slide 12.1 judgment and choice mathematicalmarketing chapter 12 judgment and choice this chapter...
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Slide 12.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
Slide 12.Slide 12.22Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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.Slide 12.33Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.44Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
But the Data Never Looked Like That
1.0
0n
Pr(Detect) .5
Slide 12.Slide 12.55Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.66Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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)
Slide 12.Slide 12.77Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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:
Slide 12.Slide 12.88Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
Graphical Picture of What We Just Did
z
)zPr(
0
1
Pr(Detection)
0z
)zPr(1
0is
)sPr( i
is
2
is
is
Slide 12.Slide 12.99Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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 12.Slide 12.1010Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
So Why Do Detection Probabilities Not Look Like a Step Function?
dims1
mediums2
brights3
Slide 12.Slide 12.1111Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
Paired Comparison Data: Pr(Row Brand > Column Brand)
A B C
A - .6 .7
B .4 - .2
C .3 .8 -
Slide 12.Slide 12.1212Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
Assumptions of the Thurstone Model
iiiess
ei ~ N(0, )
Cov(ei, ej) = ij = rij
is js
Draw siDraw sj
Is si > sj?
2
i
Slide 12.Slide 12.1313Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.1414Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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"
Slide 12.Slide 12.1515Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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?
Slide 12.Slide 12.1616Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.1717Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.1818Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
Minimum Pearson 2
.)ss(p 2
j
2
ijiij
t
i
t
ijij
2
ijij2
pn
)pnnp(ˆ
Same model:
Different objective function
Slide 12.Slide 12.1919Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.2020Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.2121Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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:
Slide 12.Slide 12.2222Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.2323Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.2424Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.2525Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
The Thresholds or Cutoffs
c1 c2 c3 (cJ-1)c0 = - c4 = +
Slide 12.Slide 12.2626Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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
Slide 12.Slide 12.2727Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
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"
Slide 12.Slide 12.2828Judgment Judgment
and Choiceand ChoiceMathematicalMathematicalMarketingMarketing
The Theory of Signal Detectability
Response
S N
RealityS Hit Miss
N False Alarm
Correct Rejection