• Understand the 3 elements of a generalized linear model

• Understand how to identify the proper distribution for a generalized linear model

• Understand the concept of a link function

1 Detection \(\rightarrow\) presence/absence

2+ Detections \(\rightarrow\) survival, movement

1 Detection \(\rightarrow\) presence/absence

2+ Detections \(\rightarrow\) survival, movement

1 Measurement \(\rightarrow\) fecundity, age, size

2+ Measurements \(\rightarrow\) growth

Detections \(\rightarrow\) presence/absence

Counts \(\rightarrow\) density or survival/movement

Sex

Age

Fecundity

Counts/Census

Survival (individual)

Given the prevalence of discrete data in ecology (and elsewhere), we seek a means for modeling them

GLMs were developed by Nelder & Wedderburn in the 1970s

They include (as special cases):

• linear regression
  
• ANOVA
  
• logit models
  
• log-linear models
  
• multinomial models

In particular, GLMs can explicitly model discrete data as outcomes

What is the distributional form of the random process(es) in my data?

The Poisson distribution is perhaps the best known

It gives the probability of a given number of events occurring in a fixed interval of time or space

• the number of Prussian soldiers killed by horse kicks per year from 1868 - 1931

• the number of new COVID-19 infections per day in the US

• the number of email messages I receive per week from students in QERM 514

It’s unique in that it has one parameter \(\lambda\) to describe both the mean and variance

\[ y_i \sim \text{Poisson}(\lambda) \] \[ \text{Mean}(y) = \text{Var}(y) = \lambda \]

Ratios (fractions) are also very important in ecology

They convey proportions such as

• survivors / tagged individuals

• infected / susceptible

• student emails / total emails

The simplest ratio has as denominator of 1 & and numerator of either 0 or 1

For an individual, this can represent

• present (1/1) or absent (0/1)

• alive (1/1) or dead (0/1)

• mature (1/1) or immature (0/1)

The Bernoulli distribution describes the probability of a single “event” \(y_i\) occurring

\[ y_i \sim \text{Bernoulli}(p) \]

where

\[ \text{Mean}(y) = p ~ ~ ~ ~ ~ \text{Var}(y) = p(1 - p) \]

The binomial distribution is closely related to the Bernoulli

It describes the number of \(k\) “successes” in a sequence of \(n\) independent Bernoulli “trials”

For example, the number of heads in 4 coin tosses

For a population, these could be

• \(k\) survivors out of \(n\) tagged individuals

• \(k\) infected individuals out of \(n\) susceptible individuals

• \(k\) counts of allele A in \(n\) total chromosomes

Three important components

1. Distribution of the data

Are they counts, proportions?

Three important components

1. Distribution of the data

2. Link function \(g\)

Specifies the relationship between mean of the distribution and some linear predictor(s)

Three important components

1. Distribution of the data

2. Link function \(g\)

3. Linear predictor \(\eta\)

\(\eta = \mathbf{X} \boldsymbol{\beta}\)

Where did we find these link functions?

For the exponential family of distributions (eg, Normal, Gamma, Poisson) we can write out the distribution of \(y\) as

\[ f(y; \theta, \phi) = \exp \left( \frac{y \theta - b(\theta)}{a(\phi)} - c(y, \phi)\right) \]

\(\theta\) is the conanical parameter of interest

\(\phi\) is a scale (variance) parameter

\[ f(y; \theta, \phi) = \exp \left( \frac{y \theta - b(\theta)}{a(\phi)} - c(y, \phi)\right) \]

We seek some canonical function \(g\) that connects \(\eta\), \(\mu\), and \(\theta\) such that

\[ g(\mu) = \eta \\ \eta \equiv \theta \]

\[ f(y; \theta, \phi) = \exp \left( \frac{y \theta - b(\theta)}{a(\phi)} - c(y, \phi)\right) \\ \Downarrow \\ f(y; \mu, \sigma) = \frac{1}{\sqrt{2 \pi \sigma}} \exp \left( \frac{(y - \mu)^2}{2 \sigma^2} \right)\\ \]

with \(\theta = \mu\) and \(\phi = \sigma^2\)

\(a(\phi) = \phi ~~~~~~ b(\theta) = \frac{\theta^2}{2} ~~~~~~ c(y, \phi) = - \frac{\tfrac{y^2}{\phi} + \log (2 \pi \phi)}{2}\)

\[ f(y; \theta, \phi) = \exp \left( \frac{y \theta - b(\theta)}{a(\phi)} - c(y, \phi)\right) \\ \Downarrow \\ f(y; \mu, \sigma) = \frac{1}{\sqrt{2 \pi \sigma}} \exp \left( \frac{(y - \mu)^2}{2 \sigma^2} \right)\\ \]

with \(\theta = 1(\mu)\) and \(\phi = \sigma^2\)

\(a(\phi) = \phi ~~~~~~ b(\theta) = \frac{\theta^2}{2} ~~~~~~ c(y, \phi) = - \frac{\tfrac{y^2}{\phi} + \log (2 \pi \phi)}{2}\)

\[ f(y; \theta, \phi) = \exp \left( \frac{y \theta - b(\theta)}{a(\phi)} - c(y, \phi)\right) \\ \Downarrow \\ f(y; \mu) = \frac{\exp (- \mu) \mu^y}{y!} \\ \]

with \(\theta = \log (\mu)\) and \(\phi = 1\)

