Given the non-linear utility specification of WTP space models, these models often diverge during estimation and can be highly sensitive to starting parameters. While many other packages support the estimation of WTP space models, in practice these packages often fail to find solutions.
Compared to most other packages, {logitr} tends to perform better and converge more often. It also has a built-in, parallelized multi-start optimization loop for searching for different local minima from different random starting points when minimizing the negative log-likelihood, which improves the odds of converging to a solution for at least some of the multi-start iterations.
This vignette illustrates the convergence issues, comparing the performance of {logitr} to the following R packages that support WTP space models:
For the same mixed logit WTP space model, only {logitr} is able to consistently converge to a solution.
First load the required packages:
library(logitr)
library(mlogit)
library(gmnl)
library(apollo)
library(mixl)
library(dplyr)
library(tidyr)
set.seed(1234)Now set the starting parameters for each package as well as the number of draws to use for the simulated log-likelihood:
numDraws_wtp <- 50
start_wtp <- c(
scalePar = 1,
feat = 0,
brandhiland = 0,
brandweight = 0,
brandyoplait = 0,
sd_feat = 0.1,
sd_brandhiland = 0.1,
sd_brandweight = 0.1,
sd_brandyoplait = 0.1
)Take only half of the yogurt data to speed things up:
Both {apollo} and {mixl} require that the user hand-specify the model to be estimated as well as several other settings. These settings are provided here.
Convert the yogurt data for into the format required for {gmnl}
data_gmnl <- mlogit.data(
data = yogurt,
shape = "long",
choice = "choice",
id.var = 'id',
alt.var = 'alt',
chid.var = 'obsID'
)Format the yogurt data to a “wide” format for
{apollo}
yogurt_price <- yogurt %>%
select(id, obsID, price, brand) %>%
mutate(price = -1*price) %>%
pivot_wider(
names_from = 'brand',
values_from = 'price') %>%
rename(
price_dannon = dannon,
price_hiland = hiland,
price_weight = weight,
price_yoplait = yoplait)
yogurt_feat <- yogurt %>%
select(id, obsID, feat, brand) %>%
pivot_wider(
names_from = 'brand',
values_from = 'feat') %>%
rename(
feat_dannon = dannon,
feat_hiland = hiland,
feat_weight = weight,
feat_yoplait = yoplait)
yogurt_choice <- yogurt %>%
filter(choice == 1) %>%
select(id, obsID, choice = alt)
data_apollo <- yogurt_price %>%
left_join(yogurt_feat, by = c('id', 'obsID')) %>%
left_join(yogurt_choice, by = c('id', 'obsID')) %>%
arrange(id, obsID) %>%
mutate(
av_dannon = 1,
av_hiland = 1,
av_weight = 1,
av_yoplait = 1
)Define the {apollo} probabilities function
apollo_probabilities_wtp <- function(
apollo_beta, apollo_inputs, functionality = "estimate"
) {
### Attach inputs and detach after function exit
apollo_attach(apollo_beta, apollo_inputs)
on.exit(apollo_detach(apollo_beta, apollo_inputs))
### Create list of probabilities P
P <- list()
### List of utilities: these must use the same names as in mnl_settings,
# order is irrelevant
V <- list()
V[["dannon"]] <- scalePar * (b_feat * feat_dannon - price_dannon)
V[["hiland"]] <- scalePar * (b_brandhiland + b_feat * feat_hiland - price_hiland)
V[["weight"]] <- scalePar * (b_brandweight + b_feat * feat_weight - price_weight)
V[["yoplait"]] <- scalePar * (b_brandyoplait + b_feat * feat_yoplait - price_yoplait)
### Define settings for MNL model component
mnl_settings <- list(
alternatives = c(dannon = 1, hiland = 2, weight = 3, yoplait = 4),
avail = list(
dannon = av_dannon,
hiland = av_hiland,
weight = av_weight,
yoplait = av_yoplait),
choiceVar = choice,
utilities = V
)
### Compute probabilities using MNL model
P[["model"]] <- apollo_mnl(mnl_settings, functionality)
### Take product across observation for same individual
P <- apollo_panelProd(P, apollo_inputs, functionality)
### Average across inter-individual draws
P = apollo_avgInterDraws(P, apollo_inputs, functionality)
### Prepare and return outputs of function
P <- apollo_prepareProb(P, apollo_inputs, functionality)
return(P)
}Define random parameters function
apollo_randCoeff <- function(apollo_beta, apollo_inputs) {
