Draw n standardized observations from the model-implied correlation matrix
\(\hat{P}\) of a fitted cpm_fit() object, using the low-rank factor
representation \(P = \Lambda\Lambda^\top + (I - D_\zeta^2)\) (Browne, 1992;
design sec. 1.3/sec. 5.4). Each observation is generated as
\(x = \Lambda z + (I - D_\zeta^2)^{1/2}\varepsilon\) with independent
standard-normal common-factor scores \(z\) and unique deviates
\(\varepsilon\), so the draws are exactly positive semidefinite by
construction (no eigenvalue clamping) and their population correlation matrix
is \(\hat{P}\) exactly.
Arguments
- object
A
circumplex_cpmobject fromcpm_fit().- n
The number of observations (rows) to simulate; a positive whole number.
Value
A numeric matrix with n rows and one column per fitted scale,
columns in the fitted scale order with colnames set to the scale names
(rownames are NULL). The population covariance is
object$matrices$Phat, so cov() of the returned matrix converges to it
as n grows. Under the default unit scaling Phat is a correlation matrix
(zero-mean, unit-variance margins), so cor() converges to it too; under
scaling = "free" the margins carry the fitted variance ratios
\(\hat\sigma^2\).
Details
This is the mean-based simulation path of the SSM CI-trustworthiness
diagnostic (ssm_ci_accuracy(); the M4 Brief-B contract): a caller draws
standardized data here and rescales it to a group's means and SDs. The
correlation-based (augmented scales-plus-measures) path is not produced
here – it reduces to the returned population block object$matrices$Phat,
from which the caller assembles and repairs its own joint matrix.
Reproducibility
cpm_simulate() consumes R's global random number stream (the
common-factor scores then the unique deviates, drawn in that fixed order),
so it follows the package's set.seed()-immediately-before convention: a
given seed reproduces the draw exactly. It is one of the package's
stochastic entry points (alongside ssm_analyze() and the bootstrap path
of cpm_fit()); the fit itself is deterministic.
References
Browne, M. W. (1992). Circumplex models for correlation matrices. Psychometrika, 57(4), 469-497.
See also
Other analysis functions:
cpm_fit(),
ssm_analyze(),
ssm_analyze_long(),
ssm_ci_accuracy(),
ssm_draws(),
ssm_parameters(),
ssm_parameters_id(),
ssm_score(),
ssm_sem(),
ssm_sem_parameters(),
summary.circumplex_ssm_id()
Examples
data("jz2017")
scales <- c("PA", "BC", "DE", "FG", "HI", "JK", "LM", "NO")
fit <- cpm_fit(cormat = cor(jz2017[scales]), scales = scales,
n = nrow(jz2017))
#> Warning: CPM Hessian is ill-conditioned (condition number 1.83e+14): angles may be clustered or parameters weakly determined.
set.seed(1)
x <- cpm_simulate(fit, n = 500)
round(cor(x) - fit$matrices$Phat, 2)
#> PA BC DE FG HI JK LM NO
#> PA 0.00 -0.03 -0.05 0.05 0.00 -0.02 -0.01 -0.07
#> BC -0.03 0.00 0.01 0.01 -0.03 -0.04 -0.03 -0.08
#> DE -0.05 0.01 0.00 0.03 -0.01 -0.01 0.03 -0.06
#> FG 0.05 0.01 0.03 0.00 0.00 -0.04 -0.03 -0.03
#> HI 0.00 -0.03 -0.01 0.00 0.00 -0.01 -0.04 0.00
#> JK -0.02 -0.04 -0.01 -0.04 -0.01 0.00 -0.02 0.01
#> LM -0.01 -0.03 0.03 -0.03 -0.04 -0.02 0.00 -0.02
#> NO -0.07 -0.08 -0.06 -0.03 0.00 0.01 -0.02 0.00