Transcript Slide 1

Random Effects Models for
Migration Attractivities: a
Bayesian Methodology
Peter Congdon, Centre for
Statistics & Dept of Geography,
QMUL
Background
• Important for planning to understand why
some areas lose population through
migration, while others are gaining
• Also interest for other reasons in area
indices of various sorts (e.g. deprivation
indices, booming town index, etc)
• For measuring in-migrant pull (attractivity) or
out-migrant push (expulsiveness) of areas,
need to correct for ‘migration context’ of a
particular area
Migration Context
• Size/proximity of nearby areas with populations
at risk of migrating to an area, or offering
potential destinations for out-migrants from that
area
• Simple in-migration and out-migrant rates
(migrant totals divided by populations) do not
correct for context
• Attractiveness of remoter rural areas not close to
large population centres is understated by
simple measures
Methodological Considerations
• Existing literature focussed on fixed
effects modelling (and classical
estimation)
• By contrast, random effects model for
area push and pull scores may have
lower effective model dimension
• Additionally, Bayesian approach assists
in estimation and assessing
distributional properties of push/pull
indices
Properties of Push-Pull Scores
• Push and pull effects may well be
spatially correlated (e.g. places with high
attractivity concentrated in certain
regions)
• Push and pull effects may be correlated
with each other within areas.
• Easier to model such correlation with a
(Bayesian) random effects approach
Model for Migrant Flows
• Consider migration flows yij from origin areas i to
destination areas j (i,j=1,..n; i≠j).
• Migration relatively rare in relation to origin
populations Pi, but considerable variability in
rates likely.
• So Poisson but with overdispersion. For English
migration flows in case study, mean of yij is 16.9
but standard deviation is 81.
• Assume Poisson-gamma mixing (marginally
negative binomial).
Negative Binomial Migration
Interaction Model
• yij ~ Po(ijij), ij~Ga(,)
• Then integrating out ij
P(yij)=kij {/(ij+)} {ij/(ij+)}yij
kij= (yij+)/[(yij+1)()]
>0
and log(ij) can be modelled as function of
attributes of areas i and j.
Gravity Model (via NegBin Regression)
• Following principle of well known gravity model,
need to allow for (a) mass effects in origin &
destination (b) distance decay.
• Population, employment or housing stocks may
measure mass effects. Here take populations Pi
and Pj as mass measures. Rather than taking
logPi as offset (with known parameter 1), may
allow for regression effect.
• So log(ij)=0+1logPi +2logPj +log(dij)
Including Accessibility
• Traditional gravity model including masses and
distance only insensitive to spatial structure (Fik
& Mulligan, 1998). So include accessibility index
Aj=rjPr/drj
Large Aj values indicate alternatives in close
proximity to other alternatives; low values for
isolated alternatives
• So log(ij)=0+1logPi +2logPj +log(dij)+log(Aj)
where 1 and 2 expected to be close to 1,  is
negative (distance decay), typically between -0.5
and -2.
Extended Gravity Model Incorporating
Random Push-Pull Effects
• However, we seek summary indices of
area specific push and pull indices after
controlling for migration context.
• Extended gravity model proposed with
log(ij)=0+1logPi +2logPj +log(dij)
+log(Aj)+s1i+s2j
Bivariate Random Push-Pull Effects
• Bivariate push-pull effects by area
si= (s1i,s2i),
random with zero means over all areas.
• They are spatially correlated, and also
potentially correlated (+ve’ly or –ve’ly)
with each other within areas.
Correlation between push-pull effects
• Negative correlation within areas
between push & pull scores anticipated
if migration plays ‘equilibrating’ role in
job markets.
• In fact many studies show +ve
association between in- and outmigration. Compositional hypothesis:
areas with high in-migration possess
large number of persons likely to move
again, so increasing out-migration.
Quality of Life vs Job-Led
• Declining relevance of job-led model:
migration attractiveness even for
working age groups increasingly
related to quality of life considerations.
• Counterurbanising migration to less
rural areas (e.g. into South West
England) may actually run counter to
economic opportunities.
Prior for random push & pull effects in
extended gravity model
• Simple to implement bivariate
spatial prior via WINBUGS using
mv.car density.
• Multivariate CarNormal Prior is
example of Markov Random Field
(Rue & Held, 2005).
WINBUGS Case Study Application
• All age migrant flows yij between n=354
English Local Authorities in 2000-2001 (from
2001 UK Census). Read flow data in
stacked form with intra flows (area i to area
i) omitted, so have n(n-1)=124962 rows.
• Three models for spatial Push-Pull effects in
extended GM using NB regression
• Assess fit using DIC and log of pseudo
marginal likelihood (based on estimates of
conditional predictive ordinates)
Models
• Model 1 Independent fixed effect N(0,100) priors
on each s1i and s2i. Also Normal N(0,100) priors
on parameters {0,1,2,,}, and U(0,1000) prior
on .
• Model 2 Separate Univariate CARs on {s1i,s2i}.
Priors as in model 1 except Gamma priors on
spatial effect precisions 1 and 2.
• Model 3 Bivariate CAR on {s1i,s2i}. Priors as in
model 1 except Wishart prior on within area
precision matrix .
