Abstract
Studies have found that human microbiome is associated with and predictive of human health and diseases. Many statistical methods developed for microbiome data focus on different distance metrics that can capture various information in microbiomes. Prediction models were also developed for microbiome data, including deep learning methods with convolutional neural networks that consider both taxa abundance profiles and taxonomic relationships among microbial taxa from a phylogenetic tree. Studies have also suggested that a health outcome could associate with multiple forms of microbiome profiles. In addition to the abundance of some taxa that are associated with a health outcome, the presence/absence of some taxa is also associated with and predictive of the same health outcome. Moreover, associated taxa may be close to each other on a phylogenetic tree or spread apart on a phylogenetic tree. No prediction models currently exist that use multiple forms of microbiome-outcome associations. To address this, we propose a multi-kernel machine regression (MKMR) method that is able to capture various types of microbiome signals when doing predictions. MKMR utilizes multiple forms of microbiome signals through multiple kernels being transformed from multiple distance metrics for microbiomes and learn an optimal conic combination of these kernels, with kernel weights helping us understand contributions of individual microbiome signal types. Simulation studies suggest a much-improved prediction performance over competing methods with mixture of microbiome signals. Real data applicants to predict multiple health outcomes using throat and gut microbiome data also suggest a better prediction of MKMR than that of competing methods.
Keywords: microbiome, distances metrics, prediction model, multi-kernel learning
INTRODUCTION
The advancement in targeted sequencing technology of 16S ribosomal RNA has allowed an unbiased characterization of bacteria composition [1–4]. Generated 16S sequence reads are usually clustered into operational taxonomic units (OTUs) with a similarity threshold, typically 97% [5]. A phylogenetic tree that captures evolutionary relationships among OTUs can be constructed based on OTU sequence divergence which provides insights on health and disease etiology as taxa that are close to each other on a phylogenetic tree are also usually functionally similar (while taxa that are far away from each other on a phylogenetic tree are functionally independent). OTUs are characterized by their read counts. However, as total numbers of sequencing reads vary between samples, OTU counts are usually normalized by total counts per sample into relative abundance levels, thus microbiome data become compositional [6].
Studies have successfully established associations between microbiomes and human health. For example, oral microbiomes are found to be associated with oral diseases and potentially with systemic disease [7]. Gut microbiomes are involved in immune system developments [8] and inflammatory bowel disease [9]. Many statistical methods developed for microbiome data focus on different distance metrics for microbiome data [10–16]. For example, Zhao et al. [15] proposed the microbiome regression-based kernel association test (MiRKAT). MiRKAT is a semi-parametric kernel machine regression framework that tests associations between entire microbiomes and health outcomes through constructing multiple kernels from multiple distance metrics for microbiome compositions, from which an optimal one is selected. Here, a kernel measures similarity of microbiome compositions between two samples and multiple kernels from different distance metrics capture different aspects of microbiome compositions between two samples. Several distance metrics have been commonly used for microbiome data. UniFrac distances [10] capture signals of taxa incorporating phylogenetic information, with unweighted UniFrac distances [11] focusing on rare taxa, weighted UniFrac distances [12] emphasizing common ones and generalized UniFrac distances [13] focusing on taxa in between. Another commonly used distance metric is the Bray–Curtis [14] distance that only uses abundance information of OTUs but not phylogenetic information. Using different distance metrics for microbiome data, either taxa abundance only or both taxa abundance/presence/absence and phylogenetic tree information are used.
Studies have also used microbiome together with covariates to predict health outcomes with some general-purpose machine learning models such as Lasso regressions [17] and random forest [18] without utilizing the phylogenetic tree structure. Prediction models specifically for microbiome data were also developed recently. Tanaseichuk et al. [19] added a tree-guided penalty function to the multinomial logistic regression model. Chen et al. [20] combined a tree-based Laplacian penalty and a minimax concave penalty to achieve a better prediction performance. Xiao et al. [21] proposed a generalized linear mixed model where the random effect of species has a correlation structure based on phylogenetic tree. Most recently, prediction models with deep learning methods were also developed using microbiome data [22–26]. Chen et al. [22] used a convolutional neural network (CNN) which can capture the spatial relationship usually with applications in images by extracting spatial correlations among neighboring pixels. Their work clearly shows the benefit of using phylogenetic tree spatial information in prediction. Grazioli et al. [23] developed a multimodal deep learning approach which represents species relative abundance and strain-level marker profiles from a phylogenetic tree through encoding for disease prediction. Wang et al. [24] developed a deep learning method that uses convolutional layers to mimic taxonomic ranks to capture the phylogenetic relationship among microbial species. Sharma et al. [25] used a two-step approach with the first step to group OTUs into phylum clusters; a CNN was then used to extract information from each cluster and an ensemble learning approach was used to combine multiple CNNs. Reiman et al. [26] also used CNN with both phylogenetic tree information and relative abundance of microbial taxa as inputs.
However, although these methods used both phylogenetic tree and taxa abundance information, they all only use one type of signals, i.e. taxa abundance and phylogenetic tree, sometimes sparse [19, 20] sometimes abundant [21]. They may not work well for a mixture of multiple forms of microbiome signals. Many studies have suggested that a health outcome could associate with multiple forms of microbiome profiles. That is, in addition to the abundance of some taxa that are associated with a health outcome, the presence/absence of some taxa is also associated with and predictive of the same health outcome [27], including obesity ([28]), irritable bowel disease ([9]) and diabetes ([29]). Furthermore, the associated taxa may be close to each other on a phylogenetic tree or are spread apart on a phylogenetic tree. The aforementioned distance metrics for microbiome data, such as UniFrac (weighted and unweighted) and Bray–Curtis distances, can capture different information in microbiome data.
To address this, we propose a multi-kernel machine regression (MKMR) model that uses multiple distance metrics for microbiome compositions to capture various forms of microbiome information (e.g. taxa that are either close to each other or far away from each other on a phylogenetic tree, taxa that are abundant or rare and mixture of these) to better predict health outcomes. Specifically, we construct multiple kernels from multiple distance metrics with each capturing a different aspect of information in microbiome, and apply a multi-kernel learning algorithm [30] under the multi-kernel regression framework to learn a sparse conic combination of multiple kernels. Thus, information from multiple types of signals can all be used for a better prediction performance. Under the proposed MKMR framework, covariates (e.g. age, sex) can be readily incorporated to further improve the prediction performance. Through extensive simulation studies, we showed that the proposed MKMR method (i) has prediction performance almost as good as that of the single-kernel regression method with the kernel that reflects the true microbiome-outcome association; and (ii) has much better prediction performance than that of all competing methods considered in almost all simulation scenarios when signals are mixtures of different forms of microbiome-outcome associations. This suggests that our proposed MKMR method is able to capture multiple forms of microbiome signals. In real data applications, we applied the proposed MKMR to the throat microbiome data to predict smoking status [31], and American Gut Project (AGP) microbiome data [32] to predict multiple health outcomes and observed much better prediction results of MKMR than that of competing methods.
MATERIALS AND METHODS
Consider a binary outcome 
for sample 
and an OTU table with relative abundances levels of Q OTUs 
, and 
covariates (e.g. age, gender) 
. Let 
be a rooted phylogenetic tree with 
branches with branch lengths 
. Figure 1 displays the schematic flow chart of the proposed MKMR method with a hypothetical rooted phylogenetic tree of seven branches and five tips and an OTU table with five OTUs for four samples. Data we observe are thus 
, 
, for 
samples, with a phylogenetic tree 
.
