Logit with multiple alternatives

Transcription

1 Logit with multiple alternatives Matthieu de Lapparent Transport and Mobility Laboratory, School of Architecture, Civil and Environmental Engineering, Ecole Polytechnique Fédérale de Lausanne Transport and Mobility Laboratory Multinomial logit 1 / 70

2 Outline 1 Components of the Logit model Random Utility Choice set Error terms 2 Systematic utility Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity 3 A case study 4 Maximum likelihood estimation 5 Simple models Transport and Mobility Laboratory Multinomial logit 2 / 70

3 Outline 1 Components of the Logit model Random Utility Choice set Error terms 2 Systematic utility Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity 3 A case study 4 Maximum likelihood estimation 5 Simple models

4 Components of the Logit model Random Utility Random Utility For each i C n Uin = Vin + εin What is C n? What is V in? What is ε in? Transport and Mobility Laboratory Multinomial logit 3 / 70

5 Components of the Logit model Choice set Choice set Universal choice set C All potential alternatives for the population Restricted to relevant alternatives Mode choice: driving alone sharing a ride taxi motorcycle bicycle walking bus rail rapid transit horse Transport and Mobility Laboratory Multinomial logit 4 / 70

6 Components of the Logit model Choice set Choice set Individual s choice set No driver s license No auto Awareness of bus services Rail transit services unreachable Walking not an option for long distance Individual s mode choice driving alone sharing a ride taxi motorcycle bicycle walking bus rail rapid transit horse Transport and Mobility Laboratory Multinomial logit 5 / 70

7 Components of the Logit model Choice set Choice set Choice set generation is tricky How to model awareness? What does unreachable mean exactly? What does long distance mean exactly? We will continue assuming a deterministic rule Transport and Mobility Laboratory Multinomial logit 6 / 70

8 Components of the Logit model Error terms Error terms Main assumption ε in are extreme value EV(0,µ), independent and identically distributed. Comments Independence: across i and n. Identical distribution: same scale parameter µ across i and n. Scale must be normalized, e.g. µ = 1 Transport and Mobility Laboratory Multinomial logit 7 / 70

9 Components of the Logit model Error terms Illustration of µ: A rising tide lifts all boats Transport and Mobility Laboratory Multinomial logit 8 / 70

10 Components of the Logit model Error terms Derivation of the logit model Reminder: binary case C n = {i, j} U in = V in + ε in ε in EV(0, µ) ε in i.i.d. Choice model P(i C n = {i, j}) = e µv in e µv in + e µv jn Transport and Mobility Laboratory Multinomial logit 9 / 70

11 Components of the Logit model Error terms Derivation of the logit model Multiple alternatives C n = {1,..., J n } U in = V in + ε in ε in EV(0, µ) ε in i.i.d. Choice model P(i C n ) = P(V in + ε in max V jn + ε jn ) j=1,...,j n Assume without loss of generality (wlog) that i = 1 P(1 C n ) = P(V 1n + ε 1n max V jn + ε jn ) j=2,...,j n Transport and Mobility Laboratory Multinomial logit 10 / 70

12 Components of the Logit model Error terms Derivation of the logit model Composite alternative Define a composite alternative as anything but alternative one Associated utility: U = max (V jn + ε jn ) j=2,...,j n From a property of the EV distribution U EV 1 J n µ ln e µv jn, µ j=2 Transport and Mobility Laboratory Multinomial logit 11 / 70

13 Components of the Logit model Error terms Derivation of the logit model From another property of the EV distribution U = V + ε where and V = 1 J n µ ln j=2 ε EV(0, µ) e µv jn Transport and Mobility Laboratory Multinomial logit 12 / 70

14 Components of the Logit model Error terms Derivation of the logit model Therefore P(1 C n ) = P(V 1n + ε 1n max j=2,...,jn V jn + ε jn ) = P(V 1n + ε 1n V + ε ) This is a binary choice model with a systematic composite alternative P(1 C n ) = e µv 1n e µv 1n + e µv where V = 1 J n µ ln j=2 e µv jn Transport and Mobility Laboratory Multinomial logit 13 / 70

15 Components of the Logit model Error terms Derivation of the logit model and can be rewritten as P(1 C n ) = = = e µv 1n e µv 1n + e µv e µv 1n e µv 1n + Jn j=2 eµv jn e µv 1n Jn j=1 eµv jn Transport and Mobility Laboratory Multinomial logit 14 / 70

