Regression for Microbiome Data:Multinomial Mixture Models

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# Regression for Microbiome Data:</br>Multinomial Mixture Models
## 📚EPID 674📚
### Brendan J. Kelly, MD, MS
### Updated: 23 June 2020

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### Dirichlet Multinomial Mixtures

### Implementating DMM in R

### ICU Community Types

### DMM & Regression

]

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# Dirichlet Multinomial Mixtures

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# High Dimensional Microbiome Data

.center[

```
##              700013549 700014386 700014403 700014409 700014412 700014415
## OTU_97.1             0         0         0         0         0         0
## OTU_97.10            0         0         6         4         1         5
## OTU_97.100           0         0       133         7         1         4
## OTU_97.1000          0         0         0         0         0         0
## OTU_97.10000         0         0         0         0         0         0
## OTU_97.10001         0         0         0         0         0         1
## OTU_97.10002         0         0         0         0         0         0
## OTU_97.10003         0         0         0         0         0         0
## OTU_97.10004         0         0         0         0         0         0
## OTU_97.10005         0         0         0         0         0         0
## OTU_97.10006         0         0         0         0         0         0
## OTU_97.10007         0         0         0         0         0         0
## OTU_97.10008         0         1         0         0         0         0
## OTU_97.10009         0         0         1         0         0         0
## OTU_97.1001          0         0         0         0         0         0
## OTU_97.10010         0         0         0         0         0         0
```

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# High Dimensional Microbiome Data

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- How to deal with high-dimensional microbiome data?

- Descriptive (e.g., heatmaps and stacked barplots)
    
- Test a priori hypotheses regarding specific OTUs/taxa

- __Reduce dimensions__:

- single summary statistic (alpha diversity)
    
    - pairwise distances (beta diversity) with PCoA or PERMANOVA
    
    - __community types (mixture modeling)__

]

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# Dirichlet-Multinomial Mixtures

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- Dirichlet-multinomial distribution:

- compound probability distribution  
    - probability vector drawn from Dirichlet distribution (generalized beta)  
    - observation drawn from multinomial distribution (generalized binomial)  
    
- D-M mixture modelling:

- each sample ~ multinomial from one Dirichlet vector
    - vector number: minimize -log(model evidence, Laplace approx)
    - Dirichlet probability vectors = “community types”

]

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.footnote[Ding & Schloss _Nature_ 2014]

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.footnote[Ding & Schloss _Nature_ 2014]

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# Implementating DMM in R

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# Preparation for DMM

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```r
# install tidyverse ...
# install.packages("tidyverse") 
*library(tidyverse)

# new package for heatmap color schemes...
# install.packages("viridis")
*library(viridis)

# install package from Bioconductor...
# install.packages("BiocManager")
# BiocManager::install("DirichletMultinomial")
*library(DirichletMultinomial)

*set.seed(16) # for consistent DMM results

icu_matrix_et <- read_rds(
  "./data/icu_ET_specimen_otu_table.rds"
  )

icu_matrix_et[1:16,1:2]
```

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.pull-right[

```
##             VAP.001.ET.20130726 VAP.001.ET.20130729
## denovo1                       0                   0
## denovo10004                   0                   0
## denovo10011                   0                   0
## denovo10015                   0                   0
## denovo10018                   0                   0
## denovo10022                   0                   0
## denovo10039                   0                   0
## denovo1004                    0                   0
## denovo10042                   0                   0
## denovo10049                   0                   0
## denovo10065                   0                   0
## denovo10078                   3                   2
## denovo10080                   0                   0
## denovo10089                   0                   3
## denovo10091                   0                   0
## denovo10092                   2                   0
```

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---

# `DirichletMultinomial`

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```r
#filter to speed DMM model fitting
icu_matrix_et[rowSums(icu_matrix_et) > 500,] ->
  small_icu_matrix_et

*dmm <- lapply(1:10,
*             dmn,
*             count = t(small_icu_matrix_et),
*             verbose = FALSE)

*lplc <- sapply(dmm, laplace)

qplot(x = seq_along(lplc),
      y = lplc,
      geom = c("point","line")) +
  theme_bw() +
  labs(x = "Dirichlet Components",
       y = "LPLC")
```

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![](data:image/png;base64,#lecture_more-microbiome-regression_files/figure-html/dmm1-out-1.png)

