# The BCHOICE Procedure

### Example 27.5 Heterogeneity Affected by Individual Characteristics

Rossi, Allenby, and McCulloch (2005) studied a scanner panel data about purchases of margarine. The data were first analyzed in Allenby and Rossi (1991) and are about purchases of ten brands of margarine. This example considers a subset of data about six margarine brands: Parkay stick, Blue Bonnet stick, Fleischmann’s stick, a house brand stick, a generic stick, and Shedd’s Spread tub. There are 313 households, which made a total of 3,405 purchases. Information about a few demographic characteristics of these households (income and family size) is expected to have effects on the central location of the distribution of heterogeneity.

The data set, which is called Sashelp.Margarin, comes from the SASHELP library.

proc print data=Sashelp.Margarin (obs=24);
by HouseID Set;
id HouseID Set;
run;


The data for the first four choice sets are shown in Output 27.5.1.

Output 27.5.1: Data for the First Four Choice Sets

HouseID Set Choice Brand LogPrice LogInc FamSize
2100016 1 1 PPk -0.41552 3.48124 2
0 PBB -0.40048 3.48124 2
0 PFl 0.08618 3.48124 2
0 PHse -0.56212 3.48124 2
0 PGen -1.02165 3.48124 2
0 PSS -0.16252 3.48124 2

HouseID Set Choice Brand LogPrice LogInc FamSize
2100016 2 1 PPk -0.46204 3.48124 2
0 PBB -0.40048 3.48124 2
0 PFl -0.01005 3.48124 2
0 PHse -0.56212 3.48124 2
0 PGen -1.02165 3.48124 2
0 PSS -0.16252 3.48124 2

HouseID Set Choice Brand LogPrice LogInc FamSize
2100016 3 1 PPk -1.23787 3.48124 2
0 PBB -0.69315 3.48124 2
0 PFl -0.01005 3.48124 2
0 PHse -0.56212 3.48124 2
0 PGen -1.02165 3.48124 2
0 PSS -0.23572 3.48124 2

HouseID Set Choice Brand LogPrice LogInc FamSize
2100016 4 1 PPk -0.47804 3.48124 2
0 PBB -0.49430 3.48124 2
0 PFl -0.01005 3.48124 2
0 PHse -0.56212 3.48124 2
0 PGen -1.02165 3.48124 2
0 PSS -0.16252 3.48124 2

The variable HouseID represents the household ID, and each household made at least five purchases, which are defined by Set. The variable Choice represents the choice made among the six margarine brands for each purchase or choice set. The variable Brand has the value PPK for Parkay stick, PBB for Blue Bonnet stick, PFL for Fleischmann’s stick, PHse for the house brand stick, PGen for the generic stick, and PSS for Shedd’s Spread tub. The variable LogPrice is the logarithm of the product price. The variables LogInc and variable FamSize provide information about household income and family size, respectively.

The following statements fit a random-effects-only logit model using Gamerman Metropolis sampling

proc bchoice data=Sashelp.Margarin seed=123 nmc=40000 thin=2 nthreads=4;
class Brand(ref='PPk') HouseID Set;
model Choice = / choiceset=(HouseID Set);
random  Brand LogPrice / subject=HouseID remean=(LogInc FamSize)
type=un monitor=(1);
run;


The REMEAN=(LOGINC FAMSIZE) option in the RANDOM statement requests estimation of the nonzero mean of the random effects, which is a function of household income and family size. No fixed effects are specified in the MODEL statement. Summary statistics for the mean matrix of the random coefficients (), the covariance of the random coefficients (), and the random coefficients () for the first household are shown in Output 27.5.2.

