_{i_j}h^\top _{i_j}\right) \nonumber \\= & {} D_i\varDelta _i^{\top }B^{\top } - (D_i{\varvec{1}})({\varvec{1}}^{\top }D_i\varDelta _i^{\top }B^{\top }) \nonumber \\= & {} \left( D_i - D_i{\varvec{1}}{\varvec{1}}^{\top }D_i\right) \varDelta _i^{\top }B^{\top } \nonumber \\= & {} F_{i}^T B^{\top } . \end{aligned}$$

(24)

The result follows since $Z=[Z_1^{\top }, \ldots , Z^{\top }_N]^{\top }$ and $F^{\top }=[F_1, \ldots , F_N]^{\top }$. $\square$