Mixed logit

From DDWiki

Mixed logit is the extension of standard logit model which has restrcitions of fixed attribute coefficients, IIA property and not allowing correlations across alternatives. Random-coefficients model and error-components model are two widely accepted methods to derive mixed logit probabilities.

Random-coefficients model considers the random term into the observed portion in utility. The approach is suitable for the choice model study when the researcher is interested in the patterns of random taste variations. On the other hand, error-components model considers the random term into unobserved attributes, which are error components. The approach introduces heteroskedasdicity and correlation across alternatives in the unobserved portion of utility. This allows the mixed logit model represent substitution patterns in observed attributes.

1 Mixed logit probability
2 Example
3 Standard logit
4 Simulations
5 External links
6 Reference

Mixed logit probability

In following, random-coefficients model is used to explain the derivation of mixed logit probabilities. When utility is linear in β, the utility of person n choosing alternative j can be written as:

$U_{nj} = \beta_n x_{nj} + \varepsilon_{nj}$

with the assumptions of mixed logit model:

$\varepsilon_{nj}$ ~ iid extreme value

$\quad \beta_n x_{nj} \sim f(\beta_n | \theta)$

where θ is the distribution parameters over the population.

Different from standard logit model, the attribute coefficients β are not necessary fixed values. If β for all alternative j and person n chooses alternative i are known, and the unobserved error components are assumed extreme-value distrbution, the choice probalities of mixed logit model can be written in intgeration form:

$P_{ni} = \int L_{ni} (\beta) f(\beta) d\beta = \int \frac{e^{\beta x_{ni}}} {\sum_J e^{\beta x_{nj}}} f(\beta) d \beta$

where logit probabity term in the equation means the persion n chooseing altenrative i over all other alternatives j=1,2,…,J and j ≠ i.

The general mixed logit probabilities can be re-written into a more useful form:

$P_{ni} = \int \frac{e^{\beta x_{ni}}} {\sum_J e^{\beta x_{nj}}} \phi(\beta | b,W) d \beta$

where $φ(β | b, W)$ is the normal distribution density with mean b and covariance W. However, it can be applied with different distribution types, such as lognormal or uniform distributions.

Example

The concept of random-coefficients can be explained through discussing the different models with the following graphs.

Assume there is only one attribute, price, in the choice model. Usually higher product price cause lower utility because people always prefer cheaper product if all other functionalities are the same. In case (a), it shows the fixed coefficient β presents a linear relation in the standard logit model. Assume the β equals -0.5, which means a consumer likes lower price. All people in this logit model have the same price preference.

In case (b), the discrete coefficients β₁, β₂, β₃ and β₄ in logit model are fitted by spline function which can be used by interpolation for demand prediction. The graph and numbers show a individual has different preferences at different price segments. It can be seen that the discrete logit model more flexibility and better approach to realistic decision-maker's taste. Again, all people in the model show the same price preference.

In case (c) random-coefficients mixed logit model, the coefficient β is not a deterministic value, but a stochastic number presenting taste variation over consumers can only be represented by distribution. If the distribution is normal and mean value is -0.5, that means major population of people generally like low price. If we pick up an individual consumer from this group of people, he will present the price preference shown in graph (c-1).

In the left end of distribution, the area means only few people love low price very much. An individual in this group may have his price taste shown in the graph (c-2). On the other hand, the right-hand section of the distribution means some few people are less sensitive to low price. In the far right end, extremely few people may perfer high price. An individual in this group may show his price preference in the way of graph (c-3) presenting.

Therefore random coefficients in mixed logit model can simulate real-world scenario better than fixed coefficients in standard logit model. Some researchers use different distributions, such as log-normal and truncated normal, to describe the random coefficients and have improved approximation.

Standard logit

Standard logit model can be seen as a special case of mixed logit. When all β are fixed coefficients and the mixing distribution density function is 1, the mixed logit becomes simple logit form:

$P_{ni} = \frac {e^{\beta x_{ni}}} {\sum_J e^{\beta x_{nj}}}$

Simulations

There are mainly two popular estimation methods, maximum likelihood estimator (MLE) and hierarchical Bayes (HB), to calculate the attribute parameters in mixed logit model.

When using MLE method, the likihood value need to be estimated first. Since mixed logit probability is not a close form, it needs to be approximated through simulation. Assume β^r is generated from the r-th random draw. There are totally R times of draws in the simulation. The average simulated probability can be expressed as:

$\bar{P}_{ni} = \frac{1}{R} \sum_{r=1}^{R} L_{ni} (\beta^r)$

The simulated log liklihood is calculated by summing up through N people:

$LL = \sum_{n=1}^{N} \sum_{j=1}^{J} y_{nj} \ln \bar{P}_{nj}$

where y_nj=1 if person n chooses j, y_nj=0 otherwise.

External links

Reference

Kenneth E. Train (2003), Discrete Choice Methods with Simulation, Cambrige University Press. PDF
Jordan J. Louviere, David A. Hensher, Joffre D. Swait (2000), Stated Choice Methods: Analysis and Applications, Cambrige University Press.

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Category: Discrete choice models