Outline. The binary choice model. The multinomial choice model. Extensions of the basic choice model

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Dennis Alvin Cain
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2 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

3 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 Choice data (0/1) Discount R1 0 0 R2 0 0 R3 0 0 R4 0 0 R5 0 0 R6 0 0 R7 0 0 R8 1 0 R R R R R R R Observations / Choice Choice data (0/1) Discount R R R R R R R R R R R R R R R

4 The binary choice model: An illustration (cont d) 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: Discount Choice=1 Choice=0 P(Choice=1) P(Choice=0)

5 Probability(Choice=1) as a function of discount size

The binary choice model: An illustration (cont d) Two issues: If the discount is larger than 30 cents, the model predicts a probability greater than 1.

6 The binary choice model: An illustration (cont d) 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.

7 Using an S-shaped function to link the probability of choice to level of discount

8 The binary (logit) choice model In general, the logit model is: P Y = 1 = exp α + β 1x 1 + β 2 x β k x k 1 + exp α + β 1 x 1 + β 2 x β k x k = exp (α + β 1 x 1 + β 2 x β k x k ) is an intercept term that determines P(Y=1) when all explanatory variables x i equal zero i is the contribution of a unit change in x i to P(Y=1) (for a given value of the other explanatory variables)

9 The binary (logit) choice model (cont d) The previous model is equivalent to: logit P Y = 1 = log P Y = 1 1 P Y = 1 = α + β 1x 1 + β 2 x β k x k 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.

10 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 + β 1 x 1 + β 2 x β k x k 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 x i to the brand s deterministic component of utility.

11 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 x i (holding the other x s constant) is given by β i P Y = 1 1 P(Y = 1)

12 Interpreting the results of a binary choice model Odds ratio interpretation: For given values of the other explanatory variables, a unit increase in x i changes the odds by a factor of exp(β i ) 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 x i is: β i 1 P(Y = 1) x i Not that this implies that elasticities are greater at lower choice probabilities.

13 Using ME to estimate a binary logit model

14 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 Const Baseline n/a n/a Elasticities [segment 1] Elasticities of coefficients. Elasticities of Discount Response Dummy (No Choice) Response Dummy (No Choice) 0 0

15 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 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?

16 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 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.

17 Interpreting the output

18 Illustrative example: ME output (Diagnosis tab) Variable Averages Averages of independent variables for each alternative. Alternative-specific constants, if added, are set to zero by definition. Variables / Alternatives Discount Response Dummy (No Choice) 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. Variables / Alternatives Discount Response Dummy (No Choice) 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

19 Illustrative example: ME output (Estimation tab) Respondents / Choice probabilities Response probability Dummy (No Choice) probability Predicted Response Predicted Dummy (No Choice) Observed Response Observed Dummy (No Choice) R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R

20 Interpreting the output 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%.

21 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 = exp (α + β 1 Q + β 2 P) We can rewrite this equation as follows: log P Y = 1 1 P Y = 1 = α + β 1 Q + β 2 P

22 Evaluating the effect of quality on choice Model Effect of a unit change in Q P Y = 1 = exp (α + β 1 Q + β 2 P) Linear approximation: β 1 Y = 1 1 P(Y = 1) Elasticity: β 1 1 P(Y = 1) Q P Y = 1 1 P Y = 1 = exp(α + β 1Q + β 2 P) exp(β 1 ) logit P Y = 1 P Y = 1 = log 1 P Y = 1 = α + β 1 Q + β 2 P β 1

23 The multinomial choice model: An illustration 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). Observations / Choice data Alternatives Choice (0/1) Discount 1 A 0 0 B 0 0 C A 1 5 B A 0 5 B 0 0 C A 1 30 B A 0 15 B 0 0 C A 0 0 B 0 0 C A 0 0 B 0 0 C A 1 15 B A 1 15 B A 1 25 B 0 0 Observations / Choice data Alternatives Choice (0/1) Discount 11 A 0 0 B 0 10 C A 0 0 B 0 10 C A 0 0 B 0 10 C A 0 0 B A 0 0 B 0 15 C A 0 0 B 0 15 C A 0 0 B A 0 0 B A 0 0 B A 0 0 B 1 30 Observations / Choice data Alternatives Choice (0/1) Discount 21 A 1 30 B A 1 20 B A 1 25 B A 1 20 B A 0 15 B A 0 5 B A 1 10 B A 0 15 B A 0 5 B 0 25 C A 0 10 B 1 30

