confmirt: Confirmatory Full-Information Item Factor Analysis for Mixed Data Formats

Description

confmirt fits a conditional (i.e., confirmatory) maximum likelihood factor analysis model to dichotomous and polychotomous data under the item response theory paradigm using Cai's (2010) Metropolis-Hastings Robbins-Monro algorithm.

Usage

confmirt(data, sem.model, guess = 0, gmeans = 0, ncycles = 2000,
	burnin = 100, SEM.cycles = 50, kdraws = 1, tol = .001, printcycles = TRUE,
    calcLL = TRUE, draws = 2000, debug = FALSE, ...)

## S3 method for class 'confmirt':
coef(object, SE = TRUE, print.gmeans = FALSE, digits = 3, ...)

## S3 method for class 'confmirt':
summary(object, digits = 3, ...)

## S3 method for class 'confmirt':
residuals(object, digits = 3, ...)

## S3 method for class 'confmirt':
anova(object, object2, ...)

Arguments

data

a matrix or data.frame that consists of numerically ordered data.

sem.model

an object resembling the sem declaration of a confirmatory factor analysis, but with omitted residual variances.

guess

starting values for the pseudo-guessing parameter. Can be entered as a single value to assign a global guessing parameter or may be entered as a numeric vector for each item. When the value is greater than zero the parameter will be estimated.

gmeans

starting values for the group means. Can be entered as a single value to assign a global parameter or may be entered as a numeric vector for each factor. When the value is not equal to zero the parameter will be estimated.

ncycles

the maximum number of iterations to be performed

burnin

number of burn-in cycles to perform before beginning the SEM stage

SEM.cycles

number of stochastic EM cycles to perform before beginning the MH-RM algorithm

kdraws

number of Metropolis-Hastings imputations of the factor scores at each iteration. Default is 1.

tol

tolerance that must be reached before the model is returned

printcycles

logical; display iteration history during estimation?

calcLL

logical; calculate the log-likelihood via Monte Carlo integration?

draws

the number of Monte Carlo draws to estimate the log-likelihood

debug

logical; turn on debugging features?

object

an object of class confmirt.

object2

an object of class confmirt.

logical; print standard errors?

print.gmeans

logical; print factor means?

digits

the number of significant digits to be rounded

...

additional arguments to be passed

Value

confmirt returns an object of class confmirt, with the following elements:
parsestimated parameters of the model, the rightmost columns being the intercepts
guessa vector of the estimated pseudo-guessing parameters
gparsmeans and covariance matrix for the group parameters
SEparsa matrix of standard errors for the parameters estimated
SEga vector of standard errors for the pseudo-guessing parameters
SEgparsa list of standard errors for the group parameters
estparsa list of logical sets used to identify which parameters were estimated
Thetalast iteration of the Metropolis-Hastings imputation of the factor scores
fulldatacomplete data complete with the original item names and values
Ka vector indicating how many unique values each item contains
cyclesnumber of MHRM cycles performed; will be less than ncycles if the tolerance is reached
itemloca vector indicating where the item location begin
convergelogical; did the model converge?
Callthe function call

Details

confmirt follows the confirmatory item factor analysis strategy by a stochastic version of maximum likelihood estimation described by Cai (2010). The general equation used for multidimensional item response theory in this function is in the logistic form with a scaling correction of 1.702. This correction is applied to allow comparison to mainstream programs such as TESTFACT (2003) and POLYFACT. Missing data are treated as 'missing at random' so that each response vector is included in the estimation. Residuals are computed using the LD statistic (Chen & Thissen, 1997) in the lower diagonal of the matrix returned by residuals, and Cramer's V above the diagonal. For computing the log-likelihood more accurately see logLik. Specification of the confirmatory item factor analysis model follows many of the rules in the SEM framework for confirmatory factor analysis. The variances of the latent factors should be fixed to a constant for identification (commonly fixed to 1), and this is the recommended practice since the code is optimized for this method. Parameters may be fixed to specific values by placing a constant in the 4th column of the specify.model function call with the parameter named NA in the 3rd column. Item slopes may also be constrained to be equal by labeling them with the same parameter name in the 3rd column. Guessing parameters may be specified for dichotomous items, but are treated as constants, and if a guessing parameter is declared for a polychotomous item it is ignored. coef displays the item parameters with their associated standard errors, while use of summary transforms the slopes into a factor loadings metric. Also, nested models may be compared by using the anova function, where a Chi-squared difference test and AIC difference values are displayed.

