modelupdate20240725.qmd

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title: "Update on trait-microbiome mismatch modeling"
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## Missing data model

- 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

## Simulation approach

- Simulate maternal microbiome with a multivariate normal distribution
  + $n_m$: number of mothers
  + $n_o$: number of offspring per mother
  + $n_t$: number of taxa
  + $\mathbf{X}_m \sim \text{MVNormal}(0, \mathbf{I}_{n_t})$
  
## Simulation approach 2

- Simulate offspring microbiome again 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:
  + $\mathbf{X}_{oi} \sim \text{MVNormal}(\mathbf{X}_{mi}, \boldsymbol\Sigma_m)$
  
## Simulation approach 2

- Assign only a few nonzero effects of taxa on trait. The rest are 0.
  + $\boldsymbol\beta = [20, 10, -10, 0, 0, 0, ...]$
- Use these, with some normally distributed random noise added, to get the offspring trait means
- For each mother $i$ her offspring trait values are:
  + $\mathbf{y}_i \sim \text{Normal}(\mathbf{X}_{oi}\boldsymbol\beta, 1)$
  
## Fit the model to the full data

- Random maternal intercepts for microbiome, random maternal intercepts for trait
- Trait has fixed effects for all microbiome taxa (no interaction effects or other covariates)
- Use regularized horseshoe priors which can handle very wide data
- I will write the formal statistical notation for this soon
- Used `brms` so I did not have to manually write out the full Stan code!

## Fit the model with missing data

- Then delete half the offspring microbiome data, and delete the trait data for the other half of the offspring
- Now fit the model again, modeling missing values in both microbiome and trait
- See if we get similar answers with and without the missing data

## Current progress

- This worked fine for univariate X
- Then it worked fine for multivariate X, with 5 taxa only
- Currently scaling up to 200 taxa, 10 mothers, 10 offspring per mother
- Made a mistake in the simulation so I am refitting the model
- Only takes 2 or 3 hours to fit at that scale