R for Marketing Research and Analytics

Chapter 13: Choice Modeling (Choice-based conjoint analysis)

Website for all data files:
http://r-marketing.r-forge.r-project.org/data.html

1. Ask customers to choose from among alternative designs (something consumers do every day).
   
2. Use a choice model to infer their preferences from the choices.
   
3. Predict whether they will buy alternative designs using the model.
   

Respondents can answer up to 50 of these types of questions, which gives us lots of data from which to infer their preferences.

Conjoint analysis is most commonly done using Sawtooth Software, which is a set of custom tools for designing a choice survey, fielding the questions on the web and analyzing the results. They have a slick new SAAS tool called Discover.

Other analysts like to use SAS which can design and estimate choice models. With SAS, fielding is on-your-own.

There are R packages that match (and in some cases exceed) the capabilities of SAS, making it the ideal platform for someone who wants to try out choice modeling without investing a lot in software.

Please follow along!

A more extensive version of the code I will go through today can be downloaded from the website for Chapman and Feit, R for Marketing Research and Analytics at http://r-marketing.r-forge.r-project.org/.

cbc.df <- 
  read.csv("http://goo.gl/5xQObB", 
           colClasses = c(seat = "factor", 
                          price = "factor", 
                          choice="integer"))
cbc.df$eng <- factor(cbc.df$eng, 
                    levels=c("gas", "hyb", "elec"))
cbc.df$carpool <- factor(cbc.df$carpool, 
                         levels=c("yes", "no"))
head(cbc.df[,c(-4, -5)])

  resp.id ques alt cargo  eng price choice
1       1    1   1   2ft  gas    35      0
2       1    1   2   3ft  hyb    30      0
3       1    1   3   3ft  gas    30      1
4       1    2   1   2ft  gas    30      0
5       1    2   2   3ft  gas    35      1
6       1    2   3   2ft elec    35      0

Note that the dependant variable in the last column is multinomial…a choice from among multiple options.

xtabs(choice ~ price, data=cbc.df)

price
  30   35   40 
1486  956  558 

xtabs(choice ~ cargo, data=cbc.df)

cargo
 2ft  3ft 
1312 1688 

xtabs(choice ~ seat, data=cbc.df)

seat
   6    7    8 
1164  854  982 

xtabs(choice ~ eng, data=cbc.df)

eng
 gas  hyb elec 
1444  948  608 

• As with any other multivariate data set, looking at univariate marginal summaries only tells part of the story.
  
• While we can see that customers tend to choose 6-seat minivans and tend to choose gas engines, it is hard to say whether seats or engines has a stronger influence on choice.
  
  • What if in the survey the 6-seat minivan options tended to have gas engines?
    
• As with many multivariate problems, the solution is a regression model (of a special type.)

library(mlogit)  

And do a bit of data formatting to tell mlogit which column is which in our choice data:

cbc.mlogit <- 
  mlogit.data(data=cbc.df, choice="choice", 
              shape="long", varying=3:6, 
              alt.levels=paste("pos",1:3), 
              id.var="resp.id")

It would be nice if mlogit used formula notation.

If \( X_{tj} \) is a vector of attributes of alternative \( j \) in choice task \( t \), the probability of choosing option \( j \) during choice \( t \) is

\( P(y_t=j) = \frac{exp(\beta X_{tj})}{\sum_{\textrm{all }j'}{exp(\beta X_{tj'})}} \)

where \( \beta \) is the vector of coefficients to be estimated.

MNL is generally estiamted by MLE or Bayesian methods. mlogit uses MLE.

predict.mnl <- function(model, data) { 
  data.model <- 
    model.matrix(
      update(model$formula, 0 ~ .), 
      data = data)[,-1]
  utility <- data.model%*%model$coef
  share <- exp(utility)/sum(exp(utility))
  cbind(share, data)
}

predict.mnl(m1, new.data)

        share seat cargo  eng price
8  0.11273356    7   2ft  hyb    30
1  0.43336911    6   2ft  gas    30
3  0.31917819    8   2ft  gas    30
41 0.07281396    7   3ft  gas    40
49 0.01669280    6   2ft elec    40
26 0.04521237    7   2ft  hyb    35

Suppose you were designing the first minivan to compete against the other five. Looks like you have a pretty good design.

