Introduction to (Bayesian) Isotope Mixing Models

Eric Ward

Northwest Fisheries Science Center Seattle WA, USA

eric.ward@noaa.gov

Isotopes & mixing models

• Overview of Bayesian jargon • Introduction to stable isotope mixing models

– Assumptions, data, etc

• Examples: • Using an informative prior • Accounting for hierarchical variation

– Grey wolves in B.C. • Dealing with prior information / small samples

– Trout in Pacific NW freshwater lakes • Including covariates • Including relative abundance data

Bayesian / frequentist differences

• Frequentist / likelihood methods assume parameters are fixed, and the data are random

• Bayesians assume that the data are fixed, and the parameters are random

• (all the parameters are probability distributions)

Bayesian Jargon

• Prior: π(θ) – this is a probability distribution representing prior knowledge

• Likelihood: L(x|θ) – this is the likelihood of your data given the parameter

• Posterior = Pr(θ|x) ≈ π(θ)×L(x|θ)

Prior

Posterior

Likelihood

= how much we ‘learn’ by observing the data

Choosing priors for mixing models

• Uninformative – Univariate

• Uniform(0,1) or Beta(1,1) = “flat priors”

– Multivariate • Dirichlet (1,1,1,…) = flat priors on each proportion

• Informative – Univariate

• Beta (3, 2), Beta (1, 8), etc.

– Multivariate • Dirichlet (3,2,3), Dirichlet (2,1,1)

Stable Isotopes (in brief) • Isotope: atomic elements come in different flavors (# neutrons)

• Stable isotopes are naturally occurring isotopes of elements. • Examples of stable isotopes:

• Nitrogen (15N/14N) • Carbon (13C/12C) • Sulfur (34S/32S) • Strontium, Oxygen, etc.

• Can quantify ratio of isotopes using mass spectrometry

• Isotope ratios (15N/14N) are conserved in predictable ways through

trophic transition!

• Mixing models are a useful tool for inferring the relative contributions of different source items to a mixture

Mixing Models • If we know:

– The isotope signatures of the things you eat

– Your isotopic signature

• We can estimate the proportions of different food items in your diet

Red paint white paint

pink paint (the mixture!)

“When you look at the isotopic ratios, we are corn chips with legs” -T. Dawson, UC Berkeley

C3 v C4 plants in human diet

d13C

C4 (Corn) C3 (Grasses, veg. crops)

USA

What data do we need to use mixing models?

• Isotopic signatures of consumers • Isotopic signatures of sources

– Either individual points, or means and variances

• Values for TEFs / fractionation (means & vars) – Ideally from experiments – But almost always from the literature

• Post, D.M. 2002. Using stable isotopes to estimate trophic position: models, methods, and assumptions. Ecology, 83:703-718.

• This is one of the BIG weaknesses of mixing models

• Optional external information: gut contents, observations, etc

C

Isot

ope

1 (e

.g. N

)

Isotope 2 (e.g. C)

Predator data and the isotope signatures of 4 prey items: the raw data

Consumer signatures trophic level higher

Step 1: Correct source means and variances for fractionation / TEFs

Moore & Semmens (2008)

These represent the sample variances

These represent the sample means

Illustration of correction • Sources without & with TEFs: shifted and

more variable

Step 2: Identify potential problems

Isotope 1

(1-p) = 66% p = 33%

• (In this 1st example, there’s nothing wrong)

• In 1-dimension, this is just a weighted average (X1 and X2 are known):

u = p*X1 + (1-p)*X2

Source 1 Source 2

Consumer

6 9 8

• Q: What is the value of mixing proportion p?

1-dimensional mixing model

• What if our consumer had a different value?

???? Source 2 Source 1

6 9 2

Isotope 1

• A: No positive value exists!! – In general, this means our data are inconsistent with

the assumptions of the model (missing source?)

1-dimensional mixing model

Isotope 1

p2 p1

• Similarly, what if we had data for a new source?

• Q: How would this affect our estimate of p1?

Source 1 Source 2

Consumer

6 9 8

Source 3

3

p3

Estimates of p[1] and p[3] are inversely correlated

Another diagnostic: posterior distributions of estimated source contributions have multiple modes (translation: predator either eats a lot, or none of item X) - The posteriors of p[1] and p[3] will both look like this!

1-dimensional mixing model

Isotope 1

p2 p1

• Similarly, what if we had data for a new source?

• Q: How would this affect our estimate of p1?

• A: Sources 1/3 totally confounded – Solution: (1) combine sources, or (2) include data from

another isotope / dimension, (3) informative priors – e.g. stomach contents, (4) bounds on parameters

Source 1 Source 2

Consumer

6 9 8

Source 3

3

p3

Geometry in 2D can also be a problem

• Phillips and Gregg (2003)

2D examples of bad geometry

• Always plot your data!

• Are consumers within source polygon? This tends to be a dealbreaker!

• Are any estimates potentially confounded? Uncertainty generally increases with # sources (Phillips & Gregg 2003)

Step 3: Running a mixing model

• (assuming you get past step 2) – Bad geometry = bad results!

• What are the basic assumptions? (1) all consumers have the same diet (2) all sources are equally available (in terms of biomass) (3) no prey missing (4) source parameters known exactly (large sample sizes, > 10)

Isot

ope

1

Isotope 2

Predator data and the isotope signatures of 4 prey items (corrected for fractionation)

f = [20%, 50%, 20% 10%]

Isot

ope

1

Isotope 2

f = [20%, 50%, 20% 10%]

There are many probable f vectors!!! MCMC will give us 1000s!

