Moderated Nonlinear Factor Analysis

General function for conducting moderated nonlinear factor analysis (Curran et al., 2014). Item slopes and item intercepts can be modeled as functions of person covariates.

Parameter regularization is allowed. For categorical covariates, group lasso can be used for regularization.

mnlfa(dat, items, weights=NULL, item_type="2PL", formula_int=~1, formula_slo=~1,
   formula_res=~0, formula_mean=~0, formula_sd=~0, theta=NULL, parm_list_init=NULL,
   parm_trait_init=NULL, prior_init=NULL, regular_lam=c(0,0,0), regular_alpha=c(0,0,0),
   regular_type=c("none", "none", "none"), maxit=1000, msteps=4, conv=1e-05,
   conv_mstep=1e-04, h=1e-04, parms_regular_types=NULL, parms_regular_lam=NULL,
   parms_regular_alpha=NULL, parms_iterations=NULL, center_parms=NULL, center_max_iter=6,
   L_max=.07, verbose=TRUE, numdiff=FALSE)

# S3 method for mnlfa
summary(object, file=NULL, ...)

Arguments

dat

Data frame with item responses

items

Vector containing item names

weights

Optional vector of sampling weights for persons

item_type

String or vector of item types. The item types "1PL" or "2PL" for dichotomous items, "GPCM" for polytomous and "NO" for continuous items can be chosen.

formula_int

String or list with formula for item intercepts

formula_slo

String or list with formula for item slopes

formula_res

String or list with formula for logarithms of residual standard deviations

formula_mean

Formula for mean of the trait distribution

formula_sd

Formula for standard deviation of the trait distribution

theta

Grid of \(\theta\) values used for approximation of normally distributed trait

parm_list_init

Optional list of initial item parameters

parm_trait_init

Optional list of initial parameters for trait distribution

prior_init

Optional matrix of prior distribution for persons

regular_lam

Vector of length 2 or 3 (for item_type="NO") containing regularization parameters \(\lambda\) for item intercepts and item slopes

regular_alpha

Vector of length 2 or 3 containing \(\alpha\) regularization parameter

regular_type

Type of regularization method. Can be "none", "lasso", "scad", "mcp", "scadL2", "ridge" or "elnet".

maxit

Maximum number of iterations

msteps

Maximum number of M-steps

conv

Convergence criterion with respect to parameters

conv_mstep

Convergence criterion in M-step

h

Numerical differentiation parameter

parms_regular_types

Optional list containing parameter specific regularization types

parms_regular_lam

Optional list containing parameter specific regularization parameters

parms_regular_alpha

Optional list containing parameter specific regularization parameters

parms_iterations

Optional list containing sequence of parameter indices used for updating

center_parms

Optional list indicating which parameters should be centered during initial iterations.

center_max_iter

Maximum number of iterations in which parameters should be centered.

L_max

Majorization parameter used in regularization

verbose

Logical indicating whether output should be printed

numdiff

Logical indicating whether numerical differentiation should be used

object

Object of class mnlfa

file

Optional file name

...

Further arguments to be passed

Details

The moderated factor analysis model for dichotomous responses defined as $$P(X_{pi}=1 | \theta_p )=invlogit( a_{pi} \theta_p - b_{pi} ) $$ The trait distribution \(\theta_p \sim N( \mu_p, \sigma_p^2)\) allows a latent regression of person covariates on the mean with \(\mu_p=\bold{X}_p \bold{\gamma}\) (to be specified in formula_mean) and the logarithm of the standard deviation \(\log \sigma_p=\bold{Z}_p \bold{\delta} \) (to be specified in formula_sd). Item intercepts and item slopes can be moderated by person covariates, i.e. \(a_{pi}=\bold{W}_{pi} \bold{\alpha}_i \) and \(b_{pi}=\bold{V}_{pi} \bold{\beta}_i \). Regularization on (some of) the \(\bold{\alpha}_i\) or \(\bold{\beta}_i\) parameters is allowed.

