customer choice - personal.psu.edu · discount r1 0 0 r2 0 0 r3 0 0 r4 0 0 ... linear approximation...

Customer choice

Customer choice

Outline The binary choice model

□ Illustration

□ Specification of the binary choice model

□ Interpreting the results of binary choice models

□ ME output

The multinomial choice model

□ Illustration

□ Specification of the multinomial choice model

□ Interpreting the results of multinomial choice models

□ ME output

□ Properties of the multinomial choice model

Extensions of the basic choice model

Customer choice

The binary choice model: An illustration

Assume we have data on a consumer’s choice of a particular brand (1 when

the brand was chosen, 0 otherwise) across 30 purchase occasions; we also

know whether the brand was on sale on a particular occasion (0, 10, 15, 20,

or 30 cents below the regular price).

Observations / Choice data

Choice (0/1)

Discount

R1 0 0

R2 0 0R3 0 0R4 0 0R5 0 0

R6 0 0R7 0 0

R8 1 0

R9 0 10R10 0 10R11 0 10R12 0 10

R13 0 10R14 0 10

R15 1 10


Choice (0/1)

Discount

R16 0 15

R17 0 15R18 0 15R19 1 15R20 1 15

R21 1 15R22 0 20

R23 1 20

R24 1 20R25 1 20R26 1 20R27 1 30

R28 1 30R29 1 30

R30 1 30

Customer choice


(cont’d)

Discount Choice=1 Choice=0 P(Choice=1) P(Choice=0)

0 1.00 7.00 0.13 0.88

10 1.00 6.00 0.14 0.86

15 3.00 3.00 0.50 0.50

20 4.00 1.00 0.80 0.20

30 4.00 0.00 1.00 0.00

From the individual-level data we can construct a table

showing the number of choices of the brand in question, or the

probability of brand choice, at each discount level:

Customer choice

Probability(Choice=1) as a function of

discount size

Customer choice

Two issues:

□ If the discount is larger than 30 cents, the model

predicts a probability greater than 1.

□ We can only compute a probability if we have multiple

0/1 observations for each level of discount.

Solution:

□ Choose an S-shaped curve that restricts the

probability of choice to the interval of 0 to 1.

□ Assume that the 0/1 variable is a crude measure of

an underlying probability of choice.


(cont’d)

Customer choice

Using an S-shaped function to link the

probability of choice to level of discount

Customer choice

In general, the logit model is:

𝑃 𝑌 = 1 =𝑒𝑥𝑝 𝛼 + 𝛽1𝑥1 + 𝛽2𝑥2 + …+ 𝛽𝑘𝑥𝑘

1 + 𝑒𝑥𝑝 𝛼 + 𝛽1𝑥1 + 𝛽2𝑥2 + …+ 𝛽𝑘𝑥𝑘

=1

1 + 𝑒𝑥𝑝 −(𝛼 + 𝛽1𝑥1 + 𝛽2𝑥2 + …+ 𝛽𝑘𝑥𝑘)

□ is an intercept term that determines P(Y=1) when all

explanatory variables xi equal zero

□ i is the contribution of a unit change in xi to P(Y=1) (for a

given value of the other explanatory variables)

The binary (logit) choice model

Customer choice

The previous model is equivalent to:

𝑙𝑜𝑔𝑖𝑡 𝑃 𝑌 = 1 = 𝑙𝑜𝑔𝑃 𝑌 = 1

1 − 𝑃 𝑌 = 1= 𝛼 + 𝛽1𝑥1 + 𝛽2𝑥2 + …+ 𝛽𝑘𝑥𝑘

■ The logit of P(Y=1) is defined as the natural logarithm of the odds

of choice and is a linear function of the explanatory variables.

■ The interpretation of the i coefficients is straightforward, but

thinking in terms of logits is not straightforward.

The binary (logit) choice model (cont’d)

Customer choice

A utility interpretation of the logit model

A consumer’s utility (u) for the brand in question consists of a

deterministic (v) and stochastic () component:

𝑢 = 𝑣 + 𝜖

The deterministic component is a linear function of observable

characteristics of the brand:

𝑣 = 𝛼 + 𝛽1𝑥1 + 𝛽2𝑥2 + …+ 𝛽𝑘𝑥𝑘

The brand is chosen if the utility of choosing the brand is

greater than the utility of not choosing the brand.

