Generalized Additive Models

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.title[
# Generalized Additive Models
]
.subtitle[
## a data-driven approach to estimating regression models
]
.author[
### Gavin Simpson
]
.institute[
### Department of Animal & Veterinary Sciences · Aarhus University
]
.date[
### 1400–2000 CET (1300–1900 UTC) Monday 20th January, 2025
]

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# Welcome

???

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# Logistics

## Slides

Slidedeck: [bit.ly/physalia-gam-1](https://bit.ly/physalia-gam-1)

Sources: [bit.ly/physalia-gam](https://bit.ly/physalia-gam)

Direct download a ZIP of everything: [bit.ly/physalia-gam-zip](https://bit.ly/physalia-gam-zip)

Unpack the zip & remember where you put it

---

# Today's topics

* Brief overview of R and the Tidyverse packages we’ll encounter throughout the course

* Recap generalised linear models

* Fitting your first GAM

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# GLMs

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# Generalized linear models

Generalised linear models (GLMs) are an extension of linear regression plus Poisson, logistic and other regression models

GLMs extend the types of data and error distributions that can be modelled beyond the Gaussian data of linear regression

With GLMs we can model count data, binary/presence absence data, and concentration data where the response variable is not continuous.

Such data have different mean-variance relationships and we would not expect errors to be Gaussian.

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# Generalized linear models

Typical uses of GLMs in ecology are

- Poisson GLM for count data
- Logistic GLM for presence absence data
- Gamma GLM for non-negative or positive continuous data

GLMs can handle many problems that appear non-linear

Not necessary to transform data as this is handled as part of the GLM process

---

# Binomial distribution

* For a fixed number of trials (*n*),
* fixed probability of “success” (*p*), &
* two outcomes per trial (heads or tails)

Flip a coin 10 times with *p* = 0.7, the probability of 7 heads is `$\sim Bin(n = 10, p = 0.7)$`, ~ 0.27

``` r
dbinom(x = 7, size = 10, prob = 0.7)
```

```
## [1] 0.2668279
```

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# Binomial distribution

![](index_files/figure-html/binomial-pdf-1.svg)

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# Poisson distribution

The Poisson gives the distribution of the number of “things” (individuals, events, counts) in a given sampling interval/effort if each event is **independent**.

Has a single parameter `$\lambda$` the average density or arrival rate

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# Poisson distribution

![](index_files/figure-html/poisson-pdf-1.svg)

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# The structure of a GLM

A GLM consists of three components, chosen/specified by the user

.small[
1. A **Random component**, specifying the conditional distribution of of the response `$y_i$` given the values of the explanatory data
2. A **Linear Predictor** `$\eta$` &mdash; the linear function of regressors
    `$$\eta_i = \alpha + \beta_1 x_{i1} + \beta_2 x_{i2} + \cdots + \beta_k x_{ik}$$`
	The `$x_{ij}$` are prescribed functions of the explanatory variables and can be transformed variables, dummy variables, polynomial terms, interactions etc.
3. A smooth and invertible **Link Function** `$g(\cdot)$`, which transforms the expectation of the response `$\mu_i \equiv \mathbb{E}(y_i)$` to the linear predictor
    `$$g(\mu_i) = \eta_i = \alpha + \beta_1 x_{i1} + \beta_2 x_{i2} + \cdots + \beta_k x_{ik}$$`
    As `$g(\cdot)$` is invertible, we can write
    `$$\mu_i = g^{-1}(\eta_i) = g^{-1}(\alpha + \beta_1 x_{i1} + \beta_2 x_{i2} + \cdots + \beta_k x_{ik})$$`
]

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# Conditional distribution of `$y_i$`

Originally GLMs were specified for error distribution functions belonging to the *exponential family* of probability distributions

- Continuous probability distributions
    - Gaussian (or normal distribution; used in linear regression)
	- Gamma (data with constant coefficient of variation)
	- Exponential (time to death, survival analysis)
	- Chi-square
	- Inverse-Gaussian
- Discrete probability distributions
    - Poisson (count data)
	- Binomial (0/1 data, counts from a total)
	- Multinomial

Choice depends on range of `$y_i$` and on the relationship between the variance and the expectation of `$y_i$` &mdash; *mean-variance relationship*

---

# Conditional distribution of `$y_i$`

Characteristics of common GLM probability distributions

|                  | Canonical Link | Range of `$Y_i$`               | Variance function              |
|------------------|----------------|------------------------------|--------------------------------|
| Gaussian         | Identity       | `$(-\infty, +\infty)$`         | `$\phi$`                         |
| Poisson          | Log            | `$0,1,2,\ldots,\infty$`        | `$\mu_i$`                        |
| Binomial         | Logit          | `$\frac{0,1,\ldots,n_i}{n_i}$` | `$\frac{\mu_i(1 - \mu_i)}{n_i}$` |
| Gamma            | Inverse        | `$(0, \infty)$`                | `$\phi \mu_i^2$`                 |
| Inverse-Gaussian | Inverse-square | `$(0, \infty)$`                | `$\phi \mu_i^3$`                 |

`$\phi$` is the dispersion parameter; `$\mu_i$` is the expectation of `$y_i$`. In the binomial family, `$n_i$` is the number of trials

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# Common probability distributions

Gaussian distribution is rarely adequate in ecology; GLMs offer ecologically meaningful alternatives

.small[
- **Poisson** &mdash; counts; integers, non-negative, variance increases with mean

- **Binomial** &mdash; observed proportions from a total; integers, non-negative, bounded at 0 and 1, variance largest at `$\pi = 0.5$`

- **Binomial** &mdash; presence absence data; discrete values, 0 and 1, models probability of success

- **Gamma** &mdash; concentrations; non-negative (strictly positive with log link) real values, variance increases with mean, many zero values and some high values
]

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# Old notation

Wrote linear model as

`$$y_i = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \beta_j x_{ij} + \varepsilon_i$$`

And assumed

$$ \varepsilon_i \sim \text{Normal}(0, \sigma^2) $$

This doesn't work out the same for GLMs &mdash; we don't have residuals in the linear predictor

Sampling variation comes from the response distribution

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# New notation

Rewrite linear model as

`\begin{align*}
y_i & \sim \text{Normal}(\mu_i, \sigma^2) \\
\mu_i = \eta_i & = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \beta_j x_{ij}
\end{align*}`

This now matches the general form for the GLM

`\begin{align*}
y_i & \sim \text{EF}(\mu_i, \boldsymbol{\theta}) \\
g(\mu_i) & = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \beta_j x_{ij}
\end{align*}`

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# New notation

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# New notation

Binomial GLM

`\begin{align*}
y_i & \sim \text{Binomial}(n, p_i) \\
\text{logit}(p_i) & = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \beta_j x_{ij}
\end{align*}`

Poisson GLM

`\begin{align*}
y_i & \sim \text{Poisson}(\lambda_i) \\
\log(\lambda_i) & = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \beta_j x_{ij}
\end{align*}`

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# New notation

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# New notation

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# Examples

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