# Mixed logit and preference heterogeneity

The mixed (random-coefficients) logit lets tastes vary across decision units. Instead of a single coefficient on each random attribute, choicer estimates a distribution of coefficients. Substitution then reflects both observed covariates and the estimated distribution of tastes. Estimation is by simulated maximum likelihood using Halton draws, with the likelihood, gradient and Hessian evaluated in parallel C++. `run_mxlogit()` currently estimates a cross-sectional simulated likelihood: each choice situation’s probability is integrated over the taste distribution separately. It does not hold a simulation draw fixed across a person’s tasks and then integrate the product of conditional probabilities, as a frequentist panel mixed-logit likelihood would.

A useful way to see the mechanism is to write the mixed-logit probability as a logit kernel averaged over the taste distribution, \(P_{ij} = \int L_{ij}(\beta)\, f(\beta)\, d\beta\), where \(L_{ij}(\beta) = \exp(X_{ij}\beta) / \sum_k \exp(X_{ik}\beta)\). Conditional on a draw \(\beta\), the kernel is an ordinary logit. The empirical content of the mixed logit is in the averaging: people with different \(\beta\)’s place different values on the same attributes, so demand leaving an alternative need not go to the same destinations for all consumers. For counterfactual work, the distinction between observed heterogeneity and the estimated mixing distribution is central: flexibility that is not disciplined by the available variation is supplied by the maintained distribution \(f(\beta)\).

    library(choicer)
    set_num_threads(2)

## Simulate correlated random coefficients

`simulate_mxl_data()` draws choices in which the coefficients on `w1` and `w2` are themselves random and correlated across the population.

    sim <- simulate_mxl_data(N = 2000, J = 4, seed = 1)
    sim
    #> <choicer_sim: mxl>
    #>   settings:
    #>     N = 2000
    #>     J = 4
    #>     K_x = 2
    #>     K_w = 2
    #>     outside_option = TRUE
    #>     vary_choice_set = TRUE
    #>   rows in $data: 7982
    #>   true_params: beta, delta, Sigma, L_params, mu, rc_dist, rc_correlation

## Fit

A robust recipe for mixed logit: warm-start from a plain MNL, scale the variables so the Hessian is well conditioned, and use enough Halton draws. Here we estimate a full (correlated) covariance of the random coefficients.

    fit <- run_mxlogit(
      data            = sim$data,
      id_col          = "id",
      alt_col         = "alt",
      choice_col      = "choice",
      covariate_cols  = c("x1", "x2"),  # fixed coefficients
      random_var_cols = c("w1", "w2"),  # random coefficients
      rc_correlation  = TRUE,           # estimate their full covariance
      S               = 100L,           # Halton draws per person
      draws           = "generate",     # generate draws on the fly (low memory)
      seed            = 7L,
      scale_vars      = "sd",           # condition the Hessian across blocks
      se_method       = "bhhh"
    )
    #> Optimization run time 0h:0m:0.56s
    summary(fit)
    #> Mixed Logit (MXL) model
    #>
    #> Parameter    Estimate  Std.Error  z-value  Pr(>|z|)
    #> x1           0.776408   0.070138  11.0697  0.00e+00  ***
    #> x2          -0.675643   0.068318  -9.8897  0.00e+00  ***
    #> Sigma_11     0.905183   0.451785   2.0036  4.51e-02  *
    #> Sigma_21     0.894700   0.310679   2.8798  3.98e-03  **
    #> Sigma_22     1.884033   0.575835   3.2718  1.07e-03  **
    #> ASC_1        0.536159   0.078972   6.7893  1.13e-11  ***
    #> ASC_2       -0.477738   0.105075  -4.5466  5.45e-06  ***
    #> ASC_3        0.591089   0.077545   7.6225  2.49e-14  ***
    #> ASC_4       -0.513880   0.105365  -4.8771  1.08e-06  ***
    #> ---
    #> Signif. codes:  '***' 0.001 '**' 0.01 '*' 0.05
    #>
    #> Random coefficient covariance (Sigma):
    #>        w1     w2
    #> w1 0.9052 0.8947
    #> w2 0.8947 1.8840
    #>
    #> Std. Errors: BHHH (OPG)
    #> Log-likelihood: -2435.84
    #> AIC: 4889.67  | BIC: 4940.08
    #> McFadden R2: 0.106 (adj: 0.102) | Hit rate: 0.452
    #> N: 2000  | Parameters: 9
    #> Optimization time: 0.56 s
    #> Convergence: 3 ( NLOPT_FTOL_REACHED: Optimization stopped because ftol_rel or ftol_abs (above) was reached. )

This vignette uses `se_method = "bhhh"` because the outer-product calculation is fast and keeps package-build time short. For final empirical work, compare it with the default analytical-Hessian standard errors; when the data come from a choice-based or otherwise weighted sample, use `se_method = "sandwich"` so the reported covariance is the robust WESML sandwich rather than the inverse weighted Hessian. If WESML observations are also dependent within people or markets, use the weighted cluster sandwich (`se_method = "cluster"`, with `cluster_col`) and have enough independent clusters for its asymptotics.