The deterministic component of the utility of not choosing the

brand is fixed at zero.

is the deterministic component of utility for the brand when

all the observable characteristics are zero.

The i are the relative contributions of the xi to the brand’s

deterministic component of utility.

Customer choice

Interpreting the results of a

binary choice model

Linear approximation interpretation of the effects of the

explanatory variables:

□ The approximate rate of change in P(Y=1) for a unit

increase in xi (holding the other x’s constant) is given by

𝛽𝑖𝑃 𝑌 = 1 1 − 𝑃(𝑌 = 1)

Customer choice

Interpreting the results of a

binary choice model

Odds ratio interpretation:

□ For given values of the other explanatory variables, a unit

increase in xi changes the odds by a factor of exp(𝛽𝑖)

Elasticity interpretation:

□ For given values of the other explanatory variables, the

percent change in the probability of choice due to a one

percent increase in xi is:

𝛽𝑖 1 − 𝑃(𝑌 = 1) 𝑥𝑖

□ Not that this implies that elasticities are greater at lower

choice probabilities.

Customer choice

Using ME to estimate a binary logit model

Customer choice

Illustrative example:

ME output (Segment tab)

Coefficient Estimates [segment 1]Coefficient estimates of the Choice model. Coefficients in bold are statistically significant.

Variables / Coefficient estimatesCoefficient estimates

Standard deviation

t-statistic

Discount 0.199905 0.075153 2.659987Const-1 -2.93693 1.138345 -2.58Baseline n/a n/a

Elasticities [segment 1]Elasticities of coefficients.

Elasticities of Discount ResponseDummy (No

Choice)

Response 0.931691 -0.71248

Dummy (No Choice) 0 0

Customer choice

How to interpret the ME output

What’s the interpretation of the intercept (-2.94)?

What’s the probability of choice at a discount of 0?

What’s the interpretation of the slope (.20)?

What is the effect of a one-cent increase in the size of

the discount on the odds of choice?

If the current discount is 12.67 cents and we increase

the discount by one cent, what is the effect on the

probability of choice?

What is the percentage change in the probability of

choice due to a 1% increase in the size of the discount?

What is the percentage change in the probability of not

choosing the brand due to a 1% increase in the size of

the discount?

Customer choice

Interpreting the output

When there is not discount, the deterministic component of the utility

for the brand is -2.94; since the deterministic component of the utility

for no choice is 0, this implies that the probability of choice is low (in

fact, it is .05).

A one cent discount increases the log of the odds of choice by .20.

A one cent discount increases the odds of choice by a factor of

exp(.20) = 1.22 (i.e., by 22%)

The probability of choice increases with the level of discount; at a

discount of 15 cents, the probability of choice is about .50.

The effect of a one cent discount on the probability of choice

depends on the price at which the brand is offered; for example, at

the mean discount of 12.67 cents, a discount of one cent will

increase the probability of choice by about .05.

The discount elasticity also depends on the price at which the brand

is offered, but the average aggregate elasticity is .93.

Customer choice


Customer choice

Variable Averages

Averages of independent variables for each alternative. Alternative-specific constants, if added, are set to zero by definition.

Variables / Alternatives Discount

Response 12.667

Dummy (No Choice) 0.000

Variable Averages for Chosen Alternatives

Averages of independent variables for each alternative where that alternative was the chosen alternative. Alternative-specific constants, if added, are set to zero by definition.


Response 19.615

Dummy (No Choice) 7.353

Confusion Matrix on Estimation Sample

Comparison of observed choices and predicted choices (based on MNL analysis).High values in the diagonal of the confusion matrix (in bold), compared to the non-diagonal values, indicate high convergence between observations and predictions.Analysis has been performed on the estimation dataset, and measures the goodness-of-fit of the model.

