MNL for Case 1 BWS: saddle point convergence
Posted: 22 Mar 2024, 10:59
Hi,
I'm specifying an MNL model for case 1 BWS. The experiment presented respondents with 8 choice sets, each with 4 attributes per choice set, and 16 attributes in total. I captured the order of choice, so the availability of alternatives is observed not assumed. I have copied the data structure and code below.
The model throws a warning (WARNING: Some eigenvalues of the Hessian are positive, indicating convergence to a saddle point!), leaving NaNs in the output. However, if I fix the first position-specific constant of the worst choices, the model converges. To me, it makes more sense that the position-specific constants for the worst choices are relative to the first position-specific constant of the best choices, but I also see it may be reasonable to assume the position-specific effects are independent between best and worst choices. In any case, I don't expect the model not to converge properly. Can you advise?
Thanks,
Jay
I'm specifying an MNL model for case 1 BWS. The experiment presented respondents with 8 choice sets, each with 4 attributes per choice set, and 16 attributes in total. I captured the order of choice, so the availability of alternatives is observed not assumed. I have copied the data structure and code below.
The model throws a warning (WARNING: Some eigenvalues of the Hessian are positive, indicating convergence to a saddle point!), leaving NaNs in the output. However, if I fix the first position-specific constant of the worst choices, the model converges. To me, it makes more sense that the position-specific constants for the worst choices are relative to the first position-specific constant of the best choices, but I also see it may be reasonable to assume the position-specific effects are independent between best and worst choices. In any case, I don't expect the model not to converge properly. Can you advise?
Thanks,
Jay
Code: Select all
Rows: 7,408
Columns: 77
$ uuid <chr> "01nemkz0cju…
$ panel <dbl> 13, 13, 13, …
$ setno <dbl> 1, 2, 3, 4, …
$ choice_best <dbl> 4, 2, 3, 3, …
$ choice_worst <dbl> 3, 1, 1, 4, …
$ T1_1 <dbl> 0, 0, 0, 0, …
$ T1_2 <dbl> 0, 0, 0, 0, …
$ T1_3 <dbl> 1, 0, 0, 0, …
$ T1_4 <dbl> 0, 0, 0, 0, …
$ T2_1 <dbl> 0, 0, 0, 0, …
$ T2_2 <dbl> 0, 0, 0, 0, …
$ T2_3 <dbl> 0, 0, 0, 0, …
$ T2_4 <dbl> 0, 1, 0, 0, …
$ T3_1 <dbl> 0, 0, 0, 0, …
$ T3_2 <dbl> 0, 0, 0, 0, …
$ T3_3 <dbl> 0, 1, 0, 0, …
$ T3_4 <dbl> 0, 0, 0, 0, …
$ T4_1 <dbl> 1, 0, 0, 0, …
$ T4_2 <dbl> 0, 0, 0, 0, …
$ T4_3 <dbl> 0, 0, 0, 0, …
$ T4_4 <dbl> 0, 0, 0, 0, …
$ T5_1 <dbl> 0, 0, 0, 0, …
$ T5_2 <dbl> 0, 0, 0, 0, …
$ T5_3 <dbl> 0, 0, 0, 0, …
$ T5_4 <dbl> 0, 0, 0, 1, …
$ T6_1 <dbl> 0, 0, 0, 0, …
$ T6_2 <dbl> 0, 0, 0, 0, …
$ T6_3 <dbl> 0, 0, 0, 0, …
$ T6_4 <dbl> 0, 0, 1, 0, …
$ T7_1 <dbl> 0, 0, 0, 0, …
$ T7_2 <dbl> 0, 0, 1, 0, …
$ T7_3 <dbl> 0, 0, 0, 0, …
$ T7_4 <dbl> 0, 0, 0, 0, …
$ T8_1 <dbl> 0, 0, 0, 0, …
$ T8_2 <dbl> 0, 0, 0, 1, …
$ T8_3 <dbl> 0, 0, 0, 0, …
$ T8_4 <dbl> 0, 0, 0, 0, …
$ T9_1 <dbl> 0, 0, 0, 0, …
$ T9_2 <dbl> 0, 0, 0, 0, …
$ T9_3 <dbl> 0, 0, 1, 0, …
$ T9_4 <dbl> 0, 0, 0, 0, …
$ T10_1 <dbl> 0, 0, 0, 1, …
$ T10_2 <dbl> 0, 0, 0, 0, …
$ T10_3 <dbl> 0, 0, 0, 0, …
$ T10_4 <dbl> 0, 0, 0, 0, …
$ T11_1 <dbl> 0, 0, 0, 0, …
$ T11_2 <dbl> 0, 0, 0, 0, …
$ T11_3 <dbl> 0, 0, 0, 1, …
$ T11_4 <dbl> 0, 0, 0, 0, …
$ T12_1 <dbl> 0, 0, 1, 0, …