> Tip. For real applications increase the number of draws (`S`) until your estimates are stable, and keep `scale_vars = "sd"`. Stability in `S` is a statistical requirement, not just numerical hygiene: simulated ML with a fixed number of draws is biased — the log of an unbiased probability simulator is not unbiased — and the classical asymptotics require `S` to grow with the sample (see the math companion for the conditions). If the solver struggles, pass an explicit `theta_init`: place the MNL slope and ASC estimates in the corresponding MXL parameter blocks and initialize the additional mean and Cholesky coordinates deliberately. Bounds can keep Cholesky diagonals away from numerically pathological regions. See `inst/simulations/mxl_simulation.R` for a fully hardened example.

## Parameter recovery

    recovery_table(fit, sim$true_params)
    #> <choicer_recovery> model=choicer_mxl level=0.95
    #>    parameter  group    true estimate     se    bias rel_bias_pct z_vs_true
    #>       <char> <char>   <num>    <num>  <num>   <num>        <num>     <num>
    #> 1:        x1   beta  0.8000   0.7764 0.0701 -0.0236       -2.949   -0.3364
    #> 2:        x2   beta -0.6000  -0.6756 0.0683 -0.0756       12.607   -1.1072
    #> 3:      L_11  sigma  0.0000  -0.0498 0.2496 -0.0498           NA   -0.1996
    #> 4:      L_21  sigma  0.5000   0.9404 0.3057  0.4404       88.078    1.4406
    #> 5:      L_22  sigma  0.1116  -0.0002 0.3566 -0.1117     -100.136   -0.3133
    #> 6:     ASC_1    asc  0.5000   0.5362 0.0790  0.0362        7.232    0.4579
    #> 7:     ASC_2    asc -0.5000  -0.4777 0.1051  0.0223       -4.452    0.2119
    #> 8:     ASC_3    asc  0.5000   0.5911 0.0775  0.0911       18.218    1.1747
    #> 9:     ASC_4    asc -0.5000  -0.5139 0.1054 -0.0139        2.776   -0.1317
    #>    lower_ci upper_ci covers
    #>       <num>    <num> <lgcl>
    #> 1:   0.6389   0.9139   TRUE
    #> 2:  -0.8095  -0.5417   TRUE
    #> 3:  -0.5389   0.4393   TRUE
    #> 4:   0.3412   1.5396   TRUE
    #> 5:  -0.6991   0.6988   TRUE
    #> 6:   0.3814   0.6909   TRUE
    #> 7:  -0.6837  -0.2718   TRUE
    #> 8:   0.4391   0.7431   TRUE
    #> 9:  -0.7204  -0.3074   TRUE

The `beta` rows are the fixed coefficients, the `sigma` rows describe the covariance of the random coefficients (its Cholesky elements), and the `asc` rows are the alternative-specific constants.

## Substitution through taste heterogeneity

With random coefficients, diversion is mediated by the distribution of tastes. If the estimated mixing distribution captures economically meaningful heterogeneity, people who value one alternative tend to value nearby substitutes on that latent margin. Diversion can therefore depend on which attribute is changing, so `diversion_ratios()` takes a `wrt_var`:

    elasticities(fit, elast_var = "x2")
    #>   0         1         2         3         4
    #> 0 0 -0.015656 -0.009793 -0.025914 -0.009414
    #> 1 0 -0.023929 -0.005506 -0.017254 -0.005718
    #> 2 0 -0.010723 -0.016649 -0.017014 -0.007051
    #> 3 0 -0.008498 -0.006243 -0.006971 -0.006645
    #> 4 0 -0.011956 -0.006111 -0.021578 -0.020221
    diversion_ratios(fit, wrt_var = "x2")
    #>        0      1      2      3      4
    #> 0 0.0000 0.3435 0.3183 0.3533 0.3177
    #> 1 0.3112 0.0000 0.2616 0.3195 0.2629
    #> 2 0.1789 0.1623 0.0000 0.1658 0.1434
    #> 3 0.3342 0.3336 0.2790 0.0000 0.2760
    #> 4 0.1757 0.1605 0.1412 0.1614 0.0000