Observed / Predicted Choice Response Dummy (No Choice)

Response 11 4

Dummy (No Choice) 2 13

Illustrative example: ME output (Diagnosis tab)

Customer choice

Respondents / Choice probabilitiesResponse

probabilityDummy (No

Choice) probabilityPredicted Response

Predicted Dummy (No Choice)

Observed Response

Observed Dummy (No Choice)

R1 0.050 0.950 0 1 0 1

R2 0.050 0.950 0 1 0 1

R3 0.050 0.950 0 1 0 1

R4 0.050 0.950 0 1 0 1

R5 0.050 0.950 0 1 0 1

R6 0.050 0.950 0 1 0 1

R7 0.050 0.950 0 1 0 1

R8 0.050 0.950 0 1 1 0

R9 0.281 0.719 0 1 0 1

R10 0.281 0.719 0 1 0 1

R11 0.281 0.719 0 1 0 1

R12 0.281 0.719 0 1 0 1

R13 0.281 0.719 0 1 0 1

R14 0.281 0.719 0 1 0 1

R15 0.281 0.719 0 1 1 0

R16 0.515 0.485 1 0 0 1

R17 0.515 0.485 1 0 0 1

R18 0.515 0.485 1 0 0 1

R19 0.515 0.485 1 0 1 0

R20 0.515 0.485 1 0 1 0

R21 0.515 0.485 1 0 1 0

R22 0.743 0.257 1 0 0 1

R23 0.743 0.257 1 0 1 0

R24 0.743 0.257 1 0 1 0

R25 0.743 0.257 1 0 1 0

R26 0.743 0.257 1 0 1 0

R27 0.955 0.045 1 0 1 0

R28 0.955 0.045 1 0 1 0

R29 0.955 0.045 1 0 1 0

R30 0.955 0.045 1 0 1 0

Illustrative example: ME output (Estimation tab)

Customer choice


The fit of the logit model can be assessed based on

the confusion matrix, which cross-classifies

observed and predicted choices.

If the predicted probability of choice exceeds .5,

then 𝑌 = 1, otherwise 𝑌 = 0.

The sum of the diagonals over the sample size gives

the hit rate (percent of observations for which the

actual choice was predicted correctly).

In the illustration, the hit rate is 80%.

Customer choice

Review: Basic idea of the

binary choice model

What determines choice when there are two choice

options?

Assume we have two possible influences on the choice

of a brand, quality and price. The model is

𝑃 𝑌 = 1 =1

1 + 𝑒𝑥𝑝 −(𝛼 + 𝛽1𝑄 + 𝛽2𝑃)

We can rewrite this equation as follows:

𝑙𝑜𝑔𝑃 𝑌 = 1

1 − 𝑃 𝑌 = 1= 𝛼 + 𝛽1𝑄 + 𝛽2𝑃

Customer choice

Evaluating the effect of quality on choice

Model Effect of a unit change in Q

𝑃 𝑌 = 1 =1

1 + 𝑒𝑥𝑝 −(𝛼 + 𝛽1𝑄 + 𝛽2𝑃)

Linear approximation:

𝛽1 𝑌 = 1 1 − 𝑃(𝑌 = 1)

Elasticity:

𝛽1 1 − 𝑃(𝑌 = 1) 𝑄

𝑃 𝑌 = 1

1 − 𝑃 𝑌 = 1= exp(𝛼 + 𝛽1𝑄 + 𝛽2𝑃) exp(𝛽1)

𝑙𝑜𝑔𝑖𝑡 𝑃 𝑌 = 1 = 𝑙𝑜𝑔𝑃 𝑌 = 1

1 − 𝑃 𝑌 = 1

= 𝛼 + 𝛽1𝑄 + 𝛽2𝑃

𝛽1

Customer choice

The multinomial choice model: An illustration


AlternativesChoice (0/1)

