Multilevel tobit models

Let's see it work

Tobit models, whether multilevel or one-level, can be used for two types of inference—for the entire population as if it were not censored and for the censored population.

We have been hired to analyze data on attendance at 500 soccer stadiums. The data are censored when the stadium is sold out. In such cases, it is likely that attendance would have been greater had there been more seats.

Clients who run stadiums, who could increase the number of seats at a cost, would be interested in analysis of the uncensored population. Clients who rent and cannot increase the number of seats are interested in analysis of the censored population.

We can use metobit to answer questions for both types of clients. In fact, we will fit the model once and use different predictions to answer different questions.

The data we have include attend, stadium attendance in thousands for each game played during the season. We will model attendance as a function of

• winp, the winning percentage of the local team;

• inter, the probability that the local team makes it to an international postseason;

• cost, the cost of a ticket for the game being played; and

• weather, whether a storm—rain or snow—was forecast on game day.

We could fit this model with a linear multilevel estimator but for the fact that each stadium has a seating limit. That limit is recorded in the variable max.

Using metobit, we will type

. metobit attend winp inter cost i.weather || stadium: winp, ul(max)

Option ul(max) specifies the upper-censoring point.

Right of the || is the level-2 ID variable, stadium. We are specifying that we want random intercepts for each stadium and random coefficients for winning percentage.

We fit our desired model:

. metobit attend winp inter cost i.weather || stadium: winp, ul(max)

Fitting fixed-effects model:

Iteration 0:  Log likelihood = -21793.676
Iteration 1:  Log likelihood = -21321.165
Iteration 2:  Log likelihood = -21239.918
Iteration 3:  Log likelihood =  -21239.16
Iteration 4:  Log likelihood = -21239.159

Refining starting values:

Grid node 0:  Log likelihood = -19826.409

Fitting full model:

Iteration 0:  Log likelihood = -19826.409  (not concave)
Iteration 1:  Log likelihood = -18956.642  (not concave)
Iteration 2:  Log likelihood = -18440.049
Iteration 3:  Log likelihood = -17938.155
Iteration 4:  Log likelihood = -17860.781
Iteration 5:  Log likelihood =  -17822.36
Iteration 6:  Log likelihood = -17820.965
Iteration 7:  Log likelihood = -17820.958
Iteration 8:  Log likelihood = -17820.958

Mixed-effects tobit regression                  Number of obs     =      8,131
                                                   Uncensored     =      5,451
Limits: Lower = -inf                               Left-censored  =          0
        Upper = max                                Right-censored =      2,680

Group variable: stadium                         Number of groups  =        500
                                                Obs per group:
                                                              min =          9
                                                              avg =       16.3
                                                              max =         20

Integration method: mvaghermite                 Integration pts.  =          7

                                                Wald chi2(4)      =   11728.37
Log likelihood = -17820.958                     Prob > chi2       =     0.0000



      attend 
 
 Coefficient  Std. err.      z    P>|z|     [95% conf. interval]

        winp 
 
   .4287463   .0708727     6.05   0.000     .2898384    .5676542
       inter 
 
   .5575962   .0051627   108.00   0.000     .5474774    .5677149
        cost 
 
  -.0053072   .0005233   -10.14   0.000    -.0063329   -.0042815
   1.weather 
 
  -.2126963   .2977593    -0.71   0.475    -.7962937    .3709011
       _cons 
 
   9.213013    .348091    26.47   0.000     8.530767    9.895259

stadium      
 

    var(winp)
 
   1.335236   .1471157                      1.075903    1.657078
   var(_cons)
 
   35.74543   2.838831                      30.59284    41.76584

var(e.attend)
 
   22.48149   .4547381                      21.60765    23.39066
LR test vs. tobit model: chi2(2) = 6836.40                Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.

From the model, we can obtain estimates of average attendance. There are many ways to calculate average attendance. What would be the uncensored average attendance if max had not been in effect? What is the predicted average attendance given max? What would be average attendance if seating was increased by 1,000 in all stadiums having more than 90% average attendance? 2,000? 3,000?

In a world where max was not relevant, average attendance would have been about 23,510:

. margins

Predictive margins                                       Number of obs = 8,131
Model VCE: OIM

Expression: Marginal linear prediction, predict()



             
 
            Delta-method                                      
             
 
     Margin   std. err.      z    P>|z|     [95% conf. interval]

       _cons 
   23.50981   .3189898    73.70   0.000      22.8846    24.13502

In the real world where the current value of max is binding, it would be about 18,712:

. margins, predict(ystar(.,max))

Predictive margins                                       Number of obs = 8,131
Model VCE: OIM

Expression: E(attend*|attend<max), predict(ystar(.,max))



             
 
            Delta-method                                      
             
 
     Margin   std. err.      z    P>|z|     [95% conf. interval]

       _cons 
    18.7123   .1790272   104.52   0.000     18.36141    19.06319

We could also use margins to answer what attendance would be if max was increased by 1,000 in stadiums with over 90% attendance.

. quietly generate new_max = max + 1000

. margins if attend_rate<.90, predict(ystar(.,new_max))

Predictive margins                                       Number of obs = 5,284
Model VCE: OIM

Expression: E(attend*|attend<new_max), predict(ystar(.,new_max))



             
 
            Delta-method                                      
             
 
     Margin   std. err.      z    P>|z|     [95% conf. interval]

       _cons 
   26.81172   .3243089    82.67   0.000     26.17608    27.44735

Average attendance would be 26,812 in stadiums with attendance rate greater than 90%. This seems like a large number, but the stadiums in our sample with more than 90% attendance are the larger stadiums with teams with the highest winning percentage.