Mixed logit
Encyclopedia
Mixed logit is a fully general statistical model for examining discrete choices
. The motivation for the mixed logit model arises from the limitations of the standard logit model. The standard logit model has three primary limitations, which mixed logit solves: "It [Mixed Logit] obviates the three limitations of standard logit by allowing for random taste variation, unrestricted substitution patterns, and correlation in unobserved factors over time." Mixed logit can also utilize any distribution for the random coefficients, unlike probit which is limited to the normal distribution. It has been shown that a mixed logit model can approximate to any degree of accuracy any true random utility model of discrete choice, given an appropriate specification of variables and distribution of coefficients." The following discussion draws from Ch. 6 of Discrete Choice Methods with Simulation, by Kenneth Train
(Cambridge University Press
), to which the reader is referred for more details and citations. See also the article on discrete choice
for information on how the mixed logit relates to discrete choice analysis in general and to other specific types of choice models.
's, are fixed, which means the 
's are the same for everyone. Mixed logit has different 
's for each person (i.e., each decision maker.)
In the standard logit model, the utility of person n for alternative i is:
with
~ iid extreme value
For the mixed logit model, this specification is generalized by allowing
to be random. The utility of person n for alternative i in the mixed logit model is:
with
~ iid extreme value
where θ are the parameters of the distribution of
's over the population, such as the mean and variance of 
.
Conditional on
, the probability that person n chooses alternative i is the standard logit formula:
However, since
is random and not known, the (unconditional) choice probability is the integral of this logit formula over the density of 
.
This model is also called the random coefficient logit model since
is a random variable. It allows the slopes of utility (i.e., the marginal utility) to be random, which is an extension of the random effects model where only the intercept was stochastic.
Any probability density function
can be specified for the distribution of the coefficients in the population, i.e., for
. The most widely used distribution is normal, mainly for its simplicity. For coefficients that take the same sign for all people, such as a price coefficient that is necessarily negative or the coefficient of a desirable attribute, distributions with support on only one side of zero, like the lognormal, are used. When coefficients cannot logically be unboundedly large or small, then bounded distributions are often used, such as the 
or triangular distributions.
(IIA) property. The percentage change in the probability for one alternative given a percentage change in the mth attribute of another alternative is
where β m is the mth element of
. It can be seen from this formula that "A ten-percent reduction for one alternative need not imply (as with logit) a ten-percent reduction in each other alternative." The relative percentages depend on correlation between the likelihood that respondent n will choose alternative i, L ni , and the likelihood that respondent n will choose alternative j, L nj , over various draws of β.
where the subscript t is the time dimension. We still make the logit assumption which is that
is i.i.d extreme value. That means that 
is independent over time, people, and alternatives. 
is essentially just white noise. However, correlation over time and over alternatives arises from the common effect of the 
's, which enter utility in each time period and each alternative.
To examine the correlation explicitly, assume that the β 's are normally distributed with mean
and variance 
. Then the utility
equation becomes:
and η is a draw from the standard normal density. Rearranging, the equation becomes:
where the unobserved factors are collected in
. Of the unobserved factors, 
is independent over time, and 
is not independent over time or alternatives.
Then the covariance between alternatives
and 
is,
and the covariance between time
and 
is
By specifying the X's appropriately, one can obtain any pattern of covariance over time and alternatives.
Conditional on
, the probability of the sequence of choices by a person is simply the product of the logit probability of each individual choice by that person:
since
is independent over time. Then the (unconditional) probability of the sequence of choices is simply the integral of this product of logits over the density of 
.
1. Take a draw from the probability density function that you specified for the 'taste' coefficients. That is, take a draw from
and label the draw 
, for 
representing the first draw.
2. Calculate
. (The conditional probability.)
3. Repeat many times, for
.
4. Average the results
Then the formula for the simulation look like the following,
where R is the total number of draws taken from the distribution, and r is one draw.
Once this is done you will have a value for the probability of each alternative i for each respondent n.
Discrete choice
In economics, discrete choice problems involve choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such choices contrast with standard consumption models in which the quantity of each good consumed is assumed...
. The motivation for the mixed logit model arises from the limitations of the standard logit model. The standard logit model has three primary limitations, which mixed logit solves: "It [Mixed Logit] obviates the three limitations of standard logit by allowing for random taste variation, unrestricted substitution patterns, and correlation in unobserved factors over time." Mixed logit can also utilize any distribution for the random coefficients, unlike probit which is limited to the normal distribution. It has been shown that a mixed logit model can approximate to any degree of accuracy any true random utility model of discrete choice, given an appropriate specification of variables and distribution of coefficients." The following discussion draws from Ch. 6 of Discrete Choice Methods with Simulation, by Kenneth Train
Kenneth E. Train
Kenneth E. Train is a Professor of Economics at the University of California, Berkeley., USA. He is also Vice President of in San Francisco, California. He received a Bachelors in Economics at Harvard and PhD from UC Berkeley...
