Conjoint analysis uses regression techniques to model consumer preferences and to estimate consumer utility functions. A problem with conventional conjoint analysis is that sometimes your estimated utilities do not make sense. Your results might suggest, for example, that the consumers would prefer to spend more on a product than to spend less. With PROC MCMC, you can specify constraints on the partworth utilities (parameter estimates). Suppose that the consumer product being analyzed is an offroad motorcycle. The relevant attributes are how large each motorcycle is (less than 300cc, 301–550cc, and more than 551cc), how much it costs (less than $5000, $5001–$6000, $6001–$7000, and more than $7000), whether or not it has an electric starter, whether or not the engine is counterbalanced, and whether the bike is from Japan or Europe. The preference variable is a ranking of the bikes. You could perform an ordinary conjoint analysis with PROC TRANSREG (see Chapter 97: The TRANSREG Procedure,) as follows:
options validvarname=any; proc format; value sizef 1 = '< 300cc' 2 = '300550cc' 3 = '> 551cc'; value pricef 1 = '< $5000' 2 = '$5000  $6000' 3 = '$6001  $7000' 4 = '> $7000'; value startf 1 = 'Electric Start' 2 = 'Kick Start'; value balf 1 = 'Counter Balanced' 2 = 'Unbalanced'; value orif 1 = 'Japanese' 2 = 'European'; run;
data bikes; input Size Price Start Balance Origin Rank @@; format size sizef. price pricef. start startf. balance balf. origin orif.; datalines; 2 1 2 1 2 3 1 4 2 2 2 7 1 2 1 1 2 6 3 3 1 1 2 1 1 3 2 1 1 5 3 4 2 2 2 12 2 3 2 2 1 9 1 1 1 2 1 8 2 2 1 2 2 10 2 4 1 1 1 4 3 1 1 2 1 11 3 2 2 1 1 2 ;
title 'Ordinary Conjoint Analysis by PROC TRANSREG'; proc transreg data=bikes utilities cprefix=0 lprefix=0; ods select Utilities; model identity(rank / reflect) = class(size price start balance origin / zero=sum); output out=coded(drop=intercept) replace; run;
The DATA step reads the experimental design and dependent variable Rank
and assigns formats to label the factor levels. PROC TRANSREG is run specifying UTILITIES, which requests a conjoint analysis.
The rank variable is reflected around its mean (1 12, 2 11, …, 12 1) so that in the analysis, larger partworth utilities correspond to higher preference. The OUT=CODED data set contains
the reflected ranks and a binary coding of the factors that can be used in other analyses. See Kuhfeld (2010) for more information about conjoint analysis and coding with PROC TRANSREG.
The Utilities table from the conjoint analysis is shown in Output 55.18.1. Notice the partworth utilities for price. The partworth utility for < $5000 is 0.25. As price increases to the $5000–$6000 range, utility decreases to –0.5. Then as price increases to the $6001–$7000 range, partworth utility increases to 0.5. Finally, for the most expensive bikes, utility decreases again to –0.25. In cases like this, you might want to impose constraints on the solution so that the partworth utility for price never increases as prices go up.
Output 55.18.1: Ordinary Conjoint Analysis by PROC TRANSREG
Ordinary Conjoint Analysis by PROC TRANSREG 
Utilities Table Based on the Usual Degrees of Freedom  

Label  Utility  Standard Error  Importance (% Utility Range) 
Variable 
Intercept  6.5000  0.95743  Intercept  
< 300cc  0.0000  1.35401  0.000  Class.< 300cc 
300550cc  0.0000  1.35401  Class.300550cc  
> 551cc  0.0000  1.35401  Class.> 551cc  
< $5000  0.2500  1.75891  13.333  Class.< $5000 
$5000  $6000  0.5000  1.75891  Class.$5000  $6000  
$6001  $7000  0.5000  1.75891  Class.$6001  $7000  
> $7000  0.2500  1.75891  Class.> $7000  
Electric Start  0.1250  1.01550  3.333  Class.Electric Start 
Kick Start  0.1250  1.01550  Class.Kick Start  
Counter Balanced  3.0000  1.01550  80.000  Class.Counter Balanced 
Unbalanced  3.0000  1.01550  Class.Unbalanced  
Japanese  0.1250  1.01550  3.333  Class.Japanese 
European  0.1250  1.01550  Class.European 
You could run PROC TRANSREG again, specifying monotonicity constraints on the partworth utilities for price:
title 'Constrained Conjoint Analysis by PROC TRANSREG'; proc transreg data=bikes utilities cprefix=0 lprefix=0; ods select ConservUtilities; model identity(rank / reflect) = monotone(price / tstandard=center) class(size start balance origin / zero=sum); run;
The output from this PROC TRANSREG step is shown in Output 55.18.2.