Figure 1.
The schematic plot of the proposed MKMR method. The input data consist of a rooted phylogenetic tree, an OTU table with OTU relative abundance levels, an outcome of interest and possibly some covariates. In this hypothetical phylogenetic tree, there are 
taxa/branches and 
OTUs, where 
are branch lengths. The proposed MKMR predicts the outcome 
using kernel matrices for microbiome and covariates.
Distance metrics for microbiome compositions
There are several commonly used distance metrics for microbiome data [11–14]. The Bray–Curtis distance [14] measures abundance distance across 
OTUs between samples 
and 
without using phylogenetic tree information: 
where 
are abundance levels of 
OTUs. UniFrac distances including unweighted and weighted versions use phylogenetic tree information [11, 12]. The unweighted UniFrac distance weights the branch lengths in a phylogenetic tree by differences in the presence/absence of the descending OTUs between two samples, thus captures signals of rare taxa: 
for 
branches and 
is an indicator function. The weighted UniFrac distance weights the branch lengths in a phylogenetic tree by differences in abundance levels of the descending OTUs, thus captures signals of abundant taxa, 
As a compromise between unweighted and weighted UniFrac distances, generalized UniFrac distances [13] offer granularity in between rare and abundant taxa, 
Here 
is a parameter that controls the weight of abundant taxa. When 
, the generalized UniFrac distance is the weighted UniFrac distance.
From distances to kernels
Although these distance metrics capture different aspects of microbiome compositions that are potentially predictive of outcomes, they suffer from the limitation of inexplicit feature mapping for each sample that is required for most prediction models. By converting distance measures into kernels, we can take advantages of the well-studied kernel machine regression framework, which has the nice property of ‘kernel trick’ that allows kernel matrices as inputs for models without model formulas explicitly.
Let 
be an 
distance matrix that summarizes pairwise distances of 
samples, whose entries 
are calculated using one of the distance metrics introduced above. For each distance matrix, we can construct the corresponding 
kernel matrix 
that measures pairwise similarities between samples via the following transformation:
 |
(1) |
where 
is the identity matrix, 
and 
is the element-wise squared matrix of 
. We denote the constructed kernel matrices as 
, 
, 
, and 
corresponding to the Bray–Curtis distance, unweighted UniFrac distance, weighted UniFrac distance and generalized UniFrac distance with parameter 
, respectively. The kernel transformation in Equation (1) reflects the relationship between kernel machine regression and distance-based regression that the distance between samples 
and 
can be recovered by 
, where 
is the 
-th element of 
[33]. To preserve the positive semi-definite property of a kernel matrix, we use a simple correction procedure as in [34]. Specifically, suppose that a kernel matrix admits the eigenvalue decomposition 
with a diagonal matrix 
. We reconstruct the non-negative definite kernel matrix as the nearest symmetric positive semidefinite matrix 
with 
[35].
The 
-th element 
of 
measures the similarity between samples 
and 
. Each kernel matrix 
is associated with a reproducing kernel function 
such that 
, with 
and the space of input features being 
. In fact, the kernel function 
implicitly specifies a feature mapping function 
from the input space 
to an 
feature space 
such that 
for all 
. Different choices of the kernel function imply different underlying feature spaces. In the simplest case, the linear kernel 
corresponds to the identity feature map 
.
The proposed multi-kernel machine regression
With 
kernels from 
distance metrics that capture 
forms of microbiome-outcome relationships, for a binary outcome 
, an MKMR model can be written as
 |
(2) |
Here 
is a link function (e.g. the logit function in logistic regression), 
are mapped features induced from kernel function 
and 
is the coefficient vector. The dimension of the mapped feature is typically unknown and potentially infinite. We can learn a conic combination of 
kernels by solving the optimization problem of the following loss function with a block 1-norm regularization term [30]
 |
(3) |
where 
is a regularization parameter, 
is the 2-norm of 
for kernel 
and 
is the loss function (e.g. 
for a logistic regression). Here, the coefficient vector 
for each kernel forms a ‘block’ of parameters, and by using block 1-norm regularization, the coefficient vector can be shrunk to 0 for those kernels that are uninformative. By reformulating Equation (3) in its dual form [36, 37], we have the following optimality conditions:
 |
(4) |
where 
is the kernel matrix associated with kernel function 
, 
is a Fenchel conjugate of the convex loss function 
(e.g. for logistic regression 
if 
and 
otherwise), 
is the dual variable where 
and 
where 
for 
are coefficients of a conic combination of 
kernels.
Using the algorithm in [30], coefficients 
and the dual variable 
can be learnt via the optimality conditions in Equation (4). The algorithm takes kernel matrices and outcome as input data without an explicit form of feature maps 
in Equation (2). This is the so-called ‘kernel trick’ that has been widely used in support vector machine and other machine learning methods.
Building a prediction tool
With dual parameters 
, kernel weights 
estimated for a binary outcome with a training set, we predict the binary outcome for a new sample 
by calculating kernel values 
between the new sample 
and sample 
in the training set, for each kernel 
. The estimated probability of having the binary outcome is
 |
Kernel matrices with entries 
are transformed from distance matrices for training and testing samples together. However, to avoid using testing samples in model fitting, only training samples are used in constructing kernels and then to solve the optimality problem in Equation (4), while training and testing samples are used together in constructing kernels to make predictions for testing samples.
Incorporating covariates
The proposed MKMR model can readily incorporate covariates in the following form:
 |
(5) |
where 
is the mapping function of the kernel matrix 
for microbiome data 
, and 
is the mapping function of the kernel matrix 
for covariates 
. That is, covariates can be incorporated by simply adding kernel matrices 
(e.g. linear, polynomial kernels) to the proposed MKMR algorithm.
RESULTS
We conducted simulation studies to evaluate the prediction performance of the proposed MKMR(3) method with three kernels, the Bray–Curtis, unweighted UniFrac and weighted UniFrac kernels. We considered several competing methods including the original random forest (RF), random forest using all taxa including internal nodes on the phylogenetic tree as input features (PAAM-RF) [38, 39], lasso regression (Lasso) [17], ridge regression (Ridge), elastic-net regression (EN) with mixing parameter 
, as well as three single-kernel machine regressions [40, 41] with Bray–Curtis kernel (K_BC), weighted UniFrac kernel (K_w) and unweighted UniFrac kernel (K_un). We also compared the prediction performance of MKMR with that of the recently developed deep learning method MDeep [24]. MDeep orders OTUs based on hierarchical clustering results conducted using pairwise patristic distances in the phylogenetic tree. Specifically, OTUs are ordered according to their orders in the dendrogram from hierarchical clustering results and the ordered OTUs are then used as inputs for a CNN for predictions thus using both the phylogenetic tree and OTU abundance levels.
Note that input features of competing methods RF, Lasso, Ridge and EN are abundance levels of OTUs (i.e. abundance levels of leaf nodes on the phylogenetic tree), while input features of PAAM-RF are abundance levels of all nodes on the phylogenetic tree.
