R: Stability selection in Structural Equation Modelling

StructuralModel {sharp}

R Documentation

Stability selection in Structural Equation Modelling

Description

Performs stability selection for Structural Equation Models. The underlying arrow selection algorithm (e.g. regularised Structural Equation Modelling) is run with different combinations of parameters controlling the sparsity (e.g. penalty parameter) and thresholds in selection proportions. These two hyper-parameters are jointly calibrated by maximisation of the stability score.

Usage

StructuralModel(
  xdata,
  adjacency,
  residual_covariance = NULL,
  Lambda = NULL,
  pi_list = seq(0.01, 0.99, by = 0.01),
  K = 100,
  tau = 0.5,
  seed = 1,
  n_cat = NULL,
  implementation = PenalisedLinearSystem,
  resampling = "subsampling",
  cpss = FALSE,
  PFER_method = "MB",
  PFER_thr = Inf,
  FDP_thr = Inf,
  Lambda_cardinal = 100,
  optimisation = c("grid_search", "nloptr"),
  n_cores = 1,
  output_data = FALSE,
  verbose = TRUE,
  ...
)

Arguments

`xdata`	matrix with observations as rows and variables as columns. Column names must be defined and in line with the row and column names of `adjacency`.
`adjacency`	binary adjacency matrix of the Directed Acyclic Graph (transpose of the asymmetric matrix A in Reticular Action Model notation). The row and column names of this matrix must be defined.
`residual_covariance`	binary and symmetric matrix encoding the nonzero entries in the residual covariance matrix (symmetric matrix S in Reticular Action Model notation). By default, this is the identity matrix (no residual covariance).
`Lambda`	matrix of parameters controlling the level of sparsity in the underlying feature selection algorithm specified in `implementation`.
`pi_list`	vector of thresholds in selection proportions. If `n_cat=NULL` or `n_cat=2`, these values must be `>0` and `<1`. If `n_cat=3`, these values must be `>0.5` and `<1`.
`K`	number of resampling iterations.
`tau`	subsample size. Only used if `resampling="subsampling"` and `cpss=FALSE`.
`seed`	value of the seed to initialise the random number generator and ensure reproducibility of the results (see `set.seed`).
`n_cat`	computation options for the stability score. Default is `NULL` to use the score based on a z test. Other possible values are 2 or 3 to use the score based on the negative log-likelihood.
`implementation`	function to use for variable selection.
`resampling`	resampling approach. Possible values are: `"subsampling"` for sampling without replacement of a proportion `tau` of the observations, or `"bootstrap"` for sampling with replacement generating a resampled dataset with as many observations as in the full sample. Alternatively, this argument can be a function to use for resampling. This function must use arguments named `data` and `tau` and return the IDs of observations to be included in the resampled dataset.
`cpss`	logical indicating if complementary pair stability selection should be done. For this, the algorithm is applied on two non-overlapping subsets of half of the observations. A feature is considered as selected if it is selected for both subsamples. With this method, the data is split `K/2` times (`K` models are fitted). Only used if `PFER_method="MB"`.
`PFER_method`	method used to compute the upper-bound of the expected number of False Positives (or Per Family Error Rate, PFER). If `PFER_method="MB"`, the method proposed by Meinshausen and Bühlmann (2010) is used. If `PFER_method="SS"`, the method proposed by Shah and Samworth (2013) under the assumption of unimodality is used.
`PFER_thr`	threshold in PFER for constrained calibration by error control. If `PFER_thr=Inf` and `FDP_thr=Inf`, unconstrained calibration is used (the default).
`FDP_thr`	threshold in the expected proportion of falsely selected features (or False Discovery Proportion) for constrained calibration by error control. If `PFER_thr=Inf` and `FDP_thr=Inf`, unconstrained calibration is used (the default).
`Lambda_cardinal`	number of values in the grid of parameters controlling the level of sparsity in the underlying algorithm. Only used if `Lambda=NULL`.
`optimisation`	character string indicating the type of optimisation method. With `optimisation="grid_search"` (the default), all values in `Lambda` are visited. Alternatively, optimisation algorithms implemented in `nloptr` can be used with `optimisation="nloptr"`. By default, we use `"algorithm"="NLOPT_GN_DIRECT_L"`, `"xtol_abs"=0.1`, `"ftol_abs"=0.1` and `"maxeval"=Lambda_cardinal`. These values can be changed by providing the argument `opts` (see `nloptr`). For stability selection using penalised regression, `optimisation="grid_search"` may be faster as it allows for warm start.
`n_cores`	number of cores to use for parallel computing (see argument `workers` in `multisession`). Using `n_cores>1` is only supported with `optimisation="grid_search"`.
`output_data`	logical indicating if the input datasets `xdata` and `ydata` should be included in the output.
`verbose`	logical indicating if a loading bar and messages should be printed.
`...`	additional parameters passed to the functions provided in `implementation` or `resampling`.

