Package 'SimCorMultRes'

Simulating Correlated Binary Responses Conditional on a Marginal Model Specification

Description

Simulates correlated binary responses assuming a regression model for the marginal probabilities.

Usage

rbin(clsize = clsize, intercepts = intercepts, betas = betas,
  xformula = formula(xdata), xdata = parent.frame(), link = "logit",
  cor.matrix = cor.matrix, rlatent = NULL)

Arguments

clsize	integer indicating the common cluster size.
intercepts	numerical (or numeric vector of length clsize) containing the intercept(s) of the marginal model.
betas	numerical vector or matrix containing the value of the marginal regression parameter vector associated with the covariates (i.e., excluding intercepts).
xformula	formula expression as in other marginal regression models but without including a response variable.
xdata	optional data frame containing the variables provided in xformula.
link	character string indicating the link function in the marginal model. Options include 'probit', 'logit', 'cloglog', 'cauchit' or 'identity'. Required when rlatent = NULL.
cor.matrix	matrix indicating the correlation matrix of the multivariate normal distribution when the NORTA method is employed (rlatent = NULL).
rlatent	matrix with clsize columns containing realizations of the latent random vectors when the NORTA method is not preferred. See details for more info.

Details

The formulae are easier to read from either the Vignette or the Reference Manual (both available here).

The assumed marginal model is

$Pr(Y_{it} = 1 |x_{it})=F(\beta_{t0} +\beta^{'}_{t} x_{it})$

where $F$ is the cumulative distribution function determined by link. For subject $i$, $Y_{it}$ is the $t$-th binary response and $x_{it}$ is the associated covariates vector. Finally, $\beta_{t0}$ and $\beta_{t}$ are the intercept and regression parameter vector at the $t$-th measurement occasion.

The binary response $Y_{it}$ is obtained by extending the approach of Emrich and Piedmonte (1991) as suggested in Touloumis (2016).

When $\beta_{t0}=\beta_{0}$ for all $t$, then intercepts should be provided as a single number. Otherwise, intercepts must be provided as a numeric vector such that the $t$-th element corresponds to the intercept at measurement occasion $t$.

betas should be provided as a numeric vector only when $\beta_{t}=\beta$ for all $t$. Otherwise, betas must be provided as a numeric matrix with clsize rows such that the $t$-th row contains the value of $\beta_{t}$. In either case, betas should reflect the order of the terms implied by xformula.

The appropriate use of xformula is xformula = ~ covariates, where covariates indicate the linear predictor as in other marginal regression models.

The optional argument xdata should be provided in “long” format.

The NORTA method is the default option for simulating the latent random vectors denoted by $e^{B}_{it}$ in Touloumis (2016). To import simulated values for the latent random vectors without utilizing the NORTA method, the user can employ the rlatent argument. In this case, element ($i,t$) of rlatent represents the realization of $e^{B}_{it}$.

Value

Returns a list that has components:

Ysim	the simulated binary responses. Element ($i$,$t$) represents the realization of $Y_{it}$.
simdata	a data frame that includes the simulated response variables (y), the covariates specified by xformula, subjects' identities (id) and the corresponding measurement occasions (time).
rlatent	the latent random variables denoted by $e^{B}_{it}$ in Touloumis (2016).

Author(s)

Anestis Touloumis

References

Cario, M. C. and Nelson, B. L. (1997) Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. Technical Report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois.

Emrich, L. J. and Piedmonte, M. R. (1991) A method for generating high-dimensional multivariate binary variates. The American Statistician 45, 302–304.

Li, S. T. and Hammond, J. L. (1975) Generation of pseudorandom numbers with specified univariate distributions and correlation coefficients. IEEE Transactions on Systems, Man and Cybernetics 5, 557–561.

Touloumis, A. (2016) Simulating Correlated Binary and Multinomial Responses under Marginal Model Specification: The SimCorMultRes Package. The R Journal 8, 79–91.

See Also

rmult.bcl for simulating correlated nominal responses, rmult.clm, rmult.crm and rmult.acl for simulating correlated ordinal responses.