randcoeff <- list()
randcoeff[['b_feat']] <- feat + d_feat * sd_feat
randcoeff[['b_brandhiland']] <- brandhiland + d_brandhiland * sd_brandhiland
randcoeff[['b_brandweight']] <- brandweight + d_brandweight * sd_brandweight
randcoeff[['b_brandyoplait']] <- brandyoplait + d_brandyoplait * sd_brandyoplait
return(randcoeff)
}Main control settings for {apollo}
apollo_control_wtp <- list(
modelName = "MXL_WTP_space",
modelDescr = "MXL model on yogurt choice SP data, in WTP space",
indivID = "id",
mixing = TRUE,
analyticGrad = TRUE,
panelData = TRUE,
nCores = 1
)
# Set parameters for generating draws
apollo_draws_n <- list(
interDrawsType = "halton",
interNDraws = numDraws_wtp,
interUnifDraws = c(),
interNormDraws = c(
"d_feat", "d_brandhiland", "d_brandweight", "d_brandyoplait"),
intraDrawsType = "halton",
intraNDraws = 0,
intraUnifDraws = c(),
intraNormDraws = c()
)
# Set input
apollo_inputs_wtp <- apollo_validateInputs(
apollo_beta = start_wtp,
apollo_fixed = NULL,
database = data_apollo,
apollo_draws = apollo_draws_n,
apollo_randCoeff = apollo_randCoeff,
apollo_control = apollo_control_wtp
)Format the yogurt data to a “wide” format for {mixl}
data_mixl <- data_apollo # Uses the same "wide" format as {apollo}
data_mixl$ID <- data_mixl$id
data_mixl$CHOICE <- data_mixl$choiceDefine the {mixl} utility function
mixl_wtp <- "
feat_RND = @feat + draw_1 * @sd_feat;
brandhiland_RND = @brandhiland + draw_2 * @sd_brandhiland;
brandweight_RND = @brandweight + draw_3 * @sd_brandweight;
brandyoplait_RND = @brandyoplait + draw_4 * @sd_brandyoplait;
U_1 = @scalePar * (feat_RND * $feat_dannon - $price_dannon);
U_2 = @scalePar * (brandhiland_RND + feat_RND * $feat_hiland - $price_hiland);
U_3 = @scalePar * (brandweight_RND + feat_RND * $feat_weight - $price_weight);
U_4 = @scalePar * (brandyoplait_RND + feat_RND * $feat_yoplait - $price_yoplait);
"
mixl_spec_wtp <- specify_model(mixl_wtp, data_mixl)
availabilities <- generate_default_availabilities(data_mixl, 4)The same model is now estimated using all five packages. We will see that only {logitr} is able to converge to a solution, while all other package either fail to converge or converge to only a local minimum.
{logitr} converges, even without running a multi-start:
model_logitr <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
panelID = 'id',
pars = c('feat', 'brand'),
scalePar = 'price',
randPars = c(feat = "n", brand = "n"),
startVals = start_wtp,
numDraws = numDraws_wtp
)
summary(model_logitr)#> =================================================
#>
#> Model estimated on: Sat Oct 01 16:46:57 2022
#>
#> Using logitr version: 0.8.0
#>
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("feat",
#> "brand"), scalePar = "price", randPars = c(feat = "n", brand = "n"),
#> panelID = "id", startVals = start_wtp, numDraws = numDraws_wtp,
#> numCores = numCores)
#>
#> Frequencies of alternatives:
#> 1 2 3 4
#> 0.415721 0.029984 0.170989 0.383306
#>
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>
#> Model Type: Mixed Logit
#> Model Space: Willingness-to-Pay
#> Model Run: 1 of 1
#> Iterations: 85
#> Elapsed Time: 0h:0m:4s
#> Algorithm: NLOPT_LD_LBFGS
#> Weights Used?: FALSE
#> Panel ID: id
#> Robust? FALSE
#>
#> Model Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> scalePar 0.388098 0.050706 7.6538 1.954e-14 ***
#> feat 1.738832 0.725107 2.3980 0.01648 *
#> brandhiland -13.536714 1.810701 -7.4760 7.661e-14 ***
#> brandweight -4.510068 1.015856 -4.4397 9.010e-06 ***
#> brandyoplait 6.663869 0.929658 7.1681 7.605e-13 ***
#> sd_feat 0.491225 1.072013 0.4582 0.64679
#> sd_brandhiland 5.730571 1.125803 5.0902 3.577e-07 ***
#> sd_brandweight 11.420500 1.727799 6.6099 3.847e-11 ***
#> sd_brandyoplait 8.872470 1.366376 6.4934 8.390e-11 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Log-Likelihood: -732.2132421
#> Null Log-Likelihood: -1710.6872416
#> AIC: 1482.4264842
#> BIC: 1528.4886000
#> McFadden R2: 0.5719771
#> Adj McFadden R2: 0.5667161
#> Number of Observations: 1234.0000000
#>
#> Summary of 10k Draws for Random Coefficients:
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> feat -Inf 1.4071720 1.738457 1.738145 2.069629 Inf
