Latent Regression Model for the Generalized Logistic Item Response Model and the Linear Model for Normal Responses

This function estimates a unidimensional latent regression model if a likelihood is specified, parameters from the generalized item response model (Stukel, 1988) or a mean and a standard error estimate for individual scores is provided as input. Item parameters are treated as fixed in the estimation.

Usage

latent.regression.em.raschtype(data=NULL, f.yi.qk=NULL, X,
    weights=rep(1, nrow(X)), beta.init=rep(0,ncol(X)),
    sigma.init=1, b=rep(0,ncol(X)), a=rep(1,length(b)),
    c=rep(0, length(b)), d=rep(1, length(b)), alpha1=0, alpha2=0,
    max.parchange=1e-04, theta.list=seq(-5, 5, len=20),
    maxiter=300, progress=TRUE )

latent.regression.em.normal(y, X, sig.e, weights=rep(1, nrow(X)),
    beta.init=rep(0, ncol(X)), sigma.init=1, max.parchange=1e-04,
    maxiter=300, progress=TRUE)

# S3 method for latent.regression
summary(object,...)

Arguments

data

An \(N \times I\) data frame of dichotomous item responses. If no data frame is supplied, then a user can input the individual likelihood f.yi.qk.

f.yi.qk

An optional matrix which contains the individual likelihood. This matrix is produced by rasch.mml2 or rasch.copula2. The use of this argument allows the estimation of the latent regression model independent of the parameters of the used item response model.

X

An \(N \times K\) matrix of \(K\) covariates in the latent regression model. Note that the intercept (i.e. a vector of ones) must be included in X.

weights

Student weights (optional).

beta.init

Initial regression coefficients (optional).

sigma.init

Initial residual standard deviation (optional).

b

Item difficulties (optional). They must only be provided if the likelihood f.yi.qk is not given as an input.

a

Item discriminations (optional).

c

Guessing parameter (lower asymptotes) (optional).

d

One minus slipping parameter (upper asymptotes) (optional).

alpha1

Upper tail parameter \(\alpha_1\) in the generalized logistic item response model. Default is 0.

alpha2

Lower tail parameter \(\alpha_2\) parameter in the generalized logistic item response model. Default is 0.

max.parchange

Maximum change in regression parameters

theta.list

Grid of person ability where theta is evaluated

maxiter

Maximum number of iterations

progress

An optional logical indicating whether computation progress should be displayed.

y

Individual scores

sig.e

Standard errors for individual scores

object

Object of class latent.regression

...

Further arguments to be passed

Details

In the output Regression Parameters the fraction of missing information (fmi) is reported which is the increase of variance in regression parameter estimates because ability is defined as a latent variable. The effective sample size pseudoN.latent corresponds to a sample size when the ability would be available with a reliability of one.

Value

A list with following entries

iterations

Number of iterations needed

maxiter

Maximal number of iterations

max.parchange

Maximum change in parameter estimates

coef

Coefficients

summary.coef

Summary of regression coefficients

sigma

Estimate of residual standard deviation

vcov.simple

Covariance parameters of estimated parameters (simplified version)

vcov.latent

Covariance parameters of estimated parameters which accounts for latent ability

post

Individual posterior distribution

EAP

Individual EAP estimates

SE.EAP

Standard error estimates of EAP

explvar

Explained variance in latent regression

totalvar

Total variance in latent regression

rsquared

Explained variance \(R^2\) in latent regression

References

Adams, R., & Wu. M. (2007). The mixed-coefficients multinomial logit model: A generalized form of the Rasch model. In M. von Davier & C. H. Carstensen (Eds.). Multivariate and mixture distribution Rasch models: Extensions and applications (pp. 57-76). New York: Springer. doi:10.1007/978-0-387-49839-3_4

Mislevy, R. J. (1991). Randomization-based inference about latent variables from complex samples. Psychometrika, 56(2), 177-196. doi:10.1007/BF02294457

Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83(402), 426-431. doi:10.1080/01621459.1988.10478613

Note

Using the defaults in a, c, d, alpha1 and alpha2 corresponds to the Rasch model.

See also

See also plausible.value.imputation.raschtype for plausible value imputation of generalized logistic item type models.