$ T12_2 <dbl> 0, 0, 0, 0, …
$ T12_3 <dbl> 0, 0, 0, 0, …
$ T12_4 <dbl> 0, 0, 0, 0, …
$ T13_1 <dbl> 0, 0, 0, 0, …
$ T13_2 <dbl> 0, 1, 0, 0, …
$ T13_3 <dbl> 0, 0, 0, 0, …
$ T13_4 <dbl> 0, 0, 0, 0, …
$ T14_1 <dbl> 0, 0, 0, 0, …
$ T14_2 <dbl> 0, 0, 0, 0, …
$ T14_3 <dbl> 0, 0, 0, 0, …
$ T14_4 <dbl> 1, 0, 0, 0, …
$ T15_1 <dbl> 0, 0, 0, 0, …
$ T15_2 <dbl> 1, 0, 0, 0, …
$ T15_3 <dbl> 0, 0, 0, 0, …
$ T15_4 <dbl> 0, 0, 0, 0, …
$ T16_1 <dbl> 0, 1, 0, 0, …
$ T16_2 <dbl> 0, 0, 0, 0, …
$ T16_3 <dbl> 0, 0, 0, 0, …
$ T16_4 <dbl> 0, 0, 0, 0, …
$ avail1B <dbl> 1, 1, 1, 1, …
$ avail1W <dbl> 1, 1, 1, 1, …
$ avail2B <dbl> 1, 1, 1, 1, …
$ avail2W <dbl> 1, 0, 1, 1, …
$ avail3B <dbl> 0, 1, 1, 1, …
$ avail3W <dbl> 1, 1, 0, 0, …
$ avail4B <dbl> 1, 1, 1, 1, …
$ avail4W <dbl> 1, 1, 1, 1, …
Code: Select all
### Initialise code
apollo_initialise()
### Set core controls
apollo_control = list(
modelName = "TEST_BW_MNL_no_covariates",
modelDescr = "MNL model on BW choice data, no covariates",
indivID = "uuid",
nCores = 4,
outputDirectory = "output",
seed = 1234
)
# ################################################################# #
#### LOAD DATA AND APPLY ANY TRANSFORMATIONS ####
# ################################################################# #
database
# ################################################################# #
#### DEFINE MODEL PARAMETERS ####
# ################################################################# #
### Vector of parameters, including any that are kept fixed in estimation
apollo_beta = c(set_names(rep(0, 24),
c(paste0("asc_alt", 1:4, "_", "B"),
paste0("asc_alt", 1:4, "_", "W"),
paste0("beta_T", 1:16))),
mu_worst = 1)
### Vector with names (in quotes) of parameters to be kept fixed at their starting value in apollo_beta, use apollo_beta_fixed = c() if none
apollo_fixed = c("asc_alt1_B", "beta_T2")
# ################################################################# #
#### GROUP AND VALIDATE INPUTS ####
# ################################################################# #
apollo_inputs = apollo_validateInputs()
# ################################################################# #
#### DEFINE MODEL AND LIKELIHOOD FUNCTION ####
# ################################################################# #
apollo_probabilities=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()
P_bw = list()
### List of utilities for the "best" choice
V_best=list()
V_best[["alt1B"]] = asc_alt1_B + beta_T1*T1_1 + beta_T2*T2_1 + beta_T3*T3_1 + beta_T4*T4_1 +
beta_T5*T5_1 + beta_T6*T6_1 + beta_T7*T7_1 +
beta_T8*T8_1 + beta_T9*T9_1 + beta_T10*T10_1 +
beta_T11*T11_1 + beta_T12*T12_1 + beta_T13*T13_1 +
beta_T14*T14_1 + beta_T15*T15_1 + beta_T16*T16_1
V_best[["alt2B"]] = asc_alt2_B + beta_T1*T1_2 + beta_T2*T2_2 + beta_T3*T3_2 + beta_T4*T4_2 +
beta_T5*T5_2 + beta_T6*T6_2 + beta_T7*T7_2 +
beta_T8*T8_2 + beta_T9*T9_2 + beta_T10*T10_2 +
beta_T11*T11_2 + beta_T12*T12_2 + beta_T13*T13_2 +
beta_T14*T14_2 + beta_T15*T15_2 + beta_T16*T16_2
V_best[["alt3B"]] = asc_alt3_B + beta_T1*T1_3 + beta_T2*T2_3 + beta_T3*T3_3 + beta_T4*T4_3 +
beta_T5*T5_3 + beta_T6*T6_3 + beta_T7*T7_3 +
beta_T8*T8_3 + beta_T9*T9_3 + beta_T10*T10_3 +
beta_T11*T11_3 + beta_T12*T12_3 + beta_T13*T13_3 +
beta_T14*T14_3 + beta_T15*T15_3 + beta_T16*T16_3
V_best[["alt4B"]] = asc_alt4_B + beta_T1*T1_4 + beta_T2*T2_4 + beta_T3*T3_4 + beta_T4*T4_4 +
beta_T5*T5_4 + beta_T6*T6_4 + beta_T7*T7_4 +