For polytomous item responses, the generalized partial credit model is parametrized as $$P(X_{pi}=k | \theta_p \propto \exp ( k a_{pi} \theta_p - k b_{pi} - b_{k} $$ with \(b_0=0\).

For normally distributed responses, the conditional distribution of item responses is defined as $$ X_{pi} | \theta_p \sim \mathrm{N} ( b_{pi} + a_{pi} \theta_p, \psi_{pi}^2 ) $$ Note that \(\log \psi_{pi}\) is modeled in this function.

The model is estimated using an EM algorithm with the coordinate descent method during the M-step (Sun et al., 2016).

Value

List with model results including

item

Summary table for item parameters

trait

Summary table for trait parameters

References

Curran, P. J., McGinley, J. S., Bauer, D. J., Hussong, A. M., Burns, A., Chassin, L., Sher, K., & Zucker, R. (2014). A moderated nonlinear factor model for the development of commensurate measures in integrative data analysis. Multivariate Behavioral Research, 49(3), 214-231. http://dx.doi.org/10.1080/00273171.2014.889594

Sun, J., Chen, Y., Liu, J., Ying, Z., & Xin, T. (2016). Latent variable selection for multidimensional item response theory models via L1 regularization. Psychometrika, 81(4), 921-939. https://doi.org/10.1007/s11336-016-9529-6

See also

See also the aMNLFA package for automatized moderated nonlinear factor analysis which provides convenient wrapper functions for automized analysis in the Mplus software.

See the GPCMlasso package for the regularized generalized partial credit model.

Examples

#############################################################################
# EXAMPLE 1: Dichotomous data, 1PL model
#############################################################################

data(data.mnlfa01, package="mnlfa")

dat <- data.mnlfa01
# extract items from dataset
items <- grep("I[0-9]", colnames(dat), value=TRUE)
I <- length(items)

# maximum number of iterations (use only few iterations for the only purpose of
# providing CRAN checks)
maxit <- 10

#***** Model 1: 1PL model without moderating parameters and without covariates for traits

# no covariates for trait
formula_mean <- ~0
formula_sd <- ~1
# no item covariates
formula_int <- ~1
formula_slo <- ~1

mod1 <- mnlfa::mnlfa( dat=dat, items, item_type="1PL", formula_int=formula_int,
             formula_slo=formula_slo, formula_mean=formula_mean, formula_sd=formula_sd,
             maxit=maxit )
summary(mod1)

# \donttest{
#***** Model 2: 1PL model without moderating parameters and with covariates for traits

# covariates for trait
formula_mean <- ~female + age
formula_sd <- ~1

mod2 <- mnlfa::mnlfa( dat=dat, items, item_type="1PL", formula_int=formula_int,
             formula_slo=formula_slo, formula_mean=formula_mean, formula_sd=formula_sd)
summary(mod2)

#***** Model 3: 1PL model with moderating parameters and with covariates for traits
#***   Regularization method 'mcp'

# covariates for trait
formula_mean <- ~female + age
formula_sd <- ~1
# moderation effects for items
formula_int <- ~1+female+age
formula_slo <- ~1

# center parameters for female and age in initial iterations for improving convergence
center_parms <- list( rep(2,I), rep(3,I) )

# regularization parameters for item intercept and item slope, respectively
regular_lam <- c(.06, .25)
regular_type <- c("mcp","none")

mod3 <- mnlfa::mnlfa( dat=dat, items, item_type="1PL", formula_int=formula_int,
            formula_slo=formula_slo, formula_mean=formula_mean, formula_sd=formula_sd,
            center_parms=center_parms, regular_lam=regular_lam, regular_type=regular_type )
summary(mod3)

# update Model 3 with initial parameters
parm_trait_init <- mod3$parm_trait
parm_list_init <- mod3$parm_list
prior_init <- mod3$prior
mod3b <- mnlfa::mnlfa( dat=dat, items, item_type="1PL", formula_int=formula_int,
            formula_slo=formula_slo, formula_mean=formula_mean, formula_sd=formula_sd,
            center_parms=center_parms, regular_lam=regular_lam,
            regular_type=regular_type, parm_trait_init=parm_trait_init,
            parm_list_init=parm_list_init, prior_init=prior_init,  )
summary(mod3b)

