# Basic Usage

## Data format

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:

• Outcome: A dummy-coded variable that identifies which alternative was chosen (1 is chosen, 0 is not chosen). Only one alternative should have a 1 per choice observation.
• Observation ID: A sequence of repeated numbers that identifies each unique choice observation. For example, if the first three choice observations had 2 alternatives each, then the first 6 rows of the obsID variable would be 1, 1, 2, 2, 3, 3.
• Covariates: Other variables that will be used as model covariates.

The “Data Formatting and Encoding” vignette has more details about the required data format.

## Model specification interface

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:

$$$u_{j} = \alpha p_{j} + \beta_1 x_{j1} + \beta_2 x_{j2} + \beta_3 x_{j3} + \beta_4 x_{j4} + \varepsilon_{j}, \label{eq:yogurtUtilityPref}$$$

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:

$$$u_{j} = \lambda \left( \omega_1 x_{j1} + \omega_1 x_{j2} + \omega_1 x_{j3} + \omega_2 x_{j4} - p_{j} \right) + \varepsilon_{j}, \label{eq:yogurtUtilityWtp}$$$

To specify this model, simply move "price" from the pars argument to the scalePar argument:

mnl_wtp <- logitr(
data     = yogurt,
outcome  = "choice",
obsID    = "obsID",
pars     = c("feat", "brand"),
scalePar = "price"
)

In the above model, the variables in pars are marginal WTPs, whereas in the preference space model they are marginal utilities. Price is separately specified with the scalePar argument because it acts as a scaling term in WTP space models. While price is the most typical scaling variable, other continuous variables can also be used, such as time.

Interactions between covariates can be entered in the pars vector separated by the * symbol. For example, an interaction between price with feat in a 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.

## Parallelized multi-start estimation

Since WTP space models are non-linear and 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"),
scalePar = "price",
numMultiStarts = 10
)

The multi-start is parallelized by default for faster estimation, and the number of cores 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. For models with larger data sets, you may need to set numCores = 1 to avoid memory overflow issues.

## Mixed logit models

See the “Estimating Mixed Logit Models” vignette for more details.

To 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"), log-normal ("ln"), and zero-censored normal ("cn") 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 recommended to use a multi-start search to run the optimization loop multiple times to search for different local minima.

## Viewing results

See the “Summarizing Results” vignette for more details.

Use the summary() function to print a summary of the results from an estimated model, e.g.

summary(mnl_pref)
#> =================================================
#>
#> Model estimated on: Sun Feb 19 07:42:02 2023
#>
#> Using logitr version: 1.0.1
#>
#> 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:0.03s
#> 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
#> Number of Observations:  2412.0000000

Use 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    0.7345195

The coefficient standard errors

se(mnl_pref)
#>        price         feat  brandhiland  brandweight brandyoplait
#>   0.02436526   0.12006175   0.14541671   0.05449794   0.08064229

The 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 5.092011e-06
#> brandhiland   0.0018517954  8.550110e-04  0.021146019 1.490080e-03
#> brandweight   0.0001249988  5.092011e-06  0.001490080 2.970026e-03
#> brandyoplait -0.0015377721 -1.821331e-03 -0.003681036 7.779428e-04
#>               brandyoplait
#> price        -0.0015377721
#> feat         -0.0018213311
#> brandhiland  -0.0036810363
#> brandweight   0.0007779427
#> brandyoplait  0.0065031782

## Computing and comparing WTP

For 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, scalePar = "price")
#>                Estimate Std. Error  z-value  Pr(>|z|)
#> scalePar       0.366555   0.024343  15.0581 < 2.2e-16 ***
#> feat           1.340699   0.359020   3.7343 0.0001882 ***
#> brandhiland  -10.136219   0.583635 -17.3674 < 2.2e-16 ***
#> brandweight   -1.749094   0.182017  -9.6095 < 2.2e-16 ***
#> brandyoplait   2.003848   0.143519  13.9623 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The wtp() function divides the non-price parameters by the negative of the scalePar parameter (usually “price”). 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, scalePar = "price")
#>                       pref           wtp  difference
#> scalePar         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  0.00000106

## Predicting probabilities and outcomes

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 yoplait

In 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           0.41598062           0.45725106
#> 50     13     0.03312986           0.02679506           0.04120446
#> 51     13     0.19155548           0.17711694           0.20684689
#> 52     13     0.33846321           0.31886922           0.35849238
#> 165    42     0.60764778           0.57419563           0.63731815
#> 166    42     0.02602007           0.01829390           0.03624457
#> 167    42     0.17803313           0.16293227           0.19463156
#> 168    42     0.18829902           0.16809741           0.20858569

The 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:

probs <- predict(
mnl_wtp,
newdata = data,
obsID   = "obsID",
ci      = 0.95
)

probs
#>     obsID predicted_prob predicted_prob_lower predicted_prob_upper
#> 49     13     0.43686141           0.41679020           0.45612149
#> 50     13     0.03312947           0.02687017           0.04139081
#> 51     13     0.19154829           0.17657574           0.20864641
#> 52     13     0.33846083           0.31991628           0.35777707
#> 165    42     0.60767120           0.57272528           0.63676036
#> 166    42     0.02601800           0.01854611           0.03691285
#> 167    42     0.17802363           0.16281059           0.19500928
#> 168    42     0.18828717           0.16990274           0.20985561

You 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 newdata, 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
)

#>   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                 0

You 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] 0.3776949

See the “Predicting Probabilities and Choices from Estimated Models” vignette for more details about making predictions.