Introduction to ALDEx2

This provides an overview of the R package ALDEx version 2 (ALDEx2) for differential (relative) abundance analysis of high throughput sequencing count compositional data¹. However, Nixon et al.² have found that the scale of the data can be modelled and ALDEx2 now includes scale modelling by default.

The package was developed and used initially for multiple-organism RNA-Seq data generated by high-throughput sequencing platforms (meta-RNA-Seq)³, but testing showed that it performed very well with traditional RNA-Seq datasets⁴, 16S rRNA gene variable region sequencing⁵ and selective growth-type (SELEX) experiments⁶, and that the underlying principles generalize to single-cell sequencing⁷. The analysis method should be applicable to any type of data that is generated by high-throughput⁸.

Most recently, Nixon et al.⁹ showed that the use of the CLR¹⁰ made strong assumptions about the scale of the underlying system. ALDEx2 now includes methods to include scale uncertainty and so incorporate both compositional and scale uncertainty together in the analysis. See the scaleSim vignette for more information and a complete walk-through.

Installation

There are two ways to download and install the most current of ALDEx2. The most recent version of the package will be found at github.com/ggloor/ALDEx_bioc. The package will run with only the base R packages, the standard tidyverse packages and a minimal Bioconductor install, and is capable of running several functions with the parallel package if installed. It is recommended that the package be run on the most up-to-date R and Bioconductor versions.

• Install stable version from Bioconductor

if (!requireNamespace("BiocManager", quietly = TRUE))
    install.packages("BiocManager")

BiocManager::install("ALDEx2")

Here all dependencies should be installed automatically.

• Install development version from github:

install.packages("devtools")
devtools::install_github("ggloor/ALDEx_bioc")

In this case, you may need to install additional packages manually.

## |------------(25%)----------(50%)----------(75%)----------|

What ALDEx2 does differently

ALDEx2 is a different tool than others used for differential abundance testing, but that does not mean it is complicated. Instead, it is incredibly simple once you understand four concepts.

• First, the data you collect is a single point-estimate of the data. That means, if you sequenced the same samples, or even the same library, you would get a slightly different answer. We know this because there are a number of early technical replication papers that show exactly this¹¹.

• Second, that the data collected are probabilistic; that the count data is a representation of the probability of sampling the gene fragment from the mixture of gene fragments in the library. Many other tools assume the data are counts from a multivariate distribution modelled by a Negative Binomial distribution.

• Third, that we can pretty accurately estimate this probabilisitic technical variation. For this, we assume the data collected are from a multivariate Poisson distribution constrained to have the total number of counts that the instrument can deliver¹². This can be estimated by sampling from the appropriate distribution, in this case a multinomial Dirichlet distribution.

• Fourth, that the use of the CLR (or any other normalization) makes strong assumptions about the scale of the sampled environment. Nixon et al.¹³ demonstrated how to incorporate scale uncertainty so that the effect of scale on the analysis can be modelled and accounted for.

More formally, the ALDEx2 package estimates per-feature technical variation within each sample using Monte-Carlo instances drawn from the Dirichlet distribution. Sampling from this distribution returns a posterior probability distribution of the observed data under a repeated sampling model. This distribution is converted to a log-ratio that linearizes the differences between features. This log-ratio can be modified to include uncertainty regarding the scale of the data when the log-ratio is calculated. Thus, the values returned by ALDEx2 are posterior estimates of test statistics calculated on log-ratio transformed distributions. ALDEx2 has been altered to ensure that the elements of the posterior statistical tests agree on their sign with the outcome that some edge cases with low predicted p-values are no longer counted as significant at high false discovery rates. Combining two-sample tests into a posterior estimate is somewhat intricate¹⁴, but in practice the outcomes should be very similar to those seen before, but edge cases with expected false discovery rates greater than 0.05 have become more rigorous.

ALDEx2 uses by default the centred log-ratio (clr) transformation which is scale invariant¹⁵. While this is a convenient assumption for many cases, this can lead to Type 1 and Type 2 errors¹⁶ and we now suggest that modest amounts of scale be built into the analysis by default.

The sub-compositional coherence property ensures that the answers obtained are consistent when parts of the dataset are removed (e.g., removal of rRNA reads from RNA-seq studies or rare OTU species from 16S rRNA gene amplicon studies). In the absence of scale, all feature abundance values are expressed relative to the geometric mean abundance of other features in a sample. This is conceptually similar to a quantitative PCR where abundances are expressed relative to a standard: in the case of the clr transformation, the standard is the per-sample geometric mean abundance. See Aitchison (1986) for a complete description. If the question at hand is relative change, then ALDEx2 can do this. If the question at hand is scale dependent, then ALDEx2 can do that too!