Details

In stability selection, a feature selection algorithm is fitted on K subsamples (or bootstrap samples) of the data with different parameters controlling the sparsity (Lambda). For a given (set of) sparsity parameter(s), the proportion out of the K models in which each feature is selected is calculated. Features with selection proportions above a threshold pi are considered stably selected. The stability selection model is controlled by the sparsity parameter(s) for the underlying algorithm, and the threshold in selection proportion:

V_{\lambda, \pi} = \{ j: p_{\lambda}(j) \ge \pi \}

In Structural Equation Modelling, "feature" refers to an arrow in the corresponding Directed Acyclic Graph.

These parameters can be calibrated by maximisation of a stability score (see ConsensusScore if n_cat=NULL or StabilityScore otherwise) calculated under the null hypothesis of equiprobability of selection.

It is strongly recommended to examine the calibration plot carefully to check that the grids of parameters Lambda and pi_list do not restrict the calibration to a region that would not include the global maximum (see CalibrationPlot). In particular, the grid Lambda may need to be extended when the maximum stability is observed on the left or right edges of the calibration heatmap. In some instances, multiple peaks of stability score can be observed. Simulation studies suggest that the peak corresponding to the largest number of selected features tend to give better selection performances. This is not necessarily the highest peak (which is automatically retained by the functions in this package). The user can decide to manually choose another peak.

To control the expected number of False Positives (Per Family Error Rate) in the results, a threshold PFER_thr can be specified. The optimisation problem is then constrained to sets of parameters that generate models with an upper-bound in PFER below PFER_thr (see Meinshausen and Bühlmann (2010) and Shah and Samworth (2013)).

Possible resampling procedures include defining (i) K subsamples of a proportion tau of the observations, (ii) K bootstrap samples with the full sample size (obtained with replacement), and (iii) K/2 splits of the data in half for complementary pair stability selection (see arguments resampling and cpss). In complementary pair stability selection, a feature is considered selected at a given resampling iteration if it is selected in the two complementary subsamples.

To ensure reproducibility of the results, the starting number of the random number generator is set to seed.

For parallelisation, stability selection with different sets of parameters can be run on n_cores cores. Using n_cores > 1 creates a multisession. Alternatively, the function can be run manually with different seeds and all other parameters equal. The results can then be combined using Combine.

Value

An object of class variable_selection. A list with:

`S`	a matrix of the best stability scores for different parameters controlling the level of sparsity in the underlying algorithm.
`Lambda`	a matrix of parameters controlling the level of sparsity in the underlying algorithm.
`Q`	a matrix of the average number of selected features by the underlying algorithm with different parameters controlling the level of sparsity.
`Q_s`	a matrix of the calibrated number of stably selected features with different parameters controlling the level of sparsity.
`P`	a matrix of calibrated thresholds in selection proportions for different parameters controlling the level of sparsity in the underlying algorithm.
`PFER`	a matrix of upper-bounds in PFER of calibrated stability selection models with different parameters controlling the level of sparsity.
`FDP`	a matrix of upper-bounds in FDP of calibrated stability selection models with different parameters controlling the level of sparsity.
`S_2d`	a matrix of stability scores obtained with different combinations of parameters. Columns correspond to different thresholds in selection proportions.
`PFER_2d`	a matrix of upper-bounds in FDP obtained with different combinations of parameters. Columns correspond to different thresholds in selection proportions.
`FDP_2d`	a matrix of upper-bounds in PFER obtained with different combinations of parameters. Columns correspond to different thresholds in selection proportions.
`selprop`	a matrix of selection proportions. Columns correspond to predictors from `xdata`.
`Beta`	an array of model coefficients. Columns correspond to predictors from `xdata`. Indices along the third dimension correspond to different resampling iterations. With multivariate outcomes, indices along the fourth dimension correspond to outcome-specific coefficients.
`method`	a list with `type="variable_selection"` and values used for arguments `implementation`, `family`, `resampling`, `cpss` and `PFER_method`.
`params`	a list with values used for arguments `K`, `pi_list`, `tau`, `n_cat`, `pk`, `n` (number of observations), `PFER_thr`, `FDP_thr` and `seed`. The datasets `xdata` and `ydata` are also included if `output_data=TRUE`.

