Conditional (individual-level) tastes from a mixed logit fit

For each decision maker, the mean and standard deviation of the random coefficients conditional on the choices they were observed to make, $E[\beta_n \mid y_n, x_n; \hat\theta]$ (Revelt and Train 2000; Train 2009, ch. 11): the frequentist counterpart of the individual-level beta_i summaries of run_hmnlogit.

Usage

conditional_tastes(object, ...)

# Default S3 method
conditional_tastes(object, ...)

# S3 method for class 'choicer_mxl'
conditional_tastes(object, ...)

Arguments

object: A choicer_mxl fit with stored data (keep_data = TRUE).
...: Unused.

Value

An object of class choicer_tastes: a list with

mean, sd: $K_w \times U$ matrices of conditional means and standard deviations; rows are the random coefficients (random_var_cols), columns the $U$ likelihood units, named by decision-maker id (person_col) or by choice-situation id.
unit: "decision maker" or "choice situation".
n_units: $U$.
S: Simulation draws per unit.
weights: Unit weights used in population.
population: A data.frame with one row per random coefficient: model_mean and model_sd, the moments of $f(\beta \mid \hat\theta)$; mean_cond and sd_cond, the weighted mean and standard deviation of the conditional means across units; revealed $= Var(E[\beta \mid y]) / Var(\beta)$; and residual $= E[Var(\beta \mid y)] / Var(\beta)$.

Details

Estimand. The fit estimates the population distribution of tastes, $f(\beta \mid \theta)$. Among decision makers who face $x_n$ and make the choices $y_n$, tastes follow the conditional distribution $$h(\beta \mid y_n, x_n; \theta) = \frac{P(y_n \mid x_n, \beta) \, f(\beta \mid \theta)} {\int P(y_n \mid x_n, \beta) \, f(\beta \mid \theta) \, d\beta},$$ where $P(y_n \mid x_n, \beta) = \prod_t P_{nt}(j_{nt} \mid \beta)$ is the product of the decision maker's logit probabilities. Its mean and standard deviation are simulated at $\hat\theta$ with the estimation draws, as weighted averages over draws $\beta_{ns}$ with weights $\omega_{ns} \propto P(y_n \mid x_n, \beta_{ns})$. Coefficients are on the scale they enter utility: $\mu_k + (L\eta)_k$ for a normal coefficient and $\exp(\mu_k) + \exp((L\eta)_k)$ for the shifted log-normal, with the $\mu$ term present only when rc_mean = TRUE. If a variable's mean is carried instead by the same variable in covariate_cols (with rc_mean = FALSE), only its random part is reported. This is the utility scale; the beta_i summaries of run_hmnlogit() are on the scale of the underlying normal, i.e. the log of the coefficient for log-normal coordinates.

What it is not. The conditional mean is not the decision maker's own $\beta_n$. It is the average taste among people who would make $n$'s choices, so it shrinks toward the population mean, the more so the fewer choice situations $n$ contributes, and its dispersion across decision makers understates the population dispersion. By the law of iterated expectations the conditional means average, in large samples, to the population mean (mean_cond vs model_mean in population), and by the law of total variance $Var(\beta) = Var(E[\beta \mid y]) + E[Var(\beta \mid y)]$: the share of taste variance revealed by choices (revealed) and the share left unresolved (residual) should sum to about one. These are numerical checks, not specification tests: for normal random coefficients with free means, both identities are the likelihood's first-order conditions for $\mu$ and $\Sigma$, so they hold at a converged maximum whatever the true taste distribution. A departure points to too few draws S or an unconverged $\hat\theta$; when a random coefficient's mean is fixed at zero (rc_mean = FALSE and no fixed coefficient on the same variable), a mean_cond far from zero can also flag that restriction. For log-normal coefficients the variance identity holds on the underlying normal scale, not the utility scale reported here, so expect departures of a few percent in either direction at moderate S. The sd is the dispersion of the conditional distribution of tastes, not a standard error: it ignores the sampling error in $\hat\theta$.

Cross-section. Without person_col each choice situation is its own decision maker. A single choice reveals little about tastes, so the conditional means stay close to the population mean (revealed small); fit with person_col to learn about tastes from repeated choices.

Choice probabilities are handled in log space, so every unit has a finite conditional distribution however improbable its choices. Only a unit whose utilities overflow at the estimates (non-finite log-likelihood) gets NA, and it is left out of the population summaries.

References

Revelt, D. and Train, K. (2000). Customer-specific taste parameters and mixed logit: households' choice of electricity supplier. Working Paper E00-274, Department of Economics, University of California, Berkeley.

Train, K. (2009). Discrete Choice Methods with Simulation, 2nd ed., Ch. 11. Cambridge University Press.

Examples

# \donttest{
sim <- simulate_mxl_data(N = 150, T = 5, J = 3, seed = 1)
fit <- run_mxlogit(sim$data, "id", "alt", "choice", c("x1", "x2"),
                   c("w1", "w2"), outside_opt_label = 0L,
                   rc_correlation = TRUE, person_col = "pid", S = 50L)
#> Optimization run time 0h:0m:0.02s
ct <- conditional_tastes(fit)
ct
#> Conditional tastes E[beta | observed choices] for 150 decision makers (S = 50 draws each)
#> Unit-level means and SDs in $mean and $sd.
#> Population distribution vs conditional means (revealed + residual should be close to 1):
#>    model_mean model_sd mean_cond sd_cond revealed residual
#> w1          0    0.924    -0.003   0.553    0.357    0.647
#> w2          0    0.984     0.049   0.602    0.374    0.616
# conditional means track the realized tastes, with shrinkage
cor(ct$mean["w1", ], sim$true_params$gamma_i["w1", colnames(ct$mean)])
#> [1] 0.646566
# }