Discount

1 A 0 0

B 0 0

C 1 0

2 A 1 5

B 0 0

C 0 0

3 A 0 5

B 0 0

C 1 0

4 A 1 30

B 0 0

C 0 0

5 A 0 15

B 0 0

C 1 0

6 A 0 0

B 0 0

C 1 0

7 A 0 0

B 0 0

C 1 0

8 A 1 15

B 0 0

C 0 0

9 A 1 15

B 0 0

C 0 0

10 A 1 25

B 0 0

C 0 0



Discount

11 A 0 0

B 0 10

C 1 0

12 A 0 0

B 0 10

C 1 0

13 A 0 0

B 0 10

C 1 0

14 A 0 0

B 1 15

C 0 0

15 A 0 0

B 0 15

C 1 0

16 A 0 0

B 0 15

C 1 0

17 A 0 0

B 1 20

C 0 0

18 A 0 0

B 1 30

C 0 0

19 A 0 0

B 1 20

C 0 0

20 A 0 0

B 1 30

C 0 0



Discount

21 A 1 30

B 0 5

C 0 0

22 A 1 20

B 0 5

C 0 0

23 A 1 25

B 0 15

C 0 0

24 A 1 20

B 0 5

C 0 0

25 A 0 15

B 1 10

C 0 0

26 A 0 5

B 1 25

C 0 0

27 A 1 10

B 0 30

C 0 0

28 A 0 15

B 1 30

C 0 0

29 A 0 5

B 0 25

C 1 0

30 A 0 10

B 1 30

C 0 0

Assume we have data on a consumer’s choice of one of three brands across 30

purchase occasions and we also know whether the brands were on sale on a

particular occasion (0 to 30 cents below the regular price).

Customer choice

In general, the multinomial model is:

𝑃 𝑌 = 𝑖 =𝑒𝑥𝑝 𝛼𝑖 + 𝛽1𝑥𝑖1 + 𝛽2𝑥𝑖2 + …+ 𝛽𝑘𝑥𝑖𝑘

𝑖 𝑒𝑥𝑝 𝛼𝑖 + 𝛽1𝑥𝑖1 + 𝛽2𝑥𝑖2 + …+ 𝛽𝑘𝑥𝑖𝑘

The probability of choice of alternative i is equal to the

share of alternative i’s exponentiated deterministic utility

component among all choice alternatives.

For identification, exp(·) is set to one for one brand.

The interpretation of the coefficients is the same as in

the binary logit model

The multinomial choice model

Customer choice

Using ME to estimate a multinomial logit model

Customer choice

Illustrative example: ME output (Diagnosis tab)

Variable AveragesAverages of independent variables for each alternative. Alternative-specific constants, if added, are set to zero by definition.


A 8.833

B 11.833C 0.000

Variable Averages for Chosen AlternativesAverages of independent variables for each alternative where that alternative was the chosen alternative. Alternative-specific constants, if added, are set to zero by definition.


A 19.500B 23.333C 0.000

Confusion Matrix on Estimation SampleComparison of observed choices and predicted choices (based on MNL analysis).

High values in the diagonal of the confusion matrix (in bold), compared to the non-diagonal values, indicate high convergence between observations and predictions.

Analysis has been performed on the estimation dataset, and measures the goodness-of-fit of the model.

Observed / Predicted Choice A B C

A 8 1 1B 1 7 1C 1 1 9

Customer choice

Illustrative example: ME output (Estimation tab)Respondents / Choice

probabilitiesA probability B probability C probability Predicted A Predicted B Predicted C Observed A Observed B Observed C

