Estimating Mixed Logit Models

This vignette demonstrates an example of how to use the logitr() function to estimate mixed logit (MXL) models with preference space and WTP space utility parameterizations.

Supported distributions

In mixed logit models, parameters are assumed to follow a particular distribution. This is implemented in logitr with the randPars argument, which should be a named vector defining which distribution to use for each random parameter. In the example below, we set randPars = c(feat = 'n', brand = 'n') so that feat and brand are normally distributed. Mixed logit models will estimate a mean and standard deviation of the underlying normal distribution for each random coefficient.

The current version of the package includes the following distributions:

Note that log-normal or zero-censored normal parameters force positivity, so when using these you may need to use the negative of a value. For example, the “price” coefficient is typically negative, so if modeling “price” with a log-normal or zero-censored normal distribution you should convert “price” to the negative of price in the data prior to estimating the model.

The data

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.

Preference space model

This example will estimate the following mixed 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. In the example below, we will model \(\beta_1\), \(\beta_2\), \(\beta_3\), and \(\beta_4\) as normally distributed across the population. As a result, the model will estimate a mean and standard deviation for each of these coefficients.

Note that since the yogurt data has a panel structure (i.e. multiple choice observations for each respondent), it is necessary to set the panelID argument to the id variable, which identifies the individual. This will use the panel version of the log-likelihood (see Train 2009 chapter 6, section 6.7 for details).

Finally, as with WTP space models, it is recommended to use a multistart search for mixed logit models as they are non-convex. This is implemented in the example below by setting numMultiStarts = 10:

library("logitr")

mxl_pref <- logitr(
  data     = yogurt,
  outcome  = 'choice',
  obsID    = 'obsID',
  panelID  = 'id',
  pars     = c('price', 'feat', 'brand'),
  randPars = c(feat = 'n', brand = 'n'),
  numMultiStarts = 10
)
#> Running multistart...
#>   Iterations: 10
#>   Cores: 3
#> Done!

Print a summary of the results:

summary(mxl_pref)
#> =================================================
#> 
#> Model estimated on: Thu Jun 16 05:48:23 2022 
#> 
#> Using logitr version: 0.6.1 
#> 
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price", 
#>     "feat", "brand"), randPars = c(feat = "n", brand = "n"), 
#>     panelID = "id", 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       -1266.550         34           3
#> 2       -1280.044         65           3
#> 3       -1248.998         39           3
#> 4       -1267.858         43           3
#> 5       -1239.295         52           3
#> 6       -1300.132         36           3
#> 7       -1250.922         46           3
#> 8       -1289.385         41           3
#> 9       -1252.705         43           3
#> 10      -1303.013         45           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:       Mixed Logit
#> Model Space:       Preference
#> Model Run:            5 of 10
#> Iterations:                52
#> Elapsed Time:        0h:0m:3s
#> Algorithm:     NLOPT_LD_LBFGS
#> Weights Used?:          FALSE
#> Panel ID:                  id
#> Robust?                 FALSE
#> 
#> Model Coefficients: 
#>                  Estimate Std. Error  z-value  Pr(>|z|)    
#> price           -0.448906   0.040004 -11.2216 < 2.2e-16 ***
#> feat             0.776342   0.193533   4.0114 6.036e-05 ***
#> brandhiland     -6.393908   0.523611 -12.2112 < 2.2e-16 ***
#> brandweight     -3.660216   0.306889 -11.9268 < 2.2e-16 ***
#> brandyoplait     1.127390   0.204158   5.5221 3.349e-08 ***
#> sd_feat          0.566220   0.225256   2.5137   0.01195 *  
#> sd_brandhiland  -3.201333   0.372903  -8.5849 < 2.2e-16 ***
#> sd_brandweight   4.093423   0.232198  17.6290 < 2.2e-16 ***
#> sd_brandyoplait  3.258847   0.219937  14.8172 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>                                      
#> Log-Likelihood:         -1239.2945922
#> Null Log-Likelihood:    -3343.7419990
#> AIC:                     2496.5891845
#> BIC:                     2548.6831000
#> McFadden R2:                0.6293690
#> Adj McFadden R2:            0.6266774
#> Number of Observations:  2412.0000000
#> 
#> Summary of 10k Draws for Random Coefficients: 
#>              Min.    1st Qu.     Median       Mean    3rd Qu. Max.
#> feat         -Inf  0.3940471  0.7759089  0.7755488  1.1576404  Inf
#> brandhiland  -Inf -8.5515625 -6.3928805 -6.3909395 -4.2335128  Inf
#> brandweight  -Inf -6.4232959 -3.6635735 -3.6665318 -0.9030747  Inf
#> brandyoplait -Inf -1.0752246  1.1218102  1.1190157  3.3194987  Inf

