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.

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")
<- logitr(
mnl_pref 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, simply move `"price"`

from the
`pars`

argument to the `scalePar`

argument:

```
<- logitr(
mnl_wtp 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.

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.:

```
<- logitr(
mnl_wtp 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.

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")`

:

```
<- logitr(
mxl_pref 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.

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
#> McFadden R2: 0.2054148
#> Adj McFadden R2: 0.2039195
#> 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
```

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
```

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:

```
<- subset(
data %in% c(42, 13),
yogurt, obsID 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:

```
<- predict(
probs
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:

```
<- predict(
probs
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.

```
<- predict(
outcomes
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 0
```

You can quickly compute the accuracy by dividing the number of correctly predicted choices by the total number of choices:

```
<- subset(outcomes, choice == 1)
chosen $correct <- chosen$choice == chosen$predicted_outcome
chosensum(chosen$correct) / nrow(chosen)
#> [1] 0.3776949
```

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