R: Bow-free covariance search and data de-correlation

SEMbap {SEMgraph}

R Documentation

Bow-free covariance search and data de-correlation

Description

SEMbap() function implements different deconfounding methods to adjust the data matrix by removing latent sources of confounding encoded in them. The selected methods are either based on: (i) Bow-free Acyclic Paths (BAP) search, (ii) LVs proxies as additional source nodes of the data matrix, Y or (iii) spectral transformation of Y.

Usage

SEMbap(
  graph,
  data,
  group = NULL,
  dalgo = "cggm",
  method = "BH",
  alpha = 0.05,
  hcount = "auto",
  cmax = Inf,
  limit = 200,
  verbose = FALSE,
  ...
)

Arguments

`graph`	An igraph object.
`data`	A matrix whith rows corresponding to subjects, and columns to graph nodes (variables).
`group`	A binary vector. This vector must be as long as the number of subjects. Each vector element must be 1 for cases and 0 for control subjects. If `NULL` (default), confouding within group will not be considered.
`dalgo`	Deconfounding method. Four algorithms are available: "cggm" (default). The algorithm make: (i) exhaustive search of bow-free significant covariances (see details) through `Shipley.test` function; (ii) estimation of the inverse of the selected covariance matrix (i.e. the precision matrix, W) through `fitConGraph` function; (iii) obtain the de-correlated data matrix, Z by multiplying the data matrix, Y rightward by the square root of the estimated precision matrix, Z=YW^(1/2) as suggested by Grassi, Palluzzi and Tarantino (2022). "glpc". The algorithm first makes an exhaustive search of bow-free significant covariances through `Shipley.test` function. Once obtained the adjacency matrix, Graph-Laplacian PCA (gLPCA) algorithm (Jiang et al., 2013) learns a low dimensional representation of the observed data matrix that incorporates bow-free structure. Then, the DAG is extended by including the confounding proxies, i.e. LVs, as additional source nodes defined by last q principal component scores of gLPCA and these LV scores are added to the data matrix, Z=cbind(LV,Y). "pc". The procedure add additional source nodes to DAG as in "glpc" algorithm, but confounding proxies are the q principal component scores extracted by Spectral decomposition (SVD) selecting only graph nodes and without graph edge information and bow-free covariance search. "trim". Ćevid et al. (2020) suggest multiplying the data matrix, Y leftward by a well selected spectrum transformation matrix, T which modifies the singular values of Y, while keeping its singular vectors intact, Z=TY. Trim transform limits all singular values to be at most some costant (t), where t = median of the singuar values.
`method`	Multiple testing correction method. One of the values available in `p.adjust`. By default, `method` is set to "BH" (i.e., Benjamini-Hochberg multiple test correction).
`alpha`	Significance level for false discovery rate (FDR) used for d-separation test. This argument is used to control data de-correlation. A higher `alpha` level includes more hidden covariances, thus considering more sources of confounding. If `alpha = 0`, data de-correlation is disabled. By default, `alpha = 0.05`.
`hcount`	The number of latent (or hidden) variables. By default `hcount="auto"`, the hidden count is determined with a permutation method (see details). Currently ignored if (dalgo ="cggm" or "trim").
`cmax`	Maximum number of parents set, C. This parameter can be used to perform only those tests where the number of conditioning variables does not exceed the given value. High-dimensional conditional independence tests can be very unreliable. By default, cmax = Inf.
`limit`	An integer value corresponding to the graph size (vcount) tolerance. Beyond this limit, the precision matrix is estimated by "glasso" algorithm (FHT, 2008) to reduce the computational burden of the exaustive BAP search of the `Shipley.test` procedure. By default, `limit = 200`.
`verbose`	A logical value. If FALSE (default), the processed graphs will not be plotted to screen.
`...`	Currently ignored.

