Linear Model

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- residual or error

Formal Definition

• , - observed values of covariate variables (i.e. temperature, precipitation)

• - observed value of the response variable (i.e. tree height)

• - y intercept:

General Linear Model

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y = 0.1983x - 0.3216R² = 0.9694

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General Linear Model

• Can transform the predictor values to linearize the relationship between the predictors and the response

• Also changes the variance so it only should be done if the variance is not uniform and is made uniform by the transform

Polynomial Regression

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Need More

• Not all phenomenon follow linear response

• Not all residuals are normally distributed• This leads:

– GLMs: Single function, specified regression distribution

– GAMs: Multiple functions– “Non-parametric” approaches: function is

determined by the computer

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. • Also requires to be from a defined

probability distribution

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)

Generalized Linear Models

– Linear model without the error• is a “link” function• = )• Random component is from a known

probability distribution

Common Functions in R

• Probability Distribution (Link Function)• Binomial (link = "logit")

– True/false, alive/dead• Gaussian (link = "identity")

– Continuous, normal• Gamma (link = "inverse")

– Seed distribution, distance from…• Poisson (link = "log")

– Counts

Normal Distribution

WikipediaAKA “Gaussian” Distribution

Binomial

Number of successes of yes/no experiments Wikipedia

Poisson

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

Gamma Distribution

Wait times, seed distribution, etc. Wikipedia

Deviance

• Where:– = Maximum log-likelihood for model– = Maximum log-likelihood for the most

complex model possible (i.e. fits observed data perfectly)

Degrees of Freedom

• number of observations• number of parameters (branches)• Degrees of freedom =