We did not include covariates in simulation studies because the purpose of simulation studies is to compare the prediction performance of the proposed MKMR method with that of competing methods. While incorporating covariates in simulation models does affect overall prediction of all methods, it does not change the relative prediction performance of the proposed method to that of competing methods.
Simulation settings
We considered binary outcomes and simulated microbiome data based on a previous study that examined upper-respiratory-tract microbiome data [15, 16, 34]. For each sample, we simulated 856 OTUs equipped with the same phylogenetic tree as in the original upper-respiratory-tract microbiome data. Specifically, for sample 
, we generated OTU counts from a multinomial-Dirichlet distribution 
, where the total count 
follows a negative binomial distribution with mean 1000 and size 25 and the parameters (
) follow a Dirichlet (
, 
, 
, 
; 
), 
. The parameters 
and the overdispersion parameter 
for the Dirichlet distribution were estimated from the upper-respiratory-tract microbiome data.
For each simulation setting with the generating model being introduced below, we simulated a population pool of 10 000 samples with roughly half cases and half controls. This is because the true probability of being a case has a mean close to 0.5 as we scaled signal OTUs with mean 0 and did not include the intercept in the generating model. We then randomly selected 100 cases and 100 controls from the population pool. Within the selected 200 samples, we further randomly selected 100 samples (thus roughly half cases and half controls) as the training set and the other 100 samples (thus roughly half cases and half controls) as the test set. We trained models using training data and evaluated model performance using testing data and repeated this simulation procedure 500 times. For random forest, we fixed the number of trees at 500 and the number of variables randomly sampled at each split as the square root of the total number of predictors. We used 5-fold cross validations within a training data to tune the regularization parameters for Lasso, Ridge, Elastic Net regressions and the proposed MKMR method. For MDeep, we used default parameter values on the authors’ GitHub repository [42]. We evaluate the prediction performance using the area under the ROC Curve (AUC) based on predicted probabilities in test sets across 500 simulations.
Simulation scenario I with informative OTUs that are close to each other on the phylogenetic tree
We generated binary outcomes 
using the following two models:
 |
(6) |
 |
(7) |
where 
denotes the set of informative OTUs, and 
is the abundance level of OTU 
. The ‘scale’ function standardizes a variable to have mean 0 and standard deviation 1, and 
is an indicator function for the presence/absence effect of an OTU. For all simulation scenarios, we set effect sizes of individual informative OTUs to be the same at 
.
In Simulation scenario I, the set of informative OTUs 
is a cluster of OTUs that are close to each other on the phylogenetic tree. Specifically, we followed the method in [15] and partitioned 856 OTUs into 20 clusters by the partitioning around medoids algorithm using pairwise patristic distances in the phylogenetic tree. For outcomes generated using Model A in Equation 6, 
is the cluster of OTUs with total abundance of 21.00%, 10.39% and 7.64%, each consisting of 142, 57 and 54 OTUs, respectively. For outcomes generated with presence/absence information of OTUs using Model B in Equation (7), 
is the cluster with total abundance 21.00%.
Simulation scenario II with informative OTUs that are far away from each other on the phylogenetic tree
In Simulation scenario II, informative OTUs are far away from each other on the phylogenetic tree that spread over the leaf nodes of the phylogenetic tree. Specifically, we selected 10 OTUs each from a different cluster. For outcomes generated using Model A in Equation (6), 
is a set of 10 OTUs with total abundance 31.13%, 11.85% and 5.19%, respectively. For outcomes generated with presence/absence information using Model B in Equation (7), 
is the set of 10 OTUs that are far away from each other on the phylogenetic tree with total abundance 31.13%.
Simulation scenario III with mixed informative OTUs
Simulation scenario III considers a mixture of informative OTUs that are close to or far away from each other on the phylogenetic tree. Outcomes were generated using the following models:
 |
(8) |
 |
(9) |
where 
is a cluster of OTUs that are close to each other on the phylogenetic tree and 
is a set of OTUs that are far away from each other on the phylogenetic tree. For outcomes generated using model C in Equation (8), we considered (1) 
as the cluster with total abundance 21.00% and 
as the set of 10 OTUs with total abundance 5.19%, (2) 
as the cluster with total abundance 10.39% and 
as the set of 10 OTUs with total abundance 11.85% and (3) 
as the cluster with total abundance 7.64% and 
as the set of 10 OTUs with total abundance 31.13%. For outcomes generated using model D in Equation (9), 
is the cluster with total abundance 21.00% and 
is the set of 10 OTUs with total abundance 5.19%.
Simulation results
Table 1 shows means with 2.5% and 97.5% quantiles of AUC and standard errors of means using test sets across 500 simulations for the proposed MKMR(3) method with three kernels, the Bray–Curtis kernel, the weighted UniFrac kernel and the unweighted UniFrac kernel and competing methods including the three single-kernel machine regression models. In each simulation setting, the best prediction results are highlighted in bold.
Table 1.
Simulation results of the proposed MKMR method with three kernels and competing methods including original random forest (RF), random forest using all taxa on the phylogenetic tree (PAAM-RF), Lasso regression (Lasso), ridge regression (Ridge), elastic-net regression (EN), microbiome-based deep learning method (MDeep) and the three single-kernel models with Bray–Curtis kernel (K_BC), weighted UniFrac kernel (K_w) and unweighted UniFrac kernel (K_un). Displayed are AUC means with 2.5% and 97.5% quantiles and standard errors of means (in parentheses) using test sets across 500 simulations for the three simulation scenarios considered. In each simulation scenario, multiple settings with different abundance levels of signal OTUs or presence/absence information of signal OTUs are considered.