The above summary table prints summaries of the estimated coefficients as well as a summary table of the distribution of 10,000 population draws for each normally-distributed model coefficient. The results show that the feat attribute has a significant standard deviation coefficient, suggesting that there is considerable heterogeneity across the population for this attribute. In contrast, the brand coefficients have small and insignificant standard deviation coefficients.

Compute the WTP implied from the preference space model:

wtp_mxl_pref <- wtp(mxl_pref, scalePar =  "price")
wtp_mxl_pref
#>                   Estimate Std. Error  z-value  Pr(>|z|)    
#> scalePar          0.448906   0.039865  11.2607 < 2.2e-16 ***
#> feat              1.729408   0.502648   3.4406 0.0005804 ***
#> brandhiland     -14.243309   1.388362 -10.2591 < 2.2e-16 ***
#> brandweight      -8.153634   0.975137  -8.3615 < 2.2e-16 ***
#> brandyoplait      2.511415   0.412994   6.0810 1.194e-09 ***
#> sd_feat           1.261333   0.503097   2.5071 0.0121714 *  
#> sd_brandhiland   -7.131410   0.957027  -7.4516 9.215e-14 ***
#> sd_brandweight    9.118663   0.945261   9.6467 < 2.2e-16 ***
#> sd_brandyoplait   7.259531   0.767964   9.4530 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

WTP space model

This example will estimate the following mixed 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. In the example below, we will model \(\omega_1\), \(\omega_2\), \(\omega_3\), and \(\omega_4\) as normally distributed across the population. Note that this is a slightly different assumption than in the preference space model. In the WTP space, we are assuming that the WTP for these features is normally-distributed (as opposed to the preference space model where the utility coefficients are assumed to follow a normal distribution).

In the example below, we also use a 10-iteration multistart. We also set the starting values for the first iteration to the computed WTP from the preference space model:

mxl_wtp <- logitr(
  data       = yogurt,
  outcome    = 'choice',
  obsID      = 'obsID',
  panelID    = 'id',
  pars       = c('feat', 'brand'),
  scalePar   = 'price',
  randPars   = c(feat = 'n', brand = 'n'),
  numMultiStarts = 10,
  startVals = wtp_mxl_pref$Estimate
)
#> Running multistart...
#>   Iterations: 10
#>   Cores: 3
#>   NOTE: Using user-provided starting values for first iteration
#> Done!

Print a summary of the results:

summary(mxl_wtp)
#> =================================================
#> 
#> Model estimated on: Thu Jun 16 05:48:57 2022 
#> 
#> Using logitr version: 0.6.1 
#> 
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("feat", 
#>     "brand"), scalePar = "price", randPars = c(feat = "n", brand = "n"), 
#>     panelID = "id", startVals = wtp_mxl_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       -1239.294        115           3
#> 2       -1285.949        100           3
#> 3       -1258.974         89           3
#> 4       -1286.371         66           3
#> 5       -1261.622         96           3
#> 6       -1250.922         70           3
#> 7       -1260.216         63           3
#> 8       -1255.878         57           3
#> 9       -1261.053         68           3
#> 10      -1263.875         77           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:           Mixed Logit
#> Model Space:   Willingness-to-Pay
#> Model Run:                1 of 10
#> Iterations:                   115
#> Elapsed Time:           0h:0m:10s
#> Algorithm:         NLOPT_LD_LBFGS
#> Weights Used?:              FALSE
#> Panel ID:                      id
#> Robust?                     FALSE
#> 
#> Model Coefficients: 
#>                   Estimate Std. Error  z-value  Pr(>|z|)    
#> scalePar          0.448572   0.039996  11.2155 < 2.2e-16 ***
#> feat              1.729654   0.491729   3.5175 0.0004356 ***
#> brandhiland     -14.227286   1.366031 -10.4151 < 2.2e-16 ***
#> brandweight      -8.171795   0.956028  -8.5477 < 2.2e-16 ***
#> brandyoplait      2.505569   0.407526   6.1482 7.834e-10 ***
#> sd_feat           1.265524   0.497617   2.5432 0.0109851 *  
#> sd_brandhiland   -7.119721   0.944786  -7.5358 4.863e-14 ***
#> sd_brandweight    9.131179   0.923738   9.8850 < 2.2e-16 ***
#> sd_brandyoplait   7.270999   0.753003   9.6560 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>                                      
#> Log-Likelihood:         -1239.2940197
#> Null Log-Likelihood:    -3343.7419990
#> AIC:                     2496.5880393
#> BIC:                     2548.6819000
#> McFadden R2:                0.6293691
#> Adj McFadden R2:            0.6266775
#> Number of Observations:  2412.0000000
#> 
#> Summary of 10k Draws for Random Coefficients: 
#>              Min.     1st Qu.     Median       Mean   3rd Qu. Max.
#> feat         -Inf   0.8752105   1.728687   1.727882  2.581872  Inf
#> brandhiland  -Inf -19.0258810 -14.225002 -14.220685 -9.422597  Inf
#> brandweight  -Inf -14.3353845  -8.179285  -8.185884 -2.021454  Inf
#> brandyoplait -Inf  -2.4088088   2.493120   2.486885  7.396507  Inf

If you want to compare the WTP from the two different model spaces, use the wtpCompare() function:

wtpCompare(mxl_pref, mxl_wtp, scalePar = 'price')
#>                          pref           wtp  difference
#> scalePar            0.4489061     0.4485716 -0.00033443
#> feat                1.7294076     1.7296537  0.00024604
#> brandhiland       -14.2433089   -14.2272860  0.01602288
#> brandweight        -8.1536341    -8.1717952 -0.01816109
#> brandyoplait        2.5114154     2.5055686 -0.00584683
#> sd_feat             1.2613332     1.2655242  0.00419104
#> sd_brandhiland     -7.1314099    -7.1197213  0.01168860
#> sd_brandweight      9.1186633     9.1311788  0.01251553
#> sd_brandyoplait     7.2595309     7.2709992  0.01146829
#> logLik          -1239.2945922 -1239.2940197  0.00057257

Note that the WTP will not necessarily be the same between preference space and WTP space MXL models. This is because the distributional assumptions in MXL models imply different distributions on WTP depending on the model space. See Train and Weeks (2005) and Sonnier, Ainslie, and Otter (2007) for details on this topic.

Correlated heterogeneity

By default, logitr assumes that mixed logit models have uncorrelated heterogeneity. However, correlated heterogeneity can be implemented by setting correlation = TRUE for models in either space (preference or WTP). The example below shows the results for the same mixed logit model in the preference space as above but now with correlated heterogeneity:

library("logitr")

mxl_pref_cor <- logitr(
  data     = yogurt,
  outcome  = 'choice',
  obsID    = 'obsID',
  panelID  = 'id',
  pars     = c('price', 'feat', 'brand'),
  randPars = c(feat = 'n', brand = 'n'),
  numMultiStarts = 10,
  correlation = TRUE
)
#> Running multistart...
#>   Iterations: 10
#>   Cores: 3
#> Done!

