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The Poisson model |
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If complete randomness in migration behaviour is assumed, then a Poisson model may be used to represent the aggregate count data. Strictly, we should use a Binomial model as each individual is only allowed one migration per year so that the total number of migrations has an upper limit. However, for a large sample and and a low migration rate the Poisson model provides a good approximation.
For a homogeneous population, the probability of obtaining ni outcomes in time ti may be written as
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Pr(ni)=(mi)ni exp(-mi)
/ ni!
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where mi is the mean (or expected) number of migrations in time ti.
For a constant annual migration rate r,
| mi=r*ti |
or
| log(mi)=log(r)+log(ti) |
This model is an example of a generalised linear model. We will see how to fit such models a little later. When this model is fitted (using log(r) as an OFFSET in SABRE), the average annual migration rate comes out as 0.049 moves per individual per year.
For the time being, we note that this figure can also be calculated by simply dividing the total number of moves in the data set by the total time exposure to migration opportunities for the sample. Thus, there are 312 moves and 6349 annual observations, giving an average of 0.049 moves per individual per year.
This implies that each year a proportion of 0.049 of the population (or 4.9%) migrates, and that a proportion of 0.951 (or 95.1%) remains.
Using this model, the projected proportion moving at least once over a period of T years is equal to [1-(0.951)T].
The projected proportion migrating over different time periods is
shown by the line on the graph. It is considerably higher
than the observed proportion calculated from the data, which is
indicated by circles.
It is evident that this model substantially and systematically overpredicts the proportion moving, and therefore underestimates population stability. This is a consequence of assuming that migration behaviour over one time period can be used to predict migration behaviour over a longer time period, and is an example of a general problem, which Coleman (1973) calls the "deficient diagonal" effect.
The assumption that all individuals have the same propensity to migrate, which is not subject to change over time, does not seem compatible with the migration processes generating the data.
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Allowing the migration rate to vary with time |
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The migration rate can be allowed to vary systematically with time in this simple model by replacing (ti) in the above equation by (ti) b1. Now the migration rate decreases through migration history if b 1 is less than 1 and increases if b1 is greater than 1. One reason why we may expect b1 to be less than 1 is due to inertia effects, with people increasingly less likely to move with duration in a specific locality.
It is convenient to write
| r=exp(b0) |
The mean number of migrations may now be written as:
| mi=exp(b0)*(ti)b1= exp(b0+b1*log(ti)) |
or
| log(mi)=b0+ b1*log(ti) |
This model is typical of a generalised linear model,
which contains:
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The model may be fitted using SABRE software as follows. To run the example interactively, you will need to download the SABRE software and data sets.
C read in variables from data file
data case n t ed
read rochmigx.dat
348 observations in dataset
C declare response variable
yvar n
C declare model
poisson yes
C calculate log(time)
transform ltime log t
C fit Poisson model with intercept
C and log(time) as explanatory variable
lfit int ltime
Iteration Deviance Reduction
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1 1299.5140
2 754.34418 545.2
3 658.72919 95.61
4 648.79228 9.937
5 648.49783 0.2945
6 648.49747 0.3547E-03
7 648.49747 0.5484E-09
C display parameter estimates
dis est
Parameter Estimate S. Error
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int -3.2884 0.35114
ltime 1.0887 0.11119
C display model fitted
dis m
X-vars Y-var
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int n
ltime
Model type: standard Poisson log-linear
Number of observations = 348
X-vars df= 2
Deviance = 648.49747 on 346 residual degrees of freedom
stop
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Results and conclusion |
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| 1 | The estimated coefficient b1 of ltime is 1.0887, with a standard error of 0.1112, and is therefore not significantly different from 1. The migration rate does not appear to decline or increase through migration history, but is constant. |
| Number of moves | 0 | 1 | 2 | 3 | 4 | 5 | >=6 |
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| Observed frequency | 228 | 34 | 42 | 17 | 9 | 8 | 10 |
| Expected frequency | 164.3 | 101.6 | 50.4 | 21.1 | 7.5 | 2.3 | 0.80 |
| 2 | The observed migration frequencies are compared in Table 2 with the values predicted by the Poisson model. The model does not seem to fit the data, with the number of individuals making no moves or making four or more moves substantially underpredicted. There appears to be a systematic variation in migration frequency over and above the variation attributable by chance. |
| 3 | The fit of the model may be assessed by comparing the value of the
sum of [(Expected frequency-Observed frequency) 2/Expected frequency] with the c 2 distribution on 5 degrees of freedom (7 cells - 2 estimated coefficients). The critical value at the 5% significance level is 11.07. The calculated value is in fact 192.5, an order of magnitude higher. |
| 4 | The degree of model misspecification may be measured by the dispersion parameter, which is the ratio of the scaled deviance and the residual degrees of freedom.(648.5/346)=1.87). If the model were well specified, this ratio would be approximately 1. |
| 5 | One explanation for the poor fit of the model is that the assumption of a homogeneous population is not valid. Individuals may vary in their likelihood of migration; the assumption of a migration rate which depends only on time may be incorrect. Thus, it may be possible to improve the model specification by including explanatory variables which distinguish between individuals. |
Next:Poisson model with explanatory variable |
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