The hardware and bandwidth for this mirror is donated by METANET, the Webhosting and Full Service-Cloud Provider.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]metanet.ch.
This vignette demonstrates an example of how to use the
logitr() function to estimate multinomial logit (MNL)
models with preference space and WTP space utility
parameterizations.
This example uses the yogurt data set from Jain et al. (1994). The data set contains 2,412 choice observations from a series of yogurt purchases by a panel of 100 households in Springfield, Missouri, over a roughly two-year period. The data were collected by optical scanners and contain information about the price, brand, and a “feature” variable, which identifies whether a newspaper advertisement was shown to the customer. There are four brands of yogurt: Yoplait, Dannon, Weight Watchers, and Hiland, with market shares of 34%, 40%, 23% and 3%, respectively.
In the utility models described below, the data variables are represented as follows:
| Symbol | Variable |
|---|---|
| \(p\) | The price in US dollars. |
| \(x_{j}^{\mathrm{Feat}}\) | Dummy variable for whether the newspaper advertisement was shown to the customer. |
| \(x_{j}^{\mathrm{Hiland}}\) | Dummy variable for the “Highland” brand. |
| \(x_{j}^{\mathrm{Weight}}\) | Dummy variable for the “Weight Watchers” brand. |
| \(x_{j}^{\mathrm{Yoplait}}\) | Dummy variable for the “Yoplait” brand. |
This example will estimate the following homogeneous multinomial logit model in the preference space:
\[\begin{equation} u_{j} = \alpha p_{j} + \beta_1 x_{j}^{\mathrm{Feat}} + \beta_2 x_{j}^{\mathrm{Hiland}} + \beta_3 x_{j}^{\mathrm{Weight}} + \beta_4 x_{j}^{\mathrm{Yoplait}} + \varepsilon_{j} \label{eq:mnlPrefExample} \end{equation}\]
where the parameters \(\alpha\), \(\beta_1\), \(\beta_2\), \(\beta_3\), and \(\beta_4\) have units of utility.
Estimate the model using the logitr() function:
library("logitr")
mnl_pref <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
pars = c('price', 'feat', 'brand')
)Print a summary of the results:
summary(mnl_pref)
#> =================================================
#>
#> Model estimated on: Wed Jul 24 05:46:56 2024
#>
#> Using logitr version: 1.1.2
#>
#> 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:0.01s
#> 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: 2412.0000000View the estimated model coefficients:
coef(mnl_pref)
#> price feat brandhiland brandweight brandyoplait
#> -0.3665546 0.4914392 -3.7154773 -0.6411384 0.7345195Compute the WTP implied from the preference space model:
wtp_mnl_pref <- wtp(mnl_pref, scalePar = "price")
wtp_mnl_pref
#> Estimate Std. Error z-value Pr(>|z|)
#> scalePar 0.36656 0.02431 15.0783 < 2.2e-16 ***
#> feat 1.34070 0.35952 3.7291 0.0001921 ***
#> brandhiland -10.13622 0.58374 -17.3642 < 2.2e-16 ***
#> brandweight -1.74909 0.18107 -9.6598 < 2.2e-16 ***
#> brandyoplait 2.00385 0.14287 14.0258 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1This example will estimate the following homogeneous multinomial logit model in the WTP space:
\[\begin{equation} u_{j} = \lambda ( \omega_1 x_{j}^{\mathrm{Feat}} + \omega_2 x_{j}^{\mathrm{Hiland}} + \omega_3 x_{j}^{\mathrm{Weight}} + \omega_4 x_{j}^{\mathrm{Yoplait}} - p_{j}) + \varepsilon_{j} \label{eq:mnlWtpExample} \end{equation}\]
where the parameters \(\omega_1\), \(\omega_2\), \(\omega_3\), and \(\omega_4\) have units of dollars and \(\lambda\) is the scale parameter.
Estimate the model using the logitr() function:
mnl_wtp <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
pars = c('feat', 'brand'),
scalePar = 'price',
# Since WTP space models are non-convex, run a multistart
numMultiStarts = 10,
# Use the computed WTP from the preference space model as the starting
# values for the first run:
startVals = wtp_mnl_pref$Estimate
)Print a summary of the results:
summary(mnl_wtp)
#> =================================================
#>
#> Model estimated on: Wed Jul 24 05:46:56 2024
#>
#> Using logitr version: 1.1.2
#>
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("feat",
#> "brand"), scalePar = "price", startVals = wtp_mnl_pref$Estimate,
#> numMultiStarts = 10)
#>
#> Frequencies of alternatives:
#> 1 2 3 4
#> 0.402156 0.029436 0.229270 0.339138
#>
#> Summary Of Multistart Runs:
#> Log Likelihood Iterations Exit Status
#> 1 -2656.888 84 3
#> 2 -2656.888 35 3
#> 3 -2804.904 78 4
#> 4 -2656.888 38 3
#> 5 -2656.888 40 3
#> 6 -2803.869 88 3
#> 7 -2656.888 45 3
#> 8 -2656.888 54 3
#> 9 -2656.888 67 3
#> 10 -2656.888 41 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: Multinomial Logit
#> Model Space: Willingness-to-Pay
#> Model Run: 8 of 10
#> Iterations: 54
#> Elapsed Time: 0h:0m:0.04s
#> Algorithm: NLOPT_LD_LBFGS
#> Weights Used?: FALSE
#> Robust? FALSE
#>
#> Model Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> scalePar 0.366584 0.024366 15.0449 < 2.2e-16 ***
#> feat 1.340572 0.355865 3.7671 0.0001652 ***
#> brandhiland -10.135737 0.576084 -17.5942 < 2.2e-16 ***
#> brandweight -1.749079 0.179897 -9.7227 < 2.2e-16 ***
#> brandyoplait 2.003822 0.142377 14.0741 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Log-Likelihood: -2656.8878779
#> Null Log-Likelihood: -3343.7419990
#> AIC: 5323.7757559
#> BIC: 5352.7168000
#> McFadden R2: 0.2054148
#> Adj McFadden R2: 0.2039195
#> Number of Observations: 2412.0000000View the estimated model coefficients:
Since WTP space models are non-convex, you cannot be certain that the model reached a global solution, even when using a multi-start. However, homogeneous models in the preference space are convex, so you are guaranteed to find the global solution in that space. Therefore, it can be useful to compute the WTP from the preference space model and compare it against the WTP from the WTP space model. If the WTP values and log-likelihood values from the two model spaces are equal, then the WTP space model is likely at a global solution.
To compare the WTP and log-likelihood values between the preference
space and WTP space models, use the wtpCompare()
function:
wtpCompare(mnl_pref, mnl_wtp, scalePar = 'price')
#> pref wtp difference
#> scalePar 0.3665546 0.366584 0.00002939
#> feat 1.3406987 1.340572 -0.00012716
#> brandhiland -10.1362190 -10.135737 0.00048198
#> brandweight -1.7490940 -1.749079 0.00001485
#> brandyoplait 2.0038476 2.003822 -0.00002547
#> logLik -2656.8878790 -2656.887878 0.00000107These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.