Slide 14.1 Nonmetric Scaling MathematicalMarketing Chapter 14 Nonmetric Scaling Measurement, perception and preference are the main themes of this section. The sequence of topics is  Reversing statistical reasoning  Additive Conjoint Measurement  Multidimensional Scaling  Geometric Preference Models

Slide 14.2 Nonmetric Scaling MathematicalMarketing Let’s Turn Things On Their Heads  ANOVA: Assume metric data, test for additivity  Nonmetric Conjoint: Assume additivity, test for metric data

Slide 14.3 Nonmetric Scaling MathematicalMarketing Additive Conjoint Measurement Algorithmic Steps 0. Inititialize a metric version of y 1. Fit an additive model to the metric version 2. Rescale the metric version to improve the fit 3. If the model does not yet fit, go back to Step 2

Slide 14.4 Nonmetric Scaling MathematicalMarketing The Steps in Detail: Step 0. Copy the dependent variable, y* = y where y contains ordinal factorial data. Since the y i are ordinal, I can apply any monotonic transformation to the y i *

Slide 14.5 Nonmetric Scaling MathematicalMarketing Step 1 – The Usual Least Squares We use least squares to fit an additive model: where X is a design matrix for example,

Slide 14.6 Nonmetric Scaling MathematicalMarketing Step 1 Scalar Representation Looking at row effect i and column effect j we might have Note that since the y ij * are optimally rescaled anyway we do not have to worry about the grand mean. We modify  I and  j so as to improve the fit of the model

Slide 14.7 Nonmetric Scaling MathematicalMarketing Step 2 – Optimal Scaling Find the monotone transformation that will improve the fit of the above model as much as possible. We modify y * so as to improve the fit of the model

Slide 14.8 Nonmetric Scaling MathematicalMarketing Constraints on H m  Assuming no ties, we arrange the original (ordinal) data in sequence:  To honor the ordinal scale assumption, we impose the following constraints on the optimally rescaled (transformed) data:

Slide 14.9 Nonmetric Scaling MathematicalMarketing The Shephard Diagram Transformed Data y* Model data Ranked Values Scaled Values

Slide Nonmetric Scaling MathematicalMarketing Minimize Stress to Optimize H m whereis the average of the

Slide Nonmetric Scaling MathematicalMarketing Multidimensional Scaling – Proximity Data Collection The respondent’s job is to rank (or rate) pairs of brands as to how similar they are. Assume three brands A, B and C. Respondent ranks the three unique pairs as to similarity: ___ AB ___ AC ___ BC

Slide Nonmetric Scaling MathematicalMarketing The Geometric Model  Brands judged highly similar (proximal) are represented near each other in a perceptual space.  d ij - similarity judgment between brand i and brand j

Slide Nonmetric Scaling MathematicalMarketing i = [1 1] j = [2 2] Nonmetric MDS proceeds identically, but the model is a distance model, rather than an additive model

Slide Nonmetric Scaling MathematicalMarketing Brands Repondents Typical element: Consumer i rating brands j and k In individual differences scaling (INDSCAL), each subject’s data are analyzed as follows:

Slide Nonmetric Scaling MathematicalMarketing ABC DE F GHI ABC DE F GHI ABC DE F GHC Subject 1 Subject 2 Group SpaceSubject Weights Subject 1’s SpaceSubject 2’s Space Each subject’s perceptual map might be different:

Slide Nonmetric Scaling MathematicalMarketing Subject i Geometric Models of Preference: The Vector Model Subject i Brand A Brand B Two consumers who disagree Isopreference lines

Slide Nonmetric Scaling MathematicalMarketing Subjects i i Brands A B C D i C B A D The Ideal Brand ModelIs also called the Unfolding Model

Slide Nonmetric Scaling MathematicalMarketing  A B C D i i Two consumers who disagree Isopreference circles The Ideal Brand Model in Two Dimensions