# Introduction

MarZIC estimates and tests the marginal mediation effects of zero-inflated compositional mediators. It can be applied to microbiome studies to estimate and test the marginal mediation effects of relative abundances (RA). Let $$Y$$, $$M$$ and $$X$$ denote the outcome, mediator (RA) and the independent variable respectively. And let $$1_{(M>0)}$$ denote the binary indicator variable indicating whether $$M$$ is positive. MarZIC decomposes the mediation effect into two components of which one is the total mediation effect going thru the continuous scale of the mediator, denoted by NIE$$_1$$ and the other is the mediation effect going thru the binary change (from zero to non-zero status) of the mediator, denoted by NIE$$_2$$. For a microbiome data, let $$M_j$$ denote the RA of the $$j$$th taxon/OTU/ASV. And let $$Z$$ denote confounder variable (if any) and $$Z$$ could contain multiple variables. The marginal outcome model for $$Y$$ is: $Y=\beta_0+\beta_1M_j+\beta_21_{M_j>0}+\beta_3X+\beta_4X1_{M_j>0}+\beta_5XM_j+\beta_6Z+\epsilon$ The mean of $$M_j$$ depends on $$X$$ and $$Z$$ through the following equation: $\log \Bigg(\frac{E(M_j)}{1-E(M_j)}\Bigg)=\alpha_0 + \alpha_1X+\alpha_2Z$ Let $$\Delta_j=P(M_j=0)$$ which is the probability of a structural zero (ie, true zero). We use the following logistic model for this probability:
$\log\bigg(\frac{\Delta_j}{1-\Delta_j}\bigg)=\gamma_0 + \gamma_1X+\gamma_2Z$

# Input for MarZIC() function

Typically, users just need to feed the first six inputs to the function: Experiment_dat, lib_name, y_name, x_name, conf_name and taxa_of_interest.

• MicrobData A dataset contains microbiome data. The microbiome data could be relative abundance or absolute abundance. Subjects with missing value will be removed during analysis.
• CovData A dataset contains outcome, library size and covariates.
• lib_name Name of library size variable.
• y_name Name of outcome variable
• x_name Name of covariate of interest
• conf_name Name of confounders Defaule is NULL, meaning no confounder.
• x4_inter Whether to include the interaction term $$\beta_4$$. Default is TRUE.
• x5_inter Whether to include the interaction term $$\beta_5$$. Default is TRUE.
• taxa_of_interest A character vector for taxa names indicating taxa that should be analyzed. Default is “all”, meaning all taxa should be included into analysis.
• mediator_mix_range Number of mixtures in mediator. Default is 1, meaning no mixture.
• transfer_to_RA Logical variable indicating whether the microbiome data should be transferred into relative abundance. Default is TRUE. If FALSE, microbiome data will not be transferred (e.g., if the input data is already RA data).
• num_cores Number of CPU cores to be used in parallelization task.
• adjust_method P value adjustment method. Same as p.adjust in R. Default is “fdr”.
• fdr_rate FDR cutoff for significance. Default is 0.05.
• taxDropThresh The threshold of dropping taxon due to high zero percentage. Default is 0.9, meaning taxon will be dropped for analysis if zero percentage is higher than 90%.
• taxDropCount The threshold of dropping taxon due to not enough non-zero observation counts. Default is 4 * (length(conf_name)+2), meaning taxon will be dropped if non-zero observation is less than four times of the total number of covariates (including the independent variable) plus intercept.
• zero_prop_NIE2 The threshold of zero percentage for calculating NIE2. Default is 0.1, meaning NIE2 will be calculated for taxon with zero percentage greater than 10%.
• zero_count_NIE2 The threshold of zero counts for calculating NIE2. Default is 4 * (length(conf_name)+2), meaning NIE2 will be calculated for taxon with zero counts greater than four times of number of covariates plus 1.
• SDThresh The threshold of dropping taxon due to low coefficient of variation (CV) to avoid constant taxon. Default is 0.05, meaning any taxon has CV less than 0.05 will be dropped.
• SDx The threshold of stopping analysis due to low CV of covariate of interest. Default is 0.05, meaning when CV of covariate of interest is less than 0.05, the analysis will be stopped.
• SDy The threshold of stopping analysis due to low CV of outcome. Default is 0.05, meaning when CV of outcome is less than 0.05, the analysis will be stopped.

