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| Model fitting |
data case move age year dur ed ch1 ch2 ch3 ch4 msb mse esb ese &
osb ose mbu mrm mfm msb1 epm eoj esb1 ops osb1 msb2 esb2 osb2 osb3
read rochmig.dat
6349 observations in dataset
yvar move
C transform age as before
transform tempage age - 30
transform trage tempage / 10
transform trage2 trage * trage
transform trage3 trage2 * trage
transform trage4 trage3 * trage
transform trage5 trage4 * trage
transform trage6 trage5 * trage
transform ldur log dur
C first fit the simple logistic model
lfit int ldur year trage trage2 trage3 trage4 trage5 trage6
Iteration Deviance Reduction
__________________________________________
1 8801.5829
2 2960.7613 5841.
3 2317.1450 643.6
4 2186.5548 130.6
5 2170.0008 16.55
6 2168.5289 1.472
7 2168.1916 0.3373
8 2168.1594 0.3224E-01
9 2168.1590 0.3511E-03
10 2168.1590 0.4787E-07
dis est
Parameter Estimate S. Error
___________________________________________________
int 1.5139 0.53900
ldur -1.0488 0.72558E-01
year -0.38518E-01 0.70233E-02
trage 0.24860 0.32199
trage2 -0.10853 0.59570
trage3 -0.81168 0.52582
trage4 0.38768 0.55271
trage5 0.57919 0.20955
trage6 -0.29282 0.15125
C fit the same model with random effects
C endpoints are fitted by default
fit .
Iteration Deviance Step End-points Orthogonality
length 0 1 criterion
________________________________________________________________________
1 2198.4881 1.0000 free free 4.6471
2 2198.2943 0.2500 free free 3.1137
3 2185.4266 0.3033 free free 13.365
4 2174.8955 0.1175 free fixed 9.2150
5 2142.0094 1.0000 free free 10.360
6 2135.1201 1.0000 free free 3.7965
7 2133.8038 1.0000 free free 11.834
8 2133.7948 1.0000 free free 37.114
9 2133.7948 1.0000 free free
dis est
Parameter Estimate S. Error
___________________________________________________
int 0.83341 0.77050
ldur -0.65918 0.10463
year -0.36521E-01 0.10873E-01
trage -0.69598E-01 0.34063
trage2 0.76814E-01 0.59487
trage3 -0.82208 0.53734
trage4 0.33146 0.54900
trage5 0.56760 0.21311
trage6 -0.27657 0.15032
scale 0.47710 0.17447
PROBABILITY
___________
end-point 0 0.56682 0.19724 0.36113
end-point 1 0.27460E-02 0.46361E-02 0.17495E-02
stop
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| Results and conclusion |
The deviance has decreased from 2168.16 to 2133.79. This is a reduction
of over 34 on 3 degrees of freedom, on adding the individual specific
random term to the model.
The extra three degrees of freedom are given by the scale
of the Normal mixing distribution and the two estimated probabilities
of the endpoints. Although the c2 test
is not strictly correct as the simple logistic model lies on the edge
of the parameter space of the mixture model, such a large change in deviance
(c2(3)=7.81)
demonstrates that there is considerable unobserved heterogeneity in the
population.
The coefficient estimate of ldur is still negative, but is
considerably smaller in magnitude than in the simple logistic model.
The estimate of this endogenous explanatory variable has changed by
allowing for residual heterogeneity; the estimates of the other variables
have changed little (by less than one standard error), and their
standard errors are almost unchanged.
The coefficient of ldur measures cumulative inertia
effects, and its value confirms that there is an
increasing disinclination to move with increasing length of residence.
However the effect is smaller than suggested by the simple logistic model;
that estimate was inflated because no account was taken of the fact that
with increasing duration the individuals most likely to migrate are
more and more underrepresented in the population. Inference about duration
effects can be misleading unless there is control for omitted variables.
(Lancaster 1979; Heckman and Singer 1985)
The probability of 0.36 associated with the left endpoint gives a measure
of the proportion of "stayers" in the population, i.e. those individuals
never likely to migrate. Examination of the parameter estimate and standard
error of the right endpoint (and corresponding probability of 0.0017) suggests that
this parameter (which estimates the proportion of the population migrating
every year) could be set to zero.
The scale parameter estimate is the standard deviation of the Normal
distribution assumed for the individual specific terms.
The probability of migration predicted by this random effects model may be
plotted on graphs to aid
interpretation of the parameter estimates. As before, the year is taken as
1985, the individual to be aged 40, and the duration of residence to be 10
years, as appropriate. As no interaction terms have been considered,
the trends shown on the graphs are generally valid.
In calculating the probabilities, the individual specific term is given
the estimated population
median value, taking into account both the Normal distribution and the
proportion of stayers.
The plot against age now shows two clear peaks at just above age 20 and
just below age 50. The relative size of the peaks has changed compared
to the simple logistic model; the size and location of the peak near age 50
has again to be interpreted with caution as the data are sparse for this
age group. The dominance of the first peak in the random effects model is
more plausible substantively as this is the age at which geographical
ties are at their minimum.
The graph against duration of stay shows the decline in migration
probability with duration for both the simple logistic and the random
effects models. When unobserved heterogeneity is taken into
account, the estimated decline is due to cumulative
inertia effects; in the simple logistic model the estimate is inflated
as discussed above.
The shapes of the graphs of migration probability against year are the
same for both models.
The levels of probability estimated by the two models are not strictly
comparable, as the simple logistic model gives the population average
value for individuals with given values of the explanatory variables
(age, year, duration of stay), whereas the random effects graphs show the
probability values for individuals with the median value of the nuisance
parameter.
Next:Model development: Adding explanatory variables |
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