| Simulation scenario I: informative OTUs close to each other on the phylogenetic tree | ||||
|---|---|---|---|---|
| Setting | I (Abundance) | II (Abundance) | III (Abundance) | IV (Presence) |
| Abundance | 21.00% | 10.39% | 7.64% | 21.00% |
| No. of OTUs | 142 | 57 | 54 | 142 |
| RF | 0.635 (0.524, 0.744) (0.056) | 0.651 (0.537, 0.762) (0.061) | 0.649 (0.531, 0.763) (0.062) | 0.555 (0.442, 0.665) (0.058) |
| PAAM-RF | 0.770 (0.666, 0.858) (0.050) | 0.770 (0.666, 0.862) (0.050) | 0.770 (0.669, 0.866) (0.051) | 0.580 (0.460, 0.698) (0.061) |
| Lasso | 0.671 (0.500, 0.797) (0.086) | 0.734 (0.585, 0.836) (0.063) | 0.724 (0.500, 0.845) (0.075) | 0.505 (0.420, 0.619) (0.040) |
| EN | 0.670 (0.500, 0.797) (0.048) | 0.728 (0.588, 0.834) (0.051) | 0.718 (0.500, 0.841) (0.048) | 0.508 (0.413, 0.615) (0.042) |
| Ridge | 0.514 (0.424, 0.633) (0.083) | 0.514 (0.422, 0.630) (0.062) | 0.516 (0.425, 0.621) (0.074) | 0.509 (0.422, 0.605) (0.044) |
| MDeep | 0.725 (0.595, 0.834) (0.061) | 0.745 (0.642, 0.843) (0.053) | 0.724 (0.591, 0.825) (0.060) | 0.517 (0.411, 0.645) (0.059) |
| K_BC | 0.714 (0.605, 0.808) (0.054) | 0.718 (0.611, 0.821) (0.052) | 0.704 (0.598, 0.811) (0.057) | 0.525 (0.415, 0.642) (0.059) |
| K_w | 0.810 (0.715, 0.892) (0.043) | 0.784 (0.687, 0.870) (0.049) | 0.784 (0.679, 0.876) (0.051) | 0.554 (0.448, 0.665) (0.059) |
| K_un | 0.515 (0.396, 0.630) (0.061) | 0.515 (0.399, 0.642) (0.063) | 0.511 (0.385, 0.630) (0.061) | 0.598 (0.487, 0.709) (0.058) |
| MKMR (3) | 0.807 (0.717, 0.887) (0.045) | 0.783 (0.685, 0.871) (0.046) | 0.787 (0.685, 0.881) (0.051) | 0.589 (0.468, 0.716) (0.061) |
| Simulation scenario II: informative OTUs far away from each other on the phylogenetic tree | ||||
| Setting | I (Abundance) | II (Abundance) | III (Abundance) | IV (Presence) |
| Abundance | 31.13% | 11.85% | 5.19% | 31.13% |
| No. of OTUs | 10 | 10 | 10 | 10 |
| RF | 0.640 (0.526, 0.748) (0.062) | 0.643 (0.524, 0.759) (0.060) | 0.622 (0.506, 0.743) (0.059) | 0.615 (0.491, 0.728) (0.064) |
| PAAM-RF | 0.649 (0.529, 0.763) (0.061) | 0.635 (0.510, 0.747) (0.062) | 0.593 (0.479, 0.708) (0.060) | 0.600 (0.466, 0.725) (0.068) |
| Lasso | 0.641 (0.500, 0.784) (0.087) | 0.630 (0.500, 0.783) (0.086) | 0.616 (0.495, 0.772) (0.087) | 0.507 (0.412, 0.633) (0.047) |
| EN | 0.644 (0.500, 0.779) (0.047) | 0.633 (0.500, 0.783) (0.049) | 0.621 (0.488, 0.770) (0.045) | 0.499 (0.407, 0.596) (0.044) |
| Ridge | 0.516 (0.419, 0.617) (0.083) | 0.515 (0.433, 0.631) (0.082) | 0.511 (0.423, 0.618) (0.084) | 0.508 (0.409, 0.630) (0.050) |
| MDeep | 0.694 (0.575, 0.800) (0.055) | 0.691 (0.571, 0.798) (0.056) | 0.694 (0.577, 0.800) (0.058) | 0.521 (0.399, 0.642) (0.061) |
| K_BC | 0.738 (0.633, 0.838) (0.052) | 0.690 (0.570, 0.798) (0.058) | 0.625 (0.515, 0.737) (0.056) | 0.538 (0.415, 0.655) (0.062) |
| K_w | 0.667 (0.543, 0.776) (0.060) | 0.633 (0.511, 0.754) (0.062) | 0.590 (0.478, 0.695) (0.061) | 0.516 (0.390, 0.635) (0.063) |
| K_un | 0.508 (0.401, 0.625) (0.058) | 0.512 (0.407, 0.617) (0.056) | 0.538 (0.427, 0.656) (0.059) | 0.527 (0.416, 0.632) (0.058) |
| MKMR (3) | 0.730 (0.595, 0.837) (0.062) | 0.672 (0.544, 0.792) (0.068) | 0.610 (0.491, 0.715) (0.060) | 0.532 (0.408, 0.644) (0.060) |
| Simulation scenario III: mixture of scenarios I and II | ||||
| Setting | I (Abundance) | II (Abundance) | III (Abundance) | IV (Presence) |
| Abundance | 21.00% + 5.19% | 10.39% + 11.85% | 7.64% + 31.13% | 21.00% + 5.19% |
| No. of OTUs | 142 + 10 | 57 + 10 | 54 + 10 | 142 + 10 |
| RF | 0.640 (0.516, 0.747) (0.059) | 0.667 (0.545, 0.784) (0.060) | 0.681 (0.580, 0.788) (0.055) | 0.613 (0.500, 0.727) (0.060) |
| PAAM-RF | 0.719 (0.601, 0.818) (0.055) | 0.730 (0.620, 0.837) (0.057) | 0.741 (0.625, 0.845) (0.058) | 0.613 (0.501, 0.727) (0.059) |
| Lasso | 0.619 (0.500, 0.761) (0.083) | 0.686 (0.500, 0.807) (0.079) | 0.730 (0.566, 0.839) (0.066) | 0.517 (0.437, 0.640) (0.049) |
| EN | 0.626 (0.500, 0.771) (0.050) | 0.686 (0.500, 0.808) (0.049) | 0.724 (0.566, 0.831) (0.052) | 0.520 (0.443, 0.644) (0.045) |
| Ridge | 0.517 (0.417, 0.626) (0.082) | 0.517 (0.416, 0.622) (0.076) | 0.521 (0.423, 0.634) (0.065) | 0.508 (0.420, 0.612) (0.049) |
| MDeep | 0.704 (0.589, 0.818) (0.058) | 0.748 (0.638, 0.848) (0.056) | 0.766 (0.654, 0.857) (0.055) | 0.529 (0.417, 0.643) (0.060) |
| K_BC | 0.701 (0.592, 0.809) (0.058) | 0.738 (0.627, 0.836) (0.051) | 0.771 (0.668, 0.857) (0.049) | 0.560 (0.443, 0.674) (0.058) |
| K_w | 0.764 (0.661, 0.857) (0.050) | 0.759 (0.648, 0.856) (0.053) | 0.773 (0.666, 0.875) (0.053) | 0.553 (0.428, 0.678) (0.062) |
| K_un | 0.532 (0.415, 0.641) (0.058) | 0.521 (0.416, 0.637) (0.058) | 0.513 (0.407, 0.625) (0.057) | 0.671 (0.553, 0.780) (0.060) |
| MKMR (3) | 0.762 (0.673, 0.846) (0.047) | 0.768 (0.673, 0.863) (0.049) | 0.786 (0.675, 0.877) (0.051) | 0.644 (0.498, 0.767) (0.068) |
We also considered additional kernels that cover more granular scenarios in between that of weighted and unweighted UniFrac distances derived from the generalized UniFrac distance with different values of the parameter 
= 0, 0.25, 0.5, 0.75. For simplicity of the presentation, we included these additional simulation results of MKMR(7) and additional single-kernel models K_g(0), K_g(0.25), K_g(0.5) and K_g(0.75) in Supplementary data Section A Table E1 but only MKMR(3) in the main text.
Overall, simulation results suggest that the proposed MKMR(3) method has superior prediction performance over that of competing methods. For simulation scenario I when signal OTUs are close to each other on the phylogenetic tree, the multi-kernel method has prediction performance almost as good as the best single kernel-based method with the kernel reflecting the true microbiome-outcome association. For simulation scenario II when signal OTUs are far away from each other on the phylogenetic tree, the multi-kernel method has the best performance in all settings, except when abundance level is relatively small when MDeep method performs the best and when signals are present/absent information when RF performs the best. We noticed that MDeep is less sensitive to abundance levels than other methods, probably due to the convolution layers that have combined some rare taxa. Since the three kernels used for MKMR do not capture the setting with signals being present/absent and are far away from each other on the phylogenetic tree, it is expected that MKMR does not perform as well as RF. If we use a kernel transformed from the Hamming distance, this scenario will be covered. However, as such OTUs are not widely observed to be associated with health outcomes, it was not included in the MKMR method. For simulation scenario II, the prediction performance of MKMR is also almost always as good as that of the best model out of the three single-kernel machine regressions that reflects the true microbiome-outcome associations.