Fit and Results
• Better Fit for Model 3 with Random Effects
Push-Pull Correlation Explicit in Prior
• Posterior means in Model 3 (all significant)
1=0.72, 2=0.62,=-1.4,=0.066, =0.68
• Highest attractivities in model 3 concentrated
in SW England, East Anglia and less urban
parts of North, though some regional centres
and university towns also figure. There is
obvious spatial correlation when scores are
mapped
• High attractivity areas are mix of less urban
areas which may offer higher quality of life
(e.g. Cornwall, East Anglia), & areas where
mobile groups (students, seasonal workers)
create high migrant turnover.
• Nevertheless in attractive areas attractivity
index exceeds push index, so high attractivity
not just a matter of flows by mobile groups
but attraction of quality of life also relevant
Twenty Highest Attractivities
REGION
Push Score
Pull Score
Pull minus
Push
Carrick
SW
1.16
1.70
0.54
Kerrier
SW
1.15
1.55
0.39
Plymouth
SW
1.10
1.50
0.40
North Cornwall
SW
0.97
1.50
0.53
Durham
NE
1.25
1.48
0.23
Restormel
SW
0.93
1.43
0.50
Penwith
SW
0.88
1.39
0.51
Torbay
SW
0.75
1.27
0.52
Newcastle upon Tyne
NE
1.07
1.22
0.15
Exeter
SW
0.91
1.19
0.29
South Hams
SW
0.82
1.19
0.37
Yorks & H
0.91
1.13
0.22
East
0.75
1.05
0.30
Yorks & H
0.80
1.05
0.24
West Devon
SW
0.65
1.03
0.39
Alnwick
NE
0.85
1.02
0.17
East
0.66
1.02
0.36
Yorks & H
0.65
1.01
0.36
Lancaster
NW
0.79
1.00
0.21
East Devon
SW
0.40
0.98
0.58
Name
Richmondshire
Norwich
York
Cambridge
Leeds
Results
• High posterior correlation (0.93) between
pull and push indices in model 3.
Correlation between two sets of effects
also 0.88 in fixed effects model 1 when
correlation not incorporated a priori.
• Compositional hypothesis (+ve push-pull
relationship due to mobile groups raising
both inflows and outflows) supported.
• Low attractivities concentrate in Midlands &
London
Twenty Lowest Attractivities
REGION
Push Score
Pull Score
Pull minus
Push
Harlow
East
-0.43
-0.83
-0.40
Gedling
E Midl
-0.69
-0.83
-0.14
South Staffordshire
W Midl
-0.63
-0.85
-0.22
Havering
London
-0.30
-0.87
-0.57
St. Helens
NW
-0.71
-0.88
-0.17
Erewash
E Midl
-0.82
-0.88
-0.06
Dudley
W Midl
-0.64
-0.90
-0.26
Redditch
W Midl
-0.58
-0.92
-0.34
Dartford
SE
-0.33
-0.94
-0.61
Oldham
NW
-0.65
-0.96
-0.32
North Warwickshire
W Midl
-0.69
-0.96
-0.27
Bexley
London
-0.37
-1.02
-0.66
SE
-0.43
-1.03
-0.60
Sandwell
W Midl
-0.66
-1.08
-0.42
Cannock Chase
W Midl
-0.83
-1.10
-0.27
Knowsley
NW
-0.83
-1.12
-0.29
Tamworth
W Midl
-0.69
-1.13
-0.44
East
-0.39
-1.15
-0.76
Walsall
W Midl
-0.74
-1.18
-0.44
Barking & Dagenham
London
-0.55
-1.22
-0.67
Area
Gravesham
Broxbourne
Region-wide Averages (9 Regions)
Average Pull
Average Push
Average Pull
minus Push
E Midl
-0.15
-0.29
0.15
East
-0.09
-0.02
-0.07
London
-0.18
-0.03
-0.15
NE
0.16
0.21
-0.05
NW
-0.18
-0.14
-0.04
SE
0.01
0.14
-0.12
SW
0.61
0.34
0.27
W Midl
-0.36
-0.31
-0.04
Yorks & Humb
0.17
0.04
0.13
England average
0.00
0.00
0.00
Correlations with Census & Other LA Indicators
Age Group Models
• Can apply same approach to age-specific
migration, e.g. young adult migration (ages 18 to
29) or retirement migration.
• Apply model 3 approach to migration by 18-29
year age group
• Still have highly correlated push & pull scores
(0.89)
• Still have high pull scores for SW, but London
also has high attractivity for this age group, esp
in terms of average (push-pull)
Young adult migrant push-pull scores; correlations with
area indices
Retirement Migration
• Model 3 applied to
migration by over
60s
• Markedly high
attractivity for
South West, and
for less urban
(and lower
housing cost)
areas
Retirement Push-Pull Scores & Area Characteristics
Final Remarks
• Other possible priors on random push & pull scores
(e.g. mixture of structured & unstructured)
• Alternative to Neg-Bin is lognormal e.g. using
transform zij=log(yij+1). Distribution of errors needs
to be checked – see Flowerdew & Aitken (J Reg Stud
1982)
• Work by Fotheringham et al (2000) using lognormal
approximation (and including Scottish areas). This
shows quality of life factors & high attractivity of
areas in less urban regions holds even for young
adult migrants (those most likely to be ‘job-led’
migrants)