Spatial Modelling 1 : Incorporating spatial modelling in a random effects structure

1
Lecture 22

• Spatial Modelling 1 Incorporating spatial
  modelling in a random effects structure

2
Lecture Contents

• Introduction to spatial modelling
• Nested random effect levels
• House price dataset
• Including distance as a fixed effect
• Direction effects
• Focused clustering (Falkirk dataset)

3
Spatial statistical modelling

• Here we require a statistical approach that
  accounts for the spatial location at which a
  response is collected. This means that the model
  that is fitted to the data needs to account for
  the spatial effects.
• This may be to account for any effects due to
  location in the model or to predict values of the
  response at other locations via some form of
  interpolation that accounts for both other
  predictor variables and/or the spatial location.

4
Types of spatial data

• There are many forms of spatial data but we can
  broadly divide these into three types (Cressie
  1993)
• Geostatistical data here measurements are taken
  at a fixed number of chosen locations in a
  geographical area.
• Lattice data here measurement are taken at on a
  regular lattice and at each point on this lattice
  a measurement is collected.
• Point process data here each observation is the
  location of a response and its co-ordinates are
  also recorded.

5
Geostatistical data

• Such data are collected in various fields,
  particularly mining and earth sciences.
• A measurement e.g. age coal ash is taken at each
  of a number of locations.
• Methods such as variograms and spatial Kriging
  are used to analyse such data.
• Other application areas include weather maps and
  agricultural field trials.
• Note such data is not ideally suited to standard
  random effect modelling.

6
Disease mapping

• One particular type of spatial modelling that is
  often linked with random effect modelling is
  disease mapping.
• Here cases of a disease (either human or animal)
  are observed over a chosen region e.g. a country.
  We then wish to infer the relative risk of the
  disease for a particular individual at a
  particular location based on the data collected.
• Both our practicals this afternoon will consider
  disease mapping datasets. The other two types of
  spatial data relate to disease data.

7
Lattice Data

• Such data is common in many fields, for example
  image analysis where the pixels in an image are
  found on a regular rectangular lattice.
• More importantly we will consider disease count
  data where counts of a disease are recorded for
  contiguous regions on a map.
• Although a map is not regular we can construct a
  lattice from a map by identifying neighbouring
  regions and linking neighbouring regions to form
  a lattice.

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Example
Here we see a map of 5 regions in the left hand
picture, and on the right it has been converted
to a lattice with connections between regions
that share boundaries.
?
9
Point process data

• This data is also commonly found in disease
  mapping although may be used in many applications
  where cases of an event are seen at particular
  locations.
• Each item of data consists of the location of an
  event, the response (type of event) and
  potentially predictor variables for the event.
• Note Rasmus has worked more extensively in this
  area and will be happy to answer questions here. ?

10
Disease point process modelling

• In disease mapping our data is typically binary
  i.e. people are infected (or die from) a disease
  or are not.
• The data occur in point process form but there
  are 2 problems with analysing them as a point
  process
• All our responses are 1 as we only observe the
  infected/dead people!
• Due to confidentiality and the sensitive nature
  of medical data the data cannot often be released
  as individual records.
• To counter point 1 we could sample control cases
  at random from the population however point 2
  means that we typically total up cases for fixed
  areas and use a Poisson model on the lattice data
  that this creates.

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Why might there be spatial effects?

• This depends on the response variable and
  application area.
• It is possible that geography is itself a
  predictor for our response or is a surrogate for
  other factors.
• Many factors can be linked to location e.g.
  weather, deprivation, altitude, pollution, wealth
  which might influence the response.
• So if our response is influenced by any of these
  factors then accounting for spatial effects many
  improve our model.

12
Nested random effects/ levels of geography

• The simplest link to random effect models is to
  consider nested random effects.
• We have considered pupils nested in schools and
  cows nested in herds.
• In some sense the schools and herds are spatial
  units in that schools generally take children
  from their locality and a herd is based on a
  particular farm. However we could also fit where
  the pupils live as another classification of the
  data which is more spatial.
• On the next slide we consider a dataset with more
  levels of geography.

13
UK house prices dataset

• An MMath student of mine (David Goodacre) studied
  a dataset of house prices in the UK. The data
  supplied by the Nationwide building society
  consists of average house prices in areas of the
  UK over a 12 year period (1992-2003). The data is
  for 753 towns in the UK and there are 3 levels of
  geography (towns nested in counties nested in
  regions.)
• Note that if we had individual house sale
  information then we could have considered point
  process approaches but here we consider random
  effect modelling.

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A 4-level VC model for the house price dataset

• The following model was fitted to the data
• where i indexes year, j indexes town, k indexes
  county and l indexes region. The response, y is
  the log of the average price.
• This model can be fitted using both frequentist
  and likelihood methods in packages that allow
  four levels in the model.

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Links with other topics

• It is worth noting that this house price dataset
  is a repeated measures dataset as you considered
  yesterday.
• It also contains missing data as in any year in
  which there were less than 50 sales in a postal
  town will lead to a missing observation.
• However we here assume MAR conditional on the
  model we are fitting.