\(a(\phi) = 1 ~~~~~~ b(\theta) = \exp (\theta) ~~~~~~ c(y, \phi) = - \log (y!)\)

For the binomial distribution there are several possible link functions

• logit

• probit

• complimentary log-log

The word generalized means these models are broadly applicable

For example, GLMs include linear regression models

\[ y_i = \alpha + \beta x_i + \epsilon_i \\ \epsilon_i \sim \text{N}(0,\sigma^2) \]

\[ y_i = \alpha + \beta x_i + \epsilon_i \\ \epsilon_i \sim \text{N}(0,\sigma^2) \\ \Downarrow \\ y_i = \mu_i + \epsilon_i \\ \mu_i = \alpha + \beta x_i \\ \epsilon_i \sim \text{N}(0,\sigma^2) \]

\[ y_i = \mu_i + \epsilon_i \\ \mu_i = \alpha + \beta x_i \\ \epsilon_i \sim \text{N}(0,\sigma^2) \\ \Downarrow \\ y_i = \epsilon_i \\ \mu_i = \alpha + \beta x_i \\ \epsilon_i \sim \text{N}(\mu_i,\sigma^2) \]

\[ y_i = \epsilon_i \\ \mu_i = \alpha + \beta x_i \\ \epsilon_i \sim \text{N}(\mu_i,\sigma^2) \\ \Downarrow \\ y_i \sim \text{N}(\mu_i,\sigma^2) \\ \mu_i = \alpha + \beta x_i \]

\[ y_i \sim \text{N}(\mu_i,\sigma^2) \\ \mu_i = \alpha + \beta x_i \\ \Downarrow \\ y_i \sim \text{N}(\mu_i,\sigma^2) \\ 1(\mu_i) = \mu_i \\ \mu_i = \alpha + \beta x_i \]

\[ y_i \sim \text{N}(\mu_i,\sigma^2) \\ 1(\mu_i) = \mu_i \\ \mu_i = \alpha + \beta x_i \\ \Downarrow \\ \begin{aligned} \text{data distribution:} & ~~ y_i \sim \text{N}(\mu_i,\sigma^2) \\ \\ \text{link function:} & ~~ 1(\mu_i) = \mu_i \\ \\ \text{linear predictor:} & ~~ \mu_i = \alpha + \beta x_i \end{aligned} \]

Log-density of live trees per unit area \(y_i\) as a function of fire intensity \(F_i\)

\[ \begin{aligned} \text{data distribution:} & ~~ y_i \sim \text{N}(\mu_i,\sigma^2) \\ \\ \text{link function:} & ~~ 1(\mu_i) = \mu_i \\ \\ \text{linear predictor:} & ~~ \mu_i = \alpha + \beta F_i \end{aligned} \]

We have been considering (log) density itself as a response

\[ \text{Density}_i = f (\text{Count}_i, \text{Area}_i) \\ \Downarrow \\ \text{Density}_i = \frac{\text{Count}_i}{\text{Area}_i} \\ \]

We have been considering (log) density itself as a response

\[ \text{Density}_i = f (\text{Count}_i, \text{Area}_i) \\ \Downarrow \\ \text{Density}_i = \frac{\text{Count}_i}{\text{Area}_i} \\ \]

With GLMs, we can shift our focus to

\[ \text{Count}_i = f (\text{Area}_i) \]

Counts of live trees \(y_i\) as a function of area surveyed \(A_i\) and fire intensity \(F_i\)

\[ \begin{aligned} \text{data distribution:} & ~~ y_i \sim \text{Poisson}(\lambda_i) \\ \\ \text{link function:} & ~~ \text{log}(\lambda_i) = \mu_i \\ \\ \text{linear predictor:} & ~~ \mu_i = \alpha + \beta_1 A_i + \beta_2 F_i \end{aligned} \]

Probability of spotting a sparrow \(p_i\) as a function of vegetation height \(H_i\)

\[ \begin{aligned} \text{data distribution:} & ~~ y_i \sim \text{Bernoulli}(p_i) \\ \\ \text{link function:} & ~~ \text{logit}(p_i) = \text{log}\left(\frac{p_i}{1-p_i}\right) = \mu_i \\ \\ \text{linear predictor:} & ~~ \mu_i = \alpha + \beta H_i \end{aligned} \]

Survival of salmon from parr to smolt \(s_i\) as a function of water temperature \(T_i\)

\[ \begin{aligned} \text{data distribution:} & ~~ y_i \sim \text{Binomial}(N_i, s_i) \\ \\ \text{link function:} & ~~ \text{logit}(s_i) = \text{log}\left(\frac{s_i}{1-s_i}\right) = \mu_i \\ \\ \text{linear predictor:} & ~~ \mu_i = \alpha + \beta T_i \end{aligned} \]

There are three important components to GLMs

1. Distribution of the data

2. Link function \(g\)

3. Linear predictor \(\eta\)