3.3.1 Winsorization

Microbiome data may contain outliers with extremely large counts. These outliers could reduce the statistical power, or more seriously increase the type I error rate. In ZicoSeq, we use winsorization to replace those extremely large counts with a certain quantile (e.g. 97%). We recommend always setting is.winsor = TRUE for real data analysis. The specific quantile used depends on the expected proportion of outliers.

3.3.2 Posterior sampling

Posterior sampling can be enabled by setting is.post.sample = TRUE. It can only be used when the data type is “count”. For microbiome data, the majority of the taxa are rare, resulting in a large number of zeros in the data given a limited sequencing depth. As the probability of zero strongly depends on the sequencing depth, zeros from different samples are not comparable. In ZicoSeq, we provide a method to infer/impute the underlying true proportions using an empirical Bayes approach by pooling information across samples. The imputed proportions depend on both the sequencing depth and the estimated (prior) distribution of the proportions across samples. To better model the prior distribution, we used beta mixtures instead a single-component beta distribution. Posterior sampling effectively reduces the type I error when the sequencing depth is associated with the covariate of interest (e.g. cases and controls differ in sequencing depth) and is more powerful than rarefaction. When the sequencing depth is not a confounding factor, is.post.sample slightly improves the power and could be disabled for analyzing large datasets.

3.3.3 Omnibus testing

The relationship between the taxa abundance and the covariate of interest may vary across taxa, a single transformation function may not be powerful enough to capture diverse relationships. ZicoSeq allows specifying multiple possible transformation functions (link.func) and performs omnibus testing. Although the log transformation is commonly used for microbiome due to its interpretability, we found that power transformations may be more powerful. The default is the square-root transformation.

3.3.4 Reference selection

ZicoSeq uses a reference-based approach to address compositional effects. The default uses 30% of the least variable/significant taxa as the reference. It starts with selecting 50% of the least variable taxa as the reference and differential abundance testing is then run multiple times excluding 20% taxa with the smallest p-values in each iteration. This procedure is to make sure the remaining 30% are more likely to be non-differential.

3.3.5. Error control

ZicoSeq uses a permutation-based approach to control false discovery rate (FDR). The permutation-based FDR control preserves the correlation structure in the abundance data and is more robust and powerful than the traditional BH-based FDR control based on raw p values. ZicoSeq also allows performing permutation-based family-wise error rate control. Users can determine the appropriate error control method to suit their needs.

3.4.1 p-values

• p.raw: raw p-values based on permutations, it will not be accurate if perm.no is small.
• p.adj.fdr: permutation-based FDR-adjusted p-values.
• p.adj.fwer: permutation-based FWER-adjusted (West-Young), only if is.fwer = T in ZicoSeq. perm.no is recommened to set at least 999 for accurate FWER-adjusted p-values.

3.4.2 R^2 (percentage of variance explained)

A matrix of R^2 values (number of features by number of transformation functions) will be provided. R^2 is an effect size measure and can be used to assess the association strength between the taxa abundance and the covariate of interest while adjusting for other covariates. When the omnibus testing is in action, i.e., multiple transformations are used, there will be multiple R^2 s, each corresponding to a specific transformation.

3.6.1 Proportion data

For some bioinformatics pipelines, the output could be proportion data. ZicoSeq can be applied to proportion data by specifying feature.dat.type = "proportion" . Posterior sampling will not be applied when analyzing the proportions and other parameter settings are similar to the count case.

comm.p <- t(t(comm) / colSums(comm))
ZicoSeq.obj.p <- ZicoSeq(meta.dat = meta.dat, feature.dat = comm.p, 
                    grp.name = 'SmokingStatus', adj.name = 'Sex', feature.dat.type = "proportion",
                    # Filter to remove rare taxa
                    prev.filter = 0.2, mean.abund.filter = 0,  max.abund.filter = 0.002, min.prop = 0, 
                    # Winsorization to replace outliers
                    is.winsor = TRUE, outlier.pct = 0.03, winsor.end = 'top',
                    # Posterior sampling will be automatically disabled
                    is.post.sample = FALSE, post.sample.no = 25, 
                    # Use the square-root transformation
                    link.func = list(function (x) x^0.5, function (x) x^0.25), stats.combine.func = max,
                    # Permutation-based multiple testing correction
                    perm.no = 99,  strata = NULL, 
                    # Reference-based multiple stage normalization
                    ref.pct = 0.5, stage.no = 6, excl.pct = 0.2,
                    # Family-wise error rate control
                    is.fwer = TRUE, verbose = TRUE, return.feature.dat = T)
suppressWarnings(ZicoSeq.plot(ZicoSeq.obj = ZicoSeq.obj.p, pvalue.type = 'p.adj.fdr', 
             cutoff = 0.1, text.size = 10, out.dir = NULL, width = 10, height = 6))                    

3.6.2 General data types

ZicoSeq as a general linear model-based permutation test can be applied to association analyses of other high-dimensional datasets such as transcriptome-, methylome-, metabolome- and proteome-wide association testing, where the omics features are treated as the outcomes. In this case, posterior sampling will be automatically disabled. The user should be responsible for properly normalizing, transforming, and filtering the data before applying ZicoSeq. The user should also decide whether to winsorize the top, bottom or both ends of the distribution. The following code is just for demonstration purposes and does not mean to be applied for differential abundance analysis.

comm.o <- comm[rowMeans(comm != 0) >= 0.2, ] + 1
comm.o <- log(t(t(comm.o) / colSums(comm.o)))
ZicoSeq.obj.o <- ZicoSeq(meta.dat = meta.dat, feature.dat = comm.o, 
        grp.name = 'SmokingStatus', adj.name = 'Sex', feature.dat.type = "other",
        # Filter will not be applied
        prev.filter = 0, mean.abund.filter = 0,  max.abund.filter = 0, min.prop = 0, 
        # Winsorization the top end 
        is.winsor = TRUE, outlier.pct = 0.03, winsor.end = 'top',
        # Posterior sampling will be automatically disabled
        is.post.sample = FALSE, post.sample.no = 25, 
        # Identity function is used
        link.func = list(function (x) x), stats.combine.func = max,
        # Permutation-based multiple testing correction
        perm.no = 99,  strata = NULL, 
        # Reference-based multiple-stage normalization will not be performed
        ref.pct = 0.5, stage.no = 6, excl.pct = 0.2,
        # Family-wise error rate control
        is.fwer = TRUE, verbose = TRUE, return.feature.dat = T)
    ZicoSeq.plot(ZicoSeq.obj = ZicoSeq.obj.o,  pvalue.type = 'p.adj.fdr', 
             cutoff = 0.1, text.size = 10, out.dir = NULL, width = 10, height = 6)    

3.6.3 Random effect model - within-subject comparisons

ZicoSeq can be applied to perform within-subject comparisons (e.g. before treatment vs. after treatment) by using the strata parameter. For example, if the subject variable is “subject_id” in the meta data file, the user can set strata = 'subject_id' .