In scenario III when outcomes are related to a mixture of both informative OTUs that are close to or far away from each other on the phylogenetic tree, the proposed MKMR method has the best performance among all methods. This is because MKMR is able to capture a combination of different types of signals by learning a conic combination of different kernels. This is very promising as in real microbiome studies, forms of microbiome-outcome associations will be unknown and mixed.
For competing methods, as expected, when comparing RF and PAAM-RF, PAAM-RF, which uses the phylogenetic tree structure information, has better prediction performance when signal OTUs are close to each other on the phylogenetic tree (scenarios I and III), and has similar or a bit worse prediction performance when signal OTUs are far away from each other on the phylogenetic tree (Scenario II). Among the regularized regression methods, Ridge barely has any prediction ability, while Lasso and EN have prediction performance as good as RF when signal OTUs are far away from each other on the phylogenetic tree (Scenario II). As the solution of Ridge is not sparse compared with Lasso and EN, it is no surprise Ridge has poor performance when signal OTUs are only a small fraction out of all OTUs (e.g. 10 out of 856 in scenario II). The deep learning method MDeep performs well in general in all three scenarios although not as well as PAAM-RF in scenario I. It is worth noticing that MDeep does not have much predictability when presence/absence information of signal OTUs is in effect.
Figure 2 summarizes kernel weights of the three kernels considered in the MKMR(3) method in simulation studies. It shows that the kernel that reflects the true signal OTUs has the largest median kernel weights across the 500 simulations. For simulation scenario I when signal OTUs are abundant and close to each other on the phylogenetic tree, the weighted UniFrac kernel that emphasizes OTU abundance and phylogenetic tree information has the largest kernel weight. For simulation Scenario II when signal OTUs are abundant but far away from each other on the phylogenetic tree, the Bray–Curtis kernel that emphasizes OTU abundance has the largest kernel weight. For simulation scenario I when signal OTUs are close to each other on the phylogenetic tree and their presence/absence information is associated with outcomes, the unweighted UniFrac kernel that emphasizes OTU presence/absence has the largest kernel weight, while for simulation scenario III with a mixture of informative OTUs that are close to or far away from each other on the phylogenetic tree, the weighted UniFrac kernel and the Bray–Curtis kernel have larger weights. This also explains the results in Table 1 that the MKMR(3) method can use multiple kernels to recover different forms of true signals. Moreover, by examining kernel weights, we are able to know the forms of microbiome-outcome association.
Figure 2.
Simulation results of kernel weights for the proposed MKMR(3) method with three kernels, Bray–Curtis kernel (K_BC), weighted UniFrac kernel (K_w) and unweighted UniFrac kernel (K_un) across 500 simulations for the three simulation scenarios considered.
Real data applications
We applied the proposed MKMR method to predict smoking status using the throat microbiome data and to predict multiple health outcomes using the gut micrbiome data from the AGP. To evaluate the prediction performance, we randomly split each dataset 500 times into equal-sized training data to fit the model, and testing data to evaluate the model performance. We reported prediction results of both MKMR(3) and MKMR(7) as well as competing methods including that of the seven single-kernel models.
Throat microbiome and smoking data
We predicted smoking status with throat microbiome data using the proposed MKMR method and several competing methods. The original study was conducted to investigate the effect of smoking on oropharyngeal and nasopharyngeal bacterial communities [31]. Using the QIIME pipeline, microbiome OTUs were constructed at 97% similarity level with swap samples from the upper airway. The final dataset has 28 smokers and 32 nonsmokers with 856 OTUs for each sample. Gender and antibiotics use within 3 months were also collected for all samples. There is no associations between smoking status and gender and antibiotics use within 3 months using Fisher’s exact test (Table E2 in the Supplementary data Section B). Therefore, we did not include gender and antibiotic use in building the prediction model.
Table 2 shows means with 2.5% and 97.5% quantiles of AUCs and standard errors of means for the proposed MKMR model (two versions of MRMK, 3-kernel MKMR(3) and 7-kernel MKMR(7)) and competing methods. We can see that using microbiome profiles, MKMR(3) has an average AUC of 0.714, which is similar to the single-kernel method with Bray–Curtis kernel (K_BC) with a mean AUC of 0.721. We also noticed that seven single-kernel methods all have relatively high AUCs ranging from 0.679 to 0.721, which indicates that abundant OTUs may be the main signals, while other forms of OTUs are also potential signals. Among other competing methods, RF has the best performance with an average AUC of 0.708, which is consistent with published work that suggest RF has good prediction performance in general. MDeep method, on the other hand, has no predictability in this application probably due to the small sample size.
Table 2.
Prediction results predicting smoking status using throat microbiome data with AUC means with 2.5% and 97.5% quantiles and standard errors of means (in parentheses) in test sets across 500 50/50 random splits. MKMR(3) is the proposed MKMR model with three kernels, and MKMR(7) has seven kernels.
| Method | AUC |
|---|---|
| RF | 0.708 (0.557, 0.831) |
| PAAM-RF | 0.697 (0.547, 0.840) |
| Lasso | 0.619 (0.500, 0.796) |
| EN | 0.649 (0.500, 0.820) |
| Ridge | 0.666 (0.500, 0.828) |
| MDeep | 0.499 (0.198, 0.768) |
| K_BC | 0.721 (0.589, 0.855) |
| K_w | 0.707 (0.561, 0.849) |
| K_un | 0.679 (0.508, 0.843) |
| K_g(0) | 0.716 (0.560, 0.864) |
| K_g(0.25) | 0.710 (0.569, 0.849) |
| K_g(0.5) | 0.707 (0.568, 0.847) |
| K_g(0.75) | 0.707 (0.570, 0.847) |
| MKMR(3) | 0.714 (0.571, 0.853) |
| MKMR(7) | 0.712 (0.571, 0.848) |
Figure 3 displays kernel weights from the proposed MKMR(3) and MKMKR(7) methods for the throat microbiome and smoking data. The added four kernels in MKMR(7) have very small weights. Thus, we focus on kernel weights from MKMR(3). Overall, the Bray–Curtis kernel has the largest median kernel weights. This is consistent with the prediction results in Table 2 that the single-kernel method with the Bray–Curtis kernel has the largest AUC. Based on the learned kernel weights, we can speculate that the abundance information of some OTUs that are far away from each other on the phylogenetic tree is the most predictive of the smoking status.
Figure 3.
Kernel weights of the proposed MKMR(3) method with three kernels (Bray–Curtis kernel (K_BC), weighted UniFrac kernel (K_w) and unweighted UniFrac kernel (K_un)) and MKMR(7) with seven kernels (K_BC, K_w, K_unand four generalized UniFrac kernels K_g(0), K_g(0.25), K_g(0.5) and K_g(0.75)) from predicting the smoking status with throat moicrobiome data across 500 50/50 random splits.