16 Components of the Logit model Error terms Derivation of the logit model The scale parameter µ is not identifiable: µ = 1. Warning: not identifiable not existing Limiting cases µ 0, that is variance goes to infinity lim µ 0 P(i C n) = 1 J n i C n µ +, that is variance goes to zero 1 lim µ P(i C n ) = lim µ 1+ j i eµ(v jn V in { ) 1 if Vin > max = j i V jn 0 if V in < max j i V jn Transport and Mobility Laboratory Multinomial logit 15 / 70

17 Components of the Logit model Error terms Another derivation of the Multinomial logit model P(i C n ) = P(j C n, j i, V in + ε in V jn + ε jn ) P(i C n ) = P(i C n ) = P(j C n, j i, V in V jn + ε in ε jn ) + j C n,j i + P(i C n ) = j C n,j i Vin V jn +ε in f (ε jn )dε jn f (ε in )dε in e ev jn V in ε in f (ε in )dε in Transport and Mobility Laboratory Multinomial logit 16 / 70

18 Components of the Logit model Error terms Another derivation of the Multinomial logit model, cont. P(i C n ) = P(i C n ) = + + e j Cn,j i ev jn V in ε in f (ε in )dε in e e ε in j Cn,j i ev jn V in e ε in e e ε in dε in y in = e ε in, dy in = e ε in dε in, y in ]0; + [ P(i C n ) = P(i C n ) = + ( e y in 1+ ) j Cn,j i ev jn V in dy in j C n,j i ev jn V in = e V in j C n e V jn Transport and Mobility Laboratory Multinomial logit 17 / 70

19 Outline 1 Components of the Logit model Random Utility Choice set Error terms 2 Systematic utility Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity 3 A case study 4 Maximum likelihood estimation 5 Simple models

20 Systematic utility Systematic part of the utility function Shift focus from ε to V where V in = V (z in, S n ) z in is a vector of attributes of alternative i for individual n S n is a vector of socio-economic characteristics of n Outline: Functional form: linear utility Explanatory variables: What exactly is contained in z in and S n? Functional form: capturing nonlinearities Interactions Transport and Mobility Laboratory Multinomial logit 18 / 70

21 Systematic utility Linear utility Functional form: linear utility Notation for explanatory variables x in = (z in, S n ) In general, linear-in-parameters utility functions are used V in = V (z in, S n ) = V (x in ) = k β k (x in ) k Not as restrictive as it may seem Transport and Mobility Laboratory Multinomial logit 19 / 70

22 Systematic utility Continuous variables Explanatory variables: attributes of alternatives Numerical and continuous (z in ) k R, i, n, k Associated with a specific unit Examples Auto in-vehicle time (in min.) Transit in-vehicle time (in min.) Auto out-of-pocket cost (in cents) Transit fare (in cents) Walking time to the bus stop (in min.) Straightforward modeling Transport and Mobility Laboratory Multinomial logit 20 / 70

23 Systematic utility Continuous variables Explanatory variables: attributes of alternatives V in is unitless Therefore, β depends on the unit of the associated attribute Example: consider two specifications V in = β 1 TT in + V in = β 1 TT in + If TT in is measured in minutes, the unit of β 1 is 1/min If TT in is measured in hours, the unit of β 1 is 1/hour Both models are equivalent, but the estimated β will be scaled differently β 1 TT in = β 1TT in = TT in TT in = β 1 β 1 = 60 Transport and Mobility Laboratory Multinomial logit 21 / 70

24 Systematic utility Continuous variables Explanatory variables: attributes of alternatives Impact of attributes on different alternatives Generic, or Alternative specific parameters V auto = β 1 TT auto V bus = β 1 TT bus V auto = β 1 TT auto V bus = β 2 TT bus Modeling assumption: a minute has/doesn t have the same marginal utility whether it is incurred on the auto or bus mode Transport and Mobility Laboratory Multinomial logit 22 / 70

25 Systematic utility Continuous variables Explanatory variables: socio-eco. characteristics Numerical and continuous Numerical and continuous (S n ) k R, n, k Associated with a specific unit Examples Annual income (in KCHF) Age (in years) Warning: S n do not depend on i Transport and Mobility Laboratory Multinomial logit 23 / 70

26 Systematic utility Continuous variables Explanatory variables: socio-eco. characteristics They cannot appear in all utility functions V 1 = β 1 x 11 + β 2 income V 2 = β 1 x 21 + β 2 income V 3 = β 1 x 31 + β 2 income Need to specify as alternative specific, e.g. V 1 = β 1 x 11 + β 2 income +β 4 age V 2 = β 1 x 21 + β 3 income +β 5 age V 3 = β 1 x 31 V 1 = β 1 x 11 V 2 = β 1 x 21 V 3 = β 1 x 31 Transport and Mobility Laboratory Multinomial logit 24 / 70

27 Systematic utility Discrete variables Functional form: dealing with nonlinearities Discrete and qualitative variables Continuous variables Categories Splines Box-Cox Power series Transport and Mobility Laboratory Multinomial logit 25 / 70

28 Systematic utility Discrete variables Discrete variables Mainly used to capture impact of qualitative attributes Level of comfort for the train Reliability of the bus Color of car etc... or discrete characteristics Sex Education Professional status etc. Transport and Mobility Laboratory Multinomial logit 26 / 70

29 Systematic utility Discrete variables Discrete variables Procedure for model specification Identify all possible levels of the attribute: Very high comfort (V), High comfort (H), Moderate comfort (M), Low comfort (L) Select a base case: Very high comfort Define numerical attributes Adopt a coding convention Transport and Mobility Laboratory Multinomial logit 27 / 70

30 Systematic utility Discrete variables Discrete variables Introduce a 0/1 attribute code for all levels except the base case z H for High comfort z M for Moderate comfort z L for Low comfort z H z M z L Very high comfort High comfort Moderate comfort Low comfort If a qualitative attribute has n levels, we introduce n 1 (0/1) variables in the model Transport and Mobility Laboratory Multinomial logit 28 / 70

31 Systematic utility Discrete variables Comparing two coding conventions Very high comfort fixed as base V = +β V z V +β H z H +β M z M +β L z L where β V = 0 β H : difference of utility between high comfort and very high comfort (supposedly negative) β M : difference of utility between moderate comfort and very high comfort (supposedly more negative) β L : difference of utility between low comfort and very high comfort (supposedly even more negative) Transport and Mobility Laboratory Multinomial logit 29 / 70

32 Systematic utility Discrete variables Comparing two ways of coding High comfort fixed as base V = +β V z V +β H z H +β M z M +β L z L where β H = 0 β V : difference of utility between very high comfort and high comfort (supposedly positive) β M : difference of utility between moderate comfort and high comfort (supposedly negative) β L : difference of utility between low comfort and high comfort (supposedly more negative) Transport and Mobility Laboratory Multinomial logit 30 / 70

33 Systematic utility Discrete variables Discrete variables Example of estimation with Biogeme: Model 1 Model 2 ASC BETA V BETA H BETA M BETA L Transport and Mobility Laboratory Multinomial logit 31 / 70

34 Systematic utility Nonlinearities Nonlinear transformations of the variables Example with travel time Compare a trip of 5 min with a trip of 10 min (+5 minutes) Compare a trip of 120 min with a trip of 125 min (+5 minutes) Behavioral assumption One additional minute of travel time is not perceived in the same way for short trips as for long trips Transport and Mobility Laboratory Multinomial logit 32 / 70

35 Systematic utility Nonlinearities Nonlinear transformations of the variables log Utility Time Transport and Mobility Laboratory Multinomial logit 33 / 70

36 Systematic utility Nonlinearities Nonlinear transformations of the variables Assumption 1: the marginal impact of travel time is constant V i = β T time i + Assumption 2: the marginal impact of travel time decreases with longer travel time V i = β T ln(time i ) + Remarks Still a linear-in-parameters form The unit, the value, and the interpretation of β T is different Transport and Mobility Laboratory Multinomial logit 34 / 70

37 Systematic utility Nonlinearities Continuous variables: split into categories Like earlier assumption: sensitivity to travel time varies with travel time level Logarithmic transformation not the only specification Another possibility is to split travel time into categories (here TT in minutes) Short: 0-90 min Medium: min Long: min Very long: over 271 min Possible specifications Categories with constants (inferior solution) Piecewise linear specification (spline) Transport and Mobility Laboratory Multinomial logit 35 / 70

