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Workshop in R & GLMs: #3 Diane Srivastava University of British Columbia srivast@zoology.ubc.ca

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

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hard~dens

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hard^0.45~dens

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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 GLMs[R 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 But wait…couldn’t we just code a categorical variable to be continuous? Yx 1.20 1.30 1.11 0.91

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Then there were… General Linear Models: a normally- distributed y fit to a continuous OR categorical x But wait…why do we force our data to be normal when often it isn’t?

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Generalized linear models No more need for tedious transformations! Proud to be Poisson ! All variances are unequal, but some are more unequal than others… Because most things in life aren’t normal ! Distribution solution !

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What linear models do: X Y X Log Y 1.Transform y 2.Fit line to transformed y 3.Back transform to linear y

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What GLMs do: X Y X Log fitted values 1.Start with an arbitrary fitted line 2.Back-transform line into linear space 3.Calculate residuals 4.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: (1)understand how the spreadsheet works (2)find through an iterative process the best slope

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Generalized linear models In least squares, we fit: y=mx + b + error In GLM, the model is fit more indirectly: y=g(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 least squares Assumes normality Based on Sum of Squares Fits model to transformed y Uses maximum likelihood Specify one of several distributions Based on deviance Fits model to untransformed y by means of a link function

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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 diane<-read.table(file.choose(),sep=“,",header=TRUE) 2. Look at dataframe. Make treat a factor (“treat”) 3. Fit this model: my.first.glm<-glm(growth~size*treat, family=poisson (link=log), data=diane) ; summary(my.first.glm) 4. Model dignostics par(mfrow=c(2,2)); plot(my.first.glm)

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Overdispersion UnderdispersedOverdispersedRandom

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Overdispersion Is your residual deviance = residual df (approx.)? If residual dev>>residual df, overdispersed. If residual dev<<residual df, underdispersed. Solution: second.glm<-glm(growth~size*treat, family = quasipoisson (link=log), data=diane); summary(second.glm)

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Options familydefault linkother links binomiallogitprobit, cloglog gaussianidentity Gamma--identity,inverse, log poissonlogidentity, 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 (value=1) or failure (value=0). 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 x y

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Logit p = proportion of successes If p = e ax+b / (1+ e ax+b ) calculate: log e (p/1-p)

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Logits continued Output from logistic regression with logit link: predicted log e (p/1-p) = a+bx To obtain any expected values of p, need to input a and b in original equation: p = e ax+b / (1+ e ax+b )

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Binomial GLMs Type 1 binomial Simply set family = binomial (link=logit) 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 1.Fit the binomial glm survival = size*treat 2. Fit the bionomial glm parasitism = size*treat 3. Predict what size has 50% parasitism in treatment “0”

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