fleish_Hessian

a vector of constants c1, c3, lambda

This function gives the second-order conditions necessary to verify that a kurtosis is a global minimum. A kurtosis solution from
 <code><a rd-options="SimMultiCorrData" href="/link/fleish_skurt_check?package=SimMultiCorrData&version=0.2.2&to=SimMultiCorrData" data-mini-rdoc="SimMultiCorrData::fleish_skurt_check">fleish_skurt_check</a></code> is a global minimum if and only if the determinant of the bordered Hessian, $H$, is
 less than zero (see Headrick &amp; Sawilowsky, 2002, 10.3102/10769986025004417), where
 $$|\bar{H}| = matrix(c(0, dg(c1, c3)/dc1, dg(c1, c3)/dc3,$$
 $$dg(c1, c3)/dc1, d^2 F(c1, c3, \lambda)/dc1^2, d^2 F(c1, c3, \lambda)/(dc3 dc1),$$
 $$dg(c1, c3)/dc3, d^2 F(c1, c3, \lambda)/(dc1 dc3), d^2 F(c1, c3, \lambda)/dc3^2), 3, 3, byrow = TRUE)$$
 Here, $F(c1, c3, \lambda) = f(c1, c3) + \lambda * [\gamma_{1} - g(c1, c3)]$ is the Fleishman Transformation Lagrangean expression
 (see <code><a rd-options="SimMultiCorrData" href="/link/fleish_skurt_check?package=SimMultiCorrData&version=0.2.2&to=SimMultiCorrData" data-mini-rdoc="SimMultiCorrData::fleish_skurt_check">fleish_skurt_check</a></code>). Headrick &amp; Sawilowsky (2002) gave equations for the second-order derivatives
 $d^2 F/dc1^2$, $d^2 F/dc3^2$, and $d^2 F/(dc1 dc3)$. These were verified and $dg/dc1$ and $dg/dc3$ were calculated
 using <code>D</code> (see <code><a rd-options="stats" href="/link/deriv?package=SimMultiCorrData&version=0.2.2&to=stats" data-mini-rdoc="stats::deriv">deriv</a></code>). This function would not ordinarily be called by the user.

kurtosis

Hessian

Fleishman

boundary

Generate continuous (normal or non-normal), binary, ordinal, and count (Poisson or Negative
Binomial) variables with a specified correlation matrix. It can also produce a single continuous
variable. This package can be used to simulate data sets that mimic real-world situations (i.e.
clinical or genetic data sets, plasmodes). All variables are generated from standard normal
variables with an imposed intermediate correlation matrix. Continuous variables are simulated
by specifying mean, variance, skewness, standardized kurtosis, and fifth and sixth standardized
cumulants using either Fleishman's third-order (<DOI:10.1007/BF02293811>) or Headrick's
fifth-order (<DOI:10.1016/S0167-9473(02)00072-5>) polynomial transformation. Binary and
ordinal variables are simulated using a modification of the ordsample() function from 'GenOrd'.
Count variables are simulated using the inverse cdf method. There are two simulation pathways
which differ primarily according to the calculation of the intermediate correlation matrix. In
Correlation Method 1, the intercorrelations involving count variables are determined using a
simulation based, logarithmic correlation correction (adapting Yahav and Shmueli's 2012 method,
<DOI:10.1002/asmb.901>). In Correlation Method 2, the count variables are treated as ordinal
(adapting Barbiero and Ferrari's 2015 modification of GenOrd, <DOI:10.1002/asmb.2072>).
There is an optional error loop that corrects the final correlation matrix to be within a
user-specified precision value of the target matrix. The package also includes functions to
calculate standardized cumulants for theoretical distributions or from real data sets, check
if a target correlation matrix is within the possible correlation bounds (given the distributions
of the simulated variables), summarize results (numerically or graphically), to verify valid power
method pdfs, and to calculate lower standardized kurtosis bounds.

Allison Fialkowski

SimMultiCorrData

Simulation of Correlated Data with Multiple Variable Types

fleish_Hessian function

This function gives the second-order conditions necessary to verify that a kurtosis is a global minimum. A kurtosis solution from
 <code><a rd-options='SimMultiCorrData' href='fleish_skurt_check'>fleish_skurt_check</a></code> is a global minimum if and only if the determinant of the bordered Hessian, $H$, is
 less than zero (see Headrick &amp; Sawilowsky, 2002, 10.3102/10769986025004417), where
 $$|\bar{H}| = matrix(c(0, dg(c1, c3)/dc1, dg(c1, c3)/dc3,$$
 $$dg(c1, c3)/dc1, d^2 F(c1, c3, \lambda)/dc1^2, d^2 F(c1, c3, \lambda)/(dc3 dc1),$$
 $$dg(c1, c3)/dc3, d^2 F(c1, c3, \lambda)/(dc1 dc3), d^2 F(c1, c3, \lambda)/dc3^2), 3, 3, byrow = TRUE)$$
 Here, $F(c1, c3, \lambda) = f(c1, c3) + \lambda * [\gamma_{1} - g(c1, c3)]$ is the Fleishman Transformation Lagrangean expression
 (see <code><a rd-options='SimMultiCorrData' href='fleish_skurt_check'>fleish_skurt_check</a></code>). Headrick &amp; Sawilowsky (2002) gave equations for the second-order derivatives
 $d^2 F/dc1^2$, $d^2 F/dc3^2$, and $d^2 F/(dc1 dc3)$. These were verified and $dg/dc1$ and $dg/dc3$ were calculated
 using <code>D</code> (see <code><a rd-options='stats' href='deriv'>deriv</a></code>). This function would not ordinarily be called by the user.

fleish_Hessian: Fleishman's Third-Order Transformation Hessian Calculation for Lower Boundary of Standardized Kurtosis in Asymmetric Distributions

Description

Usage

Arguments

Value

References

See Also