Module 7: LogLinear Modeling

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Module 7 Log-Linear Modeling

• RSF Conference Workshop 11-12 January 2003

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Summary

• What is a log-linear model?
• Log-linear models applied to resource selection
• Base rates
• White-tailed deer example

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What is a Log-Linear Model?

• Log-linear models are also called Poisson
  regression models.
• Log-linear models are regression models for count
  data

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What is a Log-Linear Model?

• Recall regular (normal theory) regression
• We assume responses Y1, Yn are independently
  and identically distributed normal random
  variables
• I.e., YiNormal(?i,?)
• We also assume
• ?i ß0 ß1xi1 ... ßpxip

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What is a Log-Linear Model?

• An alternative view of the regular regression
  equation is
• g(?i) ß0 ß1xi1 ... ßpxip
• where the function g(x) x.
• g is called the link function
• g links the linear predictor ß0 ß1xi1 ...
  ßpxip to ?i, the expected value of Yi.

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What is a Log-Linear Model?

• In regular regression, coefficients ß0, , ßp
  are estimated by minimizing the function
• D is called the sum of squares
• Estimates are said to be least squares estimates

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What is a Log-Linear Model?

• For a log-linear model
• We assume responses Y1, Yn are independently
  and identically distributed Poisson random
  variables
• I.e., YiPoisson(?i)
• We also assume
• g(?i) ß0 ß1xi1 ... ßpxip
• where g(x) log(x)

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What is a Log-Linear Model?

• In Poisson regression, coefficients ß0, , ßp are
  estimated by minimizing the function
• D is called the deviance function
• D is derived from the form of the Poisson
  distribution (left for the interested reader!)

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What is a Log-Linear Model?

• Differences between D statistics for different
  models can be used to assess the relative fit of
  those models.

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What is a Log-Linear Model?

• Log-linear model fitting demonstration

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RSF Setting

• Log-linear modelling can be used to relate
  resource selection to factors such as individual
  animals involved, or time of day. Potentially
  this approach is very versatile.
• It is assumed that data available consists of I
  independent counts Y1 to YI, where the ith count
  has a Poisson distribution with mean value
• µi exp(ß0 ß1xi1 ... ßpxip)

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Base Rates

• Sometimes it is known that µi should be
  proportional to some base rate, Bi, so that the
  equation is better taken as
• µi Biexp(ß0 ß1xi1 ... ßpxip)
  exp(log(Bi) ß0 ß1xi1 ... ßpxip)

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Base Rates

• E.g., suppose that Yi number of times that
  animal is seen in the ith type of habitat, where
  it is known that 60 of the habitat available to
  the animal is of this type. Then it is
  appropriate to set Bi 0.60 so that if there is
  no selection then the model
• µi Biexp(ß0)
• should fit exp(ß0) allows for the total number
  of observations made on the animal.

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Computer Programs

• Fitting log-linear model requires a suitable
  computer program. Many are available, including
  SAS Proc Genmod, GLIM, S-Plus glm(), and Manlys
  LOGLIN. Quattro or Excel can also be used.

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White-tailed Deer Example

• This application of log-linear modelling was
  described by Heisey (1985).
• The data were reported by Nelson (1979) from a
  radio tracking study of habitat use by
  white-tailed deer.
• Data consisted of habitat use (HU) of two
  white-tailed deer in four types of habitat, with
  the proportional availability of those habitats
  (HA).

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White-tailed Deer Example

• Data
• 
  Midday Morning and
  evening
• Deer 68 Deer 342
  Deer 68 Deer 342
• Habitat HU HA HU HA HU
  HA HU HA
• 
  
• Aspen 18 0.66 29 0.65 43
  0.66 46 0.65
• Clearcut 2 0.20 1 0.13 33
  0.20 29 0.13
• Plantation 0 0.09 4 0.13 5
  0.09 4 0.13
• Spruce 0 0.05 0 0.09 0
  0.05 2 0.09
• 
  

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White-tailed Deer Example

• This is a Design III study with sampling protocol
  A, where availability was censused and use was
  sampled for each of two animals.
• One method for analysing data
• Estimate selection ratios first using the midday
  results, and then separately for the morning and
  evening results using animals as replicates.

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White-tailed Deer Example

• Problem
• Variance estimates would be unreliable as they
  would be based on differences between only two
  animals. Hence tests for selection and for
  differences between selection ratios would also
  be unreliable.

