Rich Harris Scott Orford

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Rich HarrisScott Orford

• Theres space in them lines
• The geography of multilevel modelling
• Downloadable from www.geodemographics.info

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Introduction

• We argue that geographers should take numbers and
  statistics seriously not because they are
  necessarily natures own language, which was
  their justification in the quantitative
  revolution, but because they are a crucial
  component in the construction of social reality
  (Barnes, 2001 379).
• The still apparent dismissal of quantitative
  approaches under the label of positivism (by
  some) hides the complex spatial formations
  implicit or explicit to specific techniques.

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About multilevel modelling (MLM)

• Addresses the geographical paradox of traditional
  regression models
• Traditional regression modelling spatial
  autocorrelation (localized patterns) in the
  residuals are a violation of the regression
  assumptions.
• In contrast, multilevel modelling
• In conceptual terms, it can be seen that within a
  multilevel approach heterogeneity and difference
  both between people and contexts are seen as
  the norm, not an aberration (Duncan et al., 1998
  104, emphasis added).
• In geographical context this usually the
  difference(s) between people at one scale and
  places at another.

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Conceptual model
j
y
x
i
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which (in model terms) is equivalent to
j
y
x
i
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Spatial structure

• Vertical not horizontal linkage
• i.e. individuals are placed into groups at a
  higher level
• The physical distances between members within
  groups makes no difference in algorithmic terms
• Discrete object (vector) view of space
• Contrasts with e.g. Geographically Weighted
  Regression (GWR)
• yi ?0(ui,vi) ? ?k(ui,vi)xik ?I
• ?k(u,v) is a continuous function (at a fixed
  level)
• Regression coefficients are not assumed to be
  random but a deterministic function of location
  in geographical space
• A model of spatial interaction is assumed a
  priori
• i.e. the (x, y) location of an observation is
  another attribute (measurement) that is used in
  the calculation
• Discrete/Field distinction is not clear-cut.

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k
j
i
Wij ? 1/dc
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Uses of multilevel modelling

• Often looking for neighbourhood (/contextual)
  effects
• We would expect spatial differences anyway
• Compositional relationship
• Y ? X
• So if x1 gt x2 then y1 gt y2
• But, we are looking for second order effects
• yij ? (xi(j), J)
• These could be exogenous (e.g. regional
  effects)
• Or, endogenous (spatial interaction or
  collective effects)
• The underlying rationale of area-based policies
  is that concentrations of deprivation give rise
  to problems greater than the sum of its parts
  (McCulloch, 2001 687).

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Partitioning variance

• There is nothing magical about multilevel
  models the principle difference between them and
  simple OLS regression models is that multilevel
  models permit complex error terms (i.e., variance
  components) by using sophisticated computational
  algorithms (Oakes, 2003 1934).
• There is an individual-level, micro-model which
  represents the within-place equation, and an
  ecological, macro-model in which the parameters
  of the within-place model are the responses in
  the between-places models. This simultaneous
  specification allows for the separation, in a
  quantitative sense, of the compositional from the
  contextual (Duncan et al., 1998 102).
• yij ?0j ?1jxij eij
• ?0j ?0 u0j
• ?1j ?1 u1j
• N(0,?)

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Example of some recent research questions

• In a system where parents have constrained choice
  as to which schools their children attend
• What factors are associated with a pupil
  attending a near school?
• Is a consequence of pupils not attending a near
  school to increase observed ethnic segregation at
  the school level over and above that expected
  from the neighbourhood from which they are drawn?
• Does increased segregation (if it occurs) affect
  school performance?