Examples

## See Example 3.5 in the Vignette.
set.seed(123)
sample_size <- 5000
cluster_size <- 4
beta_intercepts <- 0
beta_coefficients <- 0.2
latent_correlation_matrix <- toeplitz(c(1, 0.9, 0.9, 0.9))
x <- rep(rnorm(sample_size), each = cluster_size)
simulated_binary_dataset <- rbin(clsize = cluster_size,
  intercepts = beta_intercepts, betas = beta_coefficients,
  xformula = ~x, cor.matrix = latent_correlation_matrix, link = "probit")
library(gee)
binary_gee_model <- gee(y ~ x, family = binomial("probit"), id = id,
  data = simulated_binary_dataset$simdata)
summary(binary_gee_model)$coefficients

## See Example 3.6 in the Vignette.
set.seed(8)
library(evd)
simulated_latent_variables1 <- rmvevd(sample_size, dep = sqrt(1 - 0.9),
  model = "log", d = cluster_size)
  simulated_latent_variables2 <- rmvevd(sample_size, dep = sqrt(1 - 0.9),
  model = "log", d = cluster_size)
simulated_latent_variables <- simulated_latent_variables1 -
  simulated_latent_variables2
simulated_binary_dataset <- rbin(clsize = cluster_size,
  intercepts = beta_intercepts, betas = beta_coefficients,
  xformula = ~x, rlatent = simulated_latent_variables)
binary_gee_model <- gee(y ~ x, family = binomial("logit"), id = id,
  data = simulated_binary_dataset$simdata)
summary(binary_gee_model)$coefficients

---

Simulating Correlated Ordinal Responses Conditional on a Marginal Adjacent-Category Logit Model Specification

Description

Simulates correlated ordinal responses assuming an adjacent-category logit model for the marginal probabilities.

Usage

rmult.acl(clsize = clsize, intercepts = intercepts, betas = betas,
  xformula = formula(xdata), xdata = parent.frame(),
  cor.matrix = cor.matrix, rlatent = NULL)

Arguments

clsize	integer indicating the common cluster size.
intercepts	numerical vector or matrix containing the intercepts of the marginal adjacent-category logit model.
betas	numerical vector or matrix containing the value of the marginal regression parameter vector.
xformula	formula expression as in other marginal regression models but without including a response variable.
xdata	optional data frame containing the variables provided in xformula.
cor.matrix	matrix indicating the correlation matrix of the multivariate normal distribution when the NORTA method is employed (rlatent = NULL).
rlatent	matrix with (clsize * ncategories) columns containing realizations of the latent random vectors when the NORTA method is not preferred. See details for more info.

Details

The formulae are easier to read from either the Vignette or the Reference Manual (both available here).

The assumed marginal adjacent-category logit model is

$log \frac{Pr(Y_{it}=j |x_{it})}{Pr(Y_{it}=j+1 |x_{it})}=\beta_{tj0} + \beta^{'}_{t} x_{it}$

For subject $i$, $Y_{it}$ is the $t$-th ordinal response and $x_{it}$ is the associated covariates vector. Also $\beta_{tj0}$ is the $j$-th category-specific intercept at the $t$-th measurement occasion and $\beta_{t}$ is the regression parameter vector at the $t$-th measurement occasion.

The ordinal response $Y_{it}$ is obtained by utilizing the threshold approach described in the Vignette. This approach is based on the connection between baseline-category and adjacent-category logit models.

When $\beta_{tj0}=\beta_{j0}$ for all $t$, then intercepts should be provided as a numerical vector. Otherwise, intercepts must be a numerical matrix such that row $t$ contains the category-specific intercepts at the $t$-th measurement occasion.

betas should be provided as a numeric vector only when $\beta_{t}=\beta$ for all $t$. Otherwise, betas must be provided as a numeric matrix with clsize rows such that the $t$-th row contains the value of $\beta_{t}$. In either case, betas should reflect the order of the terms implied by xformula.

The appropriate use of xformula is xformula = ~ covariates, where covariates indicate the linear predictor as in other marginal regression models.

The optional argument xdata should be provided in “long” format.

The NORTA method is the default option for simulating the latent random vectors denoted by $e^{O3}_{itj}$ in the Vignette. To import simulated values for the latent random vectors without utilizing the NORTA method, the user can employ the rlatent argument. In this case, row $i$ corresponds to subject $i$ and columns $(t-1)*\code{ncategories}+1,...,t*\code{ncategories}$ should contain the realization of $e^{O3}_{it1},...,e^{O3}_{itJ}$, respectively, for $t=1,\ldots,\code{clsize}$.

