WESML weights for choice-based (endogenous stratified) samples

Computes Manski-Lerman (1977) Weighted Exogenous Sample Maximum Likelihood (WESML) weights for a choice-based sample. The weight for a choice situation whose chosen alternative is \(j\) is \(w = Q(j) / H(j)\), where \(Q(j)\) is the population share of alternative \(j\) and \(H(j)\) its sample share among choosers. Using these weights in run_mxlogit restores consistency under choice-based sampling; pair them with se_method = "sandwich" for valid (robust) standard errors (the plain inverse-Hessian is invalid under weighting).

Usage

wesml_weights(
  data,
  id_col,
  alt_col,
  choice_col,
  Q,
  H = NULL,
  normalize = TRUE,
  attach = FALSE,
  weight_name = ".wesml_weight",
  outside_opt_label = NULL,
  include_outside_option = FALSE
)

Arguments

data: A long-format choice data set (data.frame or data.table), one row per alternative per choice situation.
id_col, alt_col, choice_col: Column names identifying the choice situation, the alternative, and the 0/1 chosen indicator.
Q: Named numeric vector of population shares, one entry per chosen stratum (names matched to as.character(alt)), each strictly positive. Renormalized to sum 1 if needed.
H: Optional named numeric vector of sample shares. If NULL (default) it is computed from data as the fraction of choice situations choosing each alternative.
normalize: If TRUE (default) the returned weights are scaled to mean 1. This does not affect the point estimates or the sandwich variance.
attach: If TRUE, return data with a row-level weight column appended (the per-situation weight repeated across all rows of a situation), ready to pass to run_mxlogit(weights_col = ...). If FALSE (default) return an id-keyed table of weights.
weight_name: Name of the weight column (default ".wesml_weight").
outside_opt_label, include_outside_option: Set include_outside_option = TRUE and supply outside_opt_label when the outside good is implicit (choice situations with no 1 in choice_col are treated as having chosen the outside good).

Value

Either an id-keyed data.table with columns id_col and weight_name (default), or, when attach = TRUE, a copy of data with the weight column appended. The result carries "Q", "H", and "choice_sampling" attributes recording provenance.

Details

Strata are defined by the chosen alternative and keyed by as.character(alt) so numeric and character alternative codes match supplied share names unambiguously.

References

Manski, C. F. and Lerman, S. R. (1977). The Estimation of Choice Probabilities from Choice Based Samples. Econometrica 45(8), 1977-1988. Train, K. E. (2009). Discrete Choice Methods with Simulation, Section 3.7. Cambridge University Press.

Examples

library(data.table)
set.seed(1)
N <- 300L; J <- 3L
pop <- data.table(id = rep(seq_len(N), each = J), alt = rep(1:J, N))
pop[, x1 := rnorm(.N)]
#>         id   alt         x1
#>      <int> <int>      <num>
#>   1:     1     1 -0.6264538
#>   2:     1     2  0.1836433
#>   3:     1     3 -0.8356286
#>   4:     2     1  1.5952808
#>   5:     2     2  0.3295078
#>  ---                       
#> 896:   299     2 -0.0132557
#> 897:   299     3 -0.9314803
#> 898:   300     1  1.2146891
#> 899:   300     2 -2.0883404
#> 900:   300     3 -0.5261525
pop[, w1 := rnorm(.N)]
#>         id   alt         x1         w1
#>      <int> <int>      <num>      <num>
#>   1:     1     1 -0.6264538 -1.5414026
#>   2:     1     2  0.1836433  0.1943211
#>   3:     1     3 -0.8356286  0.2644225
#>   4:     2     1  1.5952808 -1.1187352
#>   5:     2     2  0.3295078  0.6509530
#>  ---                                  
#> 896:   299     2 -0.0132557  1.9363973
#> 897:   299     3 -0.9314803 -1.4558838
#> 898:   300     1  1.2146891  1.4819057
#> 899:   300     2 -2.0883404  1.0761196
#> 900:   300     3 -0.5261525 -0.7574884
pop[, choice := as.integer(seq_len(.N) == sample.int(.N, 1L)), by = id]
#>         id   alt         x1         w1 choice
#>      <int> <int>      <num>      <num>  <int>
#>   1:     1     1 -0.6264538 -1.5414026      0
#>   2:     1     2  0.1836433  0.1943211      1
#>   3:     1     3 -0.8356286  0.2644225      0
#>   4:     2     1  1.5952808 -1.1187352      0
#>   5:     2     2  0.3295078  0.6509530      1
#>  ---                                         
#> 896:   299     2 -0.0132557  1.9363973      0
#> 897:   299     3 -0.9314803 -1.4558838      0
#> 898:   300     1  1.2146891  1.4819057      0
#> 899:   300     2 -2.0883404  1.0761196      1
#> 900:   300     3 -0.5261525 -0.7574884      0

# Population shares of the chosen alternative
Q <- prop.table(table(pop[choice == 1, alt]))
wt <- wesml_weights(pop, "id", "alt", "choice", Q = Q)
head(wt)
#> Key: <id>
#>       id .wesml_weight
#>    <int>         <num>
#> 1:     1             1
#> 2:     2             1
#> 3:     3             1
#> 4:     4             1
#> 5:     5             1
#> 6:     6             1