Output 27.5.2: Posterior Summary Statistics

The BCHOICE Procedure

Posterior Summaries and Intervals
Parameter Subject N Mean Standard
Deviation
95% HPD Interval
REMean Brand PBB   20000 -1.2079 0.6384 -2.4293 0.0686
REMean Brand PFl   20000 -3.2484 1.9276 -7.0351 0.5201
REMean Brand PGen   20000 -5.1130 1.2332 -7.6390 -2.8030
REMean Brand PHse   20000 -3.2595 0.9194 -5.0725 -1.4761
REMean Brand PSS   20000 0.0915 1.2127 -2.3015 2.4981
REMean LogPrice   20000 -3.4148 0.8359 -5.0397 -1.7620
REMean Brand PBB LogInc   20000 0.0529 0.2114 -0.3485 0.4811
REMean Brand PFl LogInc   20000 0.7596 0.6208 -0.4726 1.9749
REMean Brand PGen LogInc   20000 -0.5079 0.4019 -1.2977 0.2698
REMean Brand PHse LogInc   20000 0.0315 0.3029 -0.5949 0.5931
REMean Brand PSS LogInc   20000 -0.6315 0.4131 -1.4645 0.1555
REMean LogPrice LogInc   20000 -0.2837 0.2817 -0.8434 0.2631
REMean Brand PBB FamSize   20000 -0.0274 0.0959 -0.2180 0.1572
REMean Brand PFl FamSize   20000 -0.7357 0.3059 -1.3283 -0.1267
REMean Brand PGen FamSize   20000 0.5775 0.1824 0.2269 0.9428
REMean Brand PHse FamSize   20000 0.2365 0.1357 -0.0291 0.4997
REMean Brand PSS FamSize   20000 0.0528 0.1974 -0.3347 0.4425
REMean LogPrice FamSize   20000 0.1010 0.1273 -0.1542 0.3424
RECov Brand PBB, Brand PBB   20000 2.2081 0.3730 1.5261 2.9615
RECov Brand PFl, Brand PBB   20000 1.8598 0.9106 0.1374 3.6776
RECov Brand PFl, Brand PFl   20000 12.0894 3.9050 5.5544 20.0864
RECov Brand PGen, Brand PBB   20000 1.9842 0.5697 0.8563 3.0829
RECov Brand PGen, Brand PFl   20000 1.3300 1.9065 -2.4622 5.1401
RECov Brand PGen, Brand PGen   20000 8.3897 1.4835 5.5924 11.3433
RECov Brand PHse, Brand PBB   20000 1.5148 0.4402 0.6799 2.3928
RECov Brand PHse, Brand PFl   20000 2.1554 1.3869 -0.5797 4.9030
RECov Brand PHse, Brand PGen   20000 5.7576 0.9570 3.8799 7.6015
RECov Brand PHse, Brand PHse   20000 5.4834 0.8441 3.9355 7.1970
RECov Brand PSS, Brand PBB   20000 1.1860 0.6287 -0.0412 2.4223
RECov Brand PSS, Brand PFl   20000 0.6096 1.9471 -3.0709 4.6460
RECov Brand PSS, Brand PGen   20000 4.8189 1.1738 2.5960 7.1020
RECov Brand PSS, Brand PHse   20000 3.4484 0.8805 1.7068 5.1649
RECov Brand PSS, Brand PSS   20000 8.7098 1.8047 5.4222 12.2276
RECov LogPrice, Brand PBB   20000 -0.2260 0.3462 -0.8764 0.4813
RECov LogPrice, Brand PFl   20000 2.1909 0.9010 0.5220 4.0796
RECov LogPrice, Brand PGen   20000 -0.9989 0.6371 -2.2175 0.2793
RECov LogPrice, Brand PHse   20000 -0.4254 0.5043 -1.3751 0.6150
RECov LogPrice, Brand PSS   20000 0.1734 0.6757 -1.1897 1.4765
RECov LogPrice, LogPrice   20000 2.1279 0.5373 1.1697 3.2261
Brand PBB HouseID 2100016 20000 -2.3546 1.0548 -4.4528 -0.3948
Brand PFl HouseID 2100016 20000 -3.9792 2.7840 -9.4491 0.9694
Brand PGen HouseID 2100016 20000 -6.5984 1.6472 -9.8601 -3.4056
Brand PHse HouseID 2100016 20000 -2.9722 1.2014 -5.5148 -0.8522
Brand PSS HouseID 2100016 20000 -3.3807 2.1701 -7.4577 0.7600
LogPrice HouseID 2100016 20000 -4.6129 1.2361 -6.9996 -2.1437

Table 27.11 collects the posterior means and standard deviations of that are shown in Output 27.5.2. The first column corresponds to the parameters that are specified in the model, namely the brands and price. The second column shows the average part-worths of each brand (versus the brand, Parkay stick) and the price at LogInc=0 and FamSize=0. The LogInc and FamSize columns list the modifying effects on the preference for each brand and price by household income and family size, respectively. Larger families show more interest in the generic and house brands and tend to stay away from the Fleischmann’s brand. For example, consider the part-worth estimates for Fleischmann’s. The posterior mean for REMean Brand PFI FamSize (the Fleischmann’s row and the Famsize column) is –0.74 with a standard deviation of 0.31, meaning that an additional unit increase in family size is associated with a reduction of 0.74 in the estimated part-worth for Fleischmann’s. In general, the demographics of households are only weakly associated with preference for brand and price. These results are in good agreement with those of Rossi, Allenby, and McCulloch (2005).

Table 27.11: Posterior Mean and Standard Deviation of

Parameter

Intercept

FamSize

Blue

Name

REMean Brand PBB

REMean Brand PBB FamSize

Bonnet

Mean

–1.21

0.05

–0.03

Std

0.64

0.21

0.10

Fleisch-

Name

REMean Brand PFI

REMean Brand PFI FamSize

mann’s

Mean

–3.25

0.76

–0.74

Std

1.93

0.62

0.31

Name

REMean Brand PGen

REMean Brand PGen FamSize

Generic

Mean

–5.11

–0.51

0.58

Std

1.23

0.40

0.18

Name

REMean Brand PHse

REMean Brand PHse FamSize

House

Mean

–3.26

0.03

0.24

Std

0.92

0.30

0.14

Shedd’s

Name

REMean Brand PSS

REMean Brand PSS FamSize

Mean

0.09

–0.63

–0.05

Std

1.21

0.41

0.20

Name

REMean LogPrice

REMean LogPrice FamSize

LogPrice

Mean

–3.41

–0.28

0.10

Std

0.84

0.28

0.13

Because the demographic variables are not zero-centered, the Intercept column shows the average part-worths of each brand and price for households with LogInc=0 and FamSize=0, which are not very meaningful. It is better to center demographic variables by their means, so that the posterior means listed in the Intercept column can be interpreted as the part-worths of a household that has an average income and average size.

Nevertheless, you can obtain the utilities of households that have any income levels and sizes. For example, the average part-worth of the Fleischmann’s brand for a household with average income (LogInc=3.1) and family size (FamSize=3) would be as follows, because the estimated LogInc coefficient is 0.76 and the estimated FamSize coefficient is –0.74 for Fleischmann’s:

You can obtain part-worths for all other brands and compare their popularity among average households.

The posterior means and standard deviations of the covariance matrix of the random coefficients () are displayed by parameters that are labeled "RECov Brand PBB, Brand PBB," "RECov Brand PFI, Brand PBB," and so on. Some of the diagonal terms are fairly large, indicating that there is quite a bit of heterogeneity between households in margarine brand preference and price sensitivity. The covariance between the generic and house brands, "RECov Brand PHse, Brand PGen," is fairly large, suggesting that household preferences for these two brands are highly correlated.

The next set of parameters, which are displayed in Output 27.5.2, contain the estimates for the random effects for the first household.