Value

Returns a list that has components:

Ysim	the simulated nominal responses. Element ($i$,$t$) represents the realization of $Y_{it}$.
simdata	a data frame that includes the simulated response variables (y), the covariates specified by xformula, subjects' identities (id) and the corresponding measurement occasions (time).
rlatent	the latent random variables denoted by $e^{O3}_{itj}$ in the Vignette.

Author(s)

Anestis Touloumis

References

Cario, M. C. and Nelson, B. L. (1997) Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. Technical Report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois.

Li, S. T. and Hammond, J. L. (1975) Generation of pseudorandom numbers with specified univariate distributions and correlation coefficients. IEEE Transactions on Systems, Man and Cybernetics 5, 557–561.

Touloumis, A. (2016) Simulating Correlated Binary and Multinomial Responses under Marginal Model Specification: The SimCorMultRes Package. The R Journal 8, 79–91.

Touloumis, A., Agresti, A. and Kateri, M. (2013) GEE for multinomial responses using a local odds ratios parameterization. Biometrics 69, 633–640.

See Also

rbin for simulating correlated binary responses, rmult.clm and rmult.crm for simulating correlated ordinal responses, and rmult.bcl for simulating nominal responses.

Examples

## See Example 3.4 in the Vignette.
beta_intercepts <- c(3, 1, 2)
beta_coefficients <- c(1, 1)
sample_size <- 500
cluster_size <- 3
set.seed(321)
x1 <- rep(rnorm(sample_size), each = cluster_size)
x2 <- rnorm(sample_size * cluster_size)
xdata <- data.frame(x1, x2)
identity_matrix <- diag(4)
equicorrelation_matrix <- toeplitz(c(1, rep(0.95, cluster_size - 1)))
latent_correlation_matrix <- kronecker(equicorrelation_matrix,
  identity_matrix)
simulated_ordinal_dataset <- rmult.acl(clsize = cluster_size,
  intercepts = beta_intercepts, betas = beta_coefficients,
  xformula = ~ x1 + x2, xdata = xdata,
  cor.matrix = latent_correlation_matrix)
suppressPackageStartupMessages(library("multgee"))
ordinal_gee_model <- ordLORgee(y ~ x1 + x2,
  data = simulated_ordinal_dataset$simdata, id = id, repeated = time,
  LORstr = "time.exch", link = "acl")
round(coef(ordinal_gee_model), 2)

---

Simulating Correlated Nominal Responses Conditional on a Marginal Baseline-Category Logit Model Specification

Description

Simulates correlated nominal responses assuming a baseline-category logit model for the marginal probabilities.

Usage

rmult.bcl(clsize = clsize, ncategories = ncategories, betas = betas,
  xformula = formula(xdata), xdata = parent.frame(),
  cor.matrix = cor.matrix, rlatent = NULL)

Arguments

clsize	integer indicating the common cluster size.
ncategories	integer indicating the number of nominal response categories.
betas	numerical vector or matrix containing the value of the marginal regression parameter vector.
xformula	formula expression as in other marginal regression models but without including a response variable.
xdata	optional data frame containing the variables provided in xformula.
cor.matrix	matrix indicating the correlation matrix of the multivariate normal distribution when the NORTA method is employed (rlatent = NULL).
rlatent	matrix with (clsize * ncategories) columns containing realizations of the latent random vectors when the NORTA method is not preferred. See details for more info.

Details

The formulae are easier to read from either the Vignette or the Reference Manual (both available here).

The assumed marginal baseline category logit model is

$log \frac{Pr(Y_{it}=j |x_{it})}{Pr(Y_{it}=J |x_{it})}=(\beta_{tj0}-\beta_{tJ0}) + (\beta^{'}_{tj}-\beta^{'}_{tJ}) x_{it}=\beta^{*}_{tj0}+ \beta^{*'}_{tj} x_{it}$

For subject $i$, $Y_{it}$ is the $t$-th nominal response and $x_{it}$ is the associated covariates vector. Also $\beta_{tj0}$ is the $j$-th category-specific intercept at the $t$-th measurement occasion and $\beta_{tj}$ is the $j$-th category-specific regression parameter vector at the $t$-th measurement occasion.