(Cambridge University Press
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
), to which the reader is referred for more details and citations. See also the article on discrete choice
Discrete choice
In economics, discrete choice problems involve choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such choices contrast with standard consumption models in which the quantity of each good consumed is assumed...
for information on how the mixed logit relates to discrete choice analysis in general and to other specific types of choice models.
Random taste variation
The standard logit model's "taste" cofficients, or's, are fixed, which means the 's are the same for everyone. Mixed logit has different 's for each person (i.e., each decision maker.)In the standard logit model, the utility of person n for alternative i is:
with
~ iid extreme valueFor the mixed logit model, this specification is generalized by allowing
to be random. The utility of person n for alternative i in the mixed logit model is:with
~ iid extreme valuewhere θ are the parameters of the distribution of
's over the population, such as the mean and variance of .Conditional on
, the probability that person n chooses alternative i is the standard logit formula:However, since
is random and not known, the (unconditional) choice probability is the integral of this logit formula over the density of .This model is also called the random coefficient logit model since
is a random variable. It allows the slopes of utility (i.e., the marginal utility) to be random, which is an extension of the random effects model where only the intercept was stochastic.Any probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
can be specified for the distribution of the coefficients in the population, i.e., for
. The most widely used distribution is normal, mainly for its simplicity. For coefficients that take the same sign for all people, such as a price coefficient that is necessarily negative or the coefficient of a desirable attribute, distributions with support on only one side of zero, like the lognormal, are used. When coefficients cannot logically be unboundedly large or small, then bounded distributions are often used, such as the or triangular distributions.Unrestricted substitution patterns
The mixed logit model can represent general substitution pattern because it does not exhibit logit's restrictive independence of irrelevant alternativesIndependence of irrelevant alternatives
Independence of irrelevant alternatives is an axiom of decision theory and various social sciences.The word is used in different meanings in different contexts....
(IIA) property. The percentage change in the probability for one alternative given a percentage change in the mth attribute of another alternative is
where β m is the mth element of
. It can be seen from this formula that "A ten-percent reduction for one alternative need not imply (as with logit) a ten-percent reduction in each other alternative." The relative percentages depend on correlation between the likelihood that respondent n will choose alternative i, L ni , and the likelihood that respondent n will choose alternative j, L nj , over various draws of β.Correlation in unobserved factors over time
Standard logit does not take into account any unobserved factors that persist over time for a given decision maker. This can be a problem if you are using panel data, which represent repeated choices over time. By applying a standard logit model to panel data you are making the assumption that the unobserved factors that affect a person's choice are new every time the person makes the choice. That is a very unlikely assumption. To take into account both random taste variation and correlation in unobserved factors over time, the utility for respondent n for alternative i at time t is specified as follows:where the subscript t is the time dimension. We still make the logit assumption which is that
is i.i.d extreme value. That means that is independent over time, people, and alternatives. is essentially just white noise. However, correlation over time and over alternatives arises from the common effect of the 's, which enter utility in each time period and each alternative.To examine the correlation explicitly, assume that the β 's are normally distributed with mean
and variance . Then the utilityUtility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
equation becomes:
and η is a draw from the standard normal density. Rearranging, the equation becomes:
where the unobserved factors are collected in
. Of the unobserved factors, is independent over time, and is not independent over time or alternatives.Then the covariance between alternatives
and is,and the covariance between time
and isBy specifying the X's appropriately, one can obtain any pattern of covariance over time and alternatives.
Conditional on
, the probability of the sequence of choices by a person is simply the product of the logit probability of each individual choice by that person:since
is independent over time. Then the (unconditional) probability of the sequence of choices is simply the integral of this product of logits over the density of .Simulation
Unfortunately there is no closed form for the integral that enters the choice probability, and so the researcher must simulate Pn. Fortunately for the researcher, simulating Pn can be very simple. There are four basic steps to follow1. Take a draw from the probability density function that you specified for the 'taste' coefficients. That is, take a draw from
and label the draw , for representing the first draw.2. Calculate
. (The conditional probability.)3. Repeat many times, for
.4. Average the results
Then the formula for the simulation look like the following,
where R is the total number of draws taken from the distribution, and r is one draw.
Once this is done you will have a value for the probability of each alternative i for each respondent n.