## Derivation of the model

A mixed logit model or random parameters logit model is a logit model for which the parameters are assumed to vary from one individual to another. It is therefore a model that takes the heterogeneity of the population into account.

### The probabilities

For the standard logit model, the probability that individual \(i\) choose alternative \(j\) is:

\[ P_{il} (x_i, \beta_i)=\frac{e^{\beta'x_{il}}}{\sum_j e^{\beta'x_{ij}}}. \]

Suppose now that the coefficients are individual-specific. The probabilities are then:

\[ \frac{e^{\beta_i'x_{il}}}{\sum_j e^{\beta_i'x_{ij}}}. \]

A first approach consists on estimating the parameters for every individual. However, these parameters are identified and can be consistently estimated only if a large number of choice situations per individual is available, which is scarcely the case. A more appealing approach consists on considering the \(\beta_i\)’s as random draws from a distribution whose parameters are estimated, which leads to the mixed logit model. The probability that individual \(i\) will choose alternative \(l\), for a given value of \(\beta_i\) is:

\[ L_{il} \mid \beta_i (x_i, \beta_i)=\frac{e^{\beta_i'x_{il}}}{\sum_j e^{\beta_i'x_{ij}}}. \]

To get the unconditional probability, we have to integrate out this conditional probability, using the density function of \(\beta\). Suppose that \(V_{il}=\alpha+\beta_i x_{il}\), i.e., there is only one individual-specific coefficient and that the density of \(\beta_i\) is \(f(\beta,\theta)\), \(\theta\) being the vector of the parameters of the distribution of \(\beta\). The unconditional probability is then:

\[ P_{il} (x_i, \theta)= \mbox{E}(P_{il} \mid \beta_i) = \int_{\beta}\frac{e^{\beta^\top x_{il}}}{\sum_j e^{\beta^\top x_{ij}}}f(\beta,\theta)d\beta, \]

which is a one-dimensional integral that can be efficiently estimated by quadrature methods. If \(V_{il}=\beta_i^{\top} x_{il}\) where \(\beta_i\) is a vector of length \(K\) and \(f(\beta,\theta)\) is the joint density of the \(K\) individual-specific coefficients, the unconditional probability is:

\[ P_{il}(x_i, \theta)= \mbox{E}(P_{il} \mid \beta_i) = \int_{\beta_1}\int_{\beta_2}...\int_{\beta_K}(P_{il} \mid \beta)f(\beta,\theta)d\beta_1 d\beta_2 \ldots d\beta_K. \]

This is a \(K\)-dimensional integral which cannot easily be estimated by quadrature methods. The only practical method is then to use simulations. More precisely, \(R\) draws of the parameters are taken from the distribution of \(\beta\), the probability is computed for every draw and the unconditional probability, which is the expected value of the conditional probabilities is estimated by the average of the \(R\) probabilities.

### Individual parameters

The expected value of a random coefficient \(\mbox{E}(\beta)\) is simply estimated by the mean of the \(R\) draws on its distribution: \(\bar{\beta}=\sum_{r=1}^R \beta_r\). Individual parameters are obtained by first computing the probabilities of the observed choice of \(i\) for every value of \(\beta_r\):

\[ P_{ir}=\frac{\sum_j y_{ij} e^{\beta_r^ \top x_{ij}}}{\sum_j e^{\beta_r^\top x_{ij}}}, \]

where \(y_{ij}\) is a dummy equal to one if \(i\) has chosen alternative \(j\). The expected value of the parameter for an individual is then estimated by using these probabilities to weight the \(R\) \(\beta\) values:

\[ \hat{\beta}_i = \frac{\sum_r P_{ir} \beta_r}{\sum_r P_{ir}}. \]

### Panel data

If there are repeated observations for the same individuals, the longitudinal dimension of the data can be taken into account in the mixed logit model, assuming that the random parameters of individual \(i\) are the same for all his choice situations. Denoting \(y_{itl}\) a dummy equal to 1 if \(i\) choose alternative \(l\) for the \(t^{th}\) choice situation, the probability of the observed choice is:

\[P_{it}=\prod_j \frac{\sum_j y_{itj}e^{\beta_i x_{itl}}}{\sum_j e^{\beta_i x_{itj}}}. \]

