# Part Worths - Conjoint Analysis

Part-Worths means level utilities for conjoint attributes. When multiple attributes come together to describe the total worth of the product concept, the utility values for the separate parts of the product (assigned to the multiple attributes) are part-worths.

## How is the Part-Worths calculated ?

We use the following algorithm to calculate CBC Conjoint Part-Worths:

• NOTATION

Let there be R respondents, with individuals r = 1 ... R

Let each respondent see T tasks, with t = 1 ... T

Let each task t have C configurations (or concepts), with c = 1 ... C (C in our case is usually 3 or 4)

If we have A attributes, a = 1 to A, with each attribute having La levels, l = 1 to La, then the part-worth for a

particular attribute/level is w’(a,l). It is this (jagged array) of part worths we are solving for in this exercise. We can

simplify this to a one-dimensional array w(s), where the elements are:

`     {w’(1,1), w’(1,2) ... w’(1,L1), w’(2,1) ... w’(A,LA)} with w having S elements.`

A specific configuration x can be represented as a one-dimensional array x(s), where x(s)=1 if the specific

level/attribute is present, and 0 otherwise.

Let Xrtc represent the specific configuration of the cth configuration in the tth task for the rth respondent. Thus the

experiment design is represented by the four dimensional matrix X with size RxTxCxS

If respondent r chooses configuration c in task t then let Yrtc=1; otherwise 0.

• UTILITY OF A SPECIFIC CONFIGURATION

The Utility Ux of a specific configuration is the sum of the part-worths for those attribute/levels present in the configuration, i.e. it is the scalar product x.w

• THE MULTI-NOMIAL LOGIT MODEL

For a simple choice between two configurations, with utilities U1 and U2, the MNL model predicts that configuration 1 will be chosen

`     EXP(U1)/(EXP(U1) + EXP(U2)) of the time (a number between 0 and 1).`

For a choice between N configurations, configuration 1 will be chosen

`     EXP(U1)/(EXP(U1) + EXP(U2) + ... + EXP(UN)) of the time.`

• MODELED CHOICE PROBABILITY

Let the choice probability (using MNL model) of choosing the cth configuration in the tth task for the rth respondent be:

`     Prtc=EXP(xrtc.w)/SUM(EXP(xrt1.w), EXP(xrt2.w), ... , EXP(xrtC.w))`

• LOG-LIKELIHOOD MEASURE

The Log-Likelihood measure LL is calculated as: Prtc is a function of the part-worth vector w, which is the set of part-worths we are solving for.

• SOLVING FOR PART-WORTHS USING MAXIMUM LIKELIHOOD

We solve for the part-worth vector by finding the vector w that gives the maximum value for LL. Note that we are solving for S variables.

This is a multi-dimensional non-linear continuous maximization problem, and requires a standard solver library. We use the Nelder-Mead Simplex Algorithm.

The Log-Likelihood function should be implemented as a function LL(w, Y, X), and then optimized to find the vector w that gives us a maximum. The responses Y, and the design X are given, and constant for a specific optimization. Initial values for w can be set to the origin 0.

The final part-worths w are re-scaled so that the part-worths for any attribute have a mean of zero, simply by subtracting the mean of the part-worths for all levels of each attribute.