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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Extending GAMs with interactions</title>
<meta charset="utf-8" />
<meta name="author" content="Eric Pedersen" />
<meta name="date" content="2021-11-04" />
<script src="libs/header-attrs/header-attrs.js"></script>
<link href="libs/remark-css/default.css" rel="stylesheet" />
<link rel="stylesheet" href="https:/stackpath.bootstrapcdn.com/bootstrap/4.3.1/css/bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="slides.css" type="text/css" />
</head>
<body>
<textarea id="source">
class: inverse, middle, left, my-title-slide, title-slide
# Extending GAMs with interactions
### Eric Pedersen
### November 4, 2021
---
# Reminder from Tuesday:
A GLM is:
- a distribution for our data +
- a link function +
- a linear model of the transformed mean as a function of covariates
--
A GAM is a GLM except:
- covariate functions can be nonlinear, built from basis functions of covariates
- Estimated with penalized likelihood, penalizing overly wiggly functions
---
# So far:
- One-dimensional smooth models
- Models with multiple separate predictors
- Normally distributed response
These are good models, but there's lots more to see!
---
# New things
- distribution of the data
- `family=` argument
- visually checking model assumptions when building models
- adding dimensions to your smooths
- space and time
---
# Distributions
- `family=` argument in `gam()`
- see `?family.mgcv` for a list
- what you'd expect, as for `lm()` and `glm()`
- most useful cases:
- `binomial` ( yes/no, `\(y \in \{0, 1\}\)` )
- `poisson` ( counts, `\(y \in \{0, 1, 2, \ldots\}\)` )
- `Gamma` ( positive, `\(y>0\)` )
---
# Special count distributions
- Poisson is often not adequate for "real" counts
- Assuming `\(\text{Var}(y) = \mathbb{E}(y)\)` is usually incorrect
![](02-extending-gams_files/figure-html/species-gala-plots-1.svg)<!-- -->
---
# Special count distributions
- Poisson is often not adequate for "real" counts
- Assuming `\(\text{Var}(y) = \mathbb{E}(y)\)` is usually incorrect
![](02-extending-gams_files/figure-html/species-gala-plots2-1.svg)<!-- -->
---
# Special count distributions
Other options?
- `quasipoisson` (count-ish, `\(y \geq 0\)`)
- `\(\text{Var}(y) = \psi\times\mathbb{E}(y)\)`
- awkward to check, no likelihood
--
- `nb`/`negbin` ( `\(y \geq 0\)`)
- `\(\text{Var}(y) = \mathbb{E}(y) + \kappa\times\mathbb{E}(y)^2\)`
- models overdispersed counts
--
- `tw`/`Tweedie` ( `\(y \geq 0\)`)
- great for CPUE-type data
- works on non-count positive data as well
- `\(\text{Var}\left(\text{y}\right) = \phi\mathbb{E}(\text{y})^q\)`
---
# Tweedie distribution
.pull-left[
![](02-extending-gams_files/figure-html/tweedie-1.svg)<!-- -->
(NB there is a point mass at zero not plotted)
]
.pull-right[
- `\(\text{Var}\left(\text{y}\right) = \phi\mathbb{E}(\text{y})^q\)`
- Poisson is `\(q=1\)`
- We estimate `\(q\)` and `\(\phi\)` for `tw`
- We set `\(q\)` and estimate `\(\phi\)` for `Tweedie`
]
---
# Negative binomial distribution
.pull-left[
![](02-extending-gams_files/figure-html/negbin-1.svg)<!-- -->
]
.pull-right[
- `\(\text{Var}\left(\text{count}\right) =\)` `\(\mathbb{E}(\text{count}) + \kappa \mathbb{E}(\text{count})^2\)`
- Poisson is `\(\kappa=0\)`
- Estimate `\(\kappa\)` for `nb`
- Set `\(\kappa\)` for `negbin`
]
---
# 🐟🐠🐡 Example 🐡🐠🐟
.pull-left[
- Let's look at species richness as a function of depth
- Number of species in each trawl (counts!)
```r
trawls <- read.csv(here("data/trawl_nl.csv"))
```
]
.pull-right[
![](02-extending-gams_files/figure-html/richness-violin-1.svg)<!-- -->
]
---
# 🐟🐠🐡 Example 🐡🐠🐟
.pull-left[
- Does richness vary with depth?
- Let's try assuming the response is Poisson!