AGP microbiome data
We also applied the proposed MKMR method (both MKMR(3) and MKMR(7)) to the gut microbiome data from the AGP (www.americangut.org). AGP is a crowdfunded, crowdsourced microbiome project aiming at creating a comprehensive map of the human microbiome. It includes the 16S rRNA V4 region sequencing from participants physical samples (fecal, oral, skin, pet or environmental) using Illumina MiSeq platform and metadata including health, lifestyle and diet, as described in [32]. We used a subset of the AGP microbiome samples who were US residents by filtering using country_residence == “United States”. We further excluded OTUs with (1) fewer than three individuals having counts 
, or 2) relative abundances 
, resulting in different sample size (
) and number of OTUs (
) for each outcome. Table E3 in Supplementary data Section C summarizes descriptive statistics of variables we used in the AGP Data.
We used gut microbiome data to predict the binary small intestine bacteria overgrow (SIBO) disease status, the binary status of antibiotics use within a year and the binary status of probiotic use within a year. Table 3 displays summaries of these outcomes, as well as covariates used to predict each outcome. Covariates in AGP are either binary or ordinal (see Table E3 in Suppementary data Section C for their descriptive statistics). For simplicity, we categorized those ordinal covariates into binary variables. Tables E4, E5 and E6 in Supplementary data Section C summarize associations between binary covariates and each outcomes.
Table 3.
Summary of the three binary health outcomes (SIBO status, antibiotic use and probiotic use) predicted using AGP microbiome data. Different covariates were used for predicting each of the three outcomes of interest together with the gut microbiome data. Sample sizes and numbers of OTUs are also shown for each outcome.
| Outcome | SIBO status | Antibiotics use | Probiotic use |
|---|---|---|---|
| (1=diseased) | (1=used within a year) | (1=used within a year) | |
| Covariates used | Consume whole grains | Male | BMI between 18.5 and 30 |
| Normal formed stool | Exercise outdoors/both | Animal products ABX | |
| Lactose tolerant | Age under 40 | Consume fermented plant (daily/regularly) | |
| Appendix not removed/not sure | No seasonal allergies | ||
| Sample size | 351 | 349 | 366 |
| Number of 1’s | 19 | 121 | 86 |
| (percentage) | (5.41%) | (34.67%) | (23.50%) |
| No. of OTUs | 1815 | 1827 | 1834 |
For single-kernel models and the proposed MKMR model, we incorporated covariates using the method described in the Method Section via linear kernels based on the assumption of linear relationships between the covariates and the outcomes. For the method of Lasso, if we simply combine covariates and OTUs and conduct variable selection, covariates may be less frequently selected as they are dominated by OTUs in numbers. Therefore, we added two penalty factors on tuning parameters of covariates and tuning parameters of OTUs in the 5-fold cross-validation step to allow differential shrinkage for covariates and OTUs. Specifically, with some initial investigation, we set the penalty factor for covariates as 
and the penalty factor for OTUs as 1. These two penalty factors will multiply the tuned parameter values for pasrameters for covariates and OTUs from 5-fold cross-validation. For RF and PAAM-RF, we included covariates during every node split in addition to the randomly selected subset of OTUs.
Table 4 shows the prediction results of the proposed MKMR method and competing methods using microbiome and covariates. We also included prediction results using microbiome only and covariates only in Supplementary data Section C. For SIBO disease status, single-kernel model 
has the highest AUC of 0.772, the proposed MKMR(3) and MKMR(7) have similar AUCs of 0.767 and 0.763, respectively. This suggests that presence/absence of some taxa that are close to each other on the phylogenetic tree might be most predictive of SIBO. To predict the binary outcome antibiotics use within 3 months yes/no, single-kernel models have similar AUCs in the range 0.624–0.628 but lower than that of the proposed MKMR(3) and MKMR(7), which has the same AUC 0.648. This suggests that multiple forms of taxa are potentially contributing to predicting of antibiotic use. Similar observations can be made for probiotic use. Note that prediction performance of the proposed MKMR is also better than that of other conventional methods. These observations suggest that the proposed MKMR method is very promising and have much improved prediction performance over competing methods.
Table 4.
Prediction results using AGP microbiome data with AUC means and 2.5% and 97.5% quantiles (in parentheses) in test sets across 500 50/50 random splits.
| Outcome | SIBO disease | Antibiotics use | Probiotic use |
|---|---|---|---|
| RF | 0.740 (0.592, 0.854) | 0.637 (0.556, 0.702) | 0.641 (0.567, 0.715) |
| PAAM-RF | 0.733 (0.579, 0.856) | 0.634 (0.563, 0.701) | 0.624 (0.554, 0.693) |
| Lasso | 0.688 (0.500, 0.844) | 0.583 (0.500, 0.647) | 0.676 (0.601, 0.739) |
| EN | 0.707 (0.500, 0.853) | 0.581 (0.500 0.649) | 0.680 (0.609, 0.747) |
| Ridge | 0.647 (0.500, 0.850) | 0.554 (0.500, 0.655) | 0.578 (0.500, 0.690) |
| K_BC | 0.755 (0.590, 0.882) | 0.628 (0.563, 0.691) | 0.703 (0.635, 0.772) |
| K_w | 0.724 (0.547, 0.883) | 0.624 (0.560, 0.686) | 0.699 (0.630, 0.767) |
| K_un | 0.772 (0.617, 0.889) | 0.624 (0.558, 0.687) | 0.706 (0.634, 0.774) |
| K_g(0) | 0.741 (0.563, 0.877) | 0.624 (0.558, 0.687) | 0.707 (0.636, 0.775) |
| K_g(0.25) | 0.732 (0.530, 0.878) | 0.626 (0.561, 0.690) | 0.706 (0.635, 0.775) |
| K_g(0.5) | 0.723 (0.536, 0.876) | 0.626 (0.561, 0.689) | 0.703 (0.633, 0.772) |
| K_g(0.75) | 0.722 (0.540, 0.880) | 0.625 (0.561, 0.687) | 0.701 (0.632, 0.769) |
| MKMR(3) | 0.767 (0.639, 0.877) | 0.648 (0.574, 0.710) | 0.702 (0.631, 0.772) |
| MKMR(7) | 0.763 (0.634, 0.874) | 0.648 (0.578, 0.712) | 0.703 (0.632, 0.775) |
Among competing methods, RF has the best performance predicting SIBO disease, while ridge regression has the best performance predicting probiotic use and antibiotic use. We notice that in simulation studies, ridge regressions barely have any prediction ability, suggesting that in real microbiome studies, there may be many signal OTUs that are associated with a health outcome with small effects. Our simulation scenarios might be too simple to capture complicated real case scenarios. However, our proposed MKMR method that captures multiple forms of microbiome signals performed the best for all four outcomes considered.
We included additional prediction results in Table E7 in Supplementary data Section C, for the three binary outcomes using AGP microbiome data only and using covariates only, respectively. As the deep learning method MDeep was not implemented to handle covariates, we included MDeep prediction results using only the microbiome data without covariates for the AGP applications in Table E7. MDeep does not perform as well as several other competing methods and the proposed MKMR method.
