# The BCHOICE Procedure

### Example 27.4 A Random-Effects-Only Logit Model

This example revisits the trash can study that is described earlier in this chapter in the getting-started section A Logit Model with Random Effects.

If you want to create a random-effects-only model using the random walk Metropolis sampling as suggested in Rossi, Allenby, and McCulloch (2005), you can add the ALG =RWM option to the PROC BCHOICE statement to specify the random walk Metropolis sampling algorithm, add the REMEAN option to the RANDOM statement to request estimation of the nonzero mean of the random effects, and not specify any fixed effects in the MODEL statement as follows:

proc bchoice data=Trashcan outpost=Postsamp seed=123 nmc=50000 thin=2
model Choice = / choiceset=(ID Task);
random Touchless Steel AutoBag Price80 / sub=ID remean
monitor=(1 to 5) type=un;
run;


Summary statistics for the mean of the random coefficients (), the covariance of the random coefficients (), and the random coefficients () for the first five individuals are shown in Output 27.4.1.

Output 27.4.1: Posterior Summary Statistics

The BCHOICE Procedure

Posterior Summaries and Intervals
Parameter Subject N Mean Standard
Deviation
95% HPD Interval
REMean Touchless   25000 1.7281 0.2711 1.1983 2.2651
REMean Steel   25000 1.0663 0.2661 0.5577 1.5998
REMean AutoBag   25000 2.2014 0.3520 1.5394 2.9100
REMean Price80   25000 -4.7018 0.6545 -6.0553 -3.4883
RECov Touchless, Touchless   25000 3.1178 1.1462 1.3552 5.3414
RECov Steel, Touchless   25000 -0.4517 0.9922 -2.1999 1.2156
RECov Steel, Steel   25000 2.6809 1.0342 1.0335 4.6468
RECov AutoBag, Touchless   25000 -0.7233 0.9521 -2.6063 1.0850
RECov AutoBag, Steel   25000 0.0301 0.8003 -1.4271 1.5548
RECov AutoBag, AutoBag   25000 3.6804 1.5597 1.1236 6.7726
RECov Price80, Touchless   25000 -1.3032 1.2372 -3.8305 0.8279
RECov Price80, Steel   25000 -1.6533 1.2926 -4.4595 0.5892
RECov Price80, AutoBag   25000 -2.3293 1.7191 -5.7326 0.6730
RECov Price80, Price80   25000 7.8969 3.3306 2.1101 14.3416
Touchless ID 1 25000 2.2223 1.0613 0.1295 4.2930
Steel ID 1 25000 0.7433 1.0823 -1.3053 2.9046
AutoBag ID 1 25000 3.7970 1.4406 1.1535 6.7240
Price80 ID 1 25000 -5.2465 2.3491 -10.0027 -0.9522
Touchless ID 2 25000 -0.0612 0.8563 -1.6726 1.6508
Steel ID 2 25000 -0.3962 0.8279 -2.1014 1.1849
AutoBag ID 2 25000 2.7067 1.3353 0.1149 5.3223
Price80 ID 2 25000 0.0271 1.3024 -2.5173 2.6137
Touchless ID 3 25000 1.1029 0.9627 -0.7106 3.1099
Steel ID 3 25000 1.6602 0.9599 -0.1585 3.5981
AutoBag ID 3 25000 0.2961 1.1676 -2.0028 2.5169
Price80 ID 3 25000 -0.9568 1.5615 -4.0940 2.0521
Touchless ID 4 25000 1.1556 0.9722 -0.7835 3.0279
Steel ID 4 25000 1.3922 1.0206 -0.5989 3.4310
AutoBag ID 4 25000 3.2143 1.4620 0.4988 6.1835
Price80 ID 4 25000 -6.6490 2.5043 -11.7925 -2.0763
Touchless ID 5 25000 0.4804 1.1831 -1.7839 2.8515
Steel ID 5 25000 2.7119 1.2432 0.4436 5.2051
AutoBag ID 5 25000 3.4736 1.6239 0.5679 6.8042
Price80 ID 5 25000 -5.1156 2.5674 -10.0957 -0.0400

The average part-worths () for touchless opening, steel material, automatic trash bag replacement, and price, which are shown as "REMean Touchless," "REMean Steel," "REMean AutoBag," and "REMean Price80," are very similar to the estimates of the fixed effects in the previous model as shown in Figure 27.9, indicating the equivalence of these two setups. The covariance estimates of the random coefficients () are also very similar.

The next set of parameters show the estimates for the random effects for the first five respondents (see Output 27.4.1). However, these estimates are no longer the deviation from the overall means but are their own effects. The part-worth for touchless opening for the first respondent (ID 1) is 2.2, which is close to the part-worth that is estimated in the model as shown in Figure 27.9.

A caterpillar plot is a side-by-side bar plot of the 95% intervals for multiple variables. A caterpillar plot is usually used to visualize and compare random-effects parameters, which can exist in large numbers in certain models. You can use the %CATER autocall macro to create a caterpillar plot. The %CATER macro requires you to specify an input data set and a list of variables that you want to plot.

In this example, the output data set Postsamp contains posterior draws for all the random-effects parameters. The following statements generate a caterpillar plot for Touchless:

%CATER(data=Postsamp, var=Touchless_ID_:)


Output 27.4.2 is a caterpillar plot of the random effects of touchless opening for the 104 participants in this study. It displays the heterogeneity in preferences for touchless opening among the sample. As shown, the individuals have diverse preferences for the feature of touchless opening for a trash can. Heterogeneity of preferences is an important aspect of the model, and ignoring its presence can lead to incorrect inferences.

Output 27.4.2: Caterpillar Plot of the Random-Effects Parameters

If you want to change the display of the caterpillar plot (for example, to use a different line pattern, color, or size of the markers), you must first modify the Stat.MCMC.Graphics.Caterpillar template and then call the %CATER macro again.