#> brandhiland -Inf -17.4039453 -13.538553 -13.542027 -9.674388 Inf
#> brandweight -Inf -12.2189602 -4.519436 -4.527690 3.182253 Inf
#> brandyoplait -Inf 0.6670773 6.648678 6.641070 12.632059 InfIncluding a 10-run multi-start helps build confidence in the solution reached:
model_logitr <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
panelID = 'id',
pars = c('feat', 'brand'),
scalePar = 'price',
randPars = c(feat = "n", brand = "n"),
startVals = start_wtp,
numDraws = numDraws_wtp,
numMultiStarts = 10
)
summary(model_logitr)#> =================================================
#>
#> Model estimated on: Sun Sep 24 14:02:07 2023
#>
#> Using logitr version: 1.1.0
#>
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("feat",
#> "brand"), scalePar = "price", randPars = c(feat = "n", brand = "n"),
#> panelID = "id", startVals = start_wtp, numMultiStarts = 10,
#> numDraws = numDraws_wtp)
#>
#> Frequencies of alternatives:
#> 1 2 3 4
#> 0.415721 0.029984 0.170989 0.383306
#>
#> Summary Of Multistart Runs:
#> Log Likelihood Iterations Exit Status
#> 1 -732.2132 86 3
#> 2 -728.0567 50 3
#> 3 -734.7358 67 3
#> 4 -725.4327 68 3
#> 5 -731.0111 71 3
#> 6 -731.3684 58 3
#> 7 -749.7861 72 3
#> 8 -718.0862 95 3
#> 9 -745.6384 50 3
#> 10 -722.1385 68 3
#>
#> Use statusCodes() to view the meaning of each status code
#>
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>
#> Model Type: Mixed Logit
#> Model Space: Willingness-to-Pay
#> Model Run: 8 of 10
#> Iterations: 95
#> Elapsed Time: 0h:0m:2s
#> Algorithm: NLOPT_LD_LBFGS
#> Weights Used?: FALSE
#> Panel ID: id
#> Robust? FALSE
#>
#> Model Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> scalePar 0.378986 0.048908 7.7490 9.326e-15 ***
#> feat 1.746832 0.725116 2.4090 0.01599 *
#> brandhiland -13.708716 1.788570 -7.6646 1.799e-14 ***
#> brandweight -9.382873 1.543072 -6.0806 1.197e-09 ***
#> brandyoplait 4.251573 0.628295 6.7668 1.316e-11 ***
#> sd_feat -0.930817 0.828995 -1.1228 0.26151
#> sd_brandhiland -4.841849 0.901030 -5.3737 7.715e-08 ***
#> sd_brandweight 10.849573 1.583590 6.8513 7.321e-12 ***
#> sd_brandyoplait 7.813052 1.129764 6.9157 4.657e-12 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log-Likelihood: -718.0862026
#> Null Log-Likelihood: -1710.6872416
#> AIC: 1454.1724052
#> BIC: 1500.2346000
#> McFadden R2: 0.5802352
#> Adj McFadden R2: 0.5749742
#> Number of Observations: 1234.0000000
#>
#> Summary of 10k Draws for Random Coefficients:
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> feat -Inf 1.120009 1.747543 1.748135 2.375291 Inf
#> brandhiland -Inf -16.972056 -13.707162 -13.704227 -10.441231 Inf
#> brandweight -Inf -16.706387 -9.391773 -9.399614 -2.075102 Inf
#> brandyoplait -Inf -1.029171 4.238197 4.231497 9.507132 InfFirst attempt using same starting points as {logitr}
model_mixl <- estimate(
mixl_spec_wtp, start_wtp,
data_mixl, availabilities,
nDraws = numDraws_wtp
){mixl} converges to a local minimum:
#> [1] -718.0862 -1544.8531#> [,1] [,2]
#> scalePar 0.3789865 -2270.55731
#> feat 1.7468317 42.24281
#> brandhiland -13.7087157 24.36731
#> brandweight -9.3828735 110.99687
#> brandyoplait 4.2515734 -492.36959
#> sd_feat -0.9308170 10.81470
#> sd_brandhiland -4.8418491 10.23414
#> sd_brandweight 10.8495725 334.92761
#> sd_brandyoplait 7.8130522 915.14924Second attempt using {logitr} solution as starting points
model_mixl <- estimate(
mixl_spec_wtp, coef(model_logitr),
data_mixl, availabilities,
nDraws = numDraws_wtp
)Again, {mixl} converges to a local minimum (though it’s closer than the previous solution):
#> [1] -718.0862 -761.5228#> [,1] [,2]
#> scalePar 0.3789865 0.01959803
#> feat 1.7468317 66.46247619
#> brandhiland -13.7087157 -172.32720627
#> brandweight -9.3828735 -114.92946558
#> brandyoplait 4.2515734 6.24519288
#> sd_feat -0.9308170 -36.14686337
#> sd_brandhiland -4.8418491 -227.05348964
#> sd_brandweight 10.8495725 198.52035799
#> sd_brandyoplait 7.8130522 173.44727278First attempt using same starting points as {logitr}. Note that additional starting parameters must be added as the {gmnl} approach to estimating WTP is a slightly different model.
