Choice modelling - an Choice modelling - an introductionintroduction

Class experiment Everybody loves Chocolate-

fact:

Of the following choices:White Chewy NoNuts

Dark Chewy NoNuts

White Soft NoNuts

Dark Soft NoNuts

White Chewy Nuts

Dark Chewy Nuts

White Soft Nuts

Dark Soft Nuts

Which one do you choose?

introduction When we wish to look at launching new products or

change features of current products we would like to ascertain the impact this will have in the market place

We need to be able to model the effect of each change on a range of similar products

The way we do this is via choice modelling whereby each respondent examine a number of market

scenarios and gets to choose which product they would purchase

We need to be able to answer the marketing questions We need to be able to model this appropriately

Introduction… There are 3 components to choice modelling:

Design of experiment Analysis of data Presentation of results

Firstly an example

Multinomial logitWe begin with a very simple example. In this example, each of ten subjects

was presented with eight different chocolate candies and asked to choose one.

The eight candies consist of the 23 combinations of dark or milk chocolate, soft or chewy centre, and nuts or no nuts. Each subject saw all eight candies and made one choice.

There are m = 8 attribute vectors in this example, one for each alternative. Let x

Dark/Milk = (1 = Dark, 0 = Milk),

Soft/Chewy = (1 = Soft, 0 = Chewy),

Nuts/No Nuts =(1 = Nuts, 0 = No Nuts).

The eight attribute vectors are

x1 = (0 0 0) (Milk, Chewy, No Nuts)

x2 = (0 0 1) (Milk, Chewy, Nuts )

x3 = (0 1 0) (Milk, Soft, No Nuts)

x4 = (0 1 1) (Milk, Soft, Nuts )

x5 = (1 0 0) (Dark, Chewy, No Nuts)

x6 = (1 0 1) (Dark, Chewy, Nuts )

x7 = (1 1 0) (Dark, Soft, No Nuts)

x8 = (1 1 1) (Dark, Soft, Nuts )

Multinomial logit modelExperimental choice data such as these are typically analyzed with

a multinomial logit model.

• The Multinomial Logit ModelThe multinomial logit model assumes that the probability that an

individual will choose one of the m alternatives, ci , from choice set C is

where xi is a vector of alternative attributes and is a vector of unknown parameters. U(c i) = xi is the utility for alternative ci, which is a linear function of the attributes.

The probability that an individual will choose one of the m alternatives, ci, from choice set C is the exponential of the utility of the alternative divided by the sum of all of the exponentiated utilities.

Hypothetical calculations

Probability Choice as a function of utility

Note: for pricing Data this is usuallya negative relationship

The input data 8 choices , 10 persons so 80 observations Typically, two variables are used to identify the choice sets,

subject ID and choice set within subject (for larger studies we aggregate this over ID)

The variable Subj is the subject number, and Set identifies the choice set within subject. The chosen alternative is indicated by c=1, which means first choice.

All second and subsequent choices are unobserved, so the unchosen alternatives are indicated by c=2,

The data

Fitting the Multinomial Logit Model

The data are now in the right form for analysis. In the SAS System, the multinomial logit model is fit with the SAS/STAT procedure PHREG (proportional hazards regression), with the ties=breslow option.

The likelihood function of the multinomial logit model has the same form as a survival analysis model fit by PROC

PHREG. See Statistics 764 Survival Analysis notes – Chapter 7

The codeproc phreg data=chocs outest=betas;

strata subj set;

model c*c(2) = dark soft nuts / ties=breslow;

label dark = ’Dark Chocolate’ soft = ’Soft Centre’ nuts = ’With Nuts’;

run;

The data= option specifies the input data set. The outest= option requests an output data set called BETAS

with the parameter estimates.

The strata statement specifies that each combination of the variables Set and Subj forms a set from which a choice was made. Each term in the likelihood function is a stratum.

There is one term or stratum per choice set per subject, and each is composed of information about the chosen and all the unchosen alternatives.

SAS output

Interpretation

“Model Fit Statistics” and “Testing Global Null Hypothesis: BETA=0,” contain the overall fit of the model.

The-2 LOG L statistic under “With Covariates” is 28.727 and the Chi-Square statistic is 12.8618 with 3 df (p=0.0049),

which is used to test the null hypothesis that the attributes do not influence choice.

Note that 41.589 (-2 LOG L Without Covariates, which is -2 LOG L for a model with no explanatory variables) minus

28.727 (-2 LOG LWith Covariates, which is -2 LOG L for a model with all explanatory variables) equals 12.8618

(Model Chi-Square, which is used to test the effects of the explanatory variables).

Probability of choice The parameter estimates are used next to construct the

estimated probability that each alternative will be chosen. The DATA step program uses the following formula to

create the choice probabilities.

Probabilities

Fabric Softener Example The study involves four fictitious fabric softener brand names

Sploosh, Plumbbob, Platter, and Moosey. Each choice set consists of each of these four brands and a

constant alternative Another. Each of the brands is available at three prices, $1.49, $1.99, and $2.49. Another is only

offered at $1.99. There are 50 subjects, each of which will see the same choice

sets.

Designing the experiment In order to do any choice model we need to construct an

experimental design Using SAS

We can use the %MKTRUNS autocall macro to help us choose the number of choice sets. All of the autocall macros used in this report are documented starting on page 261. To use this macro, you specify the number of levels for each of the factors. With four brands each with three prices, you specify four 3’s.

title ’Choice of Fabric Softener’;

%mktruns( 3 3 3 3 )

Output

The design

In this problem, the %MKTRUNS macro reports ten different sizes with no violations Ideally, we would like to have a manageable number of choice sets for people to evaluate and a design that is both orthogonal and balanced.

When violations are reported, orthogonal and balanced designs are not possible. While orthogonality and balance are not required, they are nice properties to have. With 4 three-level factors, the number of choice sets in all orthogonal and balanced designs must be divisible by 3 x 3 = 9.

In this example we would go for 18 runs.

The design ….

In the next steps, an efficient experimental design is created. We will use an autocall macro %MKTDES to create most of our designs.

When you invoke the %MKTDES macro for a simple problem, you only need to specify the factors, number of levels,

and number of runs. The macro does the rest.

For just main effects we simply type (note no second order effects are asked for here – usually we ask for them):

%let n = 18; /* n choice sets */

%mktdes(factors=x1-x4=3, n=&n)

proc print;

run;

Design output

Design

For now, notice that the macro found a perfect,

orthogonal and balanced, 100% efficient design consisting of three-level factors, x1-x4. The levels are the

integers 1 to 3.

Note that we would need to randomise the order of these eventual this design to consumers.

Design ..

What consumers see

Etc…

The Data

How the data needs to be formatted

Person #1 first 3 scenarios – note this assumes the price effect is the same for each brand – usually it’s different.

The analysis

proc phreg data=coded outest=betas;title2 ’Discrete Choice Model’;model c*c(2) = / ties=breslow;strata subj set;run;

proc phreg data=coded outest=betas;

title2 ’Discrete Choice Model’;

model c*c(2) = Sploosh Plumbbob Platter Moosey Another Price / ties=breslow;

strata subj set;

run;

The analysis