Simulate data from a common-factor population model

Draws data from a population correlation matrix, given either directly or built from a factor model. The population correlation is either supplied in R, or assembled from a loading matrix Lambda, the factor intercorrelations Phi, and the unique variances Psi as \(R = Lambda\, Phi\, Lambda' + Psi\), standardized to a correlation matrix. By default (marginals = "normal") cases are drawn with normal marginals via a matrix square root of the population correlation (a Cholesky factor, or a symmetric eigen square root for a positive-semidefinite but singular population). With marginals = "empirical" the cases instead reproduce the population correlation while carrying the empirical marginal distributions of a supplied data set. With marginals = "VM" or "IG" the cases reproduce the population correlation while carrying non-normal marginals with a prescribed skewness and kurtosis. Setting categories additionally discretizes the drawn data into ordered categories, optionally so that the population polychoric correlation of the categorized data equals the target correlation. Setting target_rmsea or target_cfi perturbs the population with model error, so the factor model fits it only approximately (a more realistic simulation target).

Usage

efa_simulate(
  N = NULL,
  Lambda = NULL,
  Phi = NULL,
  Psi = NULL,
  R = NULL,
  model_error = c("CB", "TKL", "WB", "none"),
  target_rmsea = NULL,
  target_cfi = NULL,
  marginals = c("normal", "empirical", "VM", "IG"),
  marginal_data = NULL,
  n_factors = NULL,
  skewness = NULL,
  kurtosis = NULL,
  force_pd = FALSE,
  categories = NULL,
  match = NULL,
  missing = c("none", "MCAR", "MAR", "MNAR"),
  missing_prop = NULL,
  missing_strength = NULL,
  missing_predictor = NULL,
  n_datasets = 1L,
  seed = NULL,
  return_pop = FALSE
)

Arguments

N: numeric. Number of cases (rows) to draw per dataset. Required unless return_pop = TRUE.
Lambda: matrix. A p by m matrix of factor loadings. Supply this (optionally with Phi and Psi) instead of R to build the population from a factor model.
Phi: matrix. The m by m factor intercorrelation matrix. Only used with Lambda. Default is NULL, in which case the factors are orthogonal (an identity matrix).
Psi: numeric vector or matrix. The unique variances: either a length-p vector or a p by p matrix (added as the residual covariance). Only used with Lambda. Default is NULL, in which case the unique variances that standardize the population to a correlation matrix are used.
R: matrix. A p by p population correlation matrix to draw from directly. Supply this instead of Lambda/Phi/Psi.
model_error: character. The method used to perturb the population so the factor model fits it imperfectly ("model error"): one of "CB" (Cudeck-Browne, the default), "TKL" (Tucker-Koopman-Linn), "WB" (Wu-Browne), or "none". Model error is only applied when a target is supplied in target_rmsea or target_cfi; without one the population is exact, whatever model_error. Only used with a factor-model population (Lambda).
target_rmsea: numeric. The population RMSEA the factor model should have relative to the perturbed population, a single number strictly in (0, 1). Supplying it activates model error. Simulating from an exact population overstates recovery, so a realistic value (around 0.05) is recommended for simulation studies (MacCallum, 2003). Default is NULL (an exact population; do not pass 0). Required for "CB" and "WB"; optional for "TKL".
target_cfi: numeric. Only used with model_error = "TKL": the population CFI to target, a single number strictly in (0, 1), on its own or together with target_rmsea (TKL then trades the two off). Default is NULL (do not pass 1). "CB" and "WB" target the RMSEA only.
marginals: character. The marginal distribution of the drawn data: one of "normal" (the default), which draws normal marginals; "empirical", which reproduces the population correlation while preserving the empirical marginals supplied in marginal_data; or "VM" (Vale-Maurelli) and "IG" (independent generator), which draw non-normal marginals with the target skewness and kurtosis.
marginal_data: matrix or data frame. Only used with marginals = "empirical", where it is required: a data set with one numeric column per variable (p columns), each with at least two distinct values, whose per-column distributions are resampled as the marginals of the drawn data. Its correlations are ignored. Default is NULL.
n_factors: numeric. Only used with marginals = "empirical": the number of factors the rank-matching reproduction fits. Default is NULL, in which case it is the number of columns of Lambda when the population is built from a factor model; it must be given when the population is supplied via R.
skewness: numeric. Only used with marginals = "VM" or "IG": the target marginal skewness, as a single value applied to every variable or a length-p vector. Default is NULL (0, a symmetric marginal). At least one of skewness or kurtosis must be given for these marginals.
kurtosis: numeric. Only used with marginals = "VM" or "IG": the target marginal excess kurtosis (0 for a normal marginal), as a single value applied to every variable or a length-p vector. Default is NULL (0).
force_pd: logical. Used with marginals = "VM" and with Cudeck-Browne model error (model_error = "CB"). If the Vale-Maurelli intermediate correlation matrix (or, for CB, the perturbed population at a large target_rmsea) is not positive definite, FALSE (the default) rejects it with an error, while TRUE projects it to the nearest correlation matrix (with a warning). Has no effect for the "TKL" or "WB" methods.
categories: numeric or list. Requests ordinal output by discretizing each variable into ordered categories. Either a count of equally probable categories (a single value applied to every variable or a length-p vector), or a length-p list of numeric vectors giving the marginal category proportions per variable (each strictly positive and summing to 1). Default is NULL, which returns the continuous data.
match: character. Only used with categories: an assertion about how the categorization relates to the population correlation. With a normal latent, cutting at the normal-scale thresholds already leaves the population polychoric correlation of the categorized data equal to the target correlation, so both values compute the same thresholds and produce identical data whenever both are legal. "thresholds" (the default) also cuts the "VM" and "IG" draws, whose ordinal Pearson and polychoric correlations then both depart from the population; "polychoric" states that the polychoric match is required and therefore rejects non-normal marginals. Not available with marginals = "empirical". The value is matched case-insensitively. Default is NULL ("thresholds" when categories is set).
missing: character. An optional missing-data mechanism to impose on the drawn data: one of "none" (the default, complete data), "MCAR" (missing completely at random), "MAR" (missing at random, depending on another variable), or "MNAR" (missing not at random, depending on the variable's own value). Introduced values become NA. Every variable is holed, so under "MAR" each variable's predictor is itself subject to missingness: the mechanism is MAR given the complete data and is not ignorable for an analyst who sees only the observed data (see Details).
missing_prop: numeric. Only used when missing is not "none", where it is required: the target marginal proportion of missing values per variable, a single number strictly between 0 and 1. This is the expected rate; the realized rate of a given draw varies around it.
missing_strength: numeric. Only used with missing = "MAR" or "MNAR": the slope of the logistic missingness model, setting how strongly the missing probability depends on the predictor. Default is NULL (1, a moderate dependence); 0 removes the dependence (equivalent to MCAR at the same rate) and large magnitudes make missingness nearly deterministic.
missing_predictor: integer or character. Only used with missing = "MAR": which variable drives each variable's missingness, as one column index or variable name per variable. Each must reference another variable, not itself (so a single shared predictor is not allowed – it would predict its own missingness). Default is NULL, in which case each variable's missingness is driven by the next variable cyclically (so variable order matters; supply this explicitly when the order is arbitrary).
n_datasets: numeric. The number of datasets to draw. Default is 1. With more than one, a list of datasets is returned.
seed: numeric. Optional seed for reproducible draws. When supplied, the caller's random-number stream is saved and restored, so the call leaves the global RNG state unchanged. Default is NULL (no seeding).
return_pop: logical. If TRUE, return only the population correlation matrix and draw no data. Default is FALSE.

