1. Setup
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.
2. Mean-Variance trends and Gamma Regression fitting
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:
plot of chunk plot_gam
We can then estimate the uncertainty for the partitioned data as
follows:
# Fit the gamma regression
gr <- fit_gamma_regression(ups_norm)
# Get each data points uncertainty
unc_gr <- estimate_uncertainty(gr, ups_norm, id_col, ups_design)
3. Sampling from the Data and Decision model
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.
4. Inference with Latent Gamma Mixuter Regression
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:
ups_lgmr <- fit_lgmr(ups_norm, id_col, chains = 10, cores = 10, warmup = 1000, iter = 3000)
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:
plot_lgmr_regression(ups_lgmr)
plot_regression_field(ups_lgmr, rng = 25)
5. Visualization of the results
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:
gr_results %>%
plot_volcano(
alpha = .05 # Level of significance
)
lgmr_results %>%
plot_volcano(
alpha = .05 # Level of significance
)