model_gmnl <- gmnl(
data = data_gmnl,
formula = choice ~ price + feat + brand | 0 | 0 | 0 | 1,
ranp = c(
feat = "n", brandhiland = "n", brandweight = "n",
brandyoplait = "n"),
fixed = c(TRUE, rep(FALSE, 10), TRUE),
model = "gmnl",
method = "bfgs",
haltons = NA,
panel = TRUE,
start = c(start_wtp, 0.1, 0.1, 0),
R = numDraws_wtp
){gmnl} converges to a local minimum:
#> [1] -718.0862 -1710.6872#> [,1] [,2]
#> feat 0.3789865 8.1540692
#> brandhiland 1.7468317 4.4266391
#> brandweight -13.7087157 20.9581382
#> brandyoplait -9.3828735 -93.0969112
#> het.(Intercept) 4.2515734 -440.5183170
#> sd.feat -0.9308170 2.4208585
#> sd.brandhiland -4.8418491 0.1085233
#> sd.brandweight 10.8495725 53.2376327
#> sd.brandyoplait 7.8130522 95.0008933
#> tau 0.3789865 653.9956379Second attempt using {logitr} solution as starting points:
model_gmnl <- gmnl(
data = data_gmnl,
formula = choice ~ price + feat + brand | 0 | 0 | 0 | 1,
ranp = c(
feat = "n", brandhiland = "n", brandweight = "n",
brandyoplait = "n"),
fixed = c(TRUE, rep(FALSE, 10), TRUE),
model = "gmnl",
method = "bfgs",
haltons = NA,
panel = TRUE,
start = c(coef(model_logitr), 0.1, 0.1, 0),
R = numDraws_wtp
)Again, {gmnl} converges to a local minimum:
#> [1] -718.0862 -944.3339#> [,1] [,2]
#> feat 0.3789865 2.321317
#> brandhiland 1.7468317 -18.785933
#> brandweight -13.7087157 -5.977394
#> brandyoplait -9.3828735 1.761744
#> het.(Intercept) 4.2515734 6.168432
#> sd.feat -0.9308170 2.830422
#> sd.brandhiland -4.8418491 6.358857
#> sd.brandweight 10.8495725 10.600978
#> sd.brandyoplait 7.8130522 5.455189
#> tau 0.3789865 8.230516First attempt using same starting points as {logitr}:
model_apollo <- apollo_estimate(
apollo_beta = start_wtp,
apollo_fixed = NULL,
apollo_probabilities = apollo_probabilities_wtp,
apollo_inputs = apollo_inputs_wtp,
estimate_settings = list(printLevel = 0)
){apollo} fails to converge and just returns the starting coefficients:
#> model
#> -718.0862 -2928.4048 #> [,1]
#> scalePar 0.3789865
#> feat 1.7468317
#> brandhiland -13.7087157
#> brandweight -9.3828735
#> brandyoplait 4.2515734
#> sd_feat -0.9308170
#> sd_brandhiland -4.8418491
#> sd_brandweight 10.8495725
#> sd_brandyoplait 7.8130522Second attempt using {logitr} solution as starting points:
model_apollo <- apollo_estimate(
apollo_beta = coef(model_logitr),
apollo_fixed = NULL,
apollo_probabilities = apollo_probabilities_wtp,
apollo_inputs = apollo_inputs_wtp,
estimate_settings = list(printLevel = 0)
)This time {apollo} converges to a local minimum:
#> model
#> -718.0862 -769.7493 #> [,1] [,2]
#> scalePar 0.3789865 2.376434e-04
#> feat 1.7468317 5.213661e+03
#> brandhiland -13.7087157 -1.622643e+04
#> brandweight -9.3828735 -1.529634e+04
#> brandyoplait 4.2515734 4.573893e+03
#> sd_feat -0.9308170 6.131294e+02
#> sd_brandhiland -4.8418491 -9.461200e+03
#> sd_brandweight 10.8495725 1.260092e+04
#> sd_brandyoplait 7.8130522 1.347027e+04