The nominal response $Y_{it}$ is obtained by extending the principle of maximum random utility (McFadden, 1974) as suggested in Touloumis (2016).

betas should be provided as a numeric vector only when $\beta_{tj0}=\beta_{j0}$ and $\beta_{tj}=\beta_j$ for all $t$. Otherwise, betas must be provided as a numeric matrix with clsize rows such that the $t$-th row contains the value of ($\beta_{t10},\beta_{t1},\beta_{t20},\beta_{t2},...,\beta_{tJ0}, \beta_{tJ}$). In either case, betas should reflect the order of the terms implied by xformula.

The appropriate use of xformula is xformula = ~ covariates, where covariates indicate the linear predictor as in other marginal regression models.

The optional argument xdata should be provided in “long” format.

The NORTA method is the default option for simulating the latent random vectors denoted by $e^{NO}_{itj}$ in Touloumis (2016). In this case, the algorithm forces cor.matrix to respect the assumption of choice independence. To import simulated values for the latent random vectors without utilizing the NORTA method, the user can employ the rlatent argument. In this case, row $i$ corresponds to subject $i$ and columns $(t-1)*\code{ncategories}+1,...,t*\code{ncategories}$ should contain the realization of $e^{NO}_{it1},...,e^{NO}_{itJ}$, respectively, for $t=1,\ldots,\code{clsize}$.

Value

Returns a list that has components:

Ysim	the simulated nominal responses. Element ($i$,$t$) represents the realization of $Y_{it}$.
simdata	a data frame that includes the simulated response variables (y), the covariates specified by xformula, subjects' identities (id) and the corresponding measurement occasions (time).
rlatent	the latent random variables denoted by $e^{NO}_{it}$ in Touloumis (2016).

Author(s)

Anestis Touloumis

References

Cario, M. C. and Nelson, B. L. (1997) Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. Technical Report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois.

Li, S. T. and Hammond, J. L. (1975) Generation of pseudorandom numbers with specified univariate distributions and correlation coefficients. IEEE Transactions on Systems, Man and Cybernetics 5, 557–561.

McFadden, D. (1974) Conditional logit analysis of qualitative choice behavior. New York: Academic Press, 105–142.

Touloumis, A. (2016) Simulating Correlated Binary and Multinomial Responses under Marginal Model Specification: The SimCorMultRes Package. The R Journal 8, 79–91.

Touloumis, A., Agresti, A. and Kateri, M. (2013) GEE for multinomial responses using a local odds ratios parameterization. Biometrics 69, 633–640.

See Also

rbin for simulating correlated binary responses, rmult.clm, rmult.crm and rmult.acl for simulating correlated ordinal responses.

Examples

## See Example 3.1 in the Vignette.
betas <- c(1, 3, 2, 1.25, 3.25, 1.75, 0.75, 2.75, 2.25, 0, 0, 0)
sample_size <- 500
categories_no <- 4
cluster_size <- 3
set.seed(1)
x1 <- rep(rnorm(sample_size), each = cluster_size)
x2 <- rnorm(sample_size * cluster_size)
xdata <- data.frame(x1, x2)
equicorrelation_matrix <- toeplitz(c(1, rep(0.95, cluster_size - 1)))
identity_matrix <- diag(categories_no)
latent_correlation_matrix <- kronecker(equicorrelation_matrix,
  identity_matrix)
simulated_nominal_dataset <- rmult.bcl(clsize = cluster_size,
  ncategories = categories_no, betas = betas, xformula = ~ x1 + x2,
  xdata = xdata, cor.matrix = latent_correlation_matrix)
suppressPackageStartupMessages(library("multgee"))
nominal_gee_model <- nomLORgee(y ~ x1 + x2,
  data = simulated_nominal_dataset$simdata, id = id, repeated = time,
  LORstr = "time.exch")
round(coef(nominal_gee_model), 2)

---

Simulating Correlated Ordinal Responses Conditional on a Marginal Cumulative Link Model Specification

Description

Simulates correlated ordinal responses assuming a cumulative link model for the marginal probabilities.

