The BCHOICE Procedure (Experimental)

Example 27.6 Inference on Quantities of Interest

It is easy to give a model an odds interpretation in terms of a probability ratio. The odds of choosing alternative a instead of alternative b while holding all other attributes or conditions the same are

\[  \mbox{ODDS}_{ab} = \frac{P_ a}{P_ b} = \frac{\exp {(\beta _ a)}}{\exp {(\beta _ b)}} = \exp {(\beta _ a - \beta _ b)}  \]

where $P_ a$ and $P_ b$ are the probabilities of choosing alternative a and b, respectively.

In Alternative-Specific and Individual-Specific Effects, a study for travel demand was considered. Summary statistics of that study are displayed again in Output 27.6.1 for the purpose of illustration.

Output 27.6.1: PROC BCHOICE Posterior Summary Statistics

Posterior Summaries and Intervals
Parameter N Mean Standard
95% HPD Interval
Mode Auto 5000 -0.1678 0.7440 -1.7017 1.2396
Mode Plane 5000 -1.8794 1.2683 -4.6055 0.3801
TravTime 5000 -0.5695 0.2047 -0.9943 -0.2328

The overall probability ratio of choosing airplane as opposed to public transit (which is the reference category) simplifies to

\[  \mbox{ODDS}_{\mbox{(Plane vs. Transit)}} = \frac{P(\mbox{Plane})}{P(\mbox{Transit})} = \frac{\exp {(-1.88)}}{\exp {(0)}} = 0.15  \]

This ratio indicates that the likelihood of choosing planes as means of travel is 0.15 times that of choosing public transit. Public transit is a lot more favorable than planes.

The overall probability ratio of choosing a plane instead of an automobile is

\[  \mbox{ODDS}_{\mbox{(Plane vs. Auto)}} = \frac{P(\mbox{Plane})}{P(\mbox{Transit})} = \frac{\exp {(-1.88)}}{\exp {(-0.17)}} = 0.18  \]

You can derive the probabilities for any alternatives and any combinations of attributes, even imaginary ones, after you obtain the parameter estimates.

Although it is easy to obtain the point estimates, sometimes you might want to estimate other quantities, such as the standard deviations or various quantiles. If you are interested in making inference based on any quantities that are transformations of the random variables, you can do it either directly in PROC BCHOICE or by using the DATA step after you run the simulation.

Suppose you want to set the reference category as Plane so that you can directly compare the other two modes with it. You can specify REF=’PLANE’ in the CLASS statement and rerun the PROC BCHOICE simulation. You can also use the DATA step to calculate the quantities of interest. The following DATA step uses the simulated values of Mode_Auto and Mode_Plane from the output data set Bsamp to create a new series of posterior samples for Auto_Plane, a new variable that is created to directly compare airplane with automobile:

   data Transout;
      set Bsamp;
      Auto_Plane=Mode_Auto -Mode_Plane;

Then you can use some autocall macros to analyze the posterior samples of Auto_Plane. For example, the %POSTSUM macro provides summary statistics, and the %POSTINT macro provides equal-tail and HPD intervals as described in the section Autocall Macros for Postprocessing.

%postsum(data=Transout, var=Auto_Plane)
%postint(data=Transout, var=Auto_Plane)

You can also generate some ODS graphs of those posterior samples for diagnostic purposes, as described in the section Regenerating Diagnostics Plots.