Quick Start: aldex with 2 groups:

The ALDEx2 package in Bioconductor is suitable for the comparison of many different experimental designs.

The test dataset is a single gene library with 1600 sequence variants¹⁷ cloned into an expression vector at approximately equimolar amounts. The wild-type version of the gene conferred resistance to a topoisomerase toxin. Seven independent growths of the gene library were conducted under selective and non-selective conditions and the resulting abundances of each variant was read out by sequencing a pooled, barcoded library on an Illumina MiSeq. The data table is accessed by data(selex). In this data table, there are 1600 features and 14 samples. The analysis takes approximately 2 minutes and memory usage tops out at less than 1Gb of RAM on a mobile i7 class processor when we use 128 Dirichlet Monte-Carlo Instances (DMC). On an M2 Pro chip, this takes approximately 6 seconds. For speed concerns we use only the last 400 features and perform only 16 DMC. The command used for ALDEx2 is presented below:

library(ALDEx2)
#subset only the last 400 features for efficiency
data(selex)
selex.sub <- selex[1200:1600,]
conds <- c(rep("NS", 7), rep("S", 7))
x.aldex <- aldex(selex.sub, conds, mc.samples=16, test="t", effect=TRUE, 
  include.sample.summary=FALSE, denom="all", verbose=FALSE, paired.test=FALSE, gamma=NULL)

# note default is FDR of 0.05
par(mfrow=c(1,3))
aldex.plot(x.aldex, type="MA", test="welch", xlab="Log-ratio abundance",
    ylab="Difference", main='Bland-Altman plot')
aldex.plot(x.aldex, type="MW", test="welch", xlab="Dispersion",
    ylab="Difference", main='Effect plot')
aldex.plot(x.aldex, type="volcano", test="welch", xlab="Difference",
    ylab="-1(log10(q))", main='Volcano plot') 

MA, Effect and Volcano plots of ALDEx2 output. The left panel is an Bland-Altman or MA plot that shows the relationship between (relative) Abundance and Difference. The middle panel is an effect plot that shows the relationship between Difference and Dispersion; the lines are equal difference and dispersion. The right hand plot is a volcano plot; the lines represent a posterior predictive p-value of 0.001 and 1.5 fold difference. In all plots features that are not significant are in grey or black. Features that are statistically significant are in red. The log-ratio abundance axis is the clr value for the feature.

Figure @ref(fig:three-plots) shows the result of a two sample t-test and effect size calculation using aldex(). We set the test to ‘t’, and effect to TRUE. The ‘t’ option evaluates the data as a two-factor experiment using both the Welch’s t and the Wilcoxon rank tests. The t-test includes an expected value of the false discovery rate correction. The data can be plotted onto Bland-Altmann¹⁸ (MA) or effect (MW)¹⁹ or volcano²⁰ plots for for two-way tests using the aldex.plot() function. See the end of the modular section for examples of the plots.

The simple approach outlined above just calls aldex.clr, aldex.ttest, aldex.effect in turn and then merges the data into one dataframe called x.all. We will show these modules in turn, and then examine additional modules.

There is a lot more information that can be obtained than with these three plots. One of the most powerful aspect of ALDEx2 is that everything is calculated on posterior distributions. The underlying posterior distribution can be viewed if you use the individual modules explained next.

Modular ALDEx2 is way more informative

For many users, the aldex function may be sufficient. However, for power users there is a lot of value to the modular approach. There are several built-in tools that allow a much more powerful exploration of the data. In addition, all the intermediate values are exposed for the underlying entre log-ratio transformed Dirichlet Monte-Carlo replicate values. This makes it possible for anyone to add the specific R code for their experimental design — a guide to these values is outlined below.

We will use the same example and generate the same plots but include more information that can be useful. First we load the library and the included selex dataset. Then we set the comparison groups. This must be a vector of conditions in the same order as the samples in the input counts table. The aldex command is calling several other functions in the background, and each of them returns diagnostics. These diagnostics are suppressed for this vignette. The code block below shows the full set of command, but are not run. Each is run in an individual code block