Usage

rmult.clm(clsize = clsize, intercepts = intercepts, betas = betas,
  xformula = formula(xdata), xdata = parent.frame(), link = "logit",
  cor.matrix = cor.matrix, rlatent = NULL)

Arguments

clsize	integer indicating the common cluster size.
intercepts	numerical vector or matrix containing the intercepts of the marginal cumulative link model.
betas	numerical vector or matrix containing the value of the marginal regression parameter vector associated with the covariates (i.e., excluding intercepts).
xformula	formula expression as in other marginal regression models but without including a response variable.
xdata	optional data frame containing the variables provided in xformula.
link	character string indicating the link function in the marginal cumulative link model. Options include 'probit', 'logit', 'cloglog' or 'cauchit'. Required when rlatent = NULL.
cor.matrix	matrix indicating the correlation matrix of the multivariate normal distribution when the NORTA method is employed (rlatent = NULL).
rlatent	matrix with clsize columns containing realizations of the latent random vectors when the NORTA method is not preferred. See details for more info.

Details

The formulae are easier to read from either the Vignette or the Reference Manual (both available here).

The assumed marginal cumulative link model is

$Pr(Y_{it}\le j |x_{it})=F(\beta_{tj0} +\beta^{'}_{t} x_{it})$

where $F$ is the cumulative distribution function determined by link. For subject $i$, $Y_{it}$ is the $t$-th ordinal response and $x_{it}$ is the associated covariates vector. Finally, $\beta_{tj0}$ is the $j$-th category-specific intercept at the $t$-th measurement occasion and $\beta_{tj}$ is the $j$-th category-specific regression parameter vector at the $t$-th measurement occasion.

The ordinal response $Y_{it}$ is obtained by extending the approach of McCullagh (1980) as suggested in Touloumis (2016).

When $\beta_{tj0}=\beta_{j0}$ for all $t$, then intercepts should be provided as a numerical vector. Otherwise, intercepts must be a numerical matrix such that row $t$ contains the category-specific intercepts at the $t$-th measurement occasion.

betas should be provided as a numeric vector only when $\beta_{t}=\beta$ for all $t$. Otherwise, betas must be provided as a numeric matrix with clsize rows such that the $t$-th row contains the value of $\beta_{t}$. In either case, betas should reflect the order of the terms implied by xformula.

The appropriate use of xformula is xformula = ~ covariates, where covariates indicate the linear predictor as in other marginal regression models.

The optional argument xdata should be provided in “long” format.

The NORTA method is the default option for simulating the latent random vectors denoted by $e^{O1}_{it}$ in Touloumis (2016). To import simulated values for the latent random vectors without utilizing the NORTA method, the user can employ the rlatent argument. In this case, element ($i,t$) of rlatent represents the realization of $e^{O1}_{it}$.

Value

Returns a list that has components:

Ysim	the simulated ordinal responses. Element ($i$,$t$) represents the realization of $Y_{it}$.
simdata	a data frame that includes the simulated response variables (y), the covariates specified by xformula, subjects' identities (id) and the corresponding measurement occasions (time).
rlatent	the latent random variables denoted by $e^{O1}_{it}$ in Touloumis (2016).

Author(s)

Anestis Touloumis

References

Cario, M. C. and Nelson, B. L. (1997) Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. Technical Report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois.

Li, S. T. and Hammond, J. L. (1975) Generation of pseudorandom numbers with specified univariate distributions and correlation coefficients. IEEE Transactions on Systems, Man and Cybernetics 5, 557–561.

McCullagh, P. (1980) Regression models for ordinal data. Journal of the Royal Statistical Society B 42, 109–142.

Touloumis, A. (2016) Simulating Correlated Binary and Multinomial Responses under Marginal Model Specification: The SimCorMultRes Package. The R Journal 8, 79–91.

Touloumis, A., Agresti, A. and Kateri, M. (2013) GEE for multinomial responses using a local odds ratios parameterization. Biometrics 69, 633–640.

See Also

rmult.bcl for simulating correlated nominal responses, rmult.crm and rmult.acl for simulating correlated ordinal responses and rbin for simulating correlated binary responses.