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# ICU Community Types

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# DMM Assignments

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```r
*best_dmm <- dmm[[which.min(lplc)]]

*mixture(best_dmm) %>%
  as_tibble(rownames = "specimen") %>%
  rename_at(.vars = vars(contains("V")),
            .funs = function(x)
              paste0(
                gsub("V","m",x),"_prob")
            ) %>%
* mutate(assignment =
*          mixture(best_dmm,
*                  assign = TRUE)) ->
  icu_et_dmm_assignments

icu_et_dmm_assignments
```

]

.pull-right[

```
## # A tibble: 42 x 6
##    specimen             m1_prob  m2_prob   m3_prob m4_prob assignment
##    <chr>                  <dbl>    <dbl>     <dbl>   <dbl>      <int>
##  1 VAP.001.ET.20130726 3.44e-13 1.00e+ 0 0.              0          2
##  2 VAP.001.ET.20130729 2.68e-24 1.00e+ 0 0.              0          2
##  3 VAP.001.ET.20130731 1.37e-15 1.00e+ 0 0.              0          2
##  4 VAP.002.ET.20130729 1.00e+ 0 2.30e-31 0.              0          1
##  5 VAP.002.ET.20130731 1.00e+ 0 4.43e-36 0.              0          1
##  6 VAP.002.ET.20130802 1.00e+ 0 3.42e-31 0.              0          1
##  7 VAP.002.ET.20130805 3.09e-19 1.00e+ 0 0.              0          2
##  8 VAP.003.ET.20130730 1.52e-22 1.00e+ 0 5.23e-310       0          2
##  9 VAP.004.ET.20130808 1.00e+ 0 1.65e-69 0.              0          1
## 10 VAP.005.ET.20130814 9.92e-28 4.01e-12 1.00e+  0       0          3
## # … with 32 more rows
```

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# DMM Mixture Fits

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```r
*fitted(best_dmm, scale=TRUE) %>%
# scale indicates whether fits scaled by the...
# ... variability of mixturewt parameter theta
  as_tibble(rownames = "otu_id") %>%
  rename_at(.vars = vars(contains("V")),
            .funs = function(x) 
              paste0(gsub("V","m",x),"_fit")) ->
  icu_et_dmm_otu_fits

icu_et_dmm_otu_fits %>%
  gather(key = which_mix, value = mix_fit, -otu_id) %>%
  ggplot(data = .) +
  geom_tile(mapping = aes(x = which_mix,
                          y = otu_id,
                          fill = mix_fit)) +
  scale_fill_viridis()
```

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![](data:image/png;base64,#lecture_more-microbiome-regression_files/figure-html/dmm3-out-1.png)

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# Difference From Single-Mixture

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```r
*abs(fitted(best_dmm, scale=TRUE) -
*     as.vector(fitted(dmm[[1]],
*                      scale=TRUE))) %>%
# scale indicates whether fits scaled by the...
# ... variability of mixturewt parameter theta
  as_tibble(rownames = "otu_id") %>%
  rename_at(.vars = vars(contains("V")),
            .funs = function(x)
              paste0(gsub("V","m",x),"_diff_single")) ->
  icu_et_dmm_otu_diff_single

icu_et_dmm_otu_diff_single %>%
  gather(key = which_mix,
         value = diff,
         -otu_id) %>%
  ggplot(data = .) +
  geom_tile(mapping = aes(x = which_mix,
                          y = otu_id,
                          fill = diff)) +
  scale_fill_viridis()
```

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![](data:image/png;base64,#lecture_more-microbiome-regression_files/figure-html/dmm4-out-1.png)

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#  DMM & Regression

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# DMM & Regression?

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- DMM community types as exposure variable:

- easy ⇾ `lm()` or `glm()`  
    
    - (like α-diversity or β-diversity PC1)

- DMM community types as outcome variables:

- e.g., categorical logistic regression

- Biological validity of DMM community types? Reproducibility?

]

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# Questions?
### Post to the discussion board!

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# Thank you!
#### Slides available: [github.com/bjklab](https://github.com/bjklab/https://github.com/bjklab/EPID674_009_more-microbiome-regression.git)
#### [brendank@pennmedicine.upenn.edu](brendank@pennmedicine.upenn.edu)