Value

An object of class efa_simulated: a list with elements data (the simulated data – an N by p numeric matrix, an integer matrix of category codes when categories is set, or a length-n_datasets list of these when n_datasets > 1; NULL when return_pop = TRUE), population (the p by p population correlation matrix drawn from, model-error-perturbed when requested; with force_pd = TRUE and marginals = "VM" it stays the target matrix, from which the realized correlations of the draw can drift), model_error (NULL, or a list of the method and the target and achieved RMSEA/CFI when model error was applied), and settings. Printing the object shows a compact summary.

Details

Provide the population either as a ready correlation matrix in R, or through the model components Lambda, Phi, and Psi; the two ways are mutually exclusive. When the model components are used, Phi defaults to the identity matrix (orthogonal factors) and Psi defaults to the unique variances that make the population a correlation matrix (\(1 - \mathrm{diag}(Lambda\, Phi\, Lambda')\)); the assembled covariance is standardized with stats::cov2cor() so a non-standardized Psi still yields a correlation matrix. With the default Psi, a factor model whose implied communalities exceed 1 (a Heywood case) leaves no unique variance and is rejected; a Psi you supply is instead only required to give positive variances and a positive-semidefinite population.

With marginals = "empirical", the iterative rank-matching algorithm of Ruscio and Kaczetow (2008) reproduces the population correlation while each variable takes the empirical marginal distribution of the matching column of marginal_data (resampled with replacement). Only the marginals of marginal_data are used; its own correlations are ignored, and the drawn columns follow the population's variables, not those of marginal_data.

