slide 12.1 judgment and choice mathematicalmarketing chapter 12 judgment and choice this chapter...

Slide 12.11Judgment Judgment

and Choiceand ChoiceMathematicalMathematicalMarketingMarketing

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



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



The Early Concept of a “Threshold”

1.0

0n

Pr(Detect) .5

n2n1 n3

1.0

0

Pr(n Perceived > n2) .5

Absolute Detection

Difference Detection

Physical measurement



But the Data Never Looked Like That

1.0

0n

Pr(Detect) .5



A Simple Model for Detection

iii ess

ei ~ N(0, 2) so that

),s(N~s 2ii

Pr[Detect stimulus i] = Pr[si s0] .

si is the psychological impact of stimulus i

We make this assumption

which then implies

If si exceeds the threshold, you see/hear/feel it

00 sWe also assume



Our Assumptions Imply That the Probability of Detection Is…

0

i22

iii ds]2/)ss(exp[2

1p

(Note missing left bracket in Equation 12.6 in book.)

Converting to a z-score we get

is0

i

2i

i dz2

zexp

2

1p

(Note missing subscript i on the z in book)



Making the Equation Simpler

]/s[

dz2

zexp

2

1p

i

s2i

i

i

is0

i

2i

i dz2

zexp

2

1p

But since the normal distribution is symmetric about 0 we can say:



Graphical Picture of What We Just Did

z

)zPr(

0

1

Pr(Detection)

0z

)zPr(1

0is

)sPr( i

is

2

is

is



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)]



So Why Do Detection Probabilities Not Look Like a Step Function?

dims1

mediums2

brights3



Paired Comparison Data: Pr(Row Brand > Column Brand)

A B C

A - .6 .7

B .4 - .2

C .3 .8 -



Assumptions of the Thurstone Model

iiiess

ei ~ N(0, )

Cov(ei, ej) = ij = rij

is js

Draw siDraw sj

Is si > sj?

2

i



Deriving the E(si - sj) and V (si - sj)

pij = Pr(si > sj ) = Pr(si - sj > 0)

ji

jjiiji

ss

)es()es(E)ss(E

ji

2

j

2

i

ij

2

j

2

i

2

jij

ij

2

i

ji

r2

2

1

111)ss(V



ji

2

j

2

ijijiijr2)ss()ssPr(p

Predicting Choice Probabilities

)ss(Eji

)ss(Vji

For a ~ N[E(a), V(a)] we have

Pr [a 0] = [E(a) / V(a)]

Below si - sj plays the role of "a"



Thurstone Case III

2

j

2

ijijiij)ss()ssPr(p

2

1

1s

2

t

2

3

2

2t32,,,,s,,s,s

= 0 = 1

How many unknowns are there?

How many data points are there?



Unweighted Least Squares Estimation

2

j

2

iji

1

ji

1 )ss()]ss[Pr(

2

t

2

1tt1tt)1t(

2

3

2

13113

2

2

2

12112

)ss(z

)ss(z

)ss(z

21t

1i

t

1ijijij)zz(f



Conditions Needed for Minimizing f

0

0

0

0

0

0

/f

/f

/f

s/f

s/f

s/f

2

t

2

2

2

1

t

2

1



Minimum Pearson 2

.)ss(p 2

j

2

ijiij

t

i

t

ijij

2

ijij2

pn

)pnnp(ˆ

Same model:

Different objective function



Matrix Setup for Minimum Pearson 2

t)1t(1312

ppp p

t)1t(1312

pppˆ p

n

)p1(p]pp[V)p(V ijij

ijijij

V(p) = V

)ˆ()ˆ(ˆ 12 ppVpp



t

i

t

ijij

2

ijij2

np

)pnnp(ˆ

Modified Minimum Pearson 2

t

i

t

ijij

2

ijij2

pn

)pnnp(ˆ

Minimum Pearson 2

Simplifies the derivatives, and reduces the computational time required



Definitions and Background for ML Estimation

n

fp ij

ijfij = npij n

fnp1p ij

ijji

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

qp BA )Pr()Pr(

Note these definitions and identities:



ML Estimation of the Thurstone Model

1t

1i

t

1ij

ij

ij

ij

ij0)

fnp1(

fpl

1t

1i

t

1ijijijijij00)p1ln()fn(plnfL)ln(l

1t

1i

t

1ij

ij

ij

ij

ijA)

fnp1(

fpl

]LL[2ln2ˆ0A

A

02 ll

1t

1i

t

1ijijijijijAA)p1ln()fn(plnfL)ln(l

According to the Model According to the general alternative



Categorical or Absolute Judgment

Love Like Dislike Hate [ ] [ ] [ ] [ ]

s1 s3s2

1 2 3 4

Brand 1Brand 2Brand 3

Love Like Dislike Hate

.20 .30 .20 .30

.10 .10 .60 .20

.05 .10 .15 .70



Cumulated Category Probabilities

Brand 1

Brand 2

Brand 3

Love Like Dislike Hate.20 .30 .20 .30

.10 .10 .60 .20

.05 .10 .15 .70

Brand 1 .20 .50 .70 1.00

Brand 2 .10 .20 .80 1.00

Brand 3 .05 .15 .30 1.00

RawProbabilities

CumulatedProbabilities



The Thresholds or Cutoffs

c1 c2 c3 (cJ-1)c0 = - c4 = +



A Model for Categorical Data

ei ~ N(0, 2)

iiiess

]0scPr[]csPr[pijjiij

Probability that item i is placed in category j or less

Probability that the discriminal response to item i is less than the upper boundaryfor category j



The Probability of Using a Specific Category (or Less)

iijij

scp

Pr [a 0] = [E(a) / V(a)]

Below ci - sj is plays the role of "a"



The Theory of Signal Detectability

Response

S N

RealityS Hit Miss

N False Alarm

Correct Rejection

slide 12.1 judgment and choice mathematicalmarketing chapter 12 judgment and choice this chapter...

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