mismatchmodelpresentation20240919.qmd

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title: "Multilevel trait-microbiome mismatch model"
author: Quentin D. Read
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## The problem

- We have maternal plants from different populations
- We want to measure the association between seedling traits and seed microbiome in their offspring
  + And whether this association is different by population but that's just an extra wrinkle

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![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Flag_of_Afghanistan_%282013%E2%80%932021%29.svg/800px-Flag_of_Afghanistan_%282013%E2%80%932021%29.svg.png)

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![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Flag_of_Turkey.svg/800px-Flag_of_Turkey.svg.png)

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## The problem

- But it is not possible to observe seed microbiome and seedling traits for the same seed
- The seed can either be germinated to measure traits, or destroyed to measure microbiome, but not both

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![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Eierkarton_10_%28fcm%29.jpg/740px-Eierkarton_10_%28fcm%29.jpg)

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![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Fried_Egg_2.jpg/800px-Fried_Egg_2.jpg)

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## How to resolve the problem

- We can consider this to be a pathological missing data situation
- Any individuals missing $X$ (microbiome) are missing $y$ (trait), and vice versa
- But we can still estimate random intercepts for each mother plant, for each one
- Then model the relationship between $X$ and $y$ at the offspring level
- Bayesian approach imputes the missing data as parameters in the model

## Simulate some data

- The real dataset turns out to be a poor one to actually demonstrate the model!
- Simulate maternal microbiome with a multivariate normal distribution
  + $n_m$: number of mothers (20)
  + $n_o$: number of offspring per mother (10)
  + $n_t$: number of taxa (200)
  + $\mathbf{X}_m \sim \text{MVNormal}(0, \mathbf{I}_{n_t})$
  
## Simulate some data, part 2

- Simulate offspring microbiome, also with a multivariate normal distribution
- Use the maternal means $\mathbf{X}_m$ we simulated in the previous step, and the covariance matrix $\boldsymbol\Sigma_m$ of the simulated maternal means
- For each mother $i$ her offspring microbiome is:
  + $\mathbf{X}_{oi} \sim \text{MVNormal}(\mathbf{X}_{mi}, \boldsymbol\Sigma_m)$
  
## Simulate some data, part 3

- Assign only a few nonzero effects of taxa on trait. The rest are 0.
  + $\boldsymbol\beta = [50, 10, -50, 0, 0, 0, ...]$
- Use these, with some normally distributed random noise added, to get the offspring trait means
- For each mother $i$ her offspring trait values are:
  + $\mathbf{y}_i \sim \text{Normal}(\mathbf{X}_{oi}\boldsymbol\beta, 1)$
  
## Description of the model {.smaller}

- Random maternal intercepts for microbiome, random maternal intercepts for trait
- Covariance between random intercepts for each taxon set to 0
- Trait has fixed effects for all microbiome taxa (no interaction effects or other covariates)
- I will write the formal statistical notation for this soon

## Priors

- Use "regularized horseshoe priors" on the fixed effects, which can handle very wide data
  + Piironen & Vehtari 2017, Electronic Journal of Statistics
  + https://avehtari.github.io/modelselection/regularizedhorseshoe_slides.pdf
- Gamma priors (weakly informative) on variance of random effects
- Missing values follow the same prior distributions as non-missing values

## Fit the model to the data

- Fit the model to the full dataset
- Then set half the trait data to be missing, and the other half of the microbiome data to be missing, and refit
- Compare the fixed effect estimates

## Comparison of fixed effects

- Only taxa 1-3 were simulated to have nonzero fixed effect (most of the 200 taxa aren't shown here)

![](project/examplefigtaxa1to10.png)


## Future directions

- The model could be extended to multivariate trait outcomes and interactions between taxa, *at least in theory*
- A major issue is scaling up the model
  + With 200 taxa, it took several hours to sample the posterior but this will blow up as community size increases
  + Full Bayesian computation on anything much bigger than that is going to be very tough