1 0.099 0.025 0.875 0 0 1 0 0 1

2 0.234 0.022 0.744 0 0 1 1 0 0

3 0.234 0.022 0.744 0 0 1 0 0 1

4 0.980 0.001 0.019 1 0 0 1 0 0

5 0.702 0.008 0.290 1 0 0 0 0 1

6 0.099 0.025 0.875 0 0 1 0 0 1

7 0.099 0.025 0.875 0 0 1 0 0 1

8 0.702 0.008 0.290 1 0 0 1 0 0

9 0.702 0.008 0.290 1 0 0 1 0 0

10 0.948 0.001 0.051 1 0 0 1 0 0

11 0.085 0.167 0.748 0 0 1 0 0 1

12 0.085 0.167 0.748 0 0 1 0 0 1

13 0.085 0.167 0.748 0 0 1 0 0 1

14 0.066 0.357 0.577 0 0 1 0 1 0

15 0.066 0.357 0.577 0 0 1 0 0 1

16 0.066 0.357 0.577 0 0 1 0 0 1

17 0.040 0.606 0.354 0 1 0 0 1 0

18 0.008 0.922 0.070 0 1 0 0 1 0

19 0.040 0.606 0.354 0 1 0 0 1 0

20 0.008 0.922 0.070 0 1 0 0 1 0

21 0.979 0.002 0.019 1 0 0 1 0 0

22 0.861 0.010 0.128 1 0 0 1 0 0

23 0.920 0.031 0.049 1 0 0 1 0 0

24 0.861 0.010 0.128 1 0 0 1 0 0

25 0.664 0.061 0.275 1 0 0 0 1 0

26 0.052 0.783 0.165 0 1 0 0 1 0

27 0.058 0.875 0.067 0 1 0 1 0 0

28 0.146 0.794 0.060 0 1 0 0 1 0

29 0.052 0.783 0.165 0 1 0 0 0 1

30 0.058 0.875 0.067 0 1 0 0 1 0

Customer choice

Illustrative example:

ME output (Segment tab)Coefficient Estimates [segment 1]Coefficient estimates of the Choice model. Coefficients in bold are statistically significant.

Variables / Coefficient estimates Coefficient estimates Standard deviation t-statistic

Discount 0.203877 0.052848 3.857763Const-1 -2.17458 0.797709 -2.72604Const-2 -3.53879 1.158237 -3.05533Baseline n/a n/a


Elasticities of Discount A B C

A 0.547463 -0.11885 -0.40045B -0.24913 1.110283 -0.68194

C 0 0 0

Customer choice

Properties of the MNL model:

Independence of irrelevant alternatives (IIA)

This assumption implies that when a new alternative

is added to a choice set, the new alternative will

steal share from the existing alternatives in

proportion to their current choice shares.

This is unrealistic because a new alternative is likely

to steal more share from more similar alternatives

(e.g., if a new cola drink is introduced, existing cola

drinks are likely more vulnerable than non-cola

drinks).

To avoid this problem, the choice alternatives can

be grouped into sets that are similar.

Customer choice

Office Star Choice Data

Data are available for 20 respondents who made a

choice between 3 alternatives: Office Star, Paper &

Co., and Office Equipment;

Five variables are used to predict people’s choices:

the number of purchases previously made at one of

the stores, ratings of whether a given store is

expensive or convenient, and whether a store offers

good service and a large choice;

Customer choice

Office Star choice data (cont’d)Observations / Choice

dataAlternatives

Choice

(0/1)

Past

purchasesExpensive Convenient Service Large choice

Respondent 1 OfficeStar 0 0 2 1 3 4

Paper & Co 0 0 4 4 7 3

Office Equip'nt 1 0 1 3 5 1


Paper & Co 0 0 7 4 6 7



Paper & Co 0 0 7 2 1 1



Paper & Co 1 8 3 4 7 7



Paper & Co 1 0 1 3 3 4



Paper & Co 0 0 5 5 2 3



Paper & Co 1 0 4 4 3 4



Paper & Co 0 0 3 6 2 4



Paper & Co 0 0 4 3 3 6



Paper & Co 0 0 7 3 1 4


Etc.

Customer choice

Office Star data: Diagnosis tabVariable AveragesAverages of independent variables for each alternative. Alternative-specific constants, if added, are set to zero by definition.

Variables / Alternatives Past purchases Expensive Convenient Service Large choice

OfficeStar 0.000 3.150 3.950 3.300 4.550Paper & Co 0.900 3.750 3.600 4.250 4.350Office Equip'nt 0.800 3.900 4.400 3.500 4.050

Variable Averages for Chosen AlternativesAverages of independent variables for each alternative where that alternative was the chosen alternative. Alternative-specific constants, if added, are set to zero by definition.

Variables / Alternatives Past purchases Expensive Convenient Service Large choice

OfficeStar 0.000 2.700 4.400 3.400 4.900Paper & Co 2.250 2.875 3.625 4.625 5.000Office Equip'nt 4.000 3.000 4.000 4.000 3.500

Confusion Matrix on Estimation SampleComparison of observed choices and predicted choices (based on MNL analysis).High values in the diagonal of the confusion matrix (in bold), compared to the non-diagonal values, indicate high convergence between observations and predictions. Analysis has been performed on the estimation dataset, and measures the goodness-of-fit of the model.