Examples

## See Example 3.2 in the Vignette.
set.seed(12345)
sample_size <- 500
cluster_size <- 4
beta_intercepts <- c(-1.5, -0.5, 0.5, 1.5)
beta_coefficients <- matrix(c(1, 2, 3, 4), 4, 1)
x <- rep(rnorm(sample_size), each = cluster_size)
latent_correlation_matrix <- toeplitz(c(1, 0.85, 0.5, 0.15))
simulated_ordinal_dataset <- rmult.clm(clsize = cluster_size,
  intercepts = beta_intercepts, betas = beta_coefficients, xformula = ~x,
  cor.matrix = latent_correlation_matrix, link = "probit")
head(simulated_ordinal_dataset$simdata, n = 8)

## Same sampling scheme except that the parameter vector is time-stationary.
set.seed(12345)
simulated_ordinal_dataset <- rmult.clm(clsize = cluster_size, betas = 1,
  xformula = ~x, cor.matrix = latent_correlation_matrix,
  intercepts = beta_intercepts, link = "probit")
## Fit a GEE model (Touloumis et al., 2013) to estimate the regression
## coefficients.
library(multgee)
ordinal_gee_model <- ordLORgee(y ~ x, id = id, repeated = time,
  link = "probit", data = simulated_ordinal_dataset$simdata)
coef(ordinal_gee_model)

---

Simulating Correlated Ordinal Responses Conditional on a Marginal Continuation-Ratio Model Specification

Description

Simulates correlated ordinal responses assuming a continuation-ratio model for the marginal probabilities.

Usage

rmult.crm(clsize = clsize, intercepts = intercepts, betas = betas,
  xformula = formula(xdata), xdata = parent.frame(), link = "logit",
  cor.matrix = cor.matrix, rlatent = NULL)

Arguments

clsize	integer indicating the common cluster size.
intercepts	numerical vector or matrix containing the intercepts of the marginal continuation-ratio model.
betas	numerical vector or matrix containing the value of the marginal regression parameter vector associated with the covariates (i.e., excluding intercepts).
xformula	formula expression as in other marginal regression models but without including a response variable.
xdata	optional data frame containing the variables provided in xformula.
link	character string indicating the link function of the marginal continuation-ratio model. Options include 'probit', 'logit', 'cloglog' or 'cauchit'. Required when rlatent = NULL.
cor.matrix	matrix indicating the correlation matrix of the multivariate normal distribution when the NORTA method is employed (rlatent = NULL).
rlatent	matrix with clsize rows and ncategories columns containing realizations of the latent random vectors when the NORTA method is not employed. See details for more info.

Details

The formulae are easier to read from either the Vignette or the Reference Manual (both available here).

The assumed marginal continuation-ratio model is

$Pr(Y_{it}=j |Y_{it} \ge j,x_{it})=F(\beta_{tj0} +\beta^{'}_{t} x_{it})$

where $F$ is the cumulative distribution function determined by link. For subject $i$, $Y_{it}$ is the $t$-th multinomial response and $x_{it}$ is the associated covariates vector. Finally, $\beta_{tj0}$ is the $j$-th category-specific intercept at the $t$-th measurement occasion and $\beta_{tj}$ is the $j$-th category-specific regression parameter vector at the $t$-th measurement occasion.

The ordinal response $Y_{it}$ is determined by extending the latent variable threshold approach of Tutz (1991) as suggested in Touloumis (2016).

When $\beta_{tj0}=\beta_{j0}$ for all $t$, then intercepts should be provided as a numerical vector. Otherwise, intercepts must be a numerical matrix such that row $t$ contains the category-specific intercepts at the $t$-th measurement occasion.

betas should be provided as a numeric vector only when $\beta_{t}=\beta$ for all $t$. Otherwise, betas must be provided as a numeric matrix with clsize rows such that the $t$-th row contains the value of $\beta_{t}$. In either case, betas should reflect the order of the terms implied by xformula.

The appropriate use of xformula is xformula = ~ covariates, where covariates indicate the linear predictor as in other marginal regression models.

The optional argument xdata should be provided in “long” format.