The joint probability for the \(T\) observations of individual \(i\) is then:

\[L_{i}(x_i, y_i, \theta)=\prod_t \prod_j \frac{\sum_jy_{itj}e^{\beta_i^\top x_{itj}}}{\sum_j e^{\beta_i^\top x_{itj}}},\]

which leads to the following unconditional probabilities:

\[ P_{i}(x_i, y_i, \theta) = \int_{\beta_1}\int_{\beta_2}\ldots\int_{\beta_K}\prod_t\prod_i \frac{\sum_j y_{itj} e ^ {\beta ^ \top x_{itj}}}{\sum_j e ^ {\beta ^ \top x_{itj}}} f(\beta, \theta) d\beta_1 d\beta_2 \ldots d\beta_K. \]

## Application

The random parameter logit model is estimated by providing a `rpar` argument to `mlogit`. This argument is a named vector, the names being the random coefficients and the values the name of the law of distribution. Currently, the normal (`"n"`), log-normal (`"ln"`), zero-censored normal (`"cn"`), uniform (`"u"`) and triangular (`"t"`) distributions are available. For these distributions, two parameters are estimated which are, for normal related distributions, the mean and the standard-deviation of the underlying normal distribution and for the uniform and triangular distribution, the mean and the half range of the distribution. For these last two distributions, zero-bounded variants are also provided (`"zbt"` and `"zbu"`). These two distributions are defined by only one parameter (the mean) and their definition domain varies from 0 to twice the mean.

Several considerations may lead to the choice of a specific distribution:

- if correlated coefficients are required, the natural choice is a (transformed-) normal distribution, `"n"`, `"ln"`, `"tn"` and `"cn"`,
- it’s often the case that one wants to impose that the distribution of a random parameter takes only positive or negative values. For example, the price coefficient should be negative for every individual. In this case, `"zbt"` and `"zbu"` can be used. The use of `"ln"` and `"cn"` can also be relevant but, in this case, if only negative values are expected, one should consider the distribution of the opposite of the random price coefficient. This can easily be done using the `opposite` argument of `dfidx`.1,
- the use of unbounded distributions often leads to implausible values of some statistics of the random parameters, especially the mean. This is particularly the case of the log-normal distribution, which has an heavy left tail. In this case, the use of bounded distribution like the uniform and the triangular distributions can be used.

Furthermore, `R` is the number of draws, `halton` indicates whether halton draws (see Train 2009, chap. 9) should be used (`NA` and `NULL` indicate respectively that default halton draws are used and that pseudo-random numbers are used), `panel` is a Boolean which indicates if the panel data version of the log-likelihood should be used.

Correlations between random parameters can be introduced only for normal-related distributed random parameters, using the `correlation` argument. If `TRUE`, all the normal-related random parameters are correlated. The `correlation` argument can also be a character vector indicating the random parameters that are assumed to be correlated.

### Train

We use the `Train` data set, previously coerced to a `dfidx` object called `Tr`. We first estimate the multinomial model: both alternatives being virtual train trips, it is relevant to use only generic coefficients and to remove the intercept:

    library(mlogit)
    data("Train", package = "mlogit")
    Train$choiceid <- 1:nrow(Train)
    Tr <- dfidx(Train, choice = "choice", varying = 4:11, sep = "_",
                opposite = c("price", "comfort", "time", "change"),
                idx = list(c("choiceid", "id")), idnames = c("chid", "alt"))
    Tr$price <- Tr$price / 100 / 2.20371
    Tr$time <- Tr$time / 60
    Train.ml <- mlogit(choice ~ price + time + change + comfort | - 1, Tr)
    Train.ml |> gaze()

            Estimate Std. Error z-value Pr(>|z|)
    price    0.28517    0.01475  19.337  < 2e-16
    time     0.73791    0.08224   8.973  < 2e-16
    change   0.25986    0.05809   4.473 7.71e-06
    comfort  0.82417    0.06117  13.473  < 2e-16