For all matrices and arrays returned, the rows are ordered in the same way and correspond to parameter values stored in Lambda.

References

Bodinier B, Filippi S, Nøst TH, Chiquet J, Chadeau-Hyam M (2023). “Automated calibration for stability selection in penalised regression and graphical models.” Journal of the Royal Statistical Society Series C: Applied Statistics, qlad058. ISSN 0035-9254, doi:10.1093/jrsssc/qlad058, https://academic.oup.com/jrsssc/advance-article-pdf/doi/10.1093/jrsssc/qlad058/50878777/qlad058.pdf.

Meinshausen N, Bühlmann P (2010). “Stability selection.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(4), 417-473. doi:10.1111/j.1467-9868.2010.00740.x.

Shah RD, Samworth RJ (2013). “Variable selection with error control: another look at stability selection.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(1), 55-80. doi:10.1111/j.1467-9868.2011.01034.x.

Jacobucci R, Grimm KJ, McArdle JJ (2016). “Regularized structural equation modeling.” Structural equation modeling: a multidisciplinary journal, 23(4), 555–566. doi:10.1080/10705511.2016.1154793.

Examples

oldpar <- par(no.readonly = TRUE)
par(mar = rep(7, 4))


# Data simulation
set.seed(1)
pk <- c(3, 2, 3)
simul <- SimulateStructural(
  n = 500,
  pk = pk,
  nu_between = 0.5,
  v_between = 1,
  v_sign = 1
)

# Stability selection (using glmnet)
dag <- LayeredDAG(layers = pk)
stab <- StructuralModel(
  xdata = simul$data,
  adjacency = dag
)
CalibrationPlot(stab)
LinearSystemMatrix(vect = Stable(stab), adjacency = dag)

# Stability selection (using OpenMx)
if (requireNamespace("OpenMx", quietly = TRUE)) {
  stab <- StructuralModel(
    xdata = simul$data,
    implementation = PenalisedOpenMx,
    Lambda = seq(50, 500, by = 50),
    adjacency = dag
  )
  CalibrationPlot(stab)
  OpenMxMatrix(SelectedVariables(stab), adjacency = dag)
}

## Not run: 
# Data simulation with latent variables
set.seed(1)
pk <- c(3, 2, 3)
simul <- SimulateStructural(
  n = 500,
  pk = pk,
  nu_between = 0.5,
  v_sign = 1,
  v_between = 1,
  n_manifest = 3,
  ev_manifest = 0.95
)

# Stability selection (using OpenMx)
if (requireNamespace("OpenMx", quietly = TRUE)) {
  dag <- LayeredDAG(layers = pk, n_manifest = 3)
  penalised <- dag
  penalised[, seq_len(ncol(simul$data))] <- 0
  stab <- StructuralModel(
    xdata = simul$data,
    implementation = PenalisedOpenMx,
    adjacency = dag,
    penalised = penalised,
    Lambda = seq(10, 100, by = 20),
    K = 10 # to increase for real use
  )
  CalibrationPlot(stab)
  ids_latent <- grep("f", colnames(dag))
  OpenMxMatrix(SelectedVariables(stab),
    adjacency = dag
  )[ids_latent, ids_latent]
}

## End(Not run)

par(oldpar)

[Package sharp version 1.4.6 Index]