The {logitr} package requires that data be structured in a data.frame and arranged in a “long” format [@Wickham2014] where each row contains data on a single alternative from a choice observation. The choice observations do not have to be symmetric, meaning they can have a “ragged” structure where different choice observations have different numbers of alternatives. The data must also include variables for each of the following:
1 is chosen, 0 is not chosen). Only one alternative should have a 1 per choice observation.obsID variable would be 1, 1, 2, 2, 3, 3.The “Data Formatting and Encoding” vignette has more details about the required data format.
Models are specified and estimated using the logitr() function. The data argument should be set to the data frame containing the data, and the outcome and obsID arguments should be set to the column names in the data frame that correspond to the dummy-coded outcome (choice) variable and the observation ID variable, respectively. All variables to be used as model covariates should be provided as a vector of column names to the pars argument. Each variable in the vector is additively included as a covariate in the utility model, with the interpretation that they represent utilities in preference space models and WTPs in a WTP space model.
For example, consider a preference space model where the utility for yogurt is given by the following utility model:
\[\begin{equation} u_{j} = \alpha p_{j} + \beta_1 x_{j1} + \beta_2 x_{j2} + \beta_3 x_{j3} + \beta_4 x_{j4} + \varepsilon_{j}, \label{eq:yogurtUtilityPref} \end{equation}\]
where \(p_{j}\) is price, \(x_{j1}\) is feat, and \(x_{j2-4}\) are dummy-coded variables for each brand (with the fourth brand representing the reference level). This model can be estimated using the logitr() function as follows:
library("logitr")
mnl_pref <- logitr(
data = yogurt,
outcome = "choice",
obsID = "obsID",
pars = c("price", "feat", "brand")
)The equivalent model in the WTP space is given by the following utility model:
\[\begin{equation} u_{j} = \lambda \left( \omega_1 x_{j1} + \omega_1 x_{j2} + \omega_1 x_{j3} + \omega_2 x_{j4} - p_{j} \right) + \varepsilon_{j}, \label{eq:yogurtUtilityWtp} \end{equation}\]
To specify this model, the user should move "price" from the pars argument to the price argument and set the modelSpace argument to "wtp" ("pref" is the default value):
mnl_wtp <- logitr(
data = yogurt,
outcome = "choice",
obsID = "obsID",
pars = c("feat", "brand"),
price = "price",
modelSpace = "wtp"
) In the above model, the variables in pars are marginal WTPs, whereas in the preference space model they are marginal utilities. Note that in the WTP space model price is separately specified with the price argument as it acts as a scaling term and does not reflect a marginal WTP.
Interactions between covariates can be entered in the pars vector separated by the * symbol. For example, an interaction between price with feat in the above preference space model could be included by specifying pars = c("price", "feat", "brand", "price*feat"), or even more concisely just pars = c("price*feat", "brand") as the interaction between price and feat will produce individual parameters for price and feat in addition to the interaction parameter.
Both of these examples are multinomial logit models with fixed parameters. See the “Estimating Multinomial Logit Models” vignette for more details.
Since WTP space models have non-convex log-likelihood functions, it is recommended to use a multi-start search to run the optimization loop multiple times to search for different local minima. This is implemented using the numMultiStarts argument, e.g.:
mnl_wtp <- logitr(
data = yogurt,
outcome = "choice",
obsID = "obsID",
pars = c("feat", "brand"),
price = "price",
modelSpace = "wtp",
numMultiStarts = 10
)The multi-start is parallelized by default for faster estimation, and the number of cores / workers to use can be manually set using the numCores argument. If numCores is not provide, then the number of cores is set to parallel::detectCores() - 1 (the max cores allowed is capped at parallel::detectCores()).