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White-tailed Deer Example

• Another Method
• Assume counts are values from independent Poisson
  distributions. Reasonable providing that the
  individual observations on deer locations were
  far enough apart in time to be independent.
  Assume that this was the case.
• Heissey used program GLIM to analyse data. This
  automatically constructs X variables for the 4
  habitat types, 2 deer, and 2 times of day.

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White-tailed Deer Example

• Variables can be set up easily enough for use in
  a program that does not have this facility, as
  below. Will be seen that base rates and 12
  variables are needed to account for selection
  related to time of day deer.

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White-tailed Deer Example

• Data
• Base rates proportion available of each habitat
  type.
• X1 1 for aspen, otherwise 0
• X2 1 for clearcut, otherwise 0
• X3 1 for plantation, otherwise 0
• X4 1 for midday, 0 for morning afternoon
• X5 1 for deer 68, 0 for deer 342
• X6 X1X4 X7 X2X4 X8 X3X4 X9 X1X5
  X10 X2X5 X11 X3X5 X12 X4X5

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White-tailed Deer Example

• Data
• Sample Base
• count rate X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
  X12
• 
  
• 18 0.66 1 0 0 1 1 1 0 0 1 0 0
  1
• 29 0.65 1 0 0 1 0 1 0 0 0 0 0
  0
• 43 0.66 1 0 0 0 1 0 0 0 1 0 0
  0
• 46 0.65 1 0 0 0 0 0 0 0 0 0 0
  0
• 2 0.20 0 1 0 1 1 0 1 0 0 1 0
  1
• 1 0.13 0 1 0 1 0 0 1 0 0 0 0
  0
• .
• .
• 5 0.09 0 0 1 0 1 0 0 0 0 0 1
  0
• 4 0.13 0 0 1 0 0 0 0 0 0 0 0
  0
• 0 0.05 0 0 0 1 1 0 0 0 0 0 0
  1
• 0 0.09 0 0 0 1 0 0 0 0 0 0 0
  0
• 0 0.05 0 0 0 0 1 0 0 0 0 0 0
  0
• 2 0.09 0 0 0 0 0 0 0 0 0 0 0
  0

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White-tailed Deer Example

• No Selection Model
• Includes X4, X5 X12 allows the expected counts
  to depend on the deer the time of day, with
  deer effect possibly varying with the time of
  day habitat use is proportional to availability.
  No selection deviance 78.26 with 12 df.

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White-tailed Deer Example

• Selection Model 1
• Adding X1 to X3 allows some selection to take
  place, at same level for both deer and both times
  of day. Deviance 32.42 with 9 df.

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White-tailed Deer Example

• Selection Model 2
• It is possible to expand model to either allow
  selection to depend on time of day (adding X6 to
  X8) or allow selection to depend on deer (adding
  X9 to X11). Deviance is reduced to 6.44 with 6
  df if the first option is taken, but reduced
  hardly at all to 30.88 with 6 df if the second
  option is chosen. The first option is therefore
  best.

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White-tailed Deer Example

• Selection Model 3
• The next stage consists of allowing selection to
  depend on both the deer and the time of day by
  including all of variables X1 to X12. Deviance
  4.41 with 3 degrees of freedom.

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White-tailed Deer Example

• The only way that the model can be expanded now
  is by adding X variables that allow selection of
  habitat by deer to vary with the time of day.
  Then there are as many parameters as sample
  frequencies so that the model is 'saturated' with
  parameters and fits data exactly.

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White-tailed Deer Example

• The model building process is summarised in an
  analysis of deviance table. A reasonable
  conclusion is that there was selection, and that
  this depended on the time of day but not on the
  deer.

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White-tailed Deer Example

• Model building summary
• 
  
  Difference
• Model XL²
  df XL² df
• 
  
• No selection of habitat 78.26
  12
• 
  45.84 3
• Constant selection on habitat 32.42
  9
• 
  25.98 3
• Selection varies with time 6.44
  6
• 
  2.03 3
• Selection varies with time deer 4.41
  3
• 
  4.41 3
• Selection with time effect varying
  
• with deer 0.00
  0
• 
  
• Significantly large at 0.1 level.

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Summary

• Log-linear models explained
• Log-linear models applied to resource selection
• White-tailed deer example