Here we cover the tutorial of `UPS-DS`

. The
`UPS-DS`

has one more condition than the
`yeast-DS`

and has one more replicate in each condition
(`yeast-DS`

is shown in
`vignette("baldur_yeast_tutorial")`

).

```
library(baldur)
# Packages for pre-processing
library(dplyr, warn.conflicts = FALSE)
library(tidyr, warn.conflicts = FALSE)
# Setup the design matrix
ups_design <- model.matrix(~ 0 + factor(rep(1:3, each = 4)))
colnames(ups_design) <- paste0('fmol', c(25, 50, 100))
# Set id column
id_col <- colnames(ups)[1] # "identifier"
# Normalize and add M-V trend
ups_norm <- ups %>%
# Remove rows with NA for the sake of the tutorial
drop_na() %>%
# Normalize the data
psrn(id_col) %>%
calculate_mean_sd_trends(ups_design)
# For the contrast, we want to compare all conditions against each other, and in addtion, suppose we would like to know the difference between the mean of condition 1 and 3 against 2 ([fmol25 + fmol100]/2 - fmol50).
# This can easily be achieved with the following:
ups_contrast <- matrix(
c(
1, 1, 0, 1, -0.5, -0.5,
-1, 0, 1, -0.5, 1, -0.5,
0, -1, -1, -0.5, -0.5, 1
), nrow = ncol(ups_design),
byrow = TRUE
)
```

Lets go over the design and contrast matrix. First, lets not how the design matrix uses subsets of the real columns of each condition:

```
colnames(ups_norm[-1])
#> [1] "fmol25_1" "fmol25_2" "fmol25_3" "fmol25_4" "fmol50_1" "fmol50_2" "fmol50_3" "fmol50_4" "fmol100_1" "fmol100_2"
#> [11] "fmol100_3" "fmol100_4" "mean" "sd"
colnames(ups_design)
#> [1] "fmol25" "fmol50" "fmol100"
```

This lets `baldur`

easily identify what columns that are
of interest and setting up pre-cursors for the sampling. In addition,
`baldur`

gets information on the number of conditions there
are in the data (i.e., `ncol(ups_design)`

) and the number of
replicates in each condition (i.e., `colSums(ups_design)`

).
Next, the contrast matrix:

```
ups_contrast
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 1 0 1.0 -0.5 -0.5
#> [2,] -1 0 1 -0.5 1.0 -0.5
#> [3,] 0 -1 -1 -0.5 -0.5 1.0
```

First, note how the contrast only has two columns; currently
`baldur`

only allows pair-wise comparisons. Lets go over the
rows, the first row tells us that the first column of the design matrix
should be compared against the second. I.e., the first row means
`fmol25 - fmol50`

. The second row shows that the first column
should be compared against the third, i.e.,
`fmol25 - fmol100`

. The third row shows that the second
column should be compared against the third
(`fmol50 - fmol100`

). And, of course, the last row shows the
mean of condition one and three vs condition two.

First, I would like to note that reader that the remainder of the
tutorial will look very similar to
`vignette("baldur_yeast_tutorial")`

. The next step in
`baldur`

is to partition the trends in the mean and variance,
and then to estimate the uncertainty of each measurement.

The trends before and after partitioning can then be visualized with
`plot_gamma`

:

We can then estimate the uncertainty for the partitioned data as follows:

Finally we sample the posterior of each row in the data.
`baludr`

is very easy to run in parallel and this will
drastically reduce the running time. The only thing that needs to be
changed is the `clusters`

flag:

```
# Single trend
gr_results <- gr %>%
# Add hyper-priors for sigma
estimate_gamma_hyperparameters(ups_norm, id_col) %>%
infer_data_and_decision_model(
id_col,
ups_design,
ups_contrast,
unc_gr,
clusters = 10 # Change this to 1 to run sequentially
)
```

Here `err`

is the probability of error, i.e., the two
tail-density supporting the null-hypothesis, `lfc`

is the
estimated log\(_2\)-fold change,
`sigma`

is the common variance, and `lp`

is the
log-posterior. Columns without suffix shows the mean estimate from the
posterior, while the suffixes `_025`

, `_50`

, and
`_975`

, are the 2.5, 50.0, and 97.5, percentiles,
respectively. The suffixes `_eff`

and `_rhat`

are
the diagnostic variables returned by `rstan`

(please see the
Stan manual for details). In general, a larger `_eff`

indicates a better sampling efficiency, and `_rhat`

compares
the mixing within chains against between the chains and should be
smaller than 1.05. An important difference from `yeast-DS`

is
that each peptide gets three rows, one for each comparison in the
contrast matrix.

To run Baldur with the LGMR model is very similar to running it with
the GR model. First we fit the regression model using the
`fit_lgmr`

function. Here I will try to make use of my
parallel processors to speed-up the inference:

On rare occasions, the UPS-DS will have a few (<10) divergent transitions or exceed maximum tree depth, it is unlikely to have any impact on the final inference (considering that the posterior draws are 20 000). One could increase the acceptance rate and max tree depth at the expense of computational time. E.g., with:

```
ups_lgmr <- fit_lgmr(
ups_norm, id_col, chains = 10, cores = 10, warmup = 1000, iter = 3000,
control = list(adapt_delta = .95, max_treedepth = 11)
)
```

We can then estimate the uncertainties and hyperparameters for the data and decision model and run them as for the GR model:

```
# Estimate uncertainty
unc_lgmr <- estimate_uncertainty(ups_lgmr, ups_norm, id_col, ups_design)
# Sample from the data and decision model
lgmr_results <- ups_lgmr %>%
# Add hyper-priors for sigma
estimate_gamma_hyperparameters(ups_norm, id_col) %>%
infer_data_and_decision_model(
id_col,
ups_design,
ups_contrast,
unc_lgmr,
clusters = 10 # Change this to 1 to run sequentially
)
```

Plotting LGMR:

`baldur`

have two ways of visualizing the results 1)
plotting sigma vs LFC and 2) Volcano plots. To plot sigma against LFC we
use `plot_sa`

:

```
gr_results %>%
plot_sa(
alpha = .05, # Level of significance
lfc = 1 # Add LFC lines
)
lgmr_results %>%
plot_sa(
alpha = .05, # Level of significance
lfc = 1 # Add LFC lines
)
```

In general, a good decision is indicated by a lack of a trend between \(\sigma\) and LFC. We can see that Baludr with the LGMR model (second plot) has a lower trend compared to GR model (first plot) for which there is a trend for \(\sigma\) to increase with LFC.

To make a volcano plot one uses `plot_volcano`

in a
similar fashion to `plot_sa`

: