Multinomial Logit and Conditional Logit Models

Discrete choice models such as binary logit and multinomial logit are probably the most widely used methods for mapping the freight mode choice process (Greene and Hensher, 2013; Arunotayanun and Polak, 2011; Train and Wilson, 2008; Rich et al., 2009, 2011; Jong and Ben- Akiva, 2007; Danielis et al., 2005; Bergkvist., 2001; Sayed et al., 2000; Nijkamp et al., 1999; Abdelwahab et al., 1998). Typical multinomial logit forms are (Liao, 1994):

𝑃𝑟𝑜𝑏(𝑦 = 1) = 𝑒∑ 𝛽̂1𝑘𝑥𝑘 𝑘 𝑘=1 1+𝑒∑𝑘𝑘=1𝛽̂1𝑘𝑥𝑘_+𝑒∑𝑘𝑘=1𝛽̂2𝑘𝑥𝑘 , 𝑃𝑟𝑜𝑏(𝑦 = 2) = 𝑒∑𝑘𝑘=1𝛽̂2𝑘𝑥𝑘 1+𝑒∑𝑘𝑘=1𝛽̂1𝑘𝑥𝑘_+𝑒∑𝑘𝑘=1𝛽̂2𝑘𝑥𝑘 𝑃𝑟𝑜𝑏(𝑦 = 3) = 1 1+𝑒∑𝑘𝑘=1𝛽̂1𝑘𝑥𝑘+𝑒∑𝑘𝑘=1𝛽̂2𝑘𝑥𝑘 ⋯ 𝑃𝑟𝑜𝑏(𝑦 = 𝑗) = 𝑒∑ 𝛽 ̂_{𝑗𝑘𝑥𝑘} 𝑘 𝑘=1 1+ ∑𝐽−1𝑒∑𝑘𝑘=1𝛽̂𝑗𝑘𝑥𝑘 𝑗=1 (3.1)

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These approaches model the choice or market shares for the available transport modes. The functional forms are based on the utility maximizing choice process of the shipper. This approach is of a behavioural nature. The decision taker is assumed to base the choice on the characteristics of the offered transport services, such as delivery time, reliability and frequency of service. When making their decision not all possible attributes are included in the model due to random taste and variation. Thus, a random error is introduced into the model. All this is captured in the random utility theory (McFadden, 1973).

The logit model was first derived by Luce (1959), and it is the most widely used model because of the fact that the choice probabilities take a closed form and are readily interpretable. In the MNL, the probability that the choice outcome 𝑦_𝑖 is alternative 𝑗 from all alternatives available to the individual can be expressed as the logit formula:

𝑃(𝑦_𝑖 = 𝑗) = 𝑃_𝑖𝑗 = exp (𝑥𝑖𝛽𝑗)

∑𝐽𝑘=0exp (𝑥𝑖𝛽𝑘)

for 𝑗 = 0, … , 𝐽 (3.2)

The vector 𝛽𝑗 is a vector of coefficients specific to the 𝑗 th alternative, 𝑥𝑖 is a vector of characteristics specific to the 𝑖 th individual, 𝑦_𝑖 indicates the choice made. To identify the model, we assume without loss of generality that 𝛽₀= 0. The model can also be written in terms of the odds for each pair of options 𝑗 and 𝑞:

Ω_{𝑖𝑗|𝑖𝑞} = exp(𝑥_𝑖[𝛽_𝑗− 𝛽_𝑞]) (3.3)

This equation shows that the odds (Ω_𝑖𝑗) of choosing 𝑗 versus 𝑞 do not depend on any other additional alternatives on the choice set; the odds are determined only by the coefficient vectors for 𝑗 and 𝑞, 𝛽_𝑗 and 𝛽_𝑞. Assuming that unobserved utilities for each alternative are independently

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and identically distributed (IID), and are described by the Gumbel distribution, produces the MNL model (Domencich and McFadden, 1975).

The utility functions are linear in the parameter forms and the parameter 𝑥 is related to the variance of 𝜀 (Ben-Akiva and Lerman, 1985). Thus, for the MNL model, a scale parameter 𝛽2_{= 𝜋}2_/6𝜎2 _{. The key assumption of the MNL model is that the errors are independent of each} other. This independence means that the unobserved portion of utility for one alternative is unrelated to the unobserved portion of utility for another alternative.

If one thinks that the unobserved portion of utility of one alternative is correlated with that of other alternatives, then there are three options (Train, 2003): (1) use a different model that allows for correlated errors, such as the nested logit or mixed logit models, (2) re-specify the representative utility so that the source of the correlation is captured explicitly and thus the remaining errors are independent, or (3) use the logit model under the current specification of representative utility, considering the model to be an approximation.

The conditional logit (CL) model is closely related to the better-known MNL model, but it derives from different behavioural assumptions and is estimated in a different form. The CL model is similarly defined when choice-specific data are available (Maddala, 1983; Train, 2003; Garcia- Menendez et al., 2004; Hansen, 2011). Both multinomial logit and conditional logit are used to analyse the choice of an individual among a set of 𝑗 alternatives. The main distinction between the two can be put very simply: the MNL model focuses on the individual as the unit of analysis and uses the individual's characteristics as explanatory variables; in contrast, the CL model focuses on the set of alternatives for each individual and the explanatory variables are characteristics of those alternatives. Hence, the CL model is appropriate for a different class of

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models in which a choice among alternatives is treated as a function of the characteristics of the alternatives, rather than (or in addition to) the characteristics of the individual making the choice (characteristics of the chooser) which the MNL model currently uses. The models do, however, share a common likelihood function.

Using the properties of the Gumbel distribution, the probability that individual 𝑖 chooses alternative 𝑗 from among the choices in the choice set 𝑍_𝑖 is

𝑃(𝑦_𝑖 = 𝑗) = 𝑃_𝑖𝑗 = exp (𝑥𝑖𝑗𝛽)

∑𝐽_{𝑘∈𝑍𝑖}exp (𝑥𝑖𝑘𝛽) (3.4)

where 𝑥_𝑖𝑗 is a vector of attributes specific to the 𝑗 th alternative as perceived by the 𝑖 th individual. It is assumed that there are 𝑛 choices in each individual’s choice set, 𝑍𝑖.

Within the CL model, the parameters are assumed to be constant across the alternatives. As a result, the CL model can be used to predict the probability that an individual will choose a previously unavailable alternative, given knowledge of 𝛽 and the vector 𝑥_𝑖𝑗 of choice-specific characteristics. Consequently, conditional logit models are often used when the number of possible choices is large. The CL model is explained in detail in the articles by McFadden (1973) Boxall and Adamowicz (2002) and Shen and Saijo (2007).

The MNL model was the most widely used modelling methodology to measure shippers’ mode choice behaviour in the early stage of freight transport modelling (Jong and Ben-Akiva, 2007; Yannis and Golias, 2005; Catalani, 2001; McGinnis et al., 1981; Nam, 1997; Wilson et al., 1986). However, Oum (1990) gave an early warning against the use of MNL models estimated at the aggregate level, because of the many restrictions this model type imposes on the

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parameters (e.g. IIA). Nevertheless, because of the relative ease of obtaining aggregate data, the MNL model specification is still the one used most in modelling freight mode choice in practice (Jong et al., 2012). A few studies have explored the estimation of a freight transport demand function using a CL model (Train, 2003; Garcia-Menendez et al., 2004).