Fitting models to data

1
Fitting models to data IV(Yet more on Maximum
Likelihood Estimation)

• Fish 458, Lecture 11

2
The Poisson Distribution

• The density function
• rt is the expected number of events (r is a rate
  and t is time).
• k is the number of discrete events (count data).
• The Poisson distribution has only one parameter
  (rt) which is both the mean and the variance.
  However, often we find the variance is larger
  than would be expected under the Poisson model so
  assume this model with care better still, look
  at the negative binomial distribution first!

3
Poisson Model Example-I
100 longline sets are observed and the following
data are collected. What is the rate (numbers
per set) at which seabird are captured?
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Poisson Model Example-II

• The log-likelihood function (after removal of
  constants) is given by
• This equation is maximized at r0.69.
• How else could we have obtained the same estimate
  for r?

5
The Negative Binomial Distribution-I

• The negative binomial distribution extends the
  Poisson distribution by allowing the rate
  parameter to be a (gamma) random variable
• R is the expected number of observations
  (discrete or continuous)
• k is an overdispersion parameter.
• Note

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The Negative Binomial Distribution-II

• The mean of the negative binomial distribution is
  R.
• The variance of the negative binomial
  distribution is
• The negative binomial distribution collapses to
  the Poisson distribution as

7
The Negative Binomial Distribution-III
8
The Negative Binomial Distribution-IV

• Consider the case in which we monitor the catch
  of a given species (in number) as a function of
  fishing effort. If the catch occurs randomly per
  unit time we would expect the catch to be Poisson
  distributed with mean (and variance) equal to the
  product of the fishing duration and a rate of
  capture.
• For this problem, we apply the Poisson model and
  the Negative binomial model.

9
The Negative Binomial Distribution-V
k is 84 for this fit good evidence that the
Poisson model is adequate.
10
The Negative Binomial Distribution-VI
The data are now (really) overdispersed relative
to a Poisson distribution. The estimates are
again identical, but the negative binomial
indicates lesser precision than the Poisson.
11
Overdispersion

• Overdisperson implies that the variance of the
  data is greater than that expected under the
  distribution assumed (e.g. Poisson ?
  variancemean).
• If the data are overdispersed but this is
  ignored, you are overweighting the data (i.e.
  underestimating their uncertainty).

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Likelihood Cheat sheet
Data?
Continuous
Discrete
Number of outcomes
Can be negative?
2
Yes
Many
No
Normal / t
Binomial
lognormal / gamma
Poisson / Negative binomial / Multinomial
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Fitting Miscellany-II

• Robustness
• In many cases, the assumptions underlying the
  likelihood function are wrong some data points
  are too unlikely. Such data points are outliers.
• Outliers can either be left out of the analysis
  or the likelihood robustified to reduce their
  influence.
• Robustification includes minimizing the median
  residual, leaving out the largest residuals,
  downweighting large residuals.

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Fitting Miscellany-III

• Contradictory data
• All probability statements are based on the
  assumptions of the model and likelihood function,
  and these may be wrong!
• Often when we have two (or more) data sources
  they disagree!
• The problem is that (at least) one data source is
  not measuring what we think it is.
• Solutions
• Include some probability that each index doesnt
  tell us anything and
• Run separate assessments for each index in turn.

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Contradictory data(northern cod)
The northern cod dilemma two abundance indices
one increasing (and relatively precise), the
other not (and noisy). To pool or not to pool!
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Additional Readings

• Chen, Y. Fournier, D. 1999. Can. J. Fish. Aquat.
  Sci. 56 1525 1533.
• Fournier, D.A. Hampton, J. Sibert, J.R. 1998.
  Can. J. Fish. Aquat. Sci. 55 21052116.
• Schnute, J.T. Hilborn, R. 1993. Can. J. Fish.
  Aquat. Sci. 50 1916 1927.