Figure 4 displays kernel weights learned from the proposed MKMR(3) and MKMKR(7) methods (without the covariates) in predicting SIBO disease, antibiotic use and probiotic use with AGP microbiome data. For the three outcomes, the learned kernel weights are consistent with prediction results of single-kernel methods, summarized in Table E7 in Supplementary data Section C. That is, the kernel in the single-kernel method with the best prediction performance is the kernel with the largest kernel weight among kernels used in the proposed MKMR model. Specifically, for predicting SIBO disease, with MKMR(3), the unweighted UniFrac kernel has the largest kernel weight, suggesting that the presence/absence of some OTUs that are close to each other on the phylogenetic tree is likely to be related to SIBO disease. With MKMR(7), the unweighted UniFrac kernel K_un, the generalized UniFrac kernels K_g(0), K_g(0.25) all have some weights, suggesting that rare taxa are predicting SIBO disease. For predicting antibiotic use, MKMR(3) and MKMR(7) have similar learned kernel weights for the three overlapping kernels, while the four added kernels in MKMR(7) have very small weights. For this outcome, the Bray–Curtis kernel has the largest kernel weight, suggesting that the abundance of some OTUs that are far away from each other on the phylogenetic tree is probably most predictive. For predicting probiotic use, all three/seven kernels have small weights, suggesting that multiple forms of microbiome taxa are predictive of probiotic use with each having a relatively small effect.
Figure 4.
Kernel weights of the proposed MKMR(3) method with three kernels (Bray–Curtis kernel (K_BC), weighted UniFrac kernel (K_w) and unweighted UniFrac kernel (K_un)) and MKMR(7) with seven kernels (K_BC, K_w, K_un and four generalized UniFrac kernels K_g(0), K_g(0.25), K_g(0.5) and K_g(0.75)) from predicting SIBO disease (Figure 4A), antibiotic use (Figure 4B) and probiotic use (Figure 4C) with AGP gut microbiome data across 500 50/50 random splits.
DISCUSSION
In this article, we proposed an MKMR model to predict health outcomes using microbiome data. The idea of the proposed MKMR model is to ensemble multiple kernels with an optimal set of weights for an improved prediction performance. The kernel-ensemble approach enables us to capture different predictive signals that have different forms of associations with a health outcome to be predicted, while existing kernel-based prediction methods only select the optimal one. Through incorporating multiple kernels that focus on different forms of microbiome signals, as well as kernels for confounding variables, multiple prediction signals can be combined to improve prediction performance.
The proposed MKMR model first considers multiple microbiome distance matrices that capture various aspects of information in microbiome profiles including both abundant and sparse OTUs and phylogenetic information. From these multiple microbiome distance matrices, we construct corresponding kernels and implement these kernels in a regression model through learning a conic combination of these kernels from data in order to capture diverse information in microbiome profiles with health outcomes for the best prediction performance.
In simulation studies, the single-kernel machine regression model with the kernel that reflects the true form of association between microbiome profiles and a health outcome always has the best prediction performance as expected, except for the scenario when informative OTUs are presence/absence OTUs and are far away from each other on the phylogenetic tree. Although we could have used the Hamming distance and the corresponding kernel to capture these signal OTUs, this form of associations are not frequently observed in real microbiome studies. Simulation results showed that the proposed MKMR method has prediction performance almost always as good as that of the single-kernel regression method with the kernel that reflects the true microbiome-outcome association. This is very promising for real microbiome studies when microbiome could be associated with health outcomes in complex ways that are unknown to us.
We applied the proposed MKMR method and competing methods to two real microbiome datasets to predict multiple health outcomes, that is, using throat microbiome data to predict binary smoking status, and using gut microbiome data to predict binary SIBO disease status, binary status of antibiotics use within a year and binary status of probiotic use within a year. Because different kinds of information in microbiome profiles may associate with these health outcomes that are unknown to us, a method that is able to capture these diverse forms of microbiome predictive signals will improve the prediction performance. As expected, the proposed MKMR method consistently has the best prediction performance among all methods. We also want to emphasize that although we focused on predicting binary outcomes with the proposed MKMR framework, it can be extended straightforwardly for other outcomes such as continuous or count outcomes. We expect to apply the proposed multi-kernel learning method to large-scale microbiome data as they become available in order to reach better conclusions.
One common complication with predictions using microbiome data is it is possible that when doing prediction using testing samples, there are new taxa that do not exist in training samples, and vice versa. This could also result in a new phylogenetic tree using testing samples. This problem does not usually exist in predictions using other types of omics data such as genetic data or DNA methylation data when arrays with fixed number of features can be used for training and testing samples. If there are two different sets of taxa in training samples and testing samples, we will restrict to use overlap taxa and trim the phylogenetic trees to have a common sub-tree. With this sub-tree and overlap taxa, we can still calculate microbiome distances thus kernels and conduct predictions using the trained multi-kernel model. In order to minimize differences between these two sets of taxa from training samples and testing samples, we suggest to use the same sequencing protocol to generate microbiome data in training samples and testing samples.
Key Points
The proposed MKMR method extracts different forms of microbiome signals from different microbiome distance measures.
The proposed MKMR method learns an optimal conic combination of different kernels, with learned weights of multiple kernels indicating contributions of individual kernel, that is, contributions of individual forms of microbiome signals.
The proposed MKMR is robust and consistently has improved prediction performance across the four health outcomes predicted using throat microbiome data and gut microbiome data.
Supplementary Material
Author Biographies
Bing Li was a PhD student in the School of Public Health at Brown University, USA.
Tian Wang is a postdoctoral associate at Columbia University, USA.
Min Qian is an Associate Professor at Columbia University, USA.
Shuang Wang is a Professor at Columbia University, USA.
Contributor Information
Bing Li, Department of Biostatistics, School of Public Health, Brown University, Providence, Rhode Island, U.S.A.
Tian Wang, Department of Biostatistics, Mailman School of Public Health, Columbia University, 722 West 168th Street, New York, New York, 10032 U.S.A.
Min Qian, Department of Biostatistics, Mailman School of Public Health, Columbia University, 722 West 168th Street, New York, New York, 10032 U.S.A.
Shuang Wang, Department of Biostatistics, School of Public Health, Brown University, Providence, Rhode Island, U.S.A.
DATA AVAILABILITY
The Matlab and R code for the proposed MKMR method with a tutorial and a sample data set is available at https://github.com/BingLi17/MKMR.
The gut microbiome data and the corresponding clinical data were downloaded from the American Gut Project (www.americangut.org).
Funding
Part of the study was supported by the departmental fund of Department of Biostatistics, Columbia University.
