Slide 12.1 Judgment and Choice MathematicalMarketing Chapter 12 Judgment and Choice This chapter covers the mathematical models behind the way that consumer decide and choose. We will discuss  The detection of sensory information  The detection of differences between two things  Judgments where consumers compare two things  A model for the recognition of advertisements  How multiple judgments are combined to make a single decision As usual, estimation of the parameters in these models will serve as an important theme for this chapter

Slide 12.2 Judgment and Choice MathematicalMarketing There Are Two Different Types of Judgments  Absolute Judgment Do I see anything? How much do I like that?  Comparative Judgment Does this bagel taste better than that one? Do I like Country Time Lemonade better than Minute Maid? Psychologists began investigating how people answer these sorts of questions in the 19th Century

Slide 12.3 Judgment and Choice MathematicalMarketing The Early Concept of a “Threshold” n Pr(Detect).5 n2n2 n1n1 n3n Pr(n Perceived > n 2 ).5 Absolute Detection Difference Detection Physical measurement

Slide 12.4 Judgment and Choice MathematicalMarketing But the Data Never Looked Like That n Pr(Detect).5

Slide 12.5 Judgment and Choice MathematicalMarketing A Simple Model for Detection e i ~ N(0,  2 ) so that Pr[Detect stimulus i] = Pr[s i  s 0 ]. s i is the psychological impact of stimulus i We make this assumption which then implies If s i exceeds the threshold, you see/hear/feel it We also assume

Slide 12.6 Judgment and Choice MathematicalMarketing Our Assumptions Imply That the Probability of Detection Is… (Note missing left bracket in Equation 12.6 in book.) Converting to a z-score we get (Note missing subscript i on the z in book)

Slide 12.7 Judgment and Choice MathematicalMarketing Making the Equation Simpler But since the normal distribution is symmetric about 0 we can say:

Slide 12.8 Judgment and Choice MathematicalMarketing Graphical Picture of What We Just Did Pr(Detection) 0 0

Slide 12.9 Judgment and Choice MathematicalMarketing A General Rule for Pr(a > 0) Where a Is Normally Distributed For a ~ N[E(a), V(a)] we have Pr [a  0] =  [E(a) /  V(a)]

Slide Judgment and Choice MathematicalMarketing So Why Do Detection Probabilities Not Look Like a Step Function?

Slide Judgment and Choice MathematicalMarketing Paired Comparison Data: Pr(Row Brand > Column Brand) ABC A B C.3.8 -

Slide Judgment and Choice MathematicalMarketing Assumptions of the Thurstone Model e i ~ N(0, ) Cov(e i, e j ) =  ij = r  i  j Draw s i Draw s j Is s i > s j ?

Slide Judgment and Choice MathematicalMarketing Deriving the E(s i - s j ) and V (s i - s j ) p ij = Pr(s i > s j ) = Pr(s i - s j > 0)

Slide Judgment and Choice MathematicalMarketing Predicting Choice Probabilities For a ~ N[E(a), V(a)] we have Pr [a  0] =  [E(a) /  V(a)] Below s i - s j plays the role of "a"

Slide Judgment and Choice MathematicalMarketing Thurstone Case III = 0= 1 How many unknowns are there? How many data points are there?

Slide Judgment and Choice MathematicalMarketing Unweighted Least Squares Estimation

Slide Judgment and Choice MathematicalMarketing Conditions Needed for Minimizing f

Slide Judgment and Choice MathematicalMarketing Minimum Pearson  2 Same model: Different objective function

Slide Judgment and Choice MathematicalMarketing Matrix Setup for Minimum Pearson  2 V(p) = V

Slide Judgment and Choice MathematicalMarketing Modified Minimum Pearson  2 Minimum Pearson  2 Simplifies the derivatives, and reduces the computational time required

Slide Judgment and Choice MathematicalMarketing Definitions and Background for ML Estimation f ij = np ij Assume that we have two possible events A and B. The probability of A is Pr(A), and the probability of B is Pr(B). What are the odds of two A's on two independent trials? Pr(A) Pr(A) = Pr(A) 2 In general the Probability of p A's and q B's would be Note these definitions and identities:

Slide Judgment and Choice MathematicalMarketing ML Estimation of the Thurstone Model According to the ModelAccording to the general alternative

Slide Judgment and Choice MathematicalMarketing Categorical or Absolute Judgment Love Like Dislike Hate [ ] [ ] [ ] [ ] s1s1 s3s3 s2s Brand 1 Brand 2 Brand 3 LoveLikeDislikeHate

Slide Judgment and Choice MathematicalMarketing Cumulated Category Probabilities Brand 1 Brand 2 Brand 3 Love Like Dislike Hate Brand Brand Brand Raw Probabilities Cumulated Probabilities

Slide Judgment and Choice MathematicalMarketing The Thresholds or Cutoffs c1c1 c2c2 c 3 (c J-1 ) c 0 = -  c 4 = + 

Slide Judgment and Choice MathematicalMarketing A Model for Categorical Data e i ~ N(0,  2 ) Probability that item i is placed in category j or less Probability that the discriminal response to item i is less than the upper boundary for category j

Slide Judgment and Choice MathematicalMarketing The Probability of Using a Specific Category (or Less) Pr [a  0] =  [E(a) /  V(a)] Below c i - s j is plays the role of "a"

Slide Judgment and Choice MathematicalMarketing The Theory of Signal Detectability Response SN Reality SHitMiss NFalse Alarm Correct Rejection