data(selex)
#subset only the last 400 features for efficiency
selex.sub <- selex[1200:1600,]
conds <- c(rep("NS", 7), rep("S", 7))
x <- aldex.clr(selex.sub, conds, mc.samples=16, denom="all", verbose=F, gamma=1e-3)
x.tt <- aldex.ttest(x, hist.plot=F, paired.test=FALSE, verbose=FALSE,)
x.effect <- aldex.effect(x, CI=T, verbose=F, include.sample.summary=F, 
  paired.test=FALSE, glm.conds=NULL, useMC=F)
x.all <- data.frame(x.tt,x.effect)
par(mfrow=c(1,2))
aldex.plot(x.all, type="MA", test="welch")
aldex.plot(x.all, type="MW", test="welch")

x <- aldex.clr(selex.sub, conds, mc.samples=16, verbose=F, gamma=1e-3)

x.tt <- aldex.ttest(x, hist.plot=F, paired.test=FALSE, verbose=FALSE)

x.effect <- aldex.effect(x, CI=T, verbose=F, include.sample.summary=F, 
  paired.test=FALSE)

The aldex.plot module

We merge the data into one object for plotting. We see that the plotted data in Figure 1 and 2 are essentially the same.

Note that plotting for the aldex.glm function is different and is shown later.

# merge into one output for convenience
x.all <- data.frame(x.tt,x.effect)

par(mfrow=c(1,3))
aldex.plot(x.all, type="MA", test="welch", main='MA plot')
aldex.plot(x.all, type="MW", test="welch", main='effect plot')
aldex.plot(x.all, type="volcano", test="welch", main='volcano plot')

Output from aldex.plot function. The left panel is the MA plot, the right is the MW (effect) plot. In both plots red represents features called as differentially abundant with q  < 0.05; grey are abundant, but not differentially abundant; black are rare, but not differentially abundant. This function uses the combined output from the aldex.ttest and aldex.effect functions above

The aldex.plot() function gives you a lot of control. Figure @ref(fig:three2-plots) shows the three plot types; effect (type=‘MW’), MA or Bland-Altmann (type=‘MA’) and volcano (type=‘volcano). They are nearly identical to those in @ref(fig:three-plots), differences are due to the random Monte-Carlo sampling. You can choose to use statistical tests or only effect sizes (type=’welch’ or ‘effect’). You can control the title, axis labels and scales as well. See ?aldex.plot for all this can do!

ALDEx2 also allows you to plot and explore the underlying posterior distributions for each part. This is with the aldex.plotFeature() function and the output is shown in Figure @ref(fig:plot-feature). This is a great way to get intuition around the underlying distributions. We see that the posterior distributions are skewed, that there is considerable uncertainty in the difference and effect sizes, and that the tails are very long.

# merge into one output for convenience
# we start with the ouput from aldex.clr
# we provide the name of the row we wish to explore

aldex.plotFeature(x, 'S:E:G:D')

Output from aldex.plotFeature function. This plots the posterior distribution, the difference between and within distributions (dispersion) and the effect size distributions. The values reported are the medians of those distributions, but exploration shows that for all but the most separated parts, there is essentially always some overlap or uncertainty in the posteriors.

The effect confidence interval

As noted above, the ALDEx2 package generates a posterior distribution of the probability of observing the count given the data collected. Here we show the importance of this approach by examining the 95% CI of the effect size. Throughout, we use a standardized effect size, similar to the Cohen’s d metric, although our effect size is more robust and more conservative (being approximately 0.7 Cohen’s d when the data are Normally distributed)²⁴.

Examining the outputs of the effect, MA and the posterior distribution plots shows a very important point: there is a tremendous amount of latent variation in sequencing data . We observe that there are a few features that have an expected q value that is statistically significantly different, which are both relatively rare and which have a relatively small difference. Even when identifying features based on the expected effect size can be misleading. We find that the safest approach is to identify those features that where the 95% CI of the effect size does not cross 0.

Examining the figures we see that the features that are the rarest are least likely to reproduce the effect size with simple random sampling. The behaviour of the 95% CI metric is exactly in line with our intuition: the precision of estimation of rare features is poor—if you want more confidence in rare features you must spend more money to sequence more deeply. Figure @ref(fig:CI) shows that using both an effect size (or if you wish a p-value cutoff) along with the effect CI cutoff can remove some features that are not truly separable between groups.

With this approach we are accepting the biological variation in the data as received; i.e. we are not inferring any additional biological variation: i.e., the experimental design is always as given. We are identifying those features where simple random sampling of the library would be expected to give the same result every time. This is the approach that was used in (Macklaim et al. 2013), the results of which were independently validated and found to be very robust (Nelson et al. 2015).

This approach requires that the CI=T parameter be set when calculating effect sizes.