```r
rich_depthl10 <- gam(richness~s(log10(depth),k = 30),
family=poisson,
method="REML",
data=trawls)
```
]
.pull-right[
```r
plot(rich_depthl10, shade=TRUE)
```
![](02-extending-gams_files/figure-html/plot-year-rich-1.svg)<!-- -->
]
---
# `summary` output
```
##
## Family: poisson
## Link function: log
##
## Formula:
## richness ~ s(log10(depth), k = 30)
##
## Parametric coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 3.155047 0.003212 982.3 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df Chi.sq p-value
## s(log10(depth)) 12.65 15.6 972.3 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.157 Deviance explained = 15.7%
## -REML = 12983 Scale est. = 1 n = 4152
```
---
# Other `family` stuff
- can specify a `link=` argument
- (usually don't have to)
- `?family` has options for each distribution
- for a fitted model `model$family`
- details of what was used
- `model$family$linkfun()` gives link function
- `model$family$linkinv()` gives inverse link
---
# How can we check whether our assumed distribution is adequate?
```r
par(mfrow=c(2,2))
gam.check(rich_depthl10)
```
![](02-extending-gams_files/figure-html/pois-rich-check1-1.svg)<!-- -->
```
##
## Method: REML Optimizer: outer newton
## full convergence after 4 iterations.
## Gradient range [-1.49377e-07,-1.49377e-07]
## (score 12983.34 & scale 1).
## Hessian positive definite, eigenvalue range [2.515006,2.515006].
## Model rank = 30 / 30
##
## Basis dimension (k) checking results. Low p-value (k-index<1) may
## indicate that k is too low, especially if edf is close to k'.
##
## k' edf k-index p-value
## s(log10(depth)) 29.0 12.7 0.95 0.005 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
---
# How can we check whether our assumed distribution is adequate?
```r
par(mfrow=c(2,2))
gam.check(rich_depthl10)
```
```
##
## Method: REML Optimizer: outer newton
## full convergence after 4 iterations.
## Gradient range [-1.49377e-07,-1.49377e-07]
## (score 12983.34 & scale 1).
## Hessian positive definite, eigenvalue range [2.515006,2.515006].
## Model rank = 30 / 30
##
## Basis dimension (k) checking results. Low p-value (k-index<1) may
## indicate that k is too low, especially if edf is close to k'.
##
## k' edf k-index p-value
## s(log10(depth)) 29.0 12.7 0.95 0.01 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
---
# How can we check whether our assumed distribution is adequate?
```r
appraise(rich_depthl10)
```
![](02-extending-gams_files/figure-html/pois-rich-check3-1.svg)<!-- -->
---
---
# Let's pause the talk here to try this ourselves
---
class: inverse middle center big-subsection
## Beyond one dimensional smooths:<br/> space, time and more
---
# 🦐🦐 Example 🦐🦐
.pull-left[
```r
trawls <- read.csv(here("data/trawl_nl.csv"))
trawls_2010 <- filter(trawls, year==2010)
range(trawls_2010$shrimp)
```
```
## [1] 0.000 2293.699
```
```r
head(trawls_2010$shrimp, 3)
```
```
## [1] 0.1202384 0.9619069 13.6270140
```
Not normal, not integer!
]
.pull-right[
![](02-extending-gams_files/figure-html/unnamed-chunk-1-1.svg)<!-- -->
]
---
# 🦐🗺
.pull-left[
- spatial variation!
- how do we model this??
- `s(long) + s(lat)` misses the interaction
]
.pull-right[
![](02-extending-gams_files/figure-html/biom-space-plot-1.svg)<!-- -->
]
---
# Adding dimensions
- Thin plate regression splines (default basis)
- `s(x, y)`
- Assumes `x` and `y` are measured in the same units
- `x`, `y` projected coordinates 👍
- `x` temperature, `y` depth 👎
---
# 🦐🗺
Starting with a 2D smooth, assuming data is Gaussian
.pull-left[
```r
spatial_shrimp <- gam(shrimp ~ s(x, y, k =100),
data=trawls_2010,
family = gaussian,
method="REML")
```
]
.pull-right[
```r
plot(spatial_shrimp, asp=1)
```
![](02-extending-gams_files/figure-html/plot-effect-1.svg)<!-- -->
]
---
# That plot is hard to understand!
.pull-left[
```r
plot(spatial_shrimp, asp=1)
```
![](02-extending-gams_files/figure-html/plot-effect-bad-1.svg)<!-- -->
]
.pull-right[
```r
plot(spatial_shrimp, scheme=2, asp=1)
```
![](02-extending-gams_files/figure-html/plot-effect-good-1.svg)<!-- -->
]
---
# Using gratia:
.left-column[
```r
draw(spatial_shrimp)
```
]
.right-column[
![](02-extending-gams_files/figure-html/plot-effect-gratia2-1.svg)<!-- -->
]
---
# `summary` output
```
##
## Family: gaussian
## Link function: identity
##
## Formula:
## shrimp ~ s(x, y, k = 100)
##
## Parametric coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 125.390 8.658 14.48 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(x,y) 39.32 53.66 3.759 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.309 Deviance explained = 36.6%
## -REML = 3241.7 Scale est. = 36132 n = 482
```
---
class: inverse middle center big-subsection
## Now give it a try!
---
class: inverse middle center big-subsection
## Other multi-dimensional smooths
---
# `s(x,y)` doesn't always work
- Only works for `bs="tp"` or `bs="ts"`
- Covariates are isotropic
- What if we wanted to use lat/long?
- Or, more generally: interactions between covariates?