Use the summary() function to print a summary of the results from an estimated model, e.g.
summary(mnl_pref)
#> =================================================
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price",
#> "feat", "brand"))
#>
#> Frequencies of alternatives:
#> 1 2 3 4
#> 0.402156 0.029436 0.229270 0.339138
#>
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>
#> Model Type: Multinomial Logit
#> Model Space: Preference
#> Model Run: 1 of 1
#> Iterations: 21
#> Elapsed Time: 0h:0m:0.04s
#> Algorithm: NLOPT_LD_LBFGS
#> Weights Used?: FALSE
#> Robust? FALSE
#>
#> Model Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> price -0.366555 0.024365 -15.0441 < 2.2e-16 ***
#> feat 0.491439 0.120062 4.0932 4.254e-05 ***
#> brandhiland -3.715477 0.145417 -25.5506 < 2.2e-16 ***
#> brandweight -0.641138 0.054498 -11.7645 < 2.2e-16 ***
#> brandyoplait 0.734519 0.080642 9.1084 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Log-Likelihood: -2656.8878790
#> Null Log-Likelihood: -3343.7419990
#> AIC: 5323.7757580
#> BIC: 5352.7168000
#> McFadden R2: 0.2054148
#> Adj McFadden R2: 0.2039195
#> Number of Observations: 2412.0000000Use statusCodes() to print a description of each status code from the nloptr optimization routine.
You can also extract other values of interest at the solution, such as:
The estimated coefficients
coef(mnl_pref)
#> price feat brandhiland brandweight brandyoplait
#> -0.3665546 0.4914392 -3.7154773 -0.6411384 0.7345195The coefficient standard errors
se(mnl_pref)
#> price feat brandhiland brandweight brandyoplait
#> 0.02436526 0.12006175 0.14541671 0.05449794 0.08064229The log-likelihood
logLik(mnl_pref)
#> 'log Lik.' -2656.888 (df=5)The variance-covariance matrix
vcov(mnl_pref)
#> price feat brandhiland brandweight
#> price 0.0005936657 5.729619e-04 0.001851795 1.249988e-04
#> feat 0.0005729619 1.441482e-02 0.000855011 5.092010e-06
#> brandhiland 0.0018517954 8.550110e-04 0.021146019 1.490080e-03
#> brandweight 0.0001249988 5.092009e-06 0.001490080 2.970026e-03
#> brandyoplait -0.0015377720 -1.821331e-03 -0.003681036 7.779428e-04
#> brandyoplait
#> price -0.0015377720
#> feat -0.0018213311
#> brandhiland -0.0036810363
#> brandweight 0.0007779428
#> brandyoplait 0.0065031782For models in the preference space, a summary table of the computed WTP based on the estimated preference space parameters can be obtained with the wtp() function. For example, the computed WTP from the previously estimated fixed parameter model can be obtained with the following command:
wtp(mnl_pref, price = "price")
#> Estimate Std. Error z-value Pr(>|z|)
#> lambda 0.366555 0.024247 15.1177 < 2.2e-16 ***
#> feat 1.340699 0.359019 3.7343 0.0001882 ***
#> brandhiland -10.136219 0.582951 -17.3878 < 2.2e-16 ***
#> brandweight -1.749094 0.181729 -9.6247 < 2.2e-16 ***
#> brandyoplait 2.003848 0.143186 13.9947 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The wtp() function divides the non-price parameters by the negative of the price parameter. Standard errors are estimated using the Krinsky and Robb parametric bootstrapping method [@Krinsky1986]. Similarly, the wtpCompare() function can be used to compare the WTP from a WTP space model with that computed from an equivalent preference space model:
wtpCompare(mnl_pref, mnl_wtp, price = "price")
#> pref wtp difference
#> lambda 0.3665546 0.3665832 0.00002867
#> feat 1.3406987 1.3405926 -0.00010605
#> brandhiland -10.1362190 -10.1357635 0.00045548
#> brandweight -1.7490940 -1.7490826 0.00001133
#> brandyoplait 2.0038476 2.0038208 -0.00002686
#> logLik -2656.8878790 -2656.8878779 0.00000106To estimate a mixed logit model, use the randPars argument in the logitr() function to denote which parameters will be modeled with a distribution. The current package version supports normal ("n") and log-normal ("ln") distributions.