    # For a random-coefficient attribute the perturbation coordinate matters.
    elasticities(fit, elast_var = "w2", is_random_coef = TRUE)
    #>   0        1        2        3        4
    #> 0 0 -0.01809 -0.02890 -0.01330 -0.02617
    #> 1 0  0.05861 -0.02036 -0.02040 -0.01905
    #> 2 0 -0.02383  0.13609 -0.02394 -0.02000
    #> 3 0 -0.02050 -0.01863  0.04831 -0.01820
    #> 4 0 -0.02493 -0.02164 -0.02110  0.13556
    diversion_ratios(fit, wrt_var = "w2", is_random_coef = TRUE)
    #>         0       1      2        3       4
    #> 0  0.0000  0.5513 -6.311  0.53192 -0.4087
    #> 1 -2.0961  0.0000  8.744  0.62491  1.8795
    #> 2  1.3819  0.5036  0.000 -0.08724 -0.7105
    #> 3  1.4015 -0.4331  1.050  0.00000  0.2397
    #> 4  0.3127  0.3782 -2.482 -0.06959  0.0000

The rest of the toolkit — `predict()`, `wtp()`, `consumer_surplus()`, `blp()` — uses the same fitted object as in the getting-started vignette. Prediction and share inversion integrate over the fitted taste distribution; `wtp()` respects choicer’s supported coefficient parameterizations. In v0.2.0, `consumer_surplus()` requires a fixed price coefficient because inverse moments of a random price coefficient need not exist.

## Identification and tails

The mixed logit is a genuine generalization of the MNL — if tastes are in fact homogeneous, the estimator simply returns a near-zero variance and you are back to a logit. The issue is not that random coefficients are intrinsically fragile. The issue is identification: the additional substitution structure is carried by the mixing distribution \(f(\beta)\), and \(f(\beta)\) is often hardest to pin down where it matters most for welfare and diversion — in the tails.

Two consequences are worth keeping in front of you:

- The tails drive the economics you report. A lognormal price coefficient puts a slice of the population at near-zero price sensitivity, which can give a willingness-to-pay distribution with no finite mean and explosive welfare numbers. An unbounded normal coefficient implies a fraction of consumers with the wrong sign (who prefer paying more). These artifacts come from the assumed shape of \(f\), not from the data, and the estimator will happily contort a tail to match an aggregate moment.

- \(f(\beta)\) is hard to identify and estimate in practice. A single cross-section of choices — one decision per person, a fixed menu — carries little information about the spread of tastes. In mixed-logit research, reliable estimation typically uses repeated choices from the same individual (panel data), substantial variation in choice sets or attributes across markets (the BLP setting), or the rich, designed attribute variation of a stated-preference experiment. Without one of these, the random-coefficient variances are weakly identified and the estimates can be fragile. Note that on panel data `run_mxlogit()` does not exploit the first source: it still maximizes a cross-sectional simulated likelihood — each choice situation is integrated separately rather than sharing a taste draw across that person’s likelihood contributions. Cluster-robust standard errors (`cluster_col=`, or `vcov(fit, type = "cluster")`; see Which standard errors, and when) repair the inference for within-person dependence, but the model that actually uses the panel to identify the taste distribution is the hierarchical Bayesian logit, `run_hmnlogit()`.

## A research checklist

For a reportable `run_mxlogit()` application:

1. Fit a transparent MNL benchmark, scale poorly conditioned covariates, and use its estimates to inform explicit `theta_init` values.
2. Run several manually supplied starting values and retain their objective, convergence code and substantive outputs. choicer does not yet automate multistart estimation.
3. Increase `S` until coefficients, covariance elements, standard errors, WTP, elasticities and diversion ratios are stable—not merely until the optimizer reports convergence.
4. Compare choicer’s supported normal and shifted-lognormal restrictions where economically meaningful, and show what their tails imply for signs and WTP.
5. Compare the analytical-Hessian and BHHH results; use robust or WESML inference for weighting/misspecification and cluster inference for repeated sampling units as appropriate.
6. Treat clustering of panel tasks as an inference correction only. If tastes should persist within person, use `run_hmnlogit()` so the likelihood itself uses that repetition.
7. Compare substitution and welfare with the MNL or NL benchmark and state which data variation, rather than the mixing distribution alone, earns the change.

Bounded/censored mixing distributions, WTP-space estimation, and latent-class logit can be useful alternative modeling strategies, but they are not implemented in choicer v0.2.0. They require another implementation today and are roadmap candidates here; they should not be presented as options to `run_mxlogit()`. The broader tradeoff is laid out in Choosing among choice models.

## References

McFadden, D. and Train, K. (2000). Mixed MNL models for discrete response. Journal of Applied Econometrics, 15(5), 447-470.

Revelt, D. and Train, K. (1998). Mixed logit with repeated choices: households’ choices of appliance efficiency level. Review of Economics and Statistics, 80(4), 647-657.

Train, K. E. (2009). Discrete Choice Methods with Simulation (2nd ed.). Cambridge University Press.

Train, K. and Weeks, M. (2005). Discrete choice models in preference space and willingness-to-pay space. In R. Scarpa and A. Alberini (Eds.), Applications of Simulation Methods in Environmental and Resource Economics. Springer.