With marginals = "VM" (Vale-Maurelli, 1983) or "IG" (the independent-generator method; Foldnes & Olsson, 2016), the cases reproduce the population correlation while carrying non-normal marginals with the target skewness and (excess) kurtosis. The Vale-Maurelli family does not span every valid non-normal distribution (Foldnes & Grønneberg, 2015); "IG" covers distributions "VM" cannot. Both accept skewness and kurtosis as a single value (used for every variable) or one value per variable, defaulting the unset one to 0. Not every (skewness, kurtosis) pair is attainable: every distribution needs excess kurtosis of at least skewness^2 - 2, and the method covers a smaller region still, so an unreachable request is rejected. For marginals = "VM", the intermediate correlation matrix used for the draw can itself be non-positive-definite; it is rejected unless force_pd = TRUE, which projects it to the nearest correlation matrix (via psych::cor.smooth()) with a warning.

With categories, the drawn data are discretized into ordered categories (an integer code 1 to K). categories gives either the number of equally probable categories (one count for every variable, or one per variable) or, as a list of proportion vectors, the marginal category proportions per variable. The cut points are the thresholds that reproduce the requested proportions (Olsson, 1979): the standard-normal quantiles for marginals = "normal", and for marginals = "VM" those quantiles mapped through the same Fleishman cubic the draw uses, so the requested proportions are reproduced on the non-normal scale too. Under marginals = "IG" the thresholds stay on the standard-normal scale while the data do not, so the achieved proportions depart from the request systematically rather than by sampling noise; the departure grows with the non-normality, and only the number of categories is guaranteed. The same holds for a "VM" variable whose Fleishman cubic is not increasing over its own thresholds and the tails beyond them, which keeps the normal-scale ones and is reported with a warning; this arises when a turning point of the cubic sits at or near an outer threshold – under a strongly platykurtic marginal, or under substantial skewness or kurtosis combined with a small outer-category proportion. Because categorization attenuates product-moment correlations, the categorized data's Pearson correlation is smaller in magnitude than the population correlation; under non-normal marginals its polychoric correlation departs from the population as well. match changes none of this – it asserts an intent rather than selecting a computation, as described under that argument. Ordinal output is not available with marginals = "empirical". Empty categories left by a draw are reported with a warning, as they destabilize the polychoric correlation and the factor analysis.

With missing, missing values are introduced into the drawn data under a chosen mechanism (Rubin, 1976), each variable holed at a target expected rate missing_prop. "MCAR" draws an independent mask, so missingness is unrelated to the data. "MAR" and "MNAR" set each case's missing probability by a logistic model of a standardized predictor: another variable for "MAR" (chosen by missing_predictor) or the variable's own value for "MNAR", with slope missing_strength. The mechanism acts on the drawn (latent) values, so when categories also discretizes the data the missingness is keyed on the underlying value, not the category code. For "MAR" the predictor is evaluated on the complete drawn values, but every variable is holed at rate missing_prop, so a variable's MAR predictor is itself missing for roughly a missing_prop fraction of the cases whose missingness it drove. The mechanism is therefore MAR conditional on the complete data, and not ignorable for an analyst who sees only the observed data: estimators that are consistent under ignorable MAR, such as cor_method = "fiml" in efa_fit() and the multiple imputation behind efa_mi(), keep a residual bias here that grows with missing_prop and missing_strength. The returned matrix carries the NAs, which the correlation estimators handle downstream.

With model_error, the population is perturbed away from the exact factor structure so the q-factor model (q = ncol(Lambda)) fits it only approximately, at a prescribed misfit; exact factor structures are unrealistic and overstate recovery in simulation studies (MacCallum, 2003). The perturbation is applied once to the population, and the achieved fit is computed with the same fit-index formulas efa_fit() uses and returned in the model_error element. It is applied only when a target is supplied (target_rmsea and/or target_cfi), needs a factor-model population (Lambda) with residual degrees of freedom and an exact factor structure (a diagonal Psi), and is orthogonal to the marginal, ordinal, and missing-data options. Three methods are available. "CB" (Cudeck & Browne, 1992) matches the target RMSEA to numerical precision and keeps the q-factor model the exact minimizer (the CFI follows as a derived quantity). "TKL" (Tucker, Koopman & Linn, 1969) adds minor common factors tuned so the achieved RMSEA – and, optionally, CFI – match the target(s); with a single target the match is close, with both it is a compromise. "WB" (Wu & Browne, 2015) draws the population from an inverse-Wishart distribution around the model-implied correlation; its calibration applies to the best-fitting model, so the reported misfit of the generating model is systematically larger than the target – by roughly \(\sqrt{(p(p-1)/2)/df}\), about 1.4 times for 12 variables and 3 factors. Use "CB" when the reported RMSEA must equal the target. "CB" and "WB" target the RMSEA only; "TKL" can target the RMSEA and/or the CFI. The reported RMSEA/CFI is the misfit of the specified generating model.

Replicated draws (n_datasets > 1) are generated in parallel across replicates with future.apply; a parallel plan can be selected with future::plan() (the default plan runs sequentially). Each replicate is assigned its own reproducible random-number stream, so with a fixed seed the output is identical regardless of the number of workers.