The NORTA method is the default option for simulating the latent random vectors denoted by $e^{O2}_{itj}$ in Touloumis (2016). In this case, the algorithm forces cor.matrix to respect the local independence assumption. To import simulated values for the latent random vectors without utilizing the NORTA method, the user can employ the rlatent argument. In this case, row $i$ corresponds to subject $i$ and columns $(t-1)*\code{ncategories}+1,...,t*\code{ncategories}$ should contain the realization of $e^{O2}_{it1},...,e^{O2}_{itJ}$, respectively, for $t=1,\ldots,\code{clsize}$.

Value

Returns a list that has components:

Ysim	the simulated ordinal responses. Element ($i$,$t$) represents the realization of $Y_{it}$.
simdata	a data frame that includes the simulated response variables (y), the covariates specified by xformula, subjects' identities (id) and the corresponding measurement occasions (time).
rlatent	the latent random variables denoted by $e^{O2}_{it}$ in Touloumis (2016).

Author(s)

Anestis Touloumis

References

Cario, M. C. and Nelson, B. L. (1997) Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. Technical Report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois.

Li, S. T. and Hammond, J. L. (1975) Generation of pseudorandom numbers with specified univariate distributions and correlation coefficients. IEEE Transactions on Systems, Man and Cybernetics 5, 557–561.

Touloumis, A. (2016) Simulating Correlated Binary and Multinomial Responses under Marginal Model Specification: The SimCorMultRes Package. The R Journal (forthcoming).

Tutz, G. (1991) Sequential models in categorical regression. Computational Statistics & Data Analysis 11, 275–295.

See Also

rmult.bcl for simulating correlated nominal responses, rmult.clm and rmult.acl for simulating correlated ordinal responses and rbin for simulating correlated binary responses.

Examples

## See Example 3.3 in the Vignette.
set.seed(1)
sample_size <- 500
cluster_size <- 4
beta_intercepts <- c(-1.5, -0.5, 0.5, 1.5)
beta_coefficients <- 1
x <- rnorm(sample_size * cluster_size)
categories_no <- 5
identity_matrix <- diag(1, (categories_no - 1) * cluster_size)
equicorrelation_matrix <- toeplitz(c(0, rep(0.24, categories_no - 2)))
ones_matrix <- matrix(1, cluster_size, cluster_size)
latent_correlation_matrix <- identity_matrix +
  kronecker(equicorrelation_matrix, ones_matrix)
simulated_ordinal_dataset <- rmult.crm(clsize = cluster_size,
  intercepts = beta_intercepts, betas = beta_coefficients, xformula = ~x,
  cor.matrix = latent_correlation_matrix, link = "probit")
head(simulated_ordinal_dataset$Ysim)

---

Simulating Random Vectors using the NORTA Method

Description

Utility function to simulate random vectors with predefined marginal distributions via the NORTA method.

Usage

rnorta(R = R, cor.matrix = cor.matrix, distr = distr,
  qparameters = NULL)

Arguments

R	integer indicating the sample size.
cor.matrix	matrix indicating the correlation matrix of the multivariate normal distribution employed in the NORTA method.
distr	character string vector of length ncol(cor.matrix) naming the quantile functions of the desired marginal distributions.
qparameters	list of ncol(cor.matrix) lists indicating the parameter values of the quantile functions specified by distr.

Details

Checks are made to ensure that cor.matrix is a positive definite correlation matrix. The positive definiteness of cor.matrix is assessed via eigenvalues.

The $t$-th character string in distr indicates the quantile function of the $t$-th marginal distribution. See Distributions for the most common distributions. Quantile functions supported by other R packages are allowed provided that these packages have been uploaded first. However, note that no checks are made to ensure that the character strings in distr correspond to valid names of quantile functions.

If qparameters = NULL then the default parameter values for the quantile functions specified by distr are used. Otherwise, qparameters should be provided as a list of ncol(cor.matrix) lists such that the $t$-th list contains the desired parameter values of the $t$-th quantile function.

Value

Returns R random vectors of size ncol(cor.matrix) with marginal distributions specified by distr (and qparameters).

Author(s)

Anestis Touloumis

References

Cario, M. C. and Nelson, B. L. (1997) Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix. Technical Report, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois.

Li, S. T. and Hammond, J. L. (1975) Generation of pseudorandom numbers with specified univariate distributions and correlation coefficients. IEEE Transactions on Systems, Man and Cybernetics 5, 557–561.