All the coefficients are highly significant and have the predicted positive sign (remind than an increase in the variable `comfort` implies using a less comfortable class). The coefficients can’t be directly interpreted, but dividing them by the price coefficient, we get monetary values:

    coef(Train.ml)[- 1] / coef(Train.ml)[1]
    ##      time    change   comfort
    ## 2.5875726 0.9112191 2.8900676

We obtain the value of 5.2 euros for an hour of traveling, 1 euros for a change and 2.9 euros to travel in a more comfortable class. We then estimate a model with three random parameters, `time`, `change` and `comfort`. We first estimate the uncorrelated mixed logit model:

    Train.mxlu <- mlogit(choice ~ price + time + change + comfort | - 1, Tr,
    panel = TRUE, rpar = c(time = "n", change = "n", comfort = "n"), R = 100,
    correlation = FALSE, halton = NA, method = "bhhh")
    names(coef(Train.mxlu))

    [1] "price"      "time"       "change"     "comfort"    "sd.time"
    [6] "sd.change"  "sd.comfort"

Compared to the multinomial logit model, there are now three more coefficients which are the standard deviations of the distribution of the three random parameters. The correlated model is obtained by setting the `correlation` argument to `TRUE`.

    Train.mxlc <- update(Train.mxlu, correlation = TRUE)
    names(coef(Train.mxlc))

     [1] "price"                "time"
     [3] "change"               "comfort"
     [5] "chol.time:time"       "chol.time:change"
     [7] "chol.change:change"   "chol.time:comfort"
     [9] "chol.change:comfort"  "chol.comfort:comfort"

There are now 6 parameters which are the elements of the Choleski decomposition of the covariance matrix of the three random parameters. These 6 parameters are therefore the elements of the following matrix

\[ C= \left( \begin{array}{ccc} c_{11} & 0 & 0 \\ c_{12} & c_{22} & 0 \\ c_{13} & c_{23} & c_{33} \end{array} \right) \]

such that:

\[ CC^{\top}= \left( \begin{array}{ccc} c_{11}^2 & c_{11} c_{12} & c_{11}c_{13} \\ c_{11}c_{12} & c_{12}^2 + c_{22}^2 & c_{12}c_{23}+c_{22}c_{23} \\ c_{11}c_{13} & c_{12}c_{3} + c_{22}c_{23} & c_{13}^2 + c_{23}^2 c_{33}^2 \end{array} \right) = \left( \begin{array}{ccc} \sigma_{1}^2 & \sigma_{12} & \sigma_{13} \\ \sigma_{12} & \sigma_{2}^2 & \sigma_{23} \\ \sigma_{13} & \sigma_{23} & \sigma_{3}^2 \end{array} \right) \]

where \(\sigma_i^2\) and \(\sigma_{ij}\) are respectively the variance of the random parameter \(i\) and the covariance between two random parameters \(i\) and \(j\). Therefore, the first estimated parameter can be simply interpreted as the standard deviation of the first random parameter, but the five other can’t be interpreted easily.

Random parameters may be extracted using the function `rpar` which take as first argument a `mlogit` object and as second argument `par` the parameter(s) to be extracted. This function returns a `rpar` object and a `summary` method is provided to describe it:

    marg.ut.time <- rpar(Train.mxlc, "time")
    summary(marg.ut.time)
    ##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
    ##      -Inf 0.5079967 1.1082438 1.1082438 1.7084909       Inf

The estimated random parameter is in the “preference space”, which means that it is the marginal utility of time.