References

Cudeck, R., & Browne, M. W. (1992). Constructing a covariance matrix that yields a specified minimizer and a specified minimum discrepancy function value. Psychometrika, 57(3), 357-369. doi:10.1007/BF02295424

Fleishman, A. I. (1978). A method for simulating non-normal distributions. Psychometrika, 43(4), 521-532. doi:10.1007/BF02293811

Foldnes, N., & Grønneberg, S. (2015). How general is the Vale-Maurelli simulation approach? Psychometrika, 80(4), 1066-1083. doi:10.1007/s11336-014-9414-0

Foldnes, N., & Olsson, U. H. (2016). A simple simulation technique for nonnormal data with prespecified skewness, kurtosis, and covariance matrix. Multivariate Behavioral Research, 51(2-3), 207-219. doi:10.1080/00273171.2015.1133274

MacCallum, R. C. (2003). 2001 Presidential Address: Working with imperfect models. Multivariate Behavioral Research, 38(1), 113-139. doi:10.1207/S15327906MBR3801_5

Olsson, U. (1979). Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika, 44(4), 443-460. doi:10.1007/BF02296207

Olvera Astivia, O. L., & Zumbo, B. D. (2019). A note on the solution multiplicity of the Vale-Maurelli intermediate correlation equation. Journal of Educational and Behavioral Statistics, 44(2), 127-143. doi:10.3102/1076998618803381

Rubin, D. B. (1976). Inference and missing data. Biometrika, 63(3), 581-592. doi:10.1093/biomet/63.3.581

Ruscio, J., & Kaczetow, W. (2008). Simulating multivariate nonnormal data using an iterative algorithm. Multivariate Behavioral Research, 43(3), 355-381. doi:10.1080/00273170802285693

Tucker, L. R., Koopman, R. F., & Linn, R. L. (1969). Evaluation of factor analytic research procedures by means of simulated correlation matrices. Psychometrika, 34(4), 421-459. doi:10.1007/BF02290601

Vale, C. D., & Maurelli, V. A. (1983). Simulating multivariate nonnormal distributions. Psychometrika, 48(3), 465-471. doi:10.1007/BF02293687

Wu, H., & Browne, M. W. (2015). Quantifying adventitious error in a covariance structure as a random effect. Psychometrika, 80(3), 571-600. doi:10.1007/s11336-015-9451-3

Examples

# Build a population from a shipped loading pattern and factor correlations
Lambda <- population_models$loadings$baseline
Phi <- population_models$phis_3$moderate

# Draw one normal dataset of 500 cases (the data live in $data)
sim <- efa_simulate(N = 500, Lambda = Lambda, Phi = Phi, seed = 42)
dim(sim$data)
#> [1] 500  18

# Return only the population correlation matrix
R_pop <- efa_simulate(Lambda = Lambda, Phi = Phi, return_pop = TRUE)$population

# Draw several datasets at once from a supplied correlation matrix
sims <- efa_simulate(N = 500, R = R_pop, n_datasets = 3, seed = 42)
length(sims$data)
#> [1] 3

# Reproduce the population correlation but with skewed, empirical marginals
# (here from a chi-squared source with one column per variable)
src <- matrix(rchisq(200 * nrow(Lambda), df = 3), ncol = nrow(Lambda))
dat_emp <- efa_simulate(N = 500, Lambda = Lambda, Phi = Phi,
                        marginals = "empirical", marginal_data = src, seed = 42)

# Draw skewed, leptokurtic data with the Vale-Maurelli method
dat_vm <- efa_simulate(N = 500, Lambda = Lambda, Phi = Phi, marginals = "VM",
                       skewness = 1.5, kurtosis = 4, seed = 42)

# Draw five-category ordinal data whose polychoric correlation matches R
dat_ord <- efa_simulate(N = 500, Lambda = Lambda, Phi = Phi,
                        categories = 5, match = "polychoric", seed = 42)

# Draw data with 15% missing at random, driven by a neighbouring item
dat_mar <- efa_simulate(N = 500, Lambda = Lambda, Phi = Phi, missing = "MAR",
                        missing_prop = 0.15, seed = 42)
colMeans(is.na(dat_mar$data))
#>    V1    V2    V3    V4    V5    V6    V7    V8    V9   V10   V11   V12   V13 
#> 0.154 0.146 0.184 0.156 0.152 0.170 0.174 0.138 0.132 0.138 0.128 0.134 0.136 
#>   V14   V15   V16   V17   V18 
#> 0.132 0.152 0.144 0.166 0.140 

# Add realistic model error: a population the model fits with RMSEA of about .05
# (Cudeck-Browne, the default method; the achieved fit is reported)
sim_me <- efa_simulate(N = 500, Lambda = Lambda, Phi = Phi,
                       target_rmsea = 0.05, seed = 42)
sim_me$model_error$rmsea
#> [1] 0.05