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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 Mathematical Marketing Slide 14.1 Nonmetric Scaling Let’s Turn Things On Their Heads ANOVA: Assume metric data, test for additivity Nonmetric Conjoint: Assume additivity, test for metric data Mathematical Marketing Slide 14.2 Nonmetric Scaling 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 Mathematical Marketing Slide 14.3 Nonmetric Scaling The Steps in Detail: Step 0. Copy the dependent variable, y* = y where y contains ordinal factorial data. Since the yi are ordinal, I can apply any monotonic transformation to the yi* Mathematical Marketing Slide 14.4 Nonmetric Scaling Step 1 – The Usual Least Squares We use least squares to fit an additive model: y* Xβ e yˆ * e where X is a design matrix for example, 1 1 1 1 1 1 x 1 1 1 1 1 1 Mathematical Marketing Slide 14.5 Nonmetric Scaling Step 1 Scalar Representation Looking at row effect i and column effect j we might have * y ij αi β j eij * yˆ ij eij Note that since the yij* 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 Mathematical Marketing Slide 14.6 Nonmetric Scaling Step 2 – Optimal Scaling Find the monotone transformation that will improve the fit of the above model as much as possible. y H m [yij , ŷ ] * ij * ij We modify y* so as to improve the fit of the model Mathematical Marketing Slide 14.7 Nonmetric Scaling Constraints on Hm Assuming no ties, we arrange the original (ordinal) data in sequence: y1 y 2 y n To honor the ordinal scale assumption, we impose the following constraints on the optimally rescaled (transformed) data: y1* y*2 y*n Mathematical Marketing Slide 14.8 Nonmetric Scaling The Shephard Diagram Scaled Values Transformed Data y* Model data ŷ* Ranked Values 1 Mathematical Marketing 2 3 4 5 Slide 14.9 Nonmetric Scaling Minimize Stress to Optimize Hm STRESS * * 2 ( y ŷ i i) i * *)2 ( ŷ ŷ i i where Mathematical Marketing ŷ * is the average of the * ŷi Slide 14.10 Nonmetric Scaling 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 Mathematical Marketing Slide 14.11 Nonmetric Scaling The Geometric Model Brands judged highly similar (proximal) are represented near each other in a perceptual space. dij - similarity judgment between brand i and brand j Mathematical Marketing Slide 14.12 Nonmetric Scaling Nonmetric MDS proceeds identically, but the model is a distance model, rather than an additive model 3 j = [2 2] d̂*ij r (x im x jm ) 2 d̂ *ij 2 m 1 i = [1 1] (d̂ *ij ) 2 (1 2) 2 (1 2) 2 0 0 Mathematical Marketing 1 2 3 Slide 14.13 Nonmetric Scaling In individual differences scaling (INDSCAL), each subject’s data are analyzed as follows: Typical element: Consumer i rating brands j and k d (jki ) Brands d̂ (i)jk * r w im (x jm x km ) 2 m 1/2 Repondents Brands Mathematical Marketing Slide 14.14 Nonmetric Scaling Each subject’s perceptual map might be different: A B C D E F G H I Subject 1 Subject 2 Group Space A B C D E F G H C Subject 1’s Space Mathematical Marketing Subject Weights A D G B E H C F I Subject 2’s Space Slide 14.15 Nonmetric Scaling Geometric Models of Preference: The Vector Model Subject i Brand A Brand B Subject i Isopreference lines Mathematical Marketing Subject i Two consumers who disagree Slide 14.16 Nonmetric Scaling The Ideal Brand Model Is also called the Unfolding Model i Subjects i i C A B C D B D Brands Mathematical Marketing A Slide 14.17 Nonmetric Scaling The Ideal Brand Model in Two Dimensions i A B D C i Isopreference circles Mathematical Marketing Two consumers who disagree Slide 14.18 Nonmetric Scaling