Allowing for unmeasured heterogeneity: a mixture model for cross-sectional data

Omitted explanatory variables

Educational qualification accounts only in a small way for the heterogeneity (ie. the variation in migration behaviour) of the population. Other important individual differences have not been measured, or indeed may be unmeasurable.

To model heterogeneity in migration propensity due to unmeasured and unmeasurable factors, we add an individual specific term, or nuisance parameter, e_i to the linear predictor, to represent the omitted explanatory variables. This term is assumed to be constant for each individual over time. The conventional assumption is that e_i is distributed independently of the included variables. The model equation, with 5 levels of educational attainment as before, becomes:

The mixture model

SABRE assumes a Normal distribution for e_i, with mean zero and variance s², and uses a Gaussian quadrature method to fit the model. The tails of the Normal distribution cause a problem, as they assume zero probability at the extremes of the distribution. In fact, there is strong evidence that there are individuals for whom, in many situations, there will be a finite probability of never taking part in the process under investigation. These are the "stayers"; in the context of migration, these are the people who are likely never to move (over and above those who, by chance, do not move in the period covered by the study).

SABRE can allow for "stayers" by supplementing the quadrature mass points with endpoints at plus and minus infinity when this is appropriate. In this model, a nuisance parameter value of minus infinity implies zero probability of migration for that individual.

The standard SABRE mixture model is fitted using the FIT command, and includes endpoints by default. For the Poisson model, a single endpoint at minus infinity is included, which estimates the proportion of stayers. There is an option to omit the endpoints from the model and to allow the standard Poisson-Normal mixture model to be fitted, by using the ENDPOINT command. The parameterisation of the model is given in the SABRE reference guide.

We fit the log-linear Poisson-Normal mixture model for count data, first with endpoints and second without endpoints as follows:

Model with endpoints

 
data case n t ed              
read rochmigx.dat                          
                                     
        348 observations in dataset
                                      
transform ltime log t           
C reverse order of levels for ed            
transform ned ed - 6                    
transform reved ned * -1               
fac reved fed              
poisson y                       
yvar n                      
C fit random effects model 
C endpoints fitted by default  
fit int ltime fed            

    Initial Log-Linear Fit:

    Iteration        Deviance        Reduction
    __________________________________________
        1           1297.1251    
        2           748.76297        548.4    
        3           649.04377        99.72    
        4           637.92142        11.12    
        5           637.56670       0.3547    
        6           637.56619       0.5089E-03
        7           637.56619       0.1140E-08
 

    Iteration      Deviance         Step      End-point    Orthogonality
                                   length                    criterion
    ____________________________________________________________________
        1         549.93673        1.0000       free          13.255    
        2         531.94684        1.0000       free         0.28295E-01
        3         529.77935        0.0156       free          7.2279    
        4         522.42322        0.5000       free          24.948    
        5         495.13658        1.0000       free          16.832    
        6         487.98913        1.0000       free          3.9855    
        7         486.09574        1.0000       free          72.511    
        8         486.07703        1.0000       free          15.212    
        9         486.07703        1.0000       free 

dis est                                 

    Parameter              Estimate         S. Error
    ___________________________________________________
    int                    -2.6932          0.57967    
    ltime                  0.97307          0.15646    
    fed   ( 1)                  0.          ALIASED [I]
    fed   ( 2)             0.44283          0.18502    
    fed   ( 3)            -0.34053E-01      0.32219    
    fed   ( 4)             0.67497          0.32448    
    fed   ( 5)             0.32705          0.27775    
    scale                  0.45004          0.13086    
                                                           PROBABILITY
                                                           ___________
    end-point 0            0.92752          0.19029        0.48120    
 
dis m                                            

    X-vars      Y-var       Case-var
    ________________________________
    int         n           case  
    ltime 
    fed   

    Model type: standard Poisson log-linear normal mixture with end-point

    Number of observations             =    348
    Number of cases                    =    348

    X-vars df          =     6
    Scale df           =     1
    End-point df       =     1

    Deviance      =   486.07703   on   340 residual degrees of freedom
 
fit -fed                          

    Iteration      Deviance       Step      End-point      Orthogonality
                                 length                      criterion
    ____________________________________________________________________
        1         619.14491      1.0000       free            91.345    
        2         521.04026      1.0000       free            28.699    
        3         497.87490      1.0000       free            23.435    
        4         494.73843      1.0000       free            5.2864    
        5         494.52771      1.0000       free            8.9229    
        6         494.49902      1.0000       free            3.4139    
        7         494.49442      1.0000       free            5.9225    
        8         494.49442      1.0000       free 

dis m                                  

    X-vars      Y-var       Case-var
    ________________________________
    int         n           case  
    ltime 