Parameters in the “willingness to pay” (WTP) space are more easy to interpret. They can be estimated directly (a feature not supported by mlogit) or can be obtained from the marginal utility by dividing it by the coefficient of a covariate expressed in monetary value (a price for example), taken as a non random parameter. The ratio can then be interpreted as a monetary value (or willingness to pay). To obtain the distribution of the random parameters in the WTP space, one can use the `norm` argument of `rpar`:

    wtp.time <- rpar(Train.mxlc, "time", norm = "price")
    summary(wtp.time)
    ##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
    ##     -Inf 1.294733 2.824586 2.824586 4.354438      Inf

The median value (and the mean value as the distribution is symmetric) of transport time is about 6.9 euros. Several methods/functions are provided to extract the individual statistics (`mean`, `med` and `stdev` respectively for the mean, the median and the standard deviation):

    mean(rpar(Train.mxlc, "time", norm = "price"))
    ## [1] 2.824586
    med(rpar(Train.mxlc, "time", norm = "price"))
    ## [1] 2.824586
    stdev(rpar(Train.mxlc, "time", norm = "price"))
    ## [1] 2.268163

In case of correlated random parameters, as the estimated parameters can’t be directly interpreted, a `vcov` method for `mlogit` objects is provided. It has a `what` argument which default value is `coefficient`. In this case the usual covariance matrix of the coefficients is return. If `what = "rpar"`, the covariance matrix of the correlated random parameters is returned if `type = "cov"` (the default) and the correlation matrix (with standard deviations on the diagonal) is returned if `type = "cor"`. The object is of class `vcov.mlogit` and a `summary` method for this object is provided which computes, using the delta method, the standard errors of the parameters of the covariance or the correlation matrix.

    vcov(Train.mxlc, what = "rpar")

                  time     change   comfort
    time     0.7919713 -0.1831988 0.2991340
    change  -0.1831988  1.0177101 0.5056019
    comfort  0.2991340  0.5056019 2.0945211

    vcov(Train.mxlc, what = "rpar", type = "cor")

                  time     change   comfort
    time     0.8899277 -0.2040590 0.2322569
    change  -0.2040590  1.0088162 0.3463014
    comfort  0.2322569  0.3463014 1.4472460

    summary(vcov(Train.mxlc, what = "rpar", type = "cor"))

                       Estimate Std. Error z-value  Pr(>|z|)
    sd.time             0.88993    0.17244  5.1607  2.46e-07 ***
    sd.change           1.00882    0.10840  9.3062 < 2.2e-16 ***
    sd.comfort          1.44725    0.12494 11.5838 < 2.2e-16 ***
    cor.time:change    -0.20406    0.27863 -0.7324  0.463942
    cor.time:comfort    0.23226    0.17131  1.3558  0.175161
    cor.change:comfort  0.34630    0.13176  2.6282  0.008584 **
    ---
    Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    summary(vcov(Train.mxlc, what = "rpar", type = "cov"))

                       Estimate Std. Error z-value  Pr(>|z|)
    var.time            0.79197    0.30692  2.5804 0.0098697 **
    var.change          1.01771    0.21872  4.6531 3.270e-06 ***
    var.comfort         2.09452    0.36163  5.7919 6.959e-09 ***
    cov.time:change    -0.18320    0.11467 -1.5976 0.1101408
    cov.time:comfort    0.29913    0.15031  1.9901 0.0465831 *
    cov.change:comfort  0.50560    0.15333  3.2975 0.0009756 ***
    ---
    Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

In case of correlated random parameters, as the estimated parameters are not directly interpretable, further functions are provided to analyze the correlation of the coefficients:

    cor.mlogit(Train.mxlc)

                  time     change   comfort
    time     1.0000000 -0.2040590 0.2322569
    change  -0.2040590  1.0000000 0.3463014
    comfort  0.2322569  0.3463014 1.0000000

    cov.mlogit(Train.mxlc)

                  time     change   comfort
    time     0.7919713 -0.1831988 0.2991340
    change  -0.1831988  1.0177101 0.5056019
    comfort  0.2991340  0.5056019 2.0945211

    stdev(Train.mxlc)

         time    change   comfort
    0.8899277 1.0088162 1.4472460

The correlation can be restricted to a subset of random parameters by filling the `correlation` argument with a character vector indicating the corresponding covariates:

    Train.mxlc2 <- update(Train.mxlc, correlation = c("time", "comfort"))
    vcov(Train.mxlc2, what = "rpar", type = "cor")

                 time  comfort
    time    0.8698203 0.270830
    comfort 0.2708300 1.455512

The presence of random coefficients and their correlation can be investigated using any of the three tests. Actually, three nested models can be considered, a model with no random effects, a model with random but uncorrelated effects and a model with random and correlated effects. We first present the three tests of no correlated random effects:

    lrtest(Train.mxlc, Train.ml) |> gaze()
    ## Chisq = 172.197, df: 6, pval = 0.000
    waldtest(Train.mxlc) |> gaze()
    ## chisq = 176.571, df: 6, pval = 0.000
    car::lht(Train.mxlc,
             c("chol.time:time = 0", "chol.time:change =  0",
               "chol.time:comfort = 0", "chol.change:change = 0",
               "chol.change:comfort = 0", "chol.comfort:comfort = 0")) |>
        gaze()
    ## Chisq = 176.571, df: 6, pval = 0.000
    scoretest(Train.ml,
              rpar = c(time = "n", change = "n", comfort = "n"),
              R = 100, correlation = TRUE, halton = NA, panel = TRUE)
    ##
    ##  score test
    ##
    ## data:  rpar(time='n',change='n',comfort='n')
    ## chisq = 165.25, df = 6, p-value < 2.2e-16
    ## alternative hypothesis: no correlated random effects

The hypothesis of no correlated random parameters is strongly rejected. We then present the three tests of no correlation, the existence of random parameters being maintained.

    lrtest(Train.mxlc, Train.mxlu) |> gaze()
    ## Chisq = 13.396, df: 3, pval = 0.004
    car::lht(Train.mxlc,
             c("chol.time:change = 0", "chol.time:comfort = 0",
               "chol.change:comfort = 0")) |> gaze()
    ## Chisq = 29.501, df: 3, pval = 0.000
    waldtest(Train.mxlc, correlation = FALSE) |> gaze()
    ## chisq = 29.501, df: 3, pval = 0.000
    scoretest(Train.mxlu, correlation = TRUE) |> gaze()
    ## chisq = 4.133, df: 3, pval = 0.247

The hypothesis of no correlation is strongly reject with the Wald and the likelihood ratio test, only at the 5% level for the score test.

### RiskyTransport

The second example is a study by León and Miguel (2017) who consider a mode-choice model for transit from Freetown’s airport (Sierra-Leone) to downtown. Four alternatives are available: ferry, helicopter, water-taxi and hovercraft. A striking characteristic of their study is that all these alternatives experienced fatal accidents in recent years, so that the fatality risk is non-negligible and differs much from an alternative to another. For example, the probabilities of dying using the water taxi and the helicopter are respectively of 2.55 and 18.41 out of 100,000 passenger-trips. This feature enables the authors to estimate the value of a statistical life. For an individual \(i\), the utility of choosing alternative \(j\) is:

\[ U_{ij}=\beta_{il} (1 - p_j) + \beta_{ic} (c_j + w_i t_j)+\epsilon_{ij}, \]

where \(p_j\) is the probability of dying while using alternative \(j\), \(c_j\) and \(t_j\) the monetary cost and the transport time of alternative \(j\) and \(w_i\) the wage rate of individual \(i\) (which is supposed to be his valuation of transportation time). \(C_{ij} = c_j + w_i t_j\) is therefore the individual specific generalized cost for alternative \(j\). \(\beta_{il}\) and \(\beta_{ic}\) are the (individual specific) marginal utility of surviving and of expense. The value of the statistical life (VSL) is then defined by:

\[ \mbox{VSL}_i = -\frac{\beta_{il}}{\beta_{îc}} = \frac{\Delta C_{ij}}{\Delta (1-p_j)}. \]

The two covariates of interest are `cost` (the generalized cost in $PPP) and `risk` (mortality per 100,000 passenger-trips). The `risk` variable being purely alternative specific, intercepts for the alternatives cannot therefore be estimated. To avoid endogeneity problems, the authors introduce as covariates marks the individuals gave to 5 attributes of the alternatives: comfort, noise level, crowdedness, convenience and transfer location and the “quality” of the clientele. We first estimate a multinomial logit model.