Details

Missing edges in causal network inference using a directed acyclic graph (DAG) are frequently hidden by unmeasured confounding variables. A Bow-free Acyclic Paths (BAP) search is performed with d-separation tests between all pairs of variables with missing connection in the input DAG, adding a bidirected edge (i.e., bow-free covariance) to the DAG when there is an association between them. The d-separation test evaluates if two variables (Y1, Y2) in a DAG are conditionally independent for a given conditioning set, C represented in a DAG by the union of the parent sets of Y1 and Y2 (Shipley, 2000). A new bow-free covariance is added if there is a significant (Y1, Y2) association at a significance level alpha, after multiple testing correction. The selected covariance between pairs of nodes is interpreted as the effect of a latent variable (LV) acting on both nodes; i.e., the LV is an unobserved confounder. BAP-based algorithms adjust (or de-correlate) the observed data matrix by conditioning out the latent triggers responsible for the nuisance edges. For "pc" algorithm the number of hidden proxies, q is determined by a permutation method. It compares the singular values to what they would be if the variables were independent, which is estimated by permuting the columns of the data matrix, Y and selects components if their singular values are larger than those of the permuted data (for a review see Dobriban, 2020). While for "glpc" algorithm, q is determined by the number of clusters by spectral clustering through cluster_leading_eigen function. If the input graph is not acyclic, a warning message will be raised, and a cycle-breaking algorithm will be applied (see graph2dag for details).

Value

A list of four objects:

"dag", the directed acyclic graph (DAG) extracted from input graph. If (dalgo = "glpc" or "pc"), the DAG also includes LVs as source nodes.
"guu", the bow-free covariance graph, BAP = dag + guu. If (dalgo = "pc" or "trim"), guu is equal to NULL
"adj", the adjacency matrix of selected bow-free covariances; i.e, the missing edges selected after multiple testing correction. If (dalgo = "pc" or "trim"), adj matrix is equal to NULL.
"data", the adjusted (de-correlated) data matrix or if (dalgo = "glpc", or "pc"), the combined data matrix, where the first columns represent LVs scores and the other columns are the raw data.

Author(s)

Mario Grassi mario.grassi@unipv.it

References

Grassi M, Palluzzi F, Tarantino B (2022). SEMgraph: An R Package for Causal Network Analysis of High-Throughput Data with Structural Equation Models. Bioinformatics, 38(20), 4829–4830. <https://doi.org/10.1093/bioinformatics/btac567>

Shipley B (2000). A new inferential test for path models based on DAGs. Structural Equation Modeling, 7(2), 206-218. <https://doi.org/10.1207/S15328007SEM0702_4>

Jiang B, Ding C, Bin L, Tang J (2013). Graph-Laplacian PCA: Closed-Form Solution and Robustness. IEEE Conference on Computer Vision and Pattern Recognition, 3492-3498. <https://doi.org/10.1109/CVPR.2013.448>

Ćevid D, Bühlmann P, Meinshausen N (2020). Spectral deconfounding via perturbed sparse linear models. J. Mach. Learn. Res, 21(232), 1-41. <http://jmlr.org/papers/v21/19-545.html>

Dobriban E (2020). Permuatation methods for Factor Analysis and PCA. Ann. Statist. 48(5): 2824-2847 <https://doi.org/10.1214/19-AOS1907>

Friedman J, Hastie T, Tibshirani R (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3), 432-441. <https://doi.org/10.1093/biostatistics/kxm045>

Examples


#Set function param
graph <- sachs$graph
data <- log(sachs$pkc)
group <-sachs$group

# BAP decounfounding with CGGM (default)
bap <- SEMbap(graph, data, verbose = TRUE)

# SVD decounfounding with trim method
svd <- SEMbap(graph, data, dalgo = "trim")

# Model fitting (with node-perturbation)
sem1 <- SEMrun(graph, data, group)
bap1 <- SEMrun(bap$dag, bap$data, group)
svd1 <- SEMrun(svd$dag, svd$data, group)

[Package SEMgraph version 1.2.2 Index]