Observed / Predicted Choice OfficeStar Paper & Co Office Equip'nt

OfficeStar 10 1 1Paper & Co 0 7 0Office Equip'nt 0 0 1

Customer choice

Office Star Data: Estimation tabEstimation Sample DetailsChoice probabilities, predicted and observed choices, segment membership probabilities and predicted segment for the sample usedto estimate the model.

Respondents / Choice probabilities

OfficeStar probability

Paper & Co probability

Office Equip'nt

probability

Predicted OfficeStar

Predicted Paper & Co

Predicted Office

Equip'nt

Observed OfficeStar

Observed Paper & Co

Observed Office

Equip'nt

Respondent 1 0.628 0.353 0.020 1 0 0 0 0 1

Respondent 2 0.900 0.083 0.017 1 0 0 1 0 0

Respondent 3 0.998 0.001 0.002 1 0 0 1 0 0

Respondent 4 0.015 0.985 0.000 0 1 0 0 1 0

Respondent 5 0.040 0.960 0.000 0 1 0 0 1 0

Respondent 6 0.506 0.491 0.003 1 0 0 1 0 0

Respondent 7 0.251 0.388 0.361 0 1 0 0 1 0

Respondent 8 0.982 0.018 0.000 1 0 0 1 0 0

Respondent 9 0.637 0.338 0.025 1 0 0 1 0 0

Respondent 10 0.748 0.038 0.215 1 0 0 1 0 0

Respondent 11 0.039 0.960 0.001 0 1 0 0 1 0

Respondent 12 0.009 0.164 0.827 0 0 1 0 0 1

Respondent 13 0.779 0.221 0.000 1 0 0 1 0 0

Respondent 14 0.955 0.042 0.003 1 0 0 1 0 0

Respondent 15 0.719 0.000 0.281 1 0 0 1 0 0

Respondent 16 0.669 0.322 0.009 1 0 0 0 1 0

Respondent 17 0.088 0.904 0.008 0 1 0 0 1 0

Respondent 18 0.879 0.117 0.004 1 0 0 1 0 0

Respondent 19 0.051 0.725 0.224 0 1 0 0 1 0

Respondent 20 0.109 0.891 0.000 0 1 0 0 1 0

Customer choice

Office Star Data: Segment tab

Coefficient Estimates [segment 1]

Coefficient estimates of the Choice model. Coefficients in bold are statistically significant.

Variables / Coefficient estimatesCoefficient estimates

Standard deviation

t-statistic

Past purchases 0.863569 0.359493 2.402184

Expensive -0.81508 0.383513 -2.12529

Convenient 0.537433 0.373534 1.438779

Service 0.166118 0.255744 0.649551

Large choice 0.4312 0.32222 1.338216

Const-1 4.391705 1.933972 2.270821

Const-2 3.600351 1.805345 1.994273

Baseline n/a n/a

Customer choice

Office Star Data: Segment tab (cont’d)


Elasticities of Past purchases OfficeStar Paper & Co Office Equip'nt

OfficeStar 0 0 0Paper & Co -0.06699 0.084338 -0.00238Office Equip'nt -0.14645 -0.1145 1.190273

Elasticities of Expensive OfficeStar Paper & Co Office Equip'nt

OfficeStar -0.65391 0.672727 0.578669Paper & Co 0.553199 -0.82619 0.538747Office Equip'nt 0.054357 0.111399 -0.71738

Elasticities of Convenient OfficeStar Paper & Co Office Equip'nt

OfficeStar 0.494317 -0.47137 -0.5861Paper & Co -0.34876 0.581375 -0.58167Office Equip'nt -0.16315 -0.20024 1.616706

Customer choice

Office Star Data: Segment tab (cont’d)


Elasticities of Service OfficeStar Paper & Co Office Equip'nt


Elasticities of Large choice OfficeStar Paper & Co Office Equip'nt


Customer choice

Extensions of the basic choice model

The logit model assumes that the intercepts and the

effects of the explanatory variables in the

deterministic part of utility are the same across

individuals.

Two ways to get around this limitation:

□ Individual differences can be used as additional

determinants of the deterministic part of utility.

□ Different coefficients can be estimated for different

segments of consumers (so-called latent class choice

models).

customer choice - personal.psu.edu · discount r1 0 0 r2 0 0 r3 0 0 r4 0 0 ... linear approximation...

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