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General model structure

• Cross-classified, multilevel model

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Random intercepts random slope model
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p(NR2)
X
0
Grand mean
Departure from mean for given Mosaic type
Departure from mean for given school
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(for white pupils in settlement A)
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C20 Asian Enterprise
C26South Asian Industry
G42 Low Horizons
D23 Industrial Grit
G42 Low Horizons
D23 Industrial Grit
C26South Asian Industry
C20 Asian Enterprise
Crudely this is sensitivityto local whiteness
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Some (technical) benefits of MLM

• Explicitly models spatial autocorrelation and
  spatial heterogeneity (which are both forms of
  geographical context)
• Improved (precision weighted) estimates and
  adjusted standard errors
• Individual level parameters estimated using a MLM
  will differ from those estimated using a single
  level model in the presence of SA
• Hence some aspect of context (geography) is
  built-into (implicitly expressed by) the
  individual level MLM estimates (i.e. components)
• Even if there were no higher / second order
  effects, in the presence of spatial
  autocorrelation you would still need a MLM

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Critiques of multilevel modelling (1)

• The Modifiable Areal Unit / definition of
  neighbourhood hierarchy problems
• There is no single, generalisable
  interpretation of the neighbourhood (Kearns
  Parkinson, 2001 2103).
• In all applications of multilevel modelling we
  are required to believe that (a) people live out
  their lives within a fixed spatial hierarchy,
  that (b) we can identify and quantify that
  hierarchy, and that (c) this hierarchy will be
  appropriate for our whole sample population
  (Mitchell, 2001 1358).

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Critiques of multilevel modelling (2)

• The missing variable problem
• If studies of neighbourhood effects fail to
  control adequately for the influence of
  neighbourhood and household characteristics, they
  may attribute to neighbourhoods what are really
  the effects of the omitted household and
  individual variables
• Some relevant individual characteristics are
  harder to observe, however, and are generally not
  captured in empirical research (McCulloch, 2001
  670, emphasis added).

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Critiques of multilevel modelling (3)

• The slicing scales problem
• Although human geography has worked for some
  time with the truism that people make places,
  just as places make people, it is striking how
  often the story of health and place stops short
  of embracing this mutuality (Smith Easterlow,
  2005 176, emphasis added).
• Multilevel approaches ask us to make a formal
  distinction between characteristics which are
  individual and those which are area based this
  is a step backwards in terms of our understanding
  of how people and place are related (Mitchell,
  2001 1358).

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Critiques of multilevel modelling (4)

• The C-word dualism
• The distinction between composition and context
  may not be as conceptually clear or as useful as
  may appear at first glance
• Composition and context are frequently
  treated as unproblematic and obvious
  distinctions, and the underlying casual models
  are often implicit
• Context is often treated as a residual
  category, containing those factors which remain
  once individual compositional characteristics
  are taken into account (Macintyre et al., 2002
  129)

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Critiques of multilevel modelling (5)

• The confounding / selection problem
• The gaze of MLM tends to be top down (i.e. how
  the characteristics of neighbourhoods affect
  those of individuals)
• What is missing is a sense of how biographical
  outcomes are themselves influenced by health
  trajectories the possibility that people whose
  health is already compromised might actively be
  placed into deprivation is rarely entertained
  (Smith Easterlow, 2005 177).
• With respect to neighbourhood effects research,
  the trouble with observational designs is that
  people are selected into neighbourhoods they
  are not randomly distributed (Oakes, 2003 1932,
  emphasis added).
• There can be no question that social structures
  and relations impact health and that disturbing
  disparities exist. And it is patently obvious
  that health varies with neighbourhood. The
  problem is that such phenomena are, per force,
  dependent happenings and as such render
  ineffective (multilevel) regression models aiming
  to identify independent effects (Oakes, 2003
  1944).

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Critiques of multilevel modelling (6)

• The over-emphasising context problem
• The indication from these quantitative studies
  Pickett Pearl, 2001 was that area variations
  in health are incidental rather than fundamental
  that similar people have similar health
  experiences no matter where they live that,
  statistically, composition explains (much) more
  than context (Smith Easterlow, 2005 175,
  emphasis added).

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Critiques of multilevel modelling (7)

• Who or what is the underlying population?
• MLM establishes significance estimates of second
  order (neighbourhood) effects by treating the
  higher level units as a sample of an underlying
  population
• But what if your sample (size of dataset) means
  it essentially is the population?
• What does (long term) probability mean within an
  open and dynamic (social) system?

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Critiques of multilevel modelling (8)

• Theories of neighbourhood effects are
  underdeveloped
• Peer-group norms, the absence of successful role
  models, access to community-based social capital,
  real and perceived opportunity costs, and both
  personal efficacy and collective efficacy might
  all play important roles in explaining any
  neighbourhood effects (McCulloch, 2001 1367).
• Different groups of people living in the same
  places may have different experiences and
  concepts of neighbourhood (e.g.. males and
  females in a particular place may have different
  local neighbourhoods borne-out by differences
  in day-to-day experiences).

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Critiques of multilevel modelling (9)

• All these critiques build up to challenge the
  ability to establish cause
• The validity and generalisability of
  neighbourhood effects remain open to question,
  and as yet there has been little empirical
  investigation of the causal pathways by which
  social environments translate into biological
  states of health and disease (Pickett Pearl,
  2001 111).
• The causal effect of neighbourhood contexts on
  health continues to confuse and elude us (Oakes,
  2003 1930, original emphasis).

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Critical fallacy (1)

• Because there are no apparent contextual
  effects there is no geography / geography does
  not matter
• A compositional explanation for observed area
  variations in social and economic problems agues
  that areas of concentrated disadvantage arise
  solely because of the varying distribution of
  types of people whose individual characteristics
  influence their social and economic outcomes.
  That is, similar types of people will have
  similar experiences no matter where they live. It
  is therefore argued that people rather than areas
  should be targeted (McCulloch, 2001 668,
  emphasis added).
• People often dont just live anywhere
• The distribution of people is the geography
  (Mitchell, 2001 1358)
• Ironically, multilevel practice may actually
  obscure that geography by focusing on higher
  order contextual effects

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Critical fallacy (2)

• If there is no evidence of a second order
  contextual effect then a MLM is not needed
• Even without evidence of a second order
  contextual effect the modelling procedure needs
  to accommodate the geography of composition (i.e.
  the spatial autocorrelation)
• While the technique represents a powerful means
  of investigating complex forms of contextuality,
  it carries no built in assumption regarding the
  importance of context. Multilevel models are just
  as capable of showing that context does not
  matter when theoretical and empirical work has
  suggested it might, (Duncan et al., 1998 109
  after Mason, 1991).

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Critical fallacy (3)

• Multilevel modelling cant be longitudinal
  (spatio-temporal).
• Assuming the data are available the movement of
  individuals between places over time can be
  investigated

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Spatio-temporal model
j(t1)
j(t0)
i
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Conclusions

• MLM research has highlighted that geographers
  still have problems with basic concepts such as
  context, scale, neighbourhood etc.
• Problems of making operational these vague and
  contested concepts in quantitative research
  (especially health research)
• Some issues very old (ie MAUP, temporal
  dimensions, the problems with dealing with
  processes (ie dynamic v static), identifying and
  measuring cause and effect)
• Methodological Issues
• What spatial scales are appropriate (and what are
  actually available)
• What time scales are appropriate (e.g. with
  respects to effects on health)
• Better theorisation of what is meant by context
  and how it is expressed at the different levels
  of the ML hierarchy.

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• A better theoretical understanding of how
  socio-economic process operate and the scales
  (spatial and temporal) at which they operate.
• When to use MLM and when not to use MLM (eg
  discrete object v field conceptualisations)
• Critical analysis of ML research in different
  subject areas (eg health, education, housing,
  voting) in order to gain a better understanding
  of compositional / contextual effects.
• The dependency of MLM on iterative computation,
  simulation, approximation and (probability based)
  solution finding also raises interesting
  questions on research practices within
  statistical sciences, e.g. what it means to
  prove anything