---
# Enter `te()`
.pull-left[
- We can built interactions using `te()`
- Construct 2D basis from 2 1D bases
- Biomass as a function of temperature and depth?
- `te(temp_bottom, log10(depth))`
- 💭 "marginal 1Ds, join them up"
]
.pull-right[
![](02-extending-gams_files/figure-html/tensor-1.svg)<!-- -->
]
---
# Using `te()`
Just like `s()`:
```r
shrimp_te <- gam(shrimp ~ te(log10(depth), temp_bottom),
data=trawls_2010,
family=tw,
method="REML")
```
---
# `summary`
```
##
## Family: Tweedie(p=1.595)
## Link function: log
##
## Formula:
## shrimp ~ te(log10(depth), temp_bottom)
##
## Parametric coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.8484 0.2531 7.304 1.22e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## te(log10(depth),temp_bottom) 15.19 16.99 33.09 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.337 Deviance explained = 66.1%
## -REML = 1946.5 Scale est. = 8.1292 n = 482
```
---
# Things to fiddle with
- Setting `k` in two ways:
- `k=5`: 5 for all covariates (total `\(5*5=25\)`)
- `k=c(3,5)`: per basis, in order (total `\(3*5=15\)`)
- Setting `bs` in two ways:
- `bs="tp"`: tprs for all bases
- `bs=c("tp", "tp")`: tprs per basis
---
# Pulling `te()` apart: `ti()`
- Can we look at the components of the `te()`
- `te(x, y) = ti(x, y) + ti(x) + ti(y)`
```r
shrimp_ti <- gam(shrimp ~ ti(temp_bottom, depth) +
s(temp_bottom) + s(depth),
data=trawls_2010,
family=tw,
method="REML")
```
---
# `summary`
```
##
## Family: Tweedie(p=1.591)
## Link function: log
##
## Formula:
## shrimp ~ ti(temp_bottom, depth) + s(temp_bottom) + s(depth)
##
## Parametric coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.672 1.067 1.568 0.118
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## ti(temp_bottom,depth) 7.417 8.860 4.177 0.000459 ***
## s(temp_bottom) 5.661 6.850 1.703 0.081067 .
## s(depth) 6.592 7.238 10.177 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.349 Deviance explained = 66.7%
## -REML = 1951.7 Scale est. = 8.1639 n = 482
```
---
class: inverse middle center big-subsection
## Now give it a try!
---
class: inverse middle center big-subsection
## Building spatio-temporal models
---
# 🦐🗺🗓
![](02-extending-gams_files/figure-html/biom-space-time-plot-1.svg)<!-- -->
---
# Space x time
- We had a 2d spatial model, add time?
- `te(x,y,year)` ?
- Want that 2d smooth rather than `te(lat, long)`?
- `d=` groups covariates
- `te(x, y, year, d=c(2,1))` gives `x,y` smooth and `year` smooth tensor
- Assuming default `k=` and `bs=` for bases
---
# Fiddling
- Often fewer temporal replicates
- Fewer years than unique locations
- `k=` smaller for temporal covariate?
- Use cubic spline basis for time?
- simpler basis, even knot placement
- When using `ti()` everything needs to match up!
---
# 🦐🗺🗓
Putting that together:
```r
shrimp_xyt <- gam(shrimp ~ ti(y,x, year, d=c(2,1),
bs=c("tp", "cr"), k=c(20, 5)) +
s(x, y, bs="tp", k=20) +
s(year, bs="cr", k=5),
data=trawls,
family=tw,
method="REML")
```
---
# `summary`
```
##
## Family: Tweedie(p=1.686)
## Link function: log
##
## Formula:
## shrimp ~ ti(y, x, year, d = c(2, 1), bs = c("tp", "cr"), k = c(20,
## 5)) + s(x, y, bs = "tp", k = 20) + s(year, bs = "cr", k = 5)
##
## Parametric coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.89917 0.03127 124.7 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## ti(y,x,year) 47.65 58.59 8.144 <2e-16 ***
## s(x,y) 18.88 19.00 180.295 <2e-16 ***
## s(year) 3.87 3.98 108.490 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.136 Deviance explained = 45.2%
## -REML = 19084 Scale est. = 9.3801 n = 4152
```
---
# `ti(x, y)` and `ti(year)`
![](02-extending-gams_files/figure-html/ti-xyt-plot-xy-t-1.svg)<!-- -->
---
# `ti(x, y, year)`
![](02-extending-gams_files/figure-html/ti-xyt-plot-xyt-1.svg)<!-- -->
---
class: inverse middle center big-subsection
## Recap
---
# What did we learn?
- set response distribution
- `family=` argument
- see `?family`
- spatial smoothing
- `s(x,y)` for projected coordinates
- `te(lat, long)` for latitude and longitude (better to project?)
- interactions
- `te(covar1, covar2, ...)`
- `ti()` to decompose the effects
- space and time
- `te(x, y, time, d=c(2,1))`
- again `ti()` to decompose
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