Touloumis, A. (2016) Simulating Correlated Binary and Multinomial Responses under Marginal Model Specification: The SimCorMultRes Package. The R Journal 8, 79–91.

Examples

## An example with standard logistic as marginal distribution.
set.seed(1)
sample_size <- 1000
latent_correlation_matrix <- toeplitz(c(1, rep(0.8, 2)))
latent_correlation_matrix
common_marginal_distribution <- rep("qlogis", 3)
simulated_logistic_responses <- rnorta(R = sample_size,
  cor.matrix = latent_correlation_matrix,
  distr = common_marginal_distribution)

## The following lines exemplify the NORTA method.
set.seed(1)
simulated_normal_responses <- rsmvnorm(R = sample_size,
  cor.matrix = latent_correlation_matrix)
norta_simulated <- qlogis(pnorm(simulated_normal_responses))
all(simulated_logistic_responses == norta_simulated)

## Change the marginal distributions to standard normal, standard
## logistic and standard extreme value distribution.
set.seed(1)
different_marginal_distributions <- c("qnorm", "qlogis", "qgumbel")
simulated_logistic_responses <- rnorta(R = sample_size,
  cor.matrix = latent_correlation_matrix,
  distr = different_marginal_distributions)
cor(simulated_logistic_responses)
colMeans(simulated_logistic_responses)
apply(simulated_logistic_responses, 2, sd)

## Same as above but using parameter values other than the default ones.
set.seed(1)
qpars <- list(c(mean = 1, sd = 9), c(location = 2, scale = 1),
  c(loc = 3, scale = 1))
simulated_logistic_responses <- rnorta(R = sample_size,
  cor.matrix = latent_correlation_matrix,
  distr = different_marginal_distributions, qparameters = qpars)
cor(simulated_logistic_responses)
colMeans(simulated_logistic_responses)
apply(simulated_logistic_responses, 2, sd)

---

Simulating Continuous Random Vectors from a Multivariate Normal Distribution

Description

Utility function to simulate continuous random vectors from a multivariate normal distribution such that all marginal distributions are univariate standard normal.

Usage

rsmvnorm(R = R, cor.matrix = cor.matrix)

Arguments

R	integer indicating the sample size.
cor.matrix	matrix indicating the correlation matrix of the multivariate normal distribution.

Details

Checks are made to ensure that cor.matrix is a positive definite correlation matrix. The positive definiteness of cor.matrix is assessed via eigenvalues.

Value

Returns R random vectors of size ncol(cor.matrix).

Author(s)

Anestis Touloumis

Examples

## Simulating 10000 bivariate random vectors with correlation parameter
## equal to 0.4.
set.seed(1)
sample_size <- 10000
correlation_matrix <- toeplitz(c(1, 0.4))
simulated_normal_responses <- rsmvnorm(R = sample_size,
  cor.matrix = correlation_matrix)
colMeans(simulated_normal_responses)
apply(simulated_normal_responses, 2, sd)
cor(simulated_normal_responses)

---

Simulated Correlation Parameters

Description

A simulated dataset to explore the association between the correlation parameter of bivariate normally distributed random variables used in the intermediate step of the NORTA method and the correlation parameter of the resulting non-normal random responses generated by the NORTA method for all the threshold approached implemented in this package.

Usage

simulation

Format

A data frame with 100 rows and 4 columns:

rho

numeric indicating the true value of the correlation parameter.

normal

numeric indicating the simulated correlation parameter when the marginal distribution of each of the latent variables is normal.

logistic

numeric indicating the simulated correlation parameter when the marginal distribution of each of the latent variables is logistic.

gumbel

numeric indicating the simulated correlation parameter when the marginal distribution of each of the latent variables is Gumbel.

Examples

plot(rho - normal ~ rho, data = simulation, type = "l", col = "blue",
    ylim = c(0, 0.016),
    ylab = expression(rho - bar(rho)[sim]),
    xlab = expression(rho))
points(rho - logistic ~ rho, data = simulation, type = "l", col = "red")
points(rho - gumbel ~ rho, data = simulation, type = "l", col = "grey")
legend("topright", legend = c("Normal", "Logistic", "Gumbel"),
      col = c("blue", "red", "grey"), pch = "l" )

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