References
- 1. Tringe SG, Rubin EM. Metagenomics: Dna sequencing of environmental samples. Nat Rev Genet 2005; 6(11): 805–14. [DOI] [PubMed] [Google Scholar]
- 2. Lasken RS. Genomic sequencing of uncultured microorganisms from single cells. Nat Rev Microbiol 2012;10(9):631–40. [DOI] [PubMed] [Google Scholar]
- 3. Rapin A, Pattaroni C, Marsland BJ, Harris NL. Microbiota analysis using an illumina miseq platform to sequence 16s rrna genes. Current Protocol Mouse Biol 2017;7(2):100–29. [DOI] [PubMed] [Google Scholar]
- 4. Gregory Caporaso J, Kuczynski J, Stombaugh J, et al. Qiime allows analysis of high-throughput community sequencing data. Nat Methods 2010;7(5):335–6. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 5. Nguyen N-P, Warnow T, Pop M, White B. A perspective on 16s rrna operational taxonomic unit clustering using sequence similarity. NPJ Biofilms Microbiomes 2016;2(1):1–8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 6. Clemente JC, Ursell LK, Parfrey LW, Knight R. The impact of the gut microbiota on human health: an integrative view. Cell 2012;148(6):1258–70. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 7. Jing W, Peters BA, Dominianni C, et al. Cigarette smoking and the oral microbiome in a large study of american adults. ISME J 2016;10(10):2435–46. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 8. Hooper LV, Gordon JI. Commensal host-bacterial relationships in the gut. Science 2001;292(5519):1115–8. [DOI] [PubMed] [Google Scholar]
- 9. Morgan XC, Tickle TL, Sokol H, et al. Dysfunction of the intestinal microbiome in inflammatory bowel disease and treatment. Genome Biol 2012;13(9):R79. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 10. Chang Q, Luan Y, Sun F. Variance adjusted weighted unifrac: a powerful beta diversity measure for comparing communities based on phylogeny. BMC Bioinform 2011;12(1):118. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 11. Lozupone C, Knight R. Unifrac: a new phylogenetic method for comparing microbial communities. Appl Environ Microbiol 2005;71(12):8228–35. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 12. Lozupone CA, Hamady M, Kelley ST, Knight R. Quantitative and qualitative beta diversity measures lead to different insights into factors that structure microbial communities. Appl Environ Microbiol 2007;73(5):1576–85. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 13. Chen J, Bittinger K, Charlson ES, et al. Associating microbiome composition with environmental covariates using generalized unifrac distances. Bioinformatics 2012;28(16):2106–13. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 14. Roger Bray J, Curtis JT. An ordination of the upland forest communities of southern Wisconsin. Ecol Monogr 1957;27(4):326–49. [Google Scholar]
- 15. Zhao N, Chen J, Carroll IM, et al. Testing in microbiome-profiling studies with mirkat, the microbiome regression-based kernel association test. Am J Hum Genet 2015;96(5):797–807. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 16. Koh H, Blaser MJ, Li H. A powerful microbiome-based association test and a microbial taxa discovery framework for comprehensive association mapping. Microbiome 2017;5(1):45. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 17. Tibshirani R. Regression shrinkage and selection via the lasso. J R Stat Soc B Methodol 1996;58(1):267–88. [Google Scholar]
- 18. Breiman L. Random forests. Mach Learn 2001;45(1):5–32. [Google Scholar]
- 19. Tanaseichuk O, Borneman J, Jiang T. Phylogeny-based classification of microbial communities. Bioinformatics 2014;30(4):449–56. [DOI] [PubMed] [Google Scholar]
- 20. Chen L, Liu H, Kocher J-PA, et al. Glmgraph: an r package for variable selection and predictive modeling of structured genomic data. Bioinformatics 2015;31(24):3991–3. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 21. Xiao J, Chen L, Johnson S, et al. Predictive modeling of microbiome data using a phylogeny-regularized generalized linear mixed model. Front Microbiol 2018;9:1391. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 22. Chen X, Zhu Z, Zhang W, et al. Human disease prediction from microbiome data by multiple feature fusion and deep learning. Iscience 2022;25(4):104081. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 23. Grazioli F, Siarheyeu R, Alqassem I, et al. Microbiome-based disease prediction with multimodal variational information bottlenecks. PLoS Comput Biol 2022;18(4):e1010050. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 24. Wang Y, Bhattacharya T, Jiang Y, et al. A novel deep learning method for predictive modeling of microbiome data. Brief Bioinform 2021;22(3):bbaa073. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 25. Sharma D, Paterson AD, Wei X. Taxonn: ensemble of neural networks on stratified microbiome data for disease prediction. Bioinformatics 2020;36(17):4544–50. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 26. Reiman D, Metwally AA, Sun J, Dai Y. Popphy-cnn: a phylogenetic tree embedded architecture for convolutional neural networks to predict host phenotype from metagenomic data. IEEE J Biomed Health Inform 2020;24(10):2993–3001. [DOI] [PubMed] [Google Scholar]
- 27. Giliberti R, Cavaliere S, Mauriello IE, et al. Host phenotype classification from human microbiome data is mainly driven by the presence of microbial taxa. PLoS Comput Biol 2022;18(4):e1010066. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 28. Turnbaugh PJ, Hamady M, Yatsunenko T, et al. A core gut microbiome in obese and lean twins. Nature 2009;457(7228):480–4. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 29. Karlsson FH, Tremaroli V, Nookaew I, et al. Gut metagenome in european women with normal, impaired and diabetic glucose control. Nature 2013;498(7452):99–103. [DOI] [PubMed] [Google Scholar]
- 30. Bach F, R, Thibaux R, Jordan MI. Computing regularization paths for learning multiple kernels. In: Advances in neural information processing systems, ACM Digital Library, 2005, 73–80.
- 31. Charlson ES, Chen J, Custers-Allen R, et al. Disordered microbial communities in the upper respiratory tract of cigarette smokers. PloS One 2010;5(12):e15216. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 32. Liu M, Koh H, Kurtz ZD, et al. Oxalobacter formigenes-associated host features and microbial community structures examined using the american gut project. Microbiome 2017;5(1):108. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 33. Pan W. Relationship between genomic distance-based regression and kernel machine regression for multi-marker association testing. Genet Epidemiol 2011;35(4):211–6. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 34. Chen J, Li H. Kernel methods for regression analysis of microbiome compositional data. In: Topics in Applied Statistics. Springer, Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 55) 2013, 191–201. [Google Scholar]
- 35. Higham NJ. Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Appl 1988;103:103–18. [Google Scholar]
- 36. Boyd S, Vandenberghe L. Convex optimization. Cambridge university press, 2004. 10.1017/CBO9780511804441. [DOI] [Google Scholar]
- 37. Bach FR, Lanckriet GRG, and Jordan MI. Multiple kernel learning, conic duality, and the smo algorithm. In Proceedings of the twenty-first international conference on Machine learning, Banff, Canada, pp. 6, 2004.
- 38. Jyotsna Talreja Wassan Haiying Wang Fiona Browne, and Huiru Zheng. Paam-ml: A novel phylogeny and abundance aware machine learning modelling approach for microbiome classification. In 2018 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), Madrid, Spain, pp. 44–49, 2018. 10.1109/BIBM.2018.8621382. [DOI]
- 39. Wassan JT, Wang H, Browne F, Zheng H. Phy-pmrfi: phylogeny-aware prediction of metagenomic functions using random forest feature importance. IEEE Trans Nanobiosci 2019;18(3):273–82. [DOI] [PubMed] [Google Scholar]
- 40. Liu D, Lin X, Ghosh D. Semiparametric regression of multidimensional genetic pathway data: least-squares kernel machines and linear mixed models. Biometrics 2007;63(4):1079–88. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 41. Liu D, Ghosh D, Lin X. Estimation and testing for the effect of a genetic pathway on a disease outcome using logistic kernel machine regression via logistic mixed models. BMC Bioinform 2008;9(1):292. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 42. MDeep . https://github.com/lichen-lab/MDeep.
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Data Availability Statement
The Matlab and R code for the proposed MKMR method with a tutorial and a sample data set is available at https://github.com/BingLi17/MKMR.
The gut microbiome data and the corresponding clinical data were downloaded from the American Gut Project (www.americangut.org).