#***** Model 4: 1PL model with selected moderated item parameters

#* trait distribution
formula_mean <- ~0+female+age
formula_sd <- ~1

#* formulas for item intercepts
formula_int <- ~1
formula_int <- mnlfa::mnlfa_expand_to_list(x=formula_int, names_list=items)
mod_items <- c(4,5,6,7)
for (ii in mod_items){
    formula_int[[ii]] <- ~1+female+age
}
formula_slo <- ~1

mod4 <- mnlfa::mnlfa( dat=dat, items, item_type="1PL", formula_int=formula_int,
              formula_slo=formula_slo, formula_mean=formula_mean, formula_sd=formula_sd)
mod4$item
mod4$trait
summary(mod4)

#############################################################################
# EXAMPLE 2: Continuous items
#############################################################################

#-- simulate data
set.seed(78)
N <- 1000
I <- 10
z <- stats::runif(N, -1, 1)
theta <- stats::rnorm(N, mean=0.3*z - .1*z^2, sd=exp(-.5+.3*z) )
dat <- matrix(NA, nrow=N, ncol=I)
items <- colnames(dat) <- paste0("I",1:I)
dat <- as.data.frame(dat)
dat$z <- z

for (ii in 1L:I){
    f <- 0.4*(ii==1)
    dat[,ii] <- .6+-f*z^2 + (1+f*z)*theta + stats::rnorm(N, sd=exp(-.3)  )
}

#--- estimate model
formula_mean <- ~ 0 + z
formula_sd <- ~ 0 + z
formula_int <- ~1+z
formula_slo <- ~1+z
formula_res <- ~1+z

regular_type <- c("mcp","mcp","mcp")
regular_lam <- rep(1e-3,3)
maxit <- 10
item_type <- "NO"

mod1 <- mnlfa::mnlfa( dat=dat, items, item_type=item_type, formula_int=formula_int,
             formula_slo=formula_slo, formula_res=formula_res, formula_mean=formula_mean,
             formula_sd=formula_sd, maxit=maxit, regular_type=regular_type, h=1e-4,
             regular_lam=regular_lam)
summary(mod1)

#############################################################################
# EXAMPLE 3: Polytomous items
#############################################################################

#--- simulate data
set.seed(78)
N <- 2000
I <- 8
z <- stats::runif(N, -1, 1)
theta <- stats::rnorm(N, mean=0.3*z, sd=exp(-0.3*z) )
dat <- matrix(NA, nrow=N, ncol=I)
items <- colnames(dat) <- paste0("I",1:I)
dat <- as.data.frame(dat)
dat$z <- z
for (ii in 1L:I){
    f <- 0.4*(ii %in% c(1,3) )
    y <- -f*z + (1+f*z)*theta + stats::rnorm(N, sd=exp(-.3)  )
    K <- 2 + (ii %% 2==0 )
    br <- c(-Inf,seq(-1,1, len=K), Inf) + ii / I
    y <- as.numeric( cut( y, breaks=br) )-1
    dat[,ii] <- y
}

#--- estimate model
formula_mean <- ~1 + z
formula_sd <- ~1 + z
formula_int <- ~1+z
formula_slo <- ~1+z

regular_type <- rep("scadL2",2)
regular_lam <- rep(0.02,2)
regular_alpha <- rep(0.5,2)
maxit <- 10
item_type <- "GPCM"

mod1 <- mnlfa::mnlfa( dat=dat, items, item_type=item_type, formula_int=formula_int,
             formula_slo=formula_slo, formula_mean=formula_mean,
             formula_sd=formula_sd, maxit=maxit, regular_type=regular_type, h=1e-4,
             regular_lam=regular_lam, regular_alpha=regular_alpha)
summary(mod1)
# }