beta_T8*T8_4 + beta_T9*T9_4 + beta_T10*T10_4 +
beta_T11*T11_4 + beta_T12*T12_4 + beta_T13*T13_4 +
beta_T14*T14_4 + beta_T15*T15_4 + beta_T16*T16_4
### Compute probabilities for "best" choice using MNL model
mnl_settings_best = list(
alternatives = c(alt1B=1, alt2B=2, alt3B=3, alt4B=4),
avail = list(alt1B=avail1B, alt2B=avail2B, alt3B=avail3B, alt4B=avail4B),
choiceVar = choice_best,
utilities = V_best,
componentName = "best"
)
P_bw[["choice_best"]] = apollo_mnl(mnl_settings_best, functionality)
### List of utilities for the "worst" choice
V_worst = list()
V_worst[["alt1W"]] = -mu_worst * (asc_alt1_W + beta_T1*T1_1 + beta_T2*T2_1 + beta_T3*T3_1 + beta_T4*T4_1 +
beta_T5*T5_1 + beta_T6*T6_1 + beta_T7*T7_1 +
beta_T8*T8_1 + beta_T9*T9_1 + beta_T10*T10_1 +
beta_T11*T11_1 + beta_T12*T12_1 + beta_T13*T13_1 )+
beta_T14*T14_1 + beta_T15*T15_1 + beta_T16*T16_1
V_worst[["alt2W"]] = -mu_worst * (asc_alt2_W + beta_T1*T1_2 + beta_T2*T2_2 + beta_T3*T3_2 + beta_T4*T4_2 +
beta_T5*T5_2 + beta_T6*T6_2 + beta_T7*T7_2 +
beta_T8*T8_2 + beta_T9*T9_2 + beta_T10*T10_2 +
beta_T11*T11_2 + beta_T12*T12_2 + beta_T13*T13_2 +
beta_T14*T14_2 + beta_T15*T15_2 + beta_T16*T16_2)
V_worst[["alt3W"]] = -mu_worst * (asc_alt3_W + beta_T1*T1_3 + beta_T2*T2_3 + beta_T3*T3_3 + beta_T4*T4_3 +
beta_T5*T5_3 + beta_T6*T6_3 + beta_T7*T7_3 +
beta_T8*T8_3 + beta_T9*T9_3 + beta_T10*T10_3 +
beta_T11*T11_3 + beta_T12*T12_3 + beta_T13*T13_3 +
beta_T14*T14_3 + beta_T15*T15_3 + beta_T16*T16_3)
V_worst[["alt4W"]] = -mu_worst * (asc_alt4_W + beta_T1*T1_4 + beta_T2*T2_4 + beta_T3*T3_4 + beta_T4*T4_4 +
beta_T5*T5_4 + beta_T6*T6_4 + beta_T7*T7_4 +
beta_T8*T8_4 + beta_T9*T9_4 + beta_T10*T10_4 +
beta_T11*T11_4 + beta_T12*T12_4 + beta_T13*T13_4 +
beta_T14*T14_4 + beta_T15*T15_4 + beta_T16*T16_4)
### Compute probabilities for "worst" choice using MNL model
mnl_settings_worst = list(
alternatives = c(alt1W=1, alt2W=2, alt3W=3, alt4W=4),
avail = list(alt1W=avail1W, alt2W=avail2W, alt3W=avail3W, alt4W=avail4W),
choiceVar = choice_worst,
utilities = V_worst,
componentName = "worst"
)
P_bw[["choice_worst"]] = apollo_mnl(mnl_settings_worst, functionality)
### Combined model
P_bw = apollo_combineModels(P_bw, apollo_inputs, functionality)
### Take product across observation for same individual
P = apollo_panelProd(P_bw, apollo_inputs, functionality)
### Prepare and return outputs of function
P = apollo_prepareProb(P, apollo_inputs, functionality)
return(P)
}
# ################################################################# #
#### MODEL ESTIMATION ####
# ################################################################# #
### Optional starting values search
model = apollo_estimate(apollo_beta, apollo_fixed, apollo_probabilities, apollo_inputs)
# ################################################################# #
#### MODEL OUTPUTS ####
# ################################################################# #
# ----------------------------------------------------------------- #
#---- FORMATTED OUTPUT (TO SCREEN) ----
# ----------------------------------------------------------------- #
apollo_modelOutput(model, modelOutput_settings = list(printPVal=1))
# ----------------------------------------------------------------- #
#---- FORMATTED OUTPUT (TO FILE, using model name) ----
# ----------------------------------------------------------------- #
apollo_saveOutput(model)