    Model type: standard Poisson log-linear normal mixture with end-point

    Number of observations             =    348
    Number of cases                    =    348

    X-vars df          =     2
    Scale df           =     1
    End-point df       =     1

    Deviance          =  494.49442 on 344 residual degrees of freedom
    Deviance increase =  8.4173895 on   4 residual degrees of freedom
 
 

Results and conclusion

1. The addition of the individual specific random term and left endpoint to the model has reduced the deviance from 637.56 to 486.08 ie. by 151.48 on 2 degrees of freedom. Although the c² test is not strictly correct, as the standard Poisson model lies on the boundary of the parameter space of the Poisson mixture model, such a large reduction in deviance indicates a significant improvement in model fit. There appears to be considerable residual heterogeneity in the population.
2. The dispersion parameter has decreased to 486.08/340=1.43, confirming the improved fit.
3. The parameter estimates have changed little (by approximately one standard error); the standard errors of the parameter estimates have all increased. This result is typical when comparing models with and without unmeasured heterogeneity, provided all the explanatory variables are exogenous. We leave a discussion of the term exogenous until slightly later in this example.
4. Even though the standard Poisson model seems misspecified, the parameter estimates are consistent, ie. they tend to the true values when the sample size is increased. However, standard errors are underestimated and may lead us to conclude that an explanatory variable is significant, when in fact it is not. For instance, in the standard Poisson model, as the ratio of the parameter estimate to the standard error (t-ratio) for fed(5) is at about the 5% significance level of 2 , we might conclude that this factor is significant, whereas in the Poisson mixture model it is well below the 5% significance level, indicating that this factor is in fact not significant.
5. The small increase in deviance (8.42) compared to c²₍₄₎=9.49 at the 5% level, when educational qualification is removed from the model confirms that education is not significant in the Poisson mixture model.
6. The scale parameter estimate is the standard deviation of the Normal distribution assumed for the individual specific terms e_i. It is significantly different from zero and indicates considerable residual heterogeneity.
7. Note the parameter estimate for the left endpoint. The parameter value of 0.9275 (standard error 0.1903) is significantly different from zero, and the associated probability of 0.48 suggests that the sample contains a significant number of "stayers".

Model without endpoints

SABRE SESSION:CONTINUED

C put back fed fit +fed Iteration Deviance Step End-point Orthogonality length criterion ________________________________________________________________________ 1 668.03445 1.0000 free 29.768 2 528.74742 1.0000 free 22.714 3 496.35404 1.0000 free 31.771 4 487.11433 1.0000 free 3.2170 5 486.70328 1.0000 free 12.330 6 486.41714 1.0000 free 8.5039 7 486.07846 1.0000 free 5.3708 8 486.07703 1.0000 free 6.6935 9 486.07703 1.0000 free dis m X-vars Y-var Case-var ________________________________ int n case ltime fed Model type: standard Poisson log-linear normal mixture with end-point Number of observations = 348 Number of cases = 348 X-vars df = 6 Scale df = 1 End-point df = 1 Deviance = 486.07703 on 340 residual degrees of freedom Deviance increase = 8.4173895 on 4 residual degrees of freedom C fit same model without endpoints endpoint no fit . Iteration Deviance Step End-point Orthogonality length criterion _____________________________________________________________________ 1 694.22855 1.0000 fixed 26.564 2 551.25040 1.0000 fixed 25.027 3 533.52981 1.0000 fixed 16.291 4 513.44427 1.0000 fixed 6.3297 5 511.82728 1.0000 fixed 8.5030 6 511.28156 1.0000 fixed 14.935 7 511.01450 1.0000 fixed 6.4983 8 511.00122 1.0000 fixed 4.4092 9 511.00114 1.0000 fixed dis est Parameter Estimate S. Error ___________________________________________________ int -4.5013 0.58650 ltime 1.1857 0.17733 fed ( 1) 0. ALIASED [I] fed ( 2) 0.26548 0.22422 fed ( 3) 0.16689 0.35579 fed ( 4) 0.51855 0.35699 fed ( 5) 0.61071 0.45804 scale 1.1940 0.99342E-01 dis m X-vars Y-var Case-var ________________________________ int n case ltime fed Model type: standard Poisson log-linear normal mixture Number of observations = 348 Number of cases = 348 X-vars df = 6 Scale df = 1 Deviance = 511.00114 on 341 residual degrees of freedom Deviance increase = 24.924106 on 1 residual degrees of freedom

Conclusion

Next:Conclusions from cross-sectional analysis

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