Comparing the mean effect with the 95% CI. The left panel is MA plot, the right is the MW (effect) plot. In both plots red points indicate features that have an expected effect value of  > 2; those with a blue circle have the 95% CI that does not overlap 0; and grey are features that are not of interest. Both the raw effect size and the CI test exclude different features from being different. This plot uses only the output of the aldex.effect function. The grey lines in the effect plot show the effect=2 isopleth.

Complex study designs and the aldex.glm module

The aldex.glm module is included so that the probabilistic compositional approach can be used for complex study designs. This module is substantially slower than the two-comparison tests above, but we think it is worth it if you have complex study designs.

Essentially, the approach is the modular approach above but using a model matrix and covariates supplied to the glm function in R. The values returned are the expected values of the glm function given the inputs. In the example below, we are measuring the predictive value of variables A and B independently. See the documentation for the R formula function, or http://faculty.chicagobooth.edu/richard.hahn/teaching/formulanotation.pdf for more information.

Validation of features that are differential under any of the variables identified by the aldex.glm function should be performed by plotting and for this the aldex.glm.effect function as a post-hoc test.

Note that aldex.clr will accept the denom="all" or a user-defined vector of denominator offsets when a model matrix is supplied. Therefore, when it is intended that the downstream analysis is the aldex.glm function only those two denominator options are available. This will be addressed in a future release, and will likely be deprecated because the aldex.clr and all downstream functions are fully compatible with the scale modelling which is both more efficient and more easily understood.

data(selex)
selex.sub <- selex[1200:1600, ]
covariates <- data.frame("A" = sample(0:1, 14, replace = TRUE),
   "B" = c(rep(0, 7), rep(1, 7)),
   "Z" = sample(c(1,2,3), 14, replace=TRUE))
mm <- model.matrix(~ A + Z + B, covariates)
x.glm <- aldex.clr(selex.sub, mm, mc.samples=4, denom="all", verbose=T)
glm.test <- aldex.glm(x.glm, mm, fdr.method='holm')
glm.eff<- aldex.glm.effect(x.glm)

The aldex.glm.effect function will calculate the effect size for all models in the matrix that are binary. The effect size calculations for each binary predictor are output to a named list. These effect sizes, and outputs from aldex.glm can be plotted and displayed as in Figure @ref(fig:glm) using the example code below. The aldex.glm.plot function, which mirrors the adlex.plot function for glm and glm.effect outputs.

# NEW plot the glm.test and glm.eff data for particular contrasts
aldex.glm.plot(glm.test, eff=glm.eff, contrast='B', type='MW', test='fdr')

The glm output can be plotted with the aldex.glm.plot function. The values should be nearly identical to the t-test in the same dataset. Here we show an effect (MW) plot of the with significant expected p-values for variables output from the glm function with model B (actual study design groups) highlighted in red. This toy model shows how to use a generalized linear model within theALDEx2 framework. All values are the expected value of the test statistic.

x.kw <- aldex.kw(x)

Adding scale to ALDEx2

While the actual size or scale of the underlying data is lost during the process of sequencing, we can determine if any of the underlying results are dependent on scale. This theory behind this new functionality is outlined in Nixon et al.²⁵ and McGovern et al.²⁶. Essentially, we can think of the aldex.clr function being able to add both measurement error through Dirichlet Monte-Carlos sampling of the actual measurements and scale uncertainty through random sampling of the geometric mean of those measurements. Indeed, as shown below, we can infer the actual scale difference between groups. Scale uncertainty increase the robustness of the results, and can be used to properly centre the data. We recommend using scale instead of using the denominator adjustment approach given in Wu et al.²⁷. The example below is only a small sample of the scale functionality, see the scale_sim vignette for full description.

We will demonstrate the ALDEx2 scale function in two ways; with essentially no scale (gamma=1e-3) and by making a full scale model. If necessary scale can be ignored completely by setting gamma=NULL, in which case previous default behavior is replicated exactly. The minimally scaled version is simply the outputs we have been examining until now of the in vitro selection dataset. Here we have a standard of truth, and a theoretical grounding that the difference in scale is real. We have a situation where only a minority of the parts change in both relative and absolute abundance. Prior analyses(McMurrough et al. 2014) suffered from a false positive bias because of this, and were not amenable to analysis with any other tool.

set.seed(11)
data(selex)
selex.sub <- selex[1201:1600,]
conds <- c(rep("NS", 7), rep("S", 7))
# unscaled
x.u <- aldex.clr(selex.sub, conds, mc.samples=16, 
    gamma=1e-3, verbose=FALSE)
x.u.e <- aldex.effect(x.u, CI=T)