16
Estimates for house price dataset

• Below are given IGLS estimates for the model

Parameter Estimate (SE)
ß0 4.036 (0.067)
ß1 -0.020 (0.002)
ß2 0.009 (0.0001)
s2f 0.045 (0.021)
s2v 0.016 (0.004)
s2u 0.045 (0.003)
s2e 0.013 (0.0002)
Here we see that the model consists of parallel
curves with both year and year2 very significant.
The variance is greatest between regions and
between postal towns
17
Region Level Effects
Here we see that the south east of the UK and
London are the most expensive whilst Scotland the
North and Wales are the cheapest.
18
County level effects
After accounting for regions the pattern of
county effects is more sporadic. We can however
pick up 2 regions, Cheshire in the North West and
Surrey in the South East that are more expensive
than their neighbours.
19
Region level predictions

• Here we see a graph of region level predictions

20
Further Modelling

• In his project Dave looked at random slopes
  models at the various levels of the model, so
  that we could pick out whether the increase in
  prices was different in different regions.
• He also looked at fitting models of a more
  spatial nature! See next lecture.

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Why are spatial effects different?

• The main difference with spatial effects is that
  we have additional information about each
  (spatial) unit.
• For example if we observe the average house price
  of a town in Grampian, a town in Surrey and 2
  towns in Berkshire then we know something of the
  spatial relation of these towns.
• We might expect the prices in the 2 towns in
  Berkshire to be similar and to be more similar to
  Surrey which is also in the South East than
  Grampian that is in Scotland.
• In our current models we will fit an effect for
  Berkshire which will capture some of the
  relationship between its 2 towns and a South East
  effect that will capture the link with the Surrey
  town.

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Problems with the nested classification approach

• As we have seen the nested classification
  approach can capture much of the spatial
  variability however we have to decide on the
  geographic definitions of areas.
• We generally use easily available definitions
  e.g. county and region but there is no guarantee
  that these are the best classifications.
• We also have the problem of border effects, for
  example two towns on either side of a region
  border will not share either region or county
  effects but may have very similar prices.
• We will look at another approach here before
  studying more complex spatial approaches in the
  next lecture.

23
Including location in fixed effects

• It may be the case that there is a trend e.g. in
  house prices in the UK they generally fall as we
  move North and West. We could therefore add in
  two (fixed effect) predictors giving the N/S and
  E/W co-ordinates of each point.
• If the unit of observation is an area e.g. postal
  town we would generally use the co-ordinates of
  the centroid of the unit.
• If a linear relationship is not sensible then we
  could consider polynomial terms in each
  direction. For example (excluding random effects)

24
Distance effects

• Another possibility in terms of UK house prices
  is to consider the distance from London. This
  distance can be constructed from the co-ordinates
  of each point. The graph to the left gives the
  combined region and county effects and suggests a
  distance from London effect might be appropriate.

25
Distance and direction effects

• In some scenarios the direction as well as the
  distance from a particular point is important.
• This is not the case with house prices however in
  pollution data then direction can be very
  important where a dominant wind direction will
  suggest that particular directions away from the
  source will experience more pollution than
  others.
• We will next look at a dataset from Falkirk in
  Scotland that is analysed in Lawson, Browne
  Vidal Rodeiro (2003)

26
Focused Clustering

• One research area in public health looks at the
  impact of sources of pollution on the health
  status of communities. The detection of patterns
  of health events associated with pollution
  sources is known as focused clustering. The
  statistical modelling involved usually relates to
  the point process nature of such data.
• Lawson, Browne Vidal Rodeiro (2003) devote a
  whole chapter to Focused clustering and include
  some fairly complex models that can be considered
  in WinBUGS. Here we will look at some simpler
  models that can be fitted in MLwiN to a dataset
  from Falkirk in Scotland.

27
Respiratory cancer in Falkirk

• The figure to the right shows the census
  geographies of 26 regions found around a foundry
  (marked by ) in Falkirk, Scotland. It is thought
  conceivable that the foundry was an air pollution
  hazard in the early 1970s prior to the study.
  This could have an impact on the respiratory
  cancer experience of those living in the areas
  close to the foundry

28
Falkirk dataset

• The data consists of observed and expected counts
  of respiratory cancer cases in the time period
  1978-1983.
• We first compare the standardized mortality rates
  (SMRs) observed/expected against the locations
  of the centroids of the 26 areas in Falkirk
  (relative to the foundry) to look for patterns.

29
Position of the sites

• Note in the graphs to the right that the 3
  highest SMRs are close to the source both in the
  N/S and E/W directions.
• We can convert these locations to distance and
  direction measures.

30
Distance and direction

• Here we see that there appears to be a negative
  relationship between distance and SMR but no
  obvious pattern with regard to the direction
  relationship.

31
(Extra) Poisson modelling

• We have modelled the effects of deprivation,
  distance and direction in the following Poisson
  model
• Note that we have used 1st order MQL in MLwiN and
  allowed extra-Poisson variation. This shows there
  is less variation than a Poisson distribution so
  we will also try fitting SMR as a Normal
  distributed response.

32
Normal response model for SMR

• Here we see that none of the predictors has a
  significant effect which is probably because the
  dataset is so small.
• We do see however that the risk reduces as
  distance from the foundry increases and for areas
  with larger deprivation scores. (suggesting
  higher rates in less deprived areas but not
  significantly.)

33
Information for the practical

• In the practical we will return to using nested
  random effects to account for spatial effects.
• Our data is from the European community and
  consists of male deaths from malignant melanoma
  in 9 countries in the EU.
• The practical is a (modified) chapter from Browne
  (2003) and looks at MCMC methods for this
  dataset. It is also analysed using
  quasilikelihood methods in the MLwiN users guide
  and you are welcome to also try these methods.