    RT <- dfidx(RiskyTransport, choice = "choice",
                idx = list(c("chid", "id"), "mode"),
                idnames = c("chid", "alt"))
    ml.rt <- mlogit(choice ~ cost + risk  + seats + noise + crowdness +
                    convloc + clientele | 0, data = RT, weights = weight)

Note the use of the `weights` argument in order to set weights to the observations, as done in the original study.

    coef(ml.rt)[c("risk", "cost")]
    ##         risk         cost
    ## -0.093907630 -0.009540895

The ratio of the coefficients of risk and of cost is `r round(coef(ml.rt)['risk'] / coef(ml.rt)['cost'], 2)` (hundred of thousands of $), which means that the estimated value of the statistical life is a bit less than one million $. We next consider a mixed logit model. The coefficients of `cost` and `risk` are assumed to be random, following a zero-bounded triangular distribution.

    mx.rt <- mlogit(choice ~ cost + risk  + seats + noise + crowdness +
                    convloc + clientele | 0, data = RT, weights = weight,
                    rpar = c(cost = 'zbt', risk = 'zbt'), R = 100,
                    halton = NA, panel = TRUE)

The results are presented in ?@tbl-risktr.

              Multinomial logit Mixed logit
    cost               -0.00954     -0.0187
    risk               -0.09391     -0.1030
    seats               0.15169      0.1085
    noise              -0.02901      0.1423
    crowdness          -0.91860     -0.7157
    convloc            -0.37716     -0.1498
    clientele          -0.25670     -0.3314

Not that the log-likelihood is much larger for the mixed effect logit. Individual-level parameters can be extracted using the `fitted` method, with the type argument set to `parameters`.

    indpar <- fitted(mx.rt, type = "parameters")
    head(indpar)

           id        cost        risk
    1 8020605 -0.02096705 -0.10105817
    2 8260102 -0.01666475 -0.11211057
    3 8260104 -0.01728864 -0.08302831
    4 8260106 -0.01494848 -0.07319789
    5 8260108 -0.01687051 -0.06501337
    6 8260201 -0.02133116 -0.10285434

We can then compute the VSL for every individual and analyse their distribution, using quantiles and plotting on figure 1 the empirical density of VSL for African and non-African travelers (as done in León and Miguel 2017 Table 4, p.219 and Figure 5, p.223).2

    indpar$VSL <- with(indpar, risk / cost * 100)
    quantile(indpar$VSL, c(0.025, 0.975))
    ##      2.5%     97.5%
    ##  432.4199 1054.3428
    mean(indpar$VSL)
    ## [1] 608.94

Note that computing the VSL as the ratio of two random parameters which can take zero value can lead to extremely high values if the individual parameter for cost is close to 0.3

    max(indpar$cost)
    ## [1] -0.002924437
    max(indpar$VSL)
    ## [1] 3131.825

This is not the case here as (absolute) minimum value of `cost` is \(-0.003\) which leads to a maximum value of VSL of $ \(3131\).

## Bibliography

Daly, Andrew, Stephane Hess, and Kenneth Train. 2012. “Assuring Finite Moments for Willingness to Pay in Random Coefficient Models.” Transportation 39 (1): 19–31. https://doi.org/10.1007/s11116-011-9331-3.
León, Gianmarco, and Edward Miguel. 2017. “Risky Transportation Choices and the Value of a Statistical Life.” American Economic Journal: Applied Economics 9 (1): 202–28. https://doi.org/10.1257/app.20160140.
Train, Kenneth. 2009. Discrete Choice Methods with Simulation. Cambridge University Press. https://doi.org/10.1017/CBO9780511805271.

1. See vignette formula/data.↩︎

2. Note that individual-specific parameters should be interpreted with caution, as they are consistent estimates of individual parameters only if the number of choice situations for every individual is large (see Train 2009, 266).↩︎

3. See Daly, Hess, and Train (2012) for a discussion of the specifications of mixed logit models which assure finite moments of the distribution of willingness to pay.↩︎