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Choice modelling - an introduction

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Class experiment uEverybody loves Chocolate- fact: Of the following choices: WhiteChewyNoNuts DarkChewyNoNuts WhiteSoftNoNuts DarkSoftNoNuts WhiteChewyNuts DarkChewyNuts WhiteSoftNuts DarkSoftNuts Which one do you choose?

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introduction uWhen 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 uWe need to be able to model the effect of each change on a range of similar products uThe way we do this is via choice modelling nwhereby each respondent examine a number of market scenarios and gets to choose which product they would purchase nWe need to be able to answer the marketing questions nWe need to be able to model this appropriately

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Introduction… uThere are 3 components to choice modelling: n Design of experiment nAnalysis of data nPresentation of results uFirstly an example

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Multinomial logit We 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 2 3 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 )

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Multinomial logit model Experimental choice data such as these are typically analyzed with a multinomial logit model. The Multinomial Logit Model The multinomial logit model assumes that the probability that an individual will choose one of the m alternatives, c i, from choice set C is where x i is a vector of alternative attributes and is a vector of unknown parameters. U(c i ) = x i is the utility for alternative c i, which is a linear function of the attributes. The probability that an individual will choose one of the m alternatives, c i, from choice set C is the exponential of the utility of the alternative divided by the sum of all of the exponentiated utilities.

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Hypothetical calculations

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Probability Choice as a function of utility Note: for pricing Data this is usually a negative relationship

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The input data u8 choices, 10 persons so 80 observations uTypically, two variables are used to identify the choice sets, subject ID and choice set within subject (for larger studies we aggregate this over ID) uThe 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. uAll second and subsequent choices are unobserved, so the unchosen alternatives are indicated by c=2,

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The data

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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

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The code proc 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.

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SAS output

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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).

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Probability of choice uThe parameter estimates are used next to construct the estimated probability that each alternative will be chosen. uThe DATA step program uses the following formula to create the choice probabilities.

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Probabilities

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Fabric Softener Example uThe study involves four fictitious fabric softener brand names Sploosh, Plumbbob, Platter, and Moosey. uEach choice set consists of each of these four brands and a constant alternative Another. uEach of the brands is available uat three prices, $1.49, $1.99, and $2.49. Another is only offered at $1.99. uThere are 50 subjects, each of which will see the same choice sets.

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Designing the experiment uIn order to do any choice model we need to construct an experimental design uUsing 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 )

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Output

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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. uWhen 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.

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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;

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Design output

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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.

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Design..

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What consumers see Etc…

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The Data

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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.

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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;

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The analysis

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