# Output for IFAA() function

A list of 4 datasets containing the results for NIE1, NIE2, NDE, and NIE. Each dataset has row representing each taxon, 6 columns for Estimates, Standard Error, Lower bound for 95% Confidence Interval, Upper bound for 95% Confidence Interval, Adjusted p value, Significance indicator.

# Example

## A make up example with 1 taxon and 100 subjects.
set.seed(1)
nSub <- 200
nTaxa <- 10
## generate covariate of interest X
X <- rbinom(nSub, 1, 0.5)
## generate confounders
conf1<-rnorm(nSub)
conf2<-rbinom(nSub,1,0.5)

## generate mean of each taxon. All taxon are having the same mean for simplicity.
mu <- exp(-5 + X + 0.1 * conf1 + 0.1 * conf2) /
(1 + exp(-5 + X + 0.1 * conf1 + 0.1 * conf2))
phi <- 10

## generate true RA
M_taxon<-t(sapply(mu,function(x) dirmult::rdirichlet(n=1,rep(x*phi,nTaxa))))

P_zero <- exp(-3 + 0.3 * X + 0.1 * conf1 + 0.1 * conf2) /
(1 + exp(-3 + 0.3 * X + 0.1 * conf1 + 0.1 * conf2))

non_zero_ind <- t(sapply(P_zero,function(x) 1-rbinom(nTaxa,1,rep(x,nTaxa))))

True_RA<-t(apply(M_taxon*non_zero_ind,1,function(x) x/sum(x)))

## generate outcome Y based on true RA
Y <- 1 + 100 * True_RA[,1] + 5 * (True_RA[,1] > 0) + X + conf1 + conf2 + rnorm(nSub)

## library size was set to 10,000 for all subjects for simplicity.
libsize <- 10000

## generate observed RA
observed_AA <- floor(M_taxon*libsize*non_zero_ind)

observed_RA <- t(apply(observed_AA,1,function(x) x/sum(x)))
colnames(observed_RA)<-paste0("rawCount",seq_len(nTaxa))
CovData <- cbind(Y = Y, X = X, libsize = libsize, conf1 = conf1, conf2 = conf2)

Above steps could be ignored if a SummarizedExperiment object is already available. Suppose we’re interested in the mediation effects of rawCount1, rawCount2, and rawCount3, the analysis could be done as:

## run the analysis
res <- MarZIC(
MicrobData = observed_RA,
CovData = CovData,
lib_name = "libsize",
y_name = "Y",
x_name = "X",
conf_name = c("conf1","conf2"),
taxa_of_interest = c("rawCount1","rawCount2","rawCount3"),
num_cores = 1,
mediator_mix_range = 1
)

The results contain 4 datasets for NIE$$_1$$, NIE$$_2$$, NDE, NIE, respectively. The NIE$$_1$$, for example, could be extracted by:

NIE1 <- res\$NIE1_save

The significant result could be extracted by:

subset(NIE1, significance == TRUE)
#>                est       se   CI low    CI up p value unadj  p value adj
#> rawCount1 7.138202 1.666013 3.872816 10.40359  1.830673e-05 5.492018e-05
#>           significance
#> rawCount1         TRUE

From the result, we can find that rawCount1 is an significant mediator of mediating the effect of $$X$$ on $$Y$$.

# Reference

Wu et al.(2022) MarZIC: A Marginal Mediation Model for Zero-Inflated Compositional Mediators with Applications to Microbiome Data. Genes 2022, 13, 1049.

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