# scaled
# this represents and 8-fold difference in scale between groups
mu.mod <- c(rep(1,7), rep(8,7))
scale.model <- aldex.makeScaleMatrix(gamma=0.5, mu=mu.mod, conditions=conds, mc.samples=16, log=FALSE)
x.ss <- aldex.clr(selex.sub, conds, mc.samples=16, 
    gamma=scale.model, verbose=FALSE)
x.ss.e <- aldex.effect(x.ss, CI=T)
x.ss.tt <- aldex.ttest(x.ss)
x.ss.all <- cbind(x.ss.e, x.ss.tt)

First, an explanation of the scale model that was made. The aldex.makeScaleMatrix() function makes a full scale model of the data that depends on the ratios between the scales of the two groups. In the case of the SELEX dataset, we are explicitly modelling an 8-fold difference in scale between the two datasets.

Now let’s look at the unscaled posterior distributions in Figure @ref(fig:scale-pre),

aldex.plotFeature(x.u, 'S:D:S:D', pooledOnly=T)

The unscaled posterior distribution plot

and compare this to Figure @ref(fig:post-scale). The unscaled posteriors are very different, mainly because the control posterior is very narrow. The scaled posterior distributions also very different but are both more dispersed because of the additional uncertainty introduced by the scale model (note the difference in x-axis scales). In addition, the clr values are different because we are supplying the denominator values directly (log2(1) and log2(8) rather than calculating the geometric means of the samples. We stress here that the actual values of the denominator used is irrelevant, instead it is the ratio between the denominators of the two groups that is key.

aldex.plotFeature(x.ss, 'S:D:S:D', pooledOnly=T)

The scaled posterior distribution plot. Note the wider dispersion on the x axis

We can see the effect of scale if we plot the aldex.effect outputs from scaled and unscaled data. We see that the difference within groups is larger, the effect size is smaller but the difference between groups is unchanged when scale is added as shown in Figure @ref(fig:comp-scale).

par(mfrow=c(1,3))
plot(x.u.e$diff.btw, x.ss.e$diff.btw, main='diff.btw', 
 xlab='unscaled diff', ylab='scaled diff', pch=19, col='red')
abline(0,1)
plot(x.u.e$diff.win, x.ss.e$diff.win, main='diff.win', 
 xlab='unscaled dispersion', ylab='scaled dispersion', pch=19, col='red')
abline(0,1)
plot(x.u.e$effect, x.ss.e$effect, main='effect size', 
 xlab='unscaled effect', ylab='scaled effect', pch=19, col='red')
abline(0,1)

Comparing unscaled and scaled difference, dispersion and effect size.

We can also examine the effect plots, showing those parts where the 95% CI of the effect does not overlap 0 for the unscaled and scaled outputs as an example as in Figure @ref(fig:scale-CI). We see that adding scale uncertainty to the posterior model results in an increase in the dispersion of the data, but changes the difference between measurement very little.

In this dataset we have a standard of truth in an external growth assay for several of the parts displayed here. The scaled data is more similar to what we expect and the S:E:G:D part is the only one displayed on this plot that has appreciable growth in a single growth assay(McMurrough et al. 2014). Thus, adding scale uncertainty removes a large number of false positive identifications, without affecting the identification of the one true positive. Indeed, the true positive stands out more with scale than with no scale uncertainty added.

par(mfrow=c(1,3))
sgn.u <- sign(x.u.e$effect.low) == sign(x.u.e$effect.high)
sgn.s <- sign(x.ss.e$effect.low) == sign(x.ss.e$effect.high)

plot(x.u.e$diff.win, x.u.e$diff.btw, pch=19, cex=0.3,
   col="grey",xlab="Dispersion", ylab="Difference", 
  main="unscaled-CI", xlim=c(0.2,5), ylim=c(-2.5,10))
points(x.u.e$diff.win[abs(x.u.e$effect) >=2], x.u.e$diff.btw[abs(x.u.e$effect) >=2], pch=19, cex=0.5, 
   col="red")
points(x.u.e$diff.win[sgn.u], x.u.e$diff.btw[sgn.u], 
   cex=0.7, col="blue")
text(x.u.e['S:E:G:D', 'diff.win'], 
   x.u.e['S:E:G:D', 'diff.btw']-0.6, labels='S:E:G:D')
abline(0,2, lty=2, col="grey")
abline(0,-2, lty=2, col="grey")