References

Cai, L. (2010). Metropolis-Hastings Robbins-Monro Algorithm for confirmatory item factor analysis. Journal of Educational and Behavioral Statistics, 35, 307-335. Wood, R., Wilson, D. T., Gibbons, R. D., Schilling, S. G., Muraki, E., & Bock, R. D. (2003). TESTFACT 4 for Windows: Test Scoring, Item Statistics, and Full-information Item Factor Analysis [Computer software]. Lincolnwood, IL: Scientific Software International.

Examples

Run this code

#simulate data
a <- matrix(c(
1.5,NA,
0.5,NA,
1.0,NA,
1.0,0.5,
 NA,1.5,
 NA,0.5,
 NA,1.0,
 NA,1.0),ncol=2,byrow=TRUE)

d <- matrix(c(
-1.0,NA,NA,
-1.5,NA,NA,
 1.5,NA,NA,
 0.0,NA,NA,
3.0,2.0,-0.5,
2.5,1.0,-1,
2.0,0.0,NA,
1.0,NA,NA),ncol=3,byrow=TRUE)

sigma <- diag(2)
sigma[1,2] <- sigma[2,1] <- .4
dataset <- simdata(a,d,2000,sigma)

#analyses
#CIFA for 2 factor crossed structure

model.1 <- specify.model()
    F1 -> Item_1,   lam11, NA
    F1 -> Item_2,   lam12, NA
    F1 -> Item_3,   lam13, NA
    F1 -> Item_4,   lam14, NA
    F2 -> Item_4,   lam24, NA
    F2 -> Item_5,   lam25, NA
    F2 -> Item_6,   lam26, NA
    F2 -> Item_7,   lam27, NA
    F2 -> Item_8,   lam28, NA
    F1 <-> F2,      phi12, NA
    F1 <-> F1,      NA,     1
    F2 <-> F2,      NA,     1

mod1 <- confmirt(dataset,model.1)
coef(mod1)
summary(mod1)
residuals(mod1)

#fix first slope at 1.5, and set slopes 7 & 8 to be equal
model.2 <- specify.model()
    F1 -> Item_1,      NA, 1.5
    F1 -> Item_2,   lam12, NA
    F1 -> Item_3,   lam13, NA
    F1 -> Item_4,   lam14, NA
    F2 -> Item_4,   lam24, NA
    F2 -> Item_5,   lam25, NA
    F2 -> Item_6,   lam26, NA
    F2 -> Item_7,   lam27, NA
    F2 -> Item_8,   lam27, NA
    F1 <-> F2,      phi12, NA
    F1 <-> F1,      NA,     1
    F2 <-> F2,      NA,     1

mod2 <- confmirt(dataset,model.2)
anova(mod2,mod1)

#####
#bifactor 	
model.3 <- specify.model()
    F1 -> Item_1,   lam11, NA
    F1 -> Item_2,   lam12, NA
    F1 -> Item_3,   lam13, NA
    F1 -> Item_4,   lam14, NA
    F1 -> Item_5,   lam15, NA
    F1 -> Item_6,   lam16, NA
    F1 -> Item_7,   lam17, NA
    F1 -> Item_8,   lam18, NA
    F2 -> Item_1,   lam21, NA
    F2 -> Item_2,   lam22, NA
    F2 -> Item_3,   lam23, NA
    F2 -> Item_4,   lam24, NA
    F3 -> Item_5,   lam35, NA
    F3 -> Item_6,   lam36, NA                	
    F3 -> Item_7,   lam37, NA
    F3 -> Item_8,   lam38, NA
    F1 <-> F1,      NA,     1
    F2 <-> F2,      NA,     1
    F3 <-> F3,      NA,     1
	
mod3 <- confmirt(dataset,model.3)
coef(mod3)
summary(mod3)
residuals(mod3)
anova(mod1,mod3)

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