For example, assume the observed utility for yogurts was \(v_{j} = \alpha p_{j} + \beta_1 x_{j1} + \beta_2 x_{j2} + \beta_3 x_{j3} + \beta_4 x_{j4}\), where \(p_{j}\) is price, \(x_{j1}\) is feat, and \(x_{j2-4}\) are dummy-coded variables for brand. To model feat as well as each of the brands as normally-distributed, set randPars = c(feat = "n", brand = "n"):
mxl_pref <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
pars = c('price', 'feat', 'brand'),
randPars = c(feat = 'n', brand = 'n'),
numMultiStarts = 10
)Since mixed logit models also have non-convex log-likelihood functions, it is again recommended to use a multi-start search to run the optimization loop multiple times to search for different local minima.
See the “Estimating Mixed Logit Models” vignette for more details.
Estimated models can be used to predict probabilities and outcomes for a set (or multiple sets) of alternatives based on an estimated model. As an example, consider predicting probabilities for two of the choice observations from the yogurt dataset:
data <- subset(
yogurt, obsID %in% c(42, 13),
select = c('obsID', 'alt', 'choice', 'price', 'feat', 'brand')
)
data
#> obsID alt choice price feat brand
#> 49 13 1 0 8.1 0 dannon
#> 50 13 2 0 5.0 0 hiland
#> 51 13 3 1 8.6 0 weight
#> 52 13 4 0 10.8 0 yoplait
#> 165 42 1 1 6.3 0 dannon
#> 166 42 2 0 6.1 1 hiland
#> 167 42 3 0 7.9 0 weight
#> 168 42 4 0 11.5 0 yoplaitIn the example below, the probabilities for these two sets of alternatives are computed using the fixed parameter mnl_pref model using the predict() method:
probs <- predict(
mnl_pref,
newdata = data,
obsID = "obsID",
ci = 0.95
)
probs
#> obsID predicted_prob predicted_prob_lower predicted_prob_upper
#> 49 13 0.43685145 0.41607562 0.45697527
#> 50 13 0.03312986 0.02627103 0.04149784
#> 51 13 0.19155548 0.17635388 0.20813264
#> 52 13 0.33846321 0.31897562 0.35780777
#> 165 42 0.60764778 0.57370459 0.63939029
#> 166 42 0.02602007 0.01827989 0.03581829
#> 167 42 0.17803313 0.16291346 0.19540151
#> 168 42 0.18829902 0.16829570 0.20989229The resulting probs data frame contains the expected probabilities for each alternative. The lower and upper predictions reflect a 95% confidence interval (controlled by the ci argument), which are estimated using the Krinsky and Robb parametric bootstrapping method [@Krinsky1986]. The default is ci = NULL, in which case no CI predictions are made.
WTP space models can also be used to predict probabilities, but the price argument must be specified for predictions being made on a new data set:
probs <- predict(
mnl_wtp,
newdata = data,
obsID = "obsID",
price = "price"
)
probs
#> obsID predicted_prob
#> 49 13 0.43686141
#> 50 13 0.03312947
#> 51 13 0.19154829
#> 52 13 0.33846083
#> 165 42 0.60767120
#> 166 42 0.02601800
#> 167 42 0.17802363
#> 168 42 0.18828717You can also use the predict() method to predict outcomes by setting type = "outcome" (the default value is "prob" for predicting probabilities). If no new data are provided for the newdata argument, then outcomes will be predicted for every alternative in the original data used to estimate the model. In the example below the returnData argument is also set to TRUE so that the predicted outcomes can be compared to the actual ones.
outcomes <- predict(
mnl_pref,
type = "outcome",
returnData = TRUE
)
head(outcomes[c('obsID', 'choice', 'predicted_outcome')])
#> obsID choice predicted_outcome
#> 1 1 0 1
#> 2 1 0 0
#> 3 1 1 0
#> 4 1 0 0
#> 5 2 1 0
#> 6 2 0 0You can quickly compute the accuracy by dividing the number of correctly predicted choices by the total number of choices:
chosen <- subset(outcomes, choice == 1)
chosen$correct <- chosen$choice == chosen$predicted_outcome
sum(chosen$correct) / nrow(chosen)
#> [1] 0.3922056See the “Predicting Probabilities and Choices from Estimated Models” vignette for more details about making predictions.