Print a summary of the results:

summary(mxl_pref_cor)
#> =================================================
#> 
#> Model estimated on: Thu Jun 16 05:49:48 2022 
#> 
#> Using logitr version: 0.6.1 
#> 
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price", 
#>     "feat", "brand"), randPars = c(feat = "n", brand = "n"), 
#>     panelID = "id", correlation = TRUE, 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       -1249.587         47           3
#> 2       -1249.433         57           3
#> 3       -1277.641         66           3
#> 4       -1243.559        104           3
#> 5       -1256.724        149           3
#> 6       -1235.918         89           3
#> 7       -1240.373         84           3
#> 8       -1273.358         79           3
#> 9       -1257.895         90           3
#> 10      -1232.414         77           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:       Mixed Logit
#> Model Space:       Preference
#> Model Run:           10 of 10
#> Iterations:                77
#> Elapsed Time:        0h:0m:6s
#> Algorithm:     NLOPT_LD_LBFGS
#> Weights Used?:          FALSE
#> Panel ID:                  id
#> Robust?                 FALSE
#> 
#> Model Coefficients: 
#>                               Estimate Std. Error  z-value  Pr(>|z|)    
#> price                        -0.463491   0.041222 -11.2437 < 2.2e-16 ***
#> feat                          0.546099   0.263363   2.0736   0.03812 *  
#> brandhiland                  -5.725879   0.564219 -10.1483 < 2.2e-16 ***
#> brandweight                  -2.301490   0.280063  -8.2178 2.220e-16 ***
#> brandyoplait                  0.114579   0.198606   0.5769   0.56399    
#> sd_feat_feat                 -0.039768   0.263458  -0.1509   0.88002    
#> sd_feat_brandhiland          -0.367008   0.193714  -1.8946   0.05815 .  
#> sd_feat_brandweight           0.117938   0.221261   0.5330   0.59401    
#> sd_feat_brandyoplait         -0.194733   0.369675  -0.5268   0.59836    
#> sd_brandhiland_brandhiland   -1.888955   0.298073  -6.3372 2.339e-10 ***
#> sd_brandhiland_brandweight   -0.537904   0.339780  -1.5831   0.11340    
#> sd_brandhiland_brandyoplait  -1.765278   0.342346  -5.1564 2.517e-07 ***
#> sd_brandweight_brandweight    3.305476   0.233494  14.1566 < 2.2e-16 ***
#> sd_brandweight_brandyoplait  -1.794610   0.279245  -6.4267 1.304e-10 ***
#> sd_brandyoplait_brandyoplait -3.472400   0.246259 -14.1006 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>                                      
#> Log-Likelihood:         -1232.4138009
#> Null Log-Likelihood:    -3343.7419990
#> AIC:                     2494.8276019
#> BIC:                     2581.6508000
#> McFadden R2:                0.6314268
#> Adj McFadden R2:            0.6269408
#> Number of Observations:  2412.0000000
#> 
#> Summary of 10k Draws for Random Coefficients: 
#>              Min.    1st Qu.     Median       Mean    3rd Qu. Max.
#> feat         -Inf  0.2549014  0.5458057  0.5468134  0.8429505  Inf
#> brandhiland  -Inf -7.4916072 -5.7123653 -5.7187614 -3.9426542  Inf
#> brandweight  -Inf -4.8324955 -2.2957029 -2.3019793  0.2423208  Inf
#> brandyoplait -Inf -2.2211786  0.1205244  0.1235021  2.4615310  Inf

References

Jain, Dipak C, Naufel J Vilcassim, and Pradeep K Chintagunta. 1994. “A Random-Coefficients Logit Brand-Choice Model Applied to Panel Data.” Journal of Business & Economic Statistics 12 (3): 317–28.
Sonnier, Garrett, Andrew Ainslie, and Thomas Otter. 2007. “Heterogeneity distributions of willingness-to-pay in choice models.” Quant. Mark. Econ. 5 (3): 313–31. https://doi.org/10.1007/s11129-007-9024-6.
Train, Kenneth E., and Melvyn Weeks. 2005. “Discrete Choice Models in Preference and Willingness-to-Pay Space.” In Appl. Simul. Methods Environ. Resour. Econ., 1–16.