plot(x.ss.e$diff.win, x.ss.e$diff.btw, pch=19, cex=0.3, col="grey",xlab="Dispersion", ylab="Difference", main="scaled-CI",
   xlim=c(0.2,5), ylim=c(-2.5,10))
points(x.ss.e$diff.win[abs(x.ss.e$effect) >=2], 
  x.ss.e$diff.btw[abs(x.ss.e$effect) >=2], pch=19, cex=0.5, col="red")
points(x.ss.e$diff.win[sgn.u], x.ss.e$diff.btw[sgn.u], cex=0.7, 
   col="blue")
text(x.ss.e['S:E:G:D', 'diff.win'], 
   x.ss.e['S:E:G:D', 'diff.btw']-0.6, labels='S:E:G:D')
abline(0,2, lty=2, col="grey")
abline(0,-2, lty=2, col="grey")

aldex.plot(x.ss.all, test='welch', main="scaled-BH", xlim=c(0.2,6))
abline(0,2, lty=2, col="grey")
abline(0,-2, lty=2, col="grey")
text(x.ss.all['S:E:G:D', 'diff.win'], 
   x.ss.all['S:E:G:D', 'diff.btw']-0.6, labels='S:E:G:D')

Compare the unscaled and scaled plots. Here we see that adding scale uncertainty increases the dispersion with little effect on the difference. As a result, we increase our confidence in the reproducibility of the S:E:G:D feature, which has been shown to have in vitro enzymatic activity. In the third plot, which shows features with a BH valued less than 0.05, the S:E:G:D feature is the clear outlier.

ALDEx2 outputs

conds <- data.frame("A" = sample(0:1, 14, replace = TRUE), 
      "B" = c(rep(0, 7), rep(1, 7)))
mm <- model.matrix(~ A + B, conds) 
x <- aldex.clr(selex, mm, mc.samples=4)
glm.test <- aldex.glm(x)
glm.eff <- aldex.glm.effect(x)

A word about effect size and overlap

The effect size metric used by ALDEx2 is a standardized distributional effect size metric developed specifically for this package. The measure is somewhat robust, allowing up to 20% of the samples to be outliers before the value is affected, returns an effect size that is 71% the size of Cohen’s d on a Normal distribution, and requires at worst twice the number of samples as fully parametric methods (which are not robust) would to estimate values with the same precision. The metric is equally valid for Normal, random uniform and Cauchy distributions^(Andrew D. Fernandes et al. (2019) submitted).

We prefer to use the effect size whenever possible rather than statistical significance since an effect size tells the scientist what they want to know—“what is reproducibly different between groups”; this is not something that p-values deliver. We find that using the effect size returns a consistent set of true positive features irregardless of sample size, unlike p-value based methods. Furthermore, over half of the the false positive features that are observed at low sample sizes have and effect size  > 0.5 × E the chosen effect size cutoff E. This is true regardless of the source of the dataset (Andrew D. Fernandes et al. (2019) submitted).

We suggest that an effect size cutoff of 1 or greater be used when analyzing HTS datasets. If preferred the user can also set a fold-change cutoff as is commonly done with p-value based methods, and volcano plots are now included as an option to help guide the user.

Alternative plotting of outputs

The built-in aldex.plot function described above will usually be sufficient, but for more user control the example in Figure 10 shows a plot that shows which features are found by the Welchs’ or Wilcoxon test individually (blue) or by both (red).

# identify which values are significant in both the t-test and glm tests
found.by.all <- which(x.all$we.eBH < 0.05 & x.all$wi.eBH < 0.05)

# identify which values are significant in fewer than all tests
found.by.one <- which(x.all$we.eBH < 0.05 | x.all$wi.eBH < 0.05)

# plot the within and between variation of the data
plot(x.all$diff.win, x.all$diff.btw, pch=19, cex=0.3, col=rgb(0,0,0,0.3),
 xlab="Dispersion", ylab="Difference")
points(x.all$diff.win[found.by.one], x.all$diff.btw[found.by.one], pch=19,
 cex=0.7, col=rgb(0,0,1,0.5))
points(x.all$diff.win[found.by.all], x.all$diff.btw[found.by.all], pch=19,
 cex=0.7, col=rgb(1,0,0,1))
abline(0,1,lty=2)
abline(0,-1,lty=2)

Differential abundance in the selex dataset using the Welch’s t-test or Wilcoxon rank test. Features identified by both tests shown in red. Features identified by only one test are shown in blue dots. Non-significant features represent rare features if black and abundant features if grey dots.