m3 <- 
  mlogit(choice ~ 0 + seat + cargo + eng 
         + as.numeric(as.character(price)), 
         data = cbc.mlogit)

In a model like m3 where we estimate a single parameter for price, we can compute the average willingness-to-pay for a particular level of an attribute by dividing the coefficient for that level by the price coefficient.

coef(m3)["cargo3ft"]/
  (-coef(m3)["as.numeric(as.character(price))"]/1000)

cargo3ft 
2750.601 

So, on average, customers are willing to pay $2750 more for 3ft of cargo space versus 2ft.

lrtest(m1, m3)

Likelihood ratio test

Model 1: choice ~ 0 + seat + cargo + eng + price
Model 2: choice ~ 0 + seat + cargo + eng + as.numeric(as.character(price))
  #Df  LogLik Df  Chisq Pr(>Chisq)
1   7 -2581.6                     
2   6 -2582.1 -1 0.9054     0.3413

We know that different customers have different preferences. I might prefer 6-seats for my 3-person family, while you prefer the 8-seats.

We can accomodate this using a hierarchical model.

\( P(y_t=j) = \frac{exp(\beta_i X_{tj})}{\sum_{\textrm{all }j'}{exp(\beta_i X_{tj'})}} \)

where \( \beta_i \) is the coefficient vector for each customer, \( i \). We connect the customers together by assuming:

\( \beta_i \sim N_k(\mu_\beta, \Sigma_\beta) \)

This can also be estimated by MLE or Bayesian methods.

Define which parameters should be heterogeneous:

m1.rpar <- rep("n", length=length(m1$coef))
names(m1.rpar) <- names(m1$coef)
m1.rpar

   seat7    seat8 cargo3ft   enghyb  engelec  price35  price40 
     "n"      "n"      "n"      "n"      "n"      "n"      "n" 

m2.hier <- 
  mlogit(choice ~ 0 + seat + eng + cargo + price, 
                  data = cbc.mlogit, 
                  panel=TRUE, rpar = m1.rpar, 
                  correlation = TRUE)

predict.hier.mnl(m2.hier, data=new.data)

   colMeans(shares) seat cargo  eng price
8        0.08959674    7   2ft  hyb    30
1        0.46390066    6   2ft  gas    30
3        0.34231092    8   2ft  gas    30
41       0.05370156    7   3ft  gas    40
49       0.01797406    6   2ft elec    40
26       0.03251606    7   2ft  hyb    35

• These models can also be estimated by Bayesian MCMC using the ChoiceModelR package, which builds on the bayesm package.
• In the Bayesian framework, you can also introduce characteristics of the decision makers as covariates to \( \beta_i \) in the hierarchy.
  • For instance, you can allow preference for seats to be a function of whether the respondent uses their minivan for carpooling.
• The data formatting for ChoiceModelR is a little bit fussy and doesn't follow other hierarchial modeling packages like lme4.

• Designing conjoint surveys is its own topic (and one that I love to talk about!)
• Like any regression, the more data you have and the more orthogonal your attributes are, the more precise your parameter estimates will be.
• Selecting the attributes randomly works pretty well and is what I reccomend to first-timers.

Everything we've talked about can be used with any observed choices – not just choices from a survey.

• Selection of a subscription plan from a menu
• Purchases of televisions from a website
• Actual car purchases (good luck getting the data!)
  

You can also combine survey data with observational data (Feit, Beltramo and Feinberg, 2010)

• Segmentation of customers into groups based on observed behaviors or survey responses can be done using the cluster package.
• Market basket analysis helps find products that frequently occur together in customer transactions and can be done with the arules package.
• Perceptual maps that locate products in two-dimensions based on their perceived or measured attributes can be produced using principal components analysis with prcomp().
• Simple churn modeling can be done with logistic regression or hazard models using glm(). A better model might be the “buy 'til you die” models available in the btyd package.
• Hierarchical models can be extremely useful in marketing (since no two cusotmers are alike). Hierarchical linear models can be estimated using the lme4 or MCMCpack packages

Elea McDonnell Feit
Assistant Professor of Marketing Drexel University efeit@drexel.edu @eleafeit

This presentation is based on Chapter 13 of Chapman and Feit, R for Marketing Research and Analytics © 2015 Springer.

All code in the presentation is licensed under the Apache License, Version 2.0 (the “License”); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0\ Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.