38 Systematic utility Nonlinearities Continuous variables: categories with constants Same specification as for discrete variables V i = β T 1 x T 1 + β T 2 x T 2 + β T 3 x T 3 + β T 4 x T with x T 1 = 1 if TT i [0 90[, 0 otherwise x T 2 = 1 if TT i [91 180[, 0 otherwise x T 3 = 1 if TT i [ [, 0 otherwise x T 4 = 1 if TT i [271 [, 0 otherwise One β must be normalized to 0. Transport and Mobility Laboratory Multinomial logit 36 / 70

39 Systematic utility Nonlinearities Continuous variables: categories with constants Utility Time Transport and Mobility Laboratory Multinomial logit 37 / 70

40 Systematic utility Nonlinearities Continuous variables: categories with constants Drawbacks No sensitivity to travel time within the intervals Discontinuous utility function (jumps) Need for many small intervals Results may vary significantly with the definition of the intervals Appropriate when Categories have been used in the survey (income, age) Definition of categories is natural (weekday) Transport and Mobility Laboratory Multinomial logit 38 / 70

41 Systematic utility Nonlinearities Continuous variables: Piecewise linear specification Piecewise linear specification (spline) Captures the sensitivity within the intervals Enforces continuity of the utility function Transport and Mobility Laboratory Multinomial logit 39 / 70

42 Systematic utility Nonlinearities Piecewise linear specification Features Capture the sensitivity within the intervals Enforce continuity of the utility function where x T 1 = x T 3 = V i = β T 1 x T 1 + β T 2 x T 2 + β T 3 x T 3 + β T 4 x T { t if t < otherwise 0 if t < 180 t 180 if 180 t < otherwise x T 2 = x T 4 = 0 if t < 90 t 90 if 90 t < otherwise { 0 if t < 270 t 270 otherwise Transport and Mobility Laboratory Multinomial logit 40 / 70

43 Systematic utility Nonlinearities Piecewise linear specification Note: coding in Biogeme for interval [a:a+b[ 0 if t < a x Ti = t a if a t < a + b x Ti = max(0, min(t a, b)) b otherwise x T 1 = min(t, 90) x T 2 = max(0, min(t 90, 90)) x T 3 = max(0, min(t 180, 90)) x T 4 = max(0, t 270) TRAIN_TT1 = min( TRAIN_TT, 90) TRAIN_TT2 = max(0,min( TRAIN_TT - 90, 90)) TRAIN_TT3 = max(0,min( TRAIN_TT - 180, 90)) TRAIN_TT4 = max(0,train_tt - 270) Transport and Mobility Laboratory Multinomial logit 41 / 70

44 Systematic utility Nonlinearities Piecewise linear specification Examples: t TT1 TT2 TT3 TT Transport and Mobility Laboratory Multinomial logit 42 / 70

45 Systematic utility Nonlinearities Piecewise linear specification -0.5 Piecewise linear Utility Time Transport and Mobility Laboratory Multinomial logit 43 / 70

46 Systematic utility Nonlinearities Continuous variables: Box-Cox transforms Box-Cox transform Box and Cox, J. of the Royal Statistical Society (1964) V i = βx i (λ) + where x i λ 1 if λ 0 x i (λ) = λ ln x i if λ = 0. where x i > 0. Transport and Mobility Laboratory Multinomial logit 44 / 70

47 Systematic utility Nonlinearities Box-Cox transforms Box-Tukey transform If x i 0, include a constant α such that x i + α > 0 and (x i + α) λ 1 if λ 0 x i (λ, α) = λ ln(x i + α) if λ = 0. Transport and Mobility Laboratory Multinomial logit 45 / 70

48 Systematic utility Nonlinearities Box-Cox transforms (λ = 0.7) 5 Box-Cox Utility Time Transport and Mobility Laboratory Multinomial logit 46 / 70

49 Systematic utility Nonlinearities Power series Taylor expansion V i = β 1 T + β 2 T 2 + β 3 T In practice, these terms can be very correlated Difficult to interpret Risk of over fitting Transport and Mobility Laboratory Multinomial logit 47 / 70