Troubleshooting datasets

In some cases we observe that the data can be asymmetric. This occurs when the data are extremely asymmetric, such as when one group is largely composed of features that are absent in the other group or when the direction of change in the experiment is not symmetric. In this case the geometric mean will not accurately represent the appropriate basis of comparison for each group. An asymmetry can arise for many reasons: in RNA-seq it could arise because samples in one group contain a plasmid and the samples in the other group do not; in metagenomics or 16S rRNA gene sequencing it can arise when the samples in the two groups are taken from different environments; in a selex experiment it can arise because the two groups are under different selective constraints. The asymmetry can manifest either as a difference in sparsity (i.e., one group contains more 0 value features than the other) or as a systematic difference in abundance. When this occurs the geometric mean of each sample and group can be markedly different, and thus an inherent skew in the dataset can occur that leads to false positive and false negative feature calls. Asymmetry generally shows as a the centre of mass of the histogram for the x.all$diff.btw or x.all$effect being not centred around zero³². We recommend that all datasets be examined for asymmetry.

Using scale to correct for asymmetric datasets

The preferred approach is to build a full scale model of the asymmetry and to use that as an input to aldex.clr. Fundamentally, asymmetry will often arise because of a difference in size of the underlying environment. For example, one cell type may grow faster than another, or may have a different total number of mRNA molecules than another, or one environment may have a different microbial carrying capacity than another. This is the situation that is best taken care of by a scale model³³.

The example dataset below is the simulated synth2 dataset from from Wu et al(Wu et al. 2021). In this example there are 1000 features with 20 (2%) of those being asymmetrically sparse; i.e., 20 features are 0 in one group and non-zero in the other. A further 50 have an approximate 2-fold change (FC is normally distributed). The asymmetry has the effect of changing the geometric mean of the two groups. In addition, the data have an unrealistically low dispersion.

The three plots show the behaviour of unmodified and scale corrected ALDEx2.

The base output shows that some features are differentially abundant because of the inappropriate centre of the data. The approximate centre is the median difference of the points and is noted.

Adding in uncertainty about the scale of the data has a minimal effect on the median difference, but does remove all false positives. However, now there is an asymmetry in the false negatives, in that those in the ‘S’ category are more likely to be called negatives than those in the ‘N’ category.

Finally, adding in uncertainty about scale, and information on the fraction of the features that are asymmetric results in the desired output.

We recommend the scale approach whenever possible, especially if there is some underlying biology that could drive the asymmetry. See the scale_sim vignette for more information on these new capabilities.

The default ALDEx2 scale correction includes only uncertainty and not asymmetry.

set.seed(11)
data(synth2)
# basic ALDEx2
blocks <- c(rep("N", 10),rep("S", 10))
x <- aldex.clr(synth2, blocks, gamma=NULL)
x.e <- aldex.effect(x)

# Add in asymmetry and scale variance
# gamma can be smaller for the same output dispersion
# in the full scale model built here
# make a base scale for group 1 and 2
mu.vec = c(log2(rep(1,10)), log2(rep(1.02,10)))
scale_samples <- aldex.makeScaleMatrix(gamma=0.25, mu=mu.vec, 
  conditions=blocks) 
  
xs <- aldex.clr(synth2, blocks, gamma=scale_samples)
xs.e <- aldex.effect(xs)

# Add in only scale variance with sd=1
xg <- aldex.clr(synth2, blocks, gamma=0.5)
xg.e <- aldex.effect(xg)


par(mfrow=c(1,3))
aldex.plot(x.e, test='effect', main='base', xlim=c(0.1,3))
text(1, 4, labels=paste('median diff =',round(median(x.e$diff.btw),3)))
aldex.plot(xg.e, test='effect', main='ratio=1, sd=0.5', xlim=c(0.1,3))
text(1.6, 4, labels=paste('median diff =',round(median(xg.e$diff.btw),3)))
aldex.plot(xs.e, test='effect', main='ratio=1.02:1, sd=0.5', xlim=c(0.1,3))
text(1.6, 4, labels=paste('median diff =',round(median(xs.e$diff.btw),3)))

Differential abundance in a synthetic dataset using different scale parameters for the clr calculation. In this data 2% of the features are modelled to be sparse in one group but not the other and a further 5% of the features are differentially abundant. Features determined to be different between two groups are shown in red. Features that are non-significant are in black. Lines of constant effect are drawn at 0, and ± 1.

Contributors

I am grateful that ALDEx2 has taken on a life of its own.

