Workshop in R

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Workshop in R GLMs 3

• Diane Srivastava
• University of British Columbia
• srivast_at_zoology.ubc.ca

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Housekeeping

• ls() asks what variables are in the global
  environment
• rm(listls()) gets rid of EVERY variable
• q() quit, get a prompt to save workspace or
  not

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harddens
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hard0.45dens
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log(hard)dens
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Janka exercise

• Conclusion
• The best y transformation to optimize the model
  fit (highest log likelihood)
• ..is not the best y transformation for normal
  residuals

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This workshop

• Linear, general linear, and generalized linear
  models.
• Understand how GLMs work Excel simulation
• Definitions e.g. deviance, link functions
• Poisson GLMsR exercise
• Binomial distribution and logistic regression
• Fit GLMs in R! Exercise

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In the beginning there were

• Linear models a normally-distributed y fit to a
  continuous x

Y x 1.2 0 1.3 0 1.1 1 0.9 1
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Then there were

• General Linear Models a normally-distributed y
  fit to a continuous OR categorical x

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Generalized linear models
No more need for tedious transformations!
All variances are unequal, but some are more
unequal than others
Distribution solution !
Because most things in life arent normal !
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What linear models do
Y
X

• Transform y
• Fit line to transformed y
• Back transform to linear y

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What GLMs do
Log fitted values
Y
X
X

• Start with an arbitrary fitted line
• Back-transform line into linear space
• Calculate residuals
• Improve fitted line to maximize likelihood

Many iterations
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Maximum likelihood

• Means that an iterative process is used to find
  the model equation that has the highest
  probability (likelihood) of explaining the y
  values given the x values.
• Equation for likelihood depends on the error
  distribution chosen
• Least squares by contrast minimizes
  variation from the model.
• If the data are normally distributed, maximum
  likelihood gives the same answer as least squares.

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GLM simulation exercise

• Simulates fitting a model with normal errors and
  a log link to data.
• Your task
• understand how the spreadsheet works
• find through an iterative process the best slope

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Generalized linear models

• In least squares, we fit
• ymx b error
• In GLM, the model is fit more indirectly
• yg(mx b error)
• where g is a function, the inverse of which is
  called the link function
• link fn(expected y) mx b error

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LMs vs GLMs

• Uses maximum likelihood
• Specify one of several distributions
• Based on deviance
• Fits model to untransformed y by means of a link
  function

• Uses least squares
• Assumes normality
• Based on Sum of Squares
• Fits model to transformed y

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All that really matters

• By using a log link function, we do not need to
  calculate log(0).
• Be careful! A log link model predicts log y not
  y!
• Error distribution need not be normal Poisson,
  binomial, gamma, Gaussian (normal)

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Exercise

• 1. Open up the file Rlecture.csv
• dianelt-read.table(file.choose(),sep,",headerTRU
  E)
• 2. Look at dataframe. Make treat a factor
  (treat)
• 3. Fit this model
• my.first.glmlt-glm(growthsizetreat,
  familypoisson (linklog), datadiane)
  summary(my.first.glm)
• 4. Model dignostics
• par(mfrowc(2,2)) plot(my.first.glm)

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Overdispersion
Underdispersed
Overdispersed
Random
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Overdispersion

• Is your residual deviance residual df
  (approx.)?
• If residual devgtgtresidual df, overdispersed.
• If residual devltltresidual df, underdispersed.
• Solution
• second.glmlt-glm(growthsizetreat, family
  quasipoisson (linklog), datadiane)
  summary(second.glm)

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Options

• family default link other links
• binomial logit probit, cloglog
• gaussian identity
• Gamma -- identity,inverse,
• log
• poisson log identity, sqrt

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Rlecture.csv
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Binomial errors

• Variance gets constrained near limits binomial
  accounts for this
• Type 1 Classic example series of trials
  resulting in success (value1) or failure
  (value0).
• Type 2 Also continuous but bounded (e.g.
  mortality bounded between 0 and 100).

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Logistic regression

• Least squares arcsine transformations
• GLMs use logit (or probit) link with binomial
  errors

y
x
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Logit

• p proportion of successes
• If p eaxb / (1 eaxb) calculate
• loge(p/1-p)

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Logits continued

• Output from logistic regression with logit link
  predicted loge (p/1-p) abx
• To obtain any expected values of p, need to input
  a and b in original equation
• p eaxb / (1 eaxb)

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Binomial GLMs

• Type 1 binomial
• Simply set family binomial (linklogit)
• Type 2 binomial
• First create a vector of not parasitized.
• Then cbind into a matrix ( parasitized, not
  parasitized)
• Then run your binomial glm (link logit) with
  the matrix as your y.

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Homework

• Fit the binomial glm survival sizetreat
• 2. Fit the bionomial glm parasitism sizetreat
• 3. Predict what size has 50 parasitism in
  treatment 0