Output 55.18.2: Constrained Conjoint Analysis by PROC TRANSREG
Constrained Conjoint Analysis by PROC TRANSREG 
Utilities Table Based on Conservative Degrees of Freedom  

Label  Utility  Standard Error  Importance (% Utility Range) 
Variable 
Intercept  6.5000  0.97658  Intercept  
Price  0.1581  .  7.143  Monotone(Price) 
< $5000  0.2500  .  
$5000  $6000  0.0000  .  
$6001  $7000  0.0000  .  
> $7000  0.2500  .  
< 300cc  0.0000  1.38109  0.000  Class.< 300cc 
300550cc  0.0000  1.38109  Class.300550cc  
> 551cc  0.0000  1.38109  Class.> 551cc  
Electric Start  0.2083  1.00663  5.952  Class.Electric Start 
Kick Start  0.2083  1.00663  Class.Kick Start  
Counter Balanced  3.0000  0.97658  85.714  Class.Counter Balanced 
Unbalanced  3.0000  0.97658  Class.Unbalanced  
Japanese  0.0417  1.00663  1.190  Class.Japanese 
European  0.0417  1.00663  Class.European 
This monotonicity constraint is one of the few constraints on the partworth utilities that you can specify in PROC TRANSREG.
In contrast, PROC MCMC enables you to specify any constraint that can be written in the DATA step language. You can perform
the restricted conjoint analysis with PROC MCMC by using the coded factors that were output from PROC TRANSREG. The data set
is Coded
.
The likelihood is a simple regression model:
where rank
is the response, the covariates are ‘< 300cc’n
, ‘300500cc’n
, ‘< $5000’n
, ‘$5000  $6000’n
, ‘$6001  $7000’n
, ‘Electric Start’n
, ‘Counter Balanced’n
, and Japanese
. Note that OPTIONS VALIDVARNAME=ANY enables PROC TRANSREG to create names for the coded variables with blanks and special
characters. That is why the nameliteral notation (‘variablename’n) is used for the input data set variables.
Suppose that there are two constraints you want to put on some of the parameters: one is that the parameters for ‘< $5000’n
, ‘$5000  $6000’n
, and ‘$6001  $7000’n
decrease in order, and the other is that the parameter for ‘Counter Balanced’n
is strictly positive. You can consider a truncated multivariate normal prior as follows:
with the following set of constraints:






The condition that reflects an implied constraint that, by definition, 0 is the utility for the highest price range, > $7000
, which is the reference level for the binary coded price variable. The following statements fit the desired model:
title 'Bayesian Constrained Conjoint Analysis by PROC MCMC'; proc mcmc data=coded outpost=bikesout ntu=3000 nmc=50000 propcov=quanew seed=448 diag=none; ods select PostSummaries; array pw[4] pw5000 pw5000_6000 pw6001_7000 pwCounterBalanced; array sigma[4,4]; array mu[4]; begincnst; call identity(sigma); call mult(sigma, 100, sigma); call zeromatrix(mu); endcnst; parms intercept pw300cc pw300_550cc pwElectricStart pwJapanese tau 1; parms pw5000 0.3 pw5000_6000 0.2 pw6001_7000 0.1 pwCounterBalanced 1; beginnodata; prior intercept pw300: pwE: pwJ: ~ normal(0, var=100); if (pw5000 >= pw5000_6000 & pw5000_6000 >= pw6001_7000 & pw6001_7000 >= 0 & pwCounterBalanced > 0) then lp = lpdfmvn(pw, mu, sigma); else lp = .; prior pw5000 pw5000_6000 pw6001_7000 pwC: ~ general(lp); prior tau ~ gamma(0.01, iscale=0.01); endnodata; mean = intercept + pw300cc * '< 300cc'n + pw300_550cc * '300550cc'n + pw5000 * '< $5000'n + pw5000_6000 * '$5000  $6000'n + pw6001_7000 * '$6001  $7000'n + pwElectricStart * 'Electric Start'n + pwCounterBalanced * 'Counter Balanced'n + pwJapanese * Japanese; model rank ~ normal(mean, prec=tau); run;
The two ARRAY statements allocate a dimensional array for the prior covariance and an array of size 4 for the prior means. In the BEGINCNST and ENDCNST statements, the CALL IDENTITY function sets sigma
to be an identity matrix; the CALL MULT function sets sigma
’s diagonal elements to be 100 (the diagonal variance terms); and the CALL ZEROMATRIX function sets mu
to be a vector of zeros (the prior means). For matrix functions in PROC MCMC, see the section Matrix Functions in PROC MCMC.
