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GLMs

GLMs. Before we talk about GENERALIZED Linear Models (GLMS), we need to talk about GENERAL Linear Models (not-GLMs but sound really similar). Linear Model. - residual or error. Formal Definition. , - observed values of covariate variables (i.e. temperature, precipitation)

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GLMs

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  1. GLMs • Before we talk about GENERALIZED Linear Models (GLMS), we need to talk about GENERAL Linear Models (not-GLMs but sound really similar).

  2. Linear Model - residual or error

  3. Formal Definition • ,- observed values of covariate variables (i.e. temperature, precipitation) • - observed value of the response variable (i.e. tree height) • - y intercept:

  4. General Linear Model

  5. General Linear Model

  6. General Linear Model • Can transform the predictor values to linearize the relationship between the predictors and the response • Residuals do need to be from a normal distribution • Uses least squares to fit the model

  7. Polynomial Regression

  8. Need More • Not all phenomenon follow linear response • Not all residuals are normally distributed • This leads to: • GLMs: Single function, specified regression distribution • GAMs: Multiple functions • “Non-parametric” approaches: no assumptions are made about the parameters of the distribution

  9. GLM • Generalized Linear Model • Not to be confused with a general linear model • Allows a linear model to be related to the response variable via a “Link” function. • Residuals do not have to be normally distributed • Also requires to be from a defined probability distribution • Fit is based on maximum likelihood

  10. Generalized Linear Models • - a random variable with some probability distribution • Related to the response values • - error • Residuals • Linear model without the intercept • - Expected value of • Predicted value (no error)

  11. Generalized Linear Models • Linear model without the error • is a “link” function • = ) • Random component is from a known probability distribution

  12. Common Functions in R • Probability Distribution (Link Function) • Binomial (link = "logit") • True/false, alive/dead, present/absent • Gaussian (link = "identity") • Continuous, normal • Gamma (link = "inverse") • Seed distribution, distance from… • Poisson (link = "log") • Counts

  13. Normal Distribution AKA “Gaussian” Distribution Wikipedia

  14. Binomial Number of successes of yes/no experiments Wikipedia

  15. Poisson Number of events in time T, k=number of occurrences Wikipedia

  16. Gamma Distribution Wait times, seed distribution, etc. Wikipedia

  17. Deviance • Where: • = Maximum log-likelihood for model • = Maximum log-likelihood for the most complex model possible (i.e. fits observed data perfectly)

  18. Degrees of Freedom • number of observations • number of parameters • Degrees of freedom =

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