24 The multinomial choice model In general, the multinomial model is: P Y = i = exp α i + β 1 x i1 + β 2 x i2 + + β k x ik i exp α i + β 1 x i1 + β 2 x i2 + + β k x ik 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

25 Using ME to estimate a multinomial logit model

26 Illustrative example: ME output (Diagnosis tab) Variable Averages Averages of independent variables for each alternative. Alternative-specific constants, if added, are set to zero by definition. Variables / Alternatives Discount A B C 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. Variables / Alternatives Discount A B C 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 A B C A B C 1 1 9

27 Illustrative example: ME output (Estimation tab) Respondents / Choice probabilities A probability B probability C probability Predicted A Predicted B Predicted C Observed A Observed B Observed C

28 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 Const Const Baseline n/a n/a Elasticities [segment 1] Elasticities of coefficients. Elasticities of Discount A B C A B

29 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.

30 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;

31 Office Star choice data (cont d) Observations / Choice Choice Past Alternatives data (0/1) purchases Expensive Convenient Service Large choice Respondent 1 OfficeStar Paper & Co Office Equip'nt Respondent 2 OfficeStar Paper & Co Office Equip'nt Respondent 3 OfficeStar Paper & Co Office Equip'nt Respondent 4 OfficeStar Paper & Co Office Equip'nt Respondent 5 OfficeStar Paper & Co Office Equip'nt Respondent 6 OfficeStar Paper & Co Office Equip'nt Respondent 7 OfficeStar Paper & Co Office Equip'nt Respondent 8 OfficeStar Paper & Co Office Equip'nt Respondent 9 OfficeStar Paper & Co Office Equip'nt Respondent 10 OfficeStar Paper & Co Office Equip'nt Etc.

32 Office Star data: Diagnosis tab Variable Averages Averages 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 Paper & Co Office Equip'nt 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. Variables / Alternatives Past purchases Expensive Convenient Service Large choice OfficeStar Paper & Co Office Equip'nt 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 OfficeStar Paper & Co Office Equip'nt OfficeStar Paper & Co Office Equip'nt 0 0 1

33 Office Star Data: Estimation tab Estimation Sample Details Choice probabilities, predicted and observed choices, segment membership probabilities and predicted segment for the sample used to 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 Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent Respondent

34 Office Star Data: 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 Past purchases Expensive Convenient Service Large choice Const Const Baseline n/a n/a

35 Office Star Data: Segment tab (cont d) Elasticities [segment 1] Elasticities of coefficients. Elasticities of Past purchases OfficeStar Paper & Co Office Equip'nt OfficeStar Paper & Co Office Equip'nt Elasticities of Expensive OfficeStar Paper & Co Office Equip'nt OfficeStar Paper & Co Office Equip'nt Elasticities of Convenient OfficeStar Paper & Co Office Equip'nt OfficeStar Paper & Co Office Equip'nt

36 Office Star Data: Segment tab (cont d) Elasticities [segment 1] Elasticities of coefficients. Elasticities of Service OfficeStar Paper & Co Office Equip'nt OfficeStar Paper & Co Office Equip'nt Elasticities of Large choice OfficeStar Paper & Co Office Equip'nt OfficeStar Paper & Co Office Equip'nt

37 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).

Generalized logit models for nominal multinomial responses. Local odds ratios

Generalized logit models for nominal multinomial responses Categorical Data Analysis, Summer 2015 1/17 Local odds ratios Y 1 2 3 4 1 π 11 π 12 π 13 π 14 π 1+ X 2 π 21 π 22 π 23 π 24 π 2+ 3 π 31 π 32 π