There are two PARMS statements, with each of them naming a block of parameters. The first PARMS statement blocks the following: the intercept, the two size parameters, the one starttype parameter, the one origin parameter, and the precision. The second PARMS statement blocks the three price parameters and the one balance parameter, parameters that have the constraint multivariate normal prior. The second PARMS statement also specifies initial values for the parameter estimates. The initial values reflect the constraints on these parameters. The initial partworth utilities all decrease from 0.3 to 0.2 to 0.1 to 0.0 (for the implicit reference level) as the prices increase. Also, the initial partworth utility for the counterbalanced engine is set to a positive value, 1.
In the PRIOR statements, regression coefficients without constraints are given an independent normal prior with mean at 0 and variance
of 100. The next IFELSE construction imposes the constraints. When these constraints are met, pw5000, pw5000_6000, pw6001_7000, pwCounterBalanced
are jointly distributed as a multivariate normal prior with mean mu
and covariance sigma
. Otherwise, the prior is not defined and lp
is assigned a missing value. The parameter tau
is given a gamma prior, which is a conjugate prior for that parameter.
The model specification is linear. The mean is comprised of an intercept and the sum of terms like pw300cc * ‘< 300cc’n
, which is a parameter times an input data set variable. The MODEL statement specifies that the linear model for rank
is normally distributed with mean mean
and precision tau
.
The MCMC results are shown in Output 55.18.3.
Output 55.18.3: MCMC Results
Bayesian Constrained Conjoint Analysis by PROC MCMC 
Posterior Summaries  

Parameter  N  Mean  Standard Deviation 
Percentiles  
25%  50%  75%  
intercept  50000  2.2570  2.5131  0.8315  2.3661  3.8234 
pw300cc  50000  0.00983  2.4903  1.4461  0.00002  1.4821 
pw300_550cc  50000  0.0549  2.5097  1.4175  0.0106  1.4695 
pwElectricStart  50000  1.1319  2.1195  2.3710  1.0665  0.1562 
pwJapanese  50000  0.4567  2.1232  1.6836  0.4134  0.7903 
tau  50000  0.1135  0.0765  0.0578  0.0952  0.1499 
pw5000  50000  4.1614  2.1803  2.6082  3.7947  5.2823 
pw5000_6000  50000  2.6147  1.6188  1.4316  2.3400  3.4779 
pw6001_7000  50000  1.5040  1.2530  0.5432  1.1838  2.1456 
pwCounterBalanced  50000  5.8880  2.0638  4.6274  5.8946  7.1094 
The estimates of the partworth utility for the price categories are ordered as expected. This agrees with the intuition that there is a higher preference for a less expensive motor bike when all other things are equal, and that is what you see when you look at the estimated posterior means for the price partworths. The estimated standard deviations of the price partworths in this model are of approximately the same order of magnitude as the posterior means. This indicates that the partworth utilities for this subject are not significantly far from each other, and that this subject’s ranking of the options was not significantly influenced by the difference in price.
One advantage of Bayesian analysis is that you can incorporate prior information in the data analysis. Constraints on the parameter space are one possible source of information that you might have before you examine the data. This example shows that it can be accomplished in PROC MCMC.