• Andrew Fernandes wrote the original ALDEx code, and designed ALDEx2. He developed the effect size metric and the concept of reporting posterior outcomes in all cases.
• Jean Macklaim found and squished many bugs, performed unit testing, did much of the original validation. Jean also made seminal contributions to simplifying and explaining the output of ALDEx2.
• Michelle Pistner Nixon and Justin Silverman implemented the scale idea and code and the scale_sim vignette. More scale corrections will be forthcoming.
• Michelle Pistner Nixon and Justin Silverman implemented the rigorous posterior p-value idea and code
• Matt Links incorporated several ALDEx2 functions into a multicore environment.
• Adrianne Albert wrote the correlation and the one-way ANOVA modules in aldex.kw.
• Ruth Grace Wong added function definitions and made the parallel code functional with BioConducter.
• Jia Rong Wu developed and implemented the alternate denominator method to correct for asymmetric datasets.
• Andrew Fernandes, Jean Macklaim and Ruth Grace Wong contributed to the Sum-FunctionsAitchison.R code.
• Thom Quinn rewrote the t-test and Wilcoxon functions to make them substantially faster. He also wrote much of the current aldex.glm function. Tom’s propr³⁵ R package is able to integrate with the output from aldex.clr.
• Vladimir Mikryukov and everyone named above have contributed bug fixes
• Greg Gloor currently maintains ALDEx2 and had and has roles in documentation, design, testing and implementation.

Version information

. No further changes are expected for that version since it can be replicated completely within ALDEx2 by using only the aldex.clr and aldex.effect commands.

Versions 2.0 to 2.05 were development versions that enabled p-value calculations. Version 2.06 of ALDEx2 was the version used for the analysis in³⁷. This version enabled large sample comparisons by calculating effect size from a random sample of the data rather than from an exhaustive comparison.

Version 2.07 of ALDEx2 was the initial the modular version that exposed the intermediate calculations so that investigators could write functions to analyze different experimental designs. As an example, this version contains an example one-way ANOVA module. This is identical to the version submitted to Bioconductor as 0.99.1.

Future releases of ALDEx2 now use the Bioconductor versioning numbering.

sessionInfo()

## R version 4.4.1 (2024-06-14)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04.1 LTS
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
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##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
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##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## time zone: Etc/UTC
## tzcode source: system (glibc)
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] ALDEx2_1.39.0         latticeExtra_0.6-30   lattice_0.22-6       
## [4] zCompositions_1.5.0-4 truncnorm_1.0-9       NADA_1.6-1.1         
## [7] survival_3.7-0        MASS_7.3-61           BiocStyle_2.33.1     
## 
## loaded via a namespace (and not attached):
##  [1] SummarizedExperiment_1.35.5 xfun_0.48                  
##  [3] bslib_0.8.0                 Biobase_2.65.1             
##  [5] quadprog_1.5-8              tools_4.4.1                
##  [7] stats4_4.4.1                parallel_4.4.1             
##  [9] highr_0.11                  Matrix_1.7-1               
## [11] RColorBrewer_1.1-3          S4Vectors_0.43.2           
## [13] RcppParallel_5.1.9          lifecycle_1.0.4            
## [15] GenomeInfoDbData_1.2.13     compiler_4.4.1             
## [17] deldir_2.0-4                codetools_0.2-20           
## [19] GenomeInfoDb_1.41.2         htmltools_0.5.8.1          
## [21] sys_3.4.3                   buildtools_1.0.0           
## [23] sass_0.4.9                  yaml_2.3.10                
## [25] crayon_1.5.3                jquerylib_0.1.4            
## [27] BiocParallel_1.39.0         cachem_1.1.0               
## [29] DelayedArray_0.31.14        abind_1.4-8                
## [31] digest_0.6.37               maketools_1.3.1            
## [33] splines_4.4.1               fastmap_1.2.0              
## [35] grid_4.4.1                  cli_3.6.3                  
## [37] SparseArray_1.5.45          S4Arrays_1.5.11            
## [39] Rfast_2.1.0                 UCSC.utils_1.1.0           
## [41] RcppZiggurat_0.1.6          rmarkdown_2.28             
## [43] XVector_0.45.0              httr_1.4.7                 
## [45] matrixStats_1.4.1           jpeg_0.1-10                
## [47] multtest_2.61.0             interp_1.1-6               
## [49] png_0.1-8                   evaluate_1.0.1             
## [51] knitr_1.48                  GenomicRanges_1.57.2       
## [53] IRanges_2.39.2              rlang_1.1.4                
## [55] Rcpp_1.0.13                 BiocManager_1.30.25        
## [57] directlabels_2024.1.21      BiocGenerics_0.51.3        
## [59] jsonlite_1.8.9              R6_2.5.1                   
## [61] MatrixGenerics_1.17.1       zlibbioc_1.51.2