50 Systematic utility Nonlinearities Power series 0 Power series -1-2 Utility Time Transport and Mobility Laboratory Multinomial logit 48 / 70

51 Systematic utility Interactions Interactions All individuals in a population are not alike Socio-economic characteristics define segments in the population How to capture heterogeneity? Interactions of attributes and characteristics Discrete segmentation Continuous segmentation Transport and Mobility Laboratory Multinomial logit 49 / 70

52 Systematic utility Interactions Interactions of attributes and characteristics Combination of attributes cost / income fare / disposable income distance / out-of-vehicle time (=speed) warning: correlation of attributes may produce degeneracy in the model Transport and Mobility Laboratory Multinomial logit 50 / 70

53 Systematic utility Interactions Interactions: discrete segmentation Example with discrete segments Hypothesis: different sensitivities for combinations of: Gender (M,F) House location (metro, suburb, periphery areas) Each individual belongs to exactly one of the 6 segments Specification of 6 segments β M,m TT M,m + β M,s TT M,s + β M,p TT M,p + β F,m TT F,m + β F,s TT F,s + β F,p TT F,p + TT i = TT if indiv. belongs to segment i, and 0 otherwise Transport and Mobility Laboratory Multinomial logit 51 / 70

54 Systematic utility Interactions Interactions: continuous segmentation Example with continuous characteristics Hypothesis: the cost parameter varies with income ( ) inc λ β cost = ˆβ cost with λ = β cost inc ref inc Reference value is arbitrary Several characteristics can be combined: ( ) β cost = ˆβ inc λ1 ( ) age λ2 cost inc ref age ref inc β cost warning: λ must be estimated and utility is no longer linear-in-parameters Transport and Mobility Laboratory Multinomial logit 52 / 70

55 Systematic utility Heteroscedasticity Heteroscedasticity Assumption: variance of error terms is different across individuals Assume there are two different groups such that and var(ε in2 ) = α 2 var(ε in1 ) Logit is homoscedastic ε in i.i.d. across both i and n. U in1 = V in1 + ε in1 U in2 = V in2 + ε in2 How can we specify the model in order to use logit? Motivation People have different level of knowledge (e.g. taxi drivers) Different sources of data Transport and Mobility Laboratory Multinomial logit 53 / 70

56 Systematic utility Heteroscedasticity Heteroscedasticity Solution: include scale parameters αu in1 = αv in1 + αε in1 = αv in1 + ε in 1 U in2 = V in2 + ε in2 = V in2 + ε in 2 where ε in 1 and ε in 2 are i.i.d. Remarks Even if V in1 = j β jx jin1 is linear-in-parameters, αv in1 = j αβ jx jin1 is not. Normalization: a different scale parameter can be estimated for each segment of the population, except one that must be normalized. Transport and Mobility Laboratory Multinomial logit 54 / 70

57 Outline 1 Components of the Logit model Random Utility Choice set Error terms 2 Systematic utility Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity 3 A case study 4 Maximum likelihood estimation 5 Simple models

58 A case study A case study Choice of residential telephone services Household survey conducted in Pennsylvania, USA, 1984 Revealed preferences 434 observations Transport and Mobility Laboratory Multinomial logit 55 / 70

59 A case study A case study Telephone services and availability metro, suburban & some other perimeter perimeter non-metro areas areas areas Budget Measured yes yes yes Standard Measured yes yes yes Local Flat yes yes yes Extended Area Flat no yes no Metro Area Flat yes yes no Transport and Mobility Laboratory Multinomial logit 56 / 70

60 A case study A case study Universal choice set C = {BM, SM, LF, EF, MF} Specific choice sets Metro, suburban & some perimeter areas: {BM,SM,LF,MF} Other perimeter areas: C Non-metro areas: {BM,SM,LF} Transport and Mobility Laboratory Multinomial logit 57 / 70

61 A case study A case study Specification table β 1 β 2 β 3 β 4 β 5 BM ln(cost(bm)) SM ln(cost(sm)) LF ln(cost(lf)) EF ln(cost(ef)) MF ln(cost(mf)) Transport and Mobility Laboratory Multinomial logit 58 / 70