MixSIR: A Bayesian Framework For Isotopic Mixing Models

• Deal with any number of sources

• Incorporate prior information (e.g. gut contents)

http://www.ecologybox.org

Source: MixSIR manual, ecologybox.org

All the data together

What does MixSIR do?

• Implements the SIR algorithm – Sampling Importance Resampling

• Generates 1000s of independent samples from the posterior distribution of estimated source contributions (you specify # of samples)

• All vectors have to sum to 1!

In practice (Moore & Semmens 2008)

• Samples from MixSIR

• Uninformative (flat) priors

• Fishes look slightly problematic. Multiple modes??

• Solution: informative prior from gut contents

Other examples

• Individual and group variation

• Uncertainty in sources

Coping with hierarchical variation in stable isotope mixing models

Semmens, B.X.*, E.J. Ward*, J.W. Moore, and CT. Darimont. 2009. Quantifying inter- and intra-population niche variability using hierarchical Bayesian stable isotope mixing models. PLoS One. http://dx.plos.org/10.1371/journal.pone.0006187. * Equal authorship

Darimont Darimont Darimont Darimont Darimont

Step 3: Running a mixing model

• (assuming you get past step 2) – Bad geometry = bad results!

• What are the basic assumptions? (1) all consumers have the same diet (2) all sources are equally available (in terms of biomass) (3) no prey missing (4) source parameters known exactly (large sample sizes, > 10)

No uncertainty in prey proportions…

• This assumes that all predators eat the same thing.

• Different animals may have different diets!

f = [20%, 50%, 20% 10%]

Isot

ope

1

Isotope 2

Instead of this…

f = [20%, 50%, 20% 10%]

Isot

ope

1

Isotope 2

Consumers look like this…

f = [20%, 50%, 20% 10%]

Application: Coastal Gray Wolves

Outer Islands

Inner Islands

Mainland

Pack 1

Pack n

…

Individual 1

Pack n

… N = 9 N = 64

REGION PACK INDIVIDUAL

Darimont

Darimont

more terrestrial

more marine

d13C (0/00)

d15N

(0 /00

)

low high

low

hi

gh

Measure of resource use - niche

Mea

sure

of r

esou

rce

use

- nic

he

Diet of coastal wolves….

Darimont

Darimont

Darimont

Darimont

DIC Results Region Pack Individual DIC

N N N 1343

Y N N 693

Y Y N 502

N N Y 335

Y N Y 333

Y Y Y 325

How to quantify niche?

• How variable is your isotopic signature?

• How variable are the resources you consume? – Dietary niche, we can output of the hierarchical

mixing model to get this

d13C (0/00)

d15N

(0 /00

)

low high

low

hi

gh

Measure of resource use - niche

Mea

sure

of r

esou

rce

use

- nic

he

Stable Isotope Analysis (e.g. Bayesian ellipses) •data derived from tissue

Measuring Niche Variation

standard deviation parameters controlling the variation in diet across three scales

Other extensions to MixSIR

• Improving uncertainty in sources – Source signatures may be based on a small

number of samples

– We may have data from other systems to inform these

– In addition to treating the proportional contributions as parameters, the source means and variances also become parameters

Dealing with source uncertainty

Ward, E.J., Semmens, B.X., and D.E. Schindler. 2010. Including source uncertainty and prior information in the analysis of stable isotope mixing models. Environmental Science & Technology, 44(12): 4645-4650

Figure 5: Map of Urban Lakes Gradient locations.

I Isot

ope

1

Isotope 2

Graphically:

f = [20%, 50%, 20% 10%]

Design Subject of inference: 7 pred in 6 pristine lakes

One lake has no pred:

2 additional lakes: stocked pred

Design Subject of inference: 7 fish in 6 pristine lakes

One lake has no fish:

2 additional lakes: Stocked fish

PRIOR

Estimated proportions

Other extensions to MixSIR

• Including residual error / uncertainty

– Extra variation not accounted for by your model, or the variation among sources

SIAR package in R

• Good if you don’t have identifying information about individuals (groups, etc)

• May help estimation if you’re missing a minor source

• Being fused with MixSIR = MixSIAR

Also in SIAR()

• Concentration dependence:

• MixSIR assumes contribution of C/N is the same for all sources

• Some prey may be C/N enriched or depleted

Other extensions to MixSIR

• Including covariates

– Consumers might not be assigned to groups, but experience a gradient of some covariate

This example is 1D, but could easily be extended to 2D

Other extensions to MixSIR

• Including relative prey availability

– All sources might not be equally available

• Prey might be isotopically identical, but have very different abundances

• Density varies by an order of magnitude

• Yeakel et al. re-weight posterior after running MixSIR

Other extensions to MixSIR

• Ontogenetic diet shifts

– Consumers shift diet as they get older / bigger

– Collaboration w/C. Harvey (in prep)

Other extensions to MixSIR

• Incorporating movement and the Isoscape

– Isotopic signatures of consumers reflect environment they inhabit

– We can infer where animals have been based on signatures

– Collaboration w/ J. Moore & C. Phillis

Summary

• Mixing models are a rapidly growing field

• Analyses are exclusively Bayesian – Requires some basic understanding of underlying

methods / techniques

Where Can I Get These Glorious Tools?

• http://www.ecologybox.org

• Tools include: – R code to run

– Worked examples

– Step-by-step code explanations

isoecol.blogspot.com

Acknowledgements

NOAA Seattle • Brice Semmens • Eli Holmes • Mark Scheuerell • Eric Buhle • Tessa Francis UC Santa Cruz • Jon Moore • Chris Darimont UW • Daniel Schindler • Gordon Holtgrieve • Tessa Francis

• Additional collaborators • Andrew Jackson • Stu Bearhop • Andrew Parnell • Rich Inger • Don Phillips