Module 2 — Learning outcomes

By the end of this session you should be able to:

• Distinguish simple linear mixing models from Bayesian routed approaches
• Explain why isotopic routing matters for collagen-based dietary reconstruction
• Understand the ReSources workflow and its advantages over unrouted models

The two-endpoint interpolation approach

The simplest approach: linear interpolation between two known dietary endpoints.

\[ \% \text{ marine} = \frac{\delta^{13}C_{\text{consumer}} - \delta^{13}C_{\text{terrestrial}}}{\delta^{13}C_{\text{marine}} - \delta^{13}C_{\text{terrestrial}}} \times 100 \]

Two-endpoint interpolation — visualised

Two-endpoint model in biplot space

Four failure modes — overview

The Endpoint Uncertainty Problem

When you use a simple linear mixing model to estimate % marine diet from, say, δ13C, the basic approach is: % marine = (sample − terrestrial endpoint) / (marine endpoint − terrestrial endpoint) The critical assumption buried in this is that the endpoints are known constants. In practice, they are not — they are themselves estimates. —

Four failure modes — detail

What the bootstrapping histogram shows

The bootstrapping exercise shown in the figure propagates endpoint uncertainty properly. On each iteration, it:

Draws a terrestrial endpoint value from its distribution (mean ± SD) Draws a marine endpoint value from its distribution Computes % marine for the target individual using those sampled endpoints Repeats hundreds or thousands of times

The resulting distribution of % marine estimates — ranging here from roughly 38-90% — reflects the true uncertainty in the dietary estimate given realistic endpoint variability. The median lands at 66%, but the spread is enormous.

This is particularly acute in radiocarbon-based dietary mixing models (e.g.marine reservoir determinations), where the marine ¹⁴C endpoint varies spatially and temporally, but the same logic applies to stable isotope two-source mixing models using δ¹³C or δ¹⁵N.

What mixing models add

Working in two or more isotope dimensions simultaneously constrains the solution space far more tightly than a single axis.

Three generations of mixing models

Why routing matters

The Maize Example Unpacked Take a population eating:

70% of their calories from maize (a C₄ plant, so isotopically distinct, high δ¹³C) 20% of their protein from animal sources (isotopically different again)

An unrouted model assumes collagen δ¹³C simply reflects the weighted average of all dietary inputs by their proportional contribution — so it would see 70% C₄ signal and infer a heavily maize-based diet. But collagen doesn’t work that way. The amino acids that actually get incorporated into collagen come disproportionately from that 20% animal protein fraction, because that’s where the routing goes. The maize carbon, mostly burned for energy, leaves a much weaker imprint on collagen than its caloric dominance would suggest. The result is that the collagen δ¹³C sits much closer to the animal protein endpoint than to the maize endpoint — and a naive model catastrophically overestimates maize consumption.

MixSIAR — key characteristics

MixSIAR is a fully Bayesian stable isotope mixing model framework implemented in R.

It estimates:

• Posterior probability distributions of dietary source contributions
• Uncertainty propagation from all sources (isotope variability, TEF, consumer variation)

Why MixSIAR over Simpler Approaches?

MixSIAR is particularly valuable where trophic enrichment factor (TEF) uncertainty is substantial — as is typical in archaeological dietary reconstruction.

Data Requirements

Three input files are required (all .csv):

1. Consumer (mixture) data

• One row per individual
• Columns for each isotope ratio (e.g. d13C, d15N)
• Optional: grouping factors, continuous covariates

2. Source data

• Mean and SD of each isotope per source
• Sample size per source
• Can be aggregated from raw source data in R

3. Discrimination (TEF) data

• Mean and SD of trophic enrichment per source per isotope
• Often the greatest source of uncertainty in the model

R Workflow: Loading Data

library(MixSIAR)
library(R2jags)

# Load consumer (mixture) data
mix <- load_mix_data(
  filename     = "consumer.csv",
  iso_names    = c("d13C", "d15N"),
  factors      = NULL,
  fac_random   = NULL,
  fac_nested   = NULL,
  cont_effects = NULL
)

# Aggregate and load source data
source <- load_source_data(
  filename       = "sources.csv",
  source_factors = NULL,
  conc_dep       = FALSE,
  data_type      = "means",
  mix
)

# Load discrimination (TEF) data
d <- load_discr_data("discrimination.csv", mix)

Interpreting Outputs: Posterior Distributions

Dietary source contributions are expressed as full posterior probability distributions, not point estimates.

Key quantities to report:

• 95% credible interval (2.5–97.5%): uncertainty bounds
• Distributions skewed toward zero are expected for minor dietary sources

ReSources — key characteristics

Based on FRUITS Fernandes et al. (2016)

Key improvement on previous models: the routing concept

Key insight: A diet 70% maize by calories but 30% animal protein produces collagen reflecting more protein than maize.

Dataset — late medieval urban population

Dietary source end-members

Protein sources (from faunal assemblage and historical records):

Energy sources:

Step 3 — Interpret results

Example posterior for Burial 12 (high-status):

Visualising posterior distributions

Assumption 1 — All sources known and measured

Assumption 2 — Isotopic distinctness of sources

When sources overlap: (1) add a third tracer (e.g., δ³⁴S), (2) pool similar sources, or (3) apply informative priors from zooarchaeological abundance.

Assumption 3 — Routing parameter (λ)

ReSources requires a prior distribution for λ from controlled feeding experiments:

Problem 4 — Equifinality

Three protein sources all at δ¹³C ≈ −21 ‰ but different δ¹⁵N:

A consumer at δ¹³C = −20.2 ‰, δ¹⁵N = +10.8 ‰ is consistent with:

• Solution A: 60% sheep, 20% cattle, 20% pig
• Solution B: 20% sheep, 60% cattle, 20% pig
• Solution C: 20% sheep, 20% cattle, 60% pig

ReSources handles this by reporting very wide posteriors — which is the correct answer, not a failure.

Problem 5 — Sample size effects

Source reference specimens:

Consumer groups for comparison:

• n ≥ 10 per group — exploratory analyses
• n ≥ 20 per group — publication-quality comparisons

Software comparison

Module 2 — what you should now be able to do

1. Explain why two-endpoint interpolation fails — name ≥ 4 specific problems
2. Distinguish simple linear / Bayesian unrouted / Bayesian routed models and explain why routing matters for collagen
3. Explain why SIAR/early MixSIAR can produce biased estimates for bone collagen
4. Critically evaluate a mixing model’s validity: are all sources known? isotopically distinct? λ appropriate?