62 A case study A case study Utility functions V BM = β 5 ln(cost BM ) V SM = β 1 + β 5 ln(cost SM ) V LF = β 2 + β 5 ln(cost LF ) V EF = β 3 + β 5 ln(cost EF ) V MF = β 4 + β 5 ln(cost MF ) Transport and Mobility Laboratory Multinomial logit 59 / 70

63 A case study A case study Specification table II β 1 β 2 β 3 β 4 β 5 β 6 β 7 BM ln(cost(bm)) users 0 SM ln(cost(sm)) users 0 LF ln(cost(lf)) 0 1 if metro/suburb EF ln(cost(ef)) 0 0 MF ln(cost(mf)) 0 0 Transport and Mobility Laboratory Multinomial logit 60 / 70

64 A case study A case study Utility functions V BM = β 5 ln(cost BM ) + β 6 users V SM = β 1 + β 5 ln(cost SM ) + β 6 users V LF = β 2 + β 5 ln(cost LF ) + β 7 MS V EF = β 3 + β 5 ln(cost EF ) V MF = β 4 + β 5 ln(cost MF ) Transport and Mobility Laboratory Multinomial logit 61 / 70

65 Outline 1 Components of the Logit model Random Utility Choice set Error terms 2 Systematic utility Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity 3 A case study 4 Maximum likelihood estimation 5 Simple models

66 Maximum likelihood estimation Maximum likelihood estimation Logit Model P n (i C n ) = e V in j C n e V jn Log-likelihood of a sample N J L(β 1,..., β K ) = y jn ln P n (j C n ) n=1 j=1 where y jn = 1 if ind. n has chosen alt. j, 0 otherwise Transport and Mobility Laboratory Multinomial logit 62 / 70

67 Maximum likelihood estimation Maximum likelihood estimation Logit model e ln P n (i C n ) = ln V in j Cn ev jn = V in ln( j C n e V jn) Log-likelihood of a sample for logit N J L(β 1,..., β K ) = n=1 i=1 y in V in ln e V jn j Cn Transport and Mobility Laboratory Multinomial logit 63 / 70

68 Maximum likelihood estimation Maximum likelihood estimation The maximum likelihood estimation problem max L(β) β R K Nonlinear optimization If the V s are linear-in-parameters, the function is concave Transport and Mobility Laboratory Multinomial logit 64 / 70

69 Maximum likelihood estimation Maximum likelihood estimation Numerical issue P n (i C n ) = e V in j C n e V jn Largest value that can be stored in a computer , that is It is equivalent to compute P n (i C n ) = e e V in V in j C n e V jn V in = 1 j C n e V jn V in Transport and Mobility Laboratory Multinomial logit 65 / 70

70 Outline 1 Components of the Logit model Random Utility Choice set Error terms 2 Systematic utility Linear utility Continuous variables Discrete variables Nonlinearities Interactions Heteroscedasticity 3 A case study 4 Maximum likelihood estimation 5 Simple models

71 Simple models Simple models Null model U i = ε i i e V in P n (i C n ) = j C n e V = e0 jn j C n e 0 = 1 #C n L = n ln 1 #C n = n ln(#c n ) Transport and Mobility Laboratory Multinomial logit 66 / 70

72 Simple models Simple models Constants only [Assume C n = C, n] U i = c i + ε i i In the sample of size n, there are n i persons choosing alt. i. ln P(i) = c i ln( j e c j ) If C n is the same for all people choosing i, the log-likelihood for this part of the sample is L i = n i c i n i ln( j e c j ) Transport and Mobility Laboratory Multinomial logit 67 / 70

73 Simple models Simple models Constants only (ctd) The total log-likelihood is L = j n j c j n ln( j e c j ) At the maximum, the derivatives must be zero L = n 1 n c 1 ec1 j ec j = n 1 np(1) = 0. Transport and Mobility Laboratory Multinomial logit 68 / 70

74 Simple models Simple models Constants only (ctd.) Therefore, P(1) = n 1 n Conclusion If all alternatives are always available, a model with only Alternative Specific Constants reproduces exactly the market shares in the sample Transport and Mobility Laboratory Multinomial logit 69 / 70

75 Simple models Back to the case study Alt. n i n i /n c i e c i P(i) BM SM LF EF MF Null-model: L = -434 ln(5) = Warning: results have been obtained assuming that all alternatives are always available Transport and Mobility Laboratory Multinomial logit 70 / 70