Examples

#############################################################################
#  EXAMPLE 1: PISA Reading | Rasch model for dichotomous data
#############################################################################

data(data.pisaRead, package="sirt")
dat <- data.pisaRead$data
items <- grep("R", colnames(dat))
# define matrix of covariates
X <- cbind( 1, dat[, c("female","hisei","migra" ) ] )

#***
# Model 1: Latent regression model in the Rasch model
# estimate Rasch model
mod1 <- sirt::rasch.mml2( dat[,items] )
# latent regression model
lm1 <- sirt::latent.regression.em.raschtype( data=dat[,items ], X=X, b=mod1$item$b )

if (FALSE) {
#***
# Model 2: Latent regression with generalized link function
# estimate alpha parameters for link function
mod2 <- sirt::rasch.mml2( dat[,items], est.alpha=TRUE)
# use model estimated likelihood for latent regression model
lm2 <- sirt::latent.regression.em.raschtype( f.yi.qk=mod2$f.yi.qk,
            X=X, theta.list=mod2$theta.k)

#***
# Model 3: Latent regression model based on Rasch copula model
testlets <- paste( data.pisaRead$item$testlet)
itemclusters <- match( testlets, unique(testlets) )
# estimate Rasch copula model
mod3 <- sirt::rasch.copula2( dat[,items], itemcluster=itemclusters )
# use model estimated likelihood for latent regression model
lm3 <- sirt::latent.regression.em.raschtype( f.yi.qk=mod3$f.yi.qk,
                X=X, theta.list=mod3$theta.k)

#############################################################################
# EXAMPLE 2: Simulated data according to the Rasch model
#############################################################################

set.seed(899)
I <- 21     # number of items
b <- seq(-2,2, len=I)   # item difficulties
n <- 2000       # number of students

# simulate theta and covariates
theta <- stats::rnorm( n )
x <- .7 * theta + stats::rnorm( n, .5 )
y <- .2 * x+ .3*theta + stats::rnorm( n, .4 )
dfr <- data.frame( theta, 1, x, y )

# simulate Rasch model
dat1 <- sirt::sim.raschtype( theta=theta, b=b )

# estimate latent regression
mod <- sirt::latent.regression.em.raschtype( data=dat1, X=dfr[,-1], b=b )
  ## Regression Parameters
  ##
  ##        est se.simple     se        t p   beta    fmi N.simple pseudoN.latent
  ## X1 -0.2554    0.0208 0.0248 -10.2853 0 0.0000 0.2972     2000       1411.322
  ## x   0.4113    0.0161 0.0193  21.3037 0 0.4956 0.3052     2000       1411.322
  ## y   0.1715    0.0179 0.0213   8.0438 0 0.1860 0.2972     2000       1411.322
  ##
  ## Residual Variance=0.685
  ## Explained Variance=0.3639
  ## Total Variance=1.049
  ##                 R2=0.3469

# compare with linear model (based on true scores)
summary( stats::lm( theta  ~ x + y, data=dfr ) )
  ## Coefficients:
  ##             Estimate Std. Error t value Pr(>|t|)
  ## (Intercept) -0.27821    0.01984  -14.02   <2e-16 ***
  ## x            0.40747    0.01534   26.56   <2e-16 ***
  ## y            0.18189    0.01704   10.67   <2e-16 ***
  ## ---
  ##
  ## Residual standard error: 0.789 on 1997 degrees of freedom
  ## Multiple R-squared: 0.3713,     Adjusted R-squared: 0.3707

#***********
# define guessing parameters (lower asymptotes) and
# upper asymptotes ( 1 minus slipping parameters)
cI <- rep(.2, I)        # all items get a guessing parameter of .2
cI[ c(7,9) ] <- .25     # 7th and 9th get a guessing parameter of .25
dI <- rep( .95, I )    # upper asymptote of .95
dI[ c(7,11) ] <- 1        # 7th and 9th item have an asymptote of 1

# latent regression model
mod1 <- sirt::latent.regression.em.raschtype( data=dat1, X=dfr[,-1],
           b=b, c=cI, d=dI    )
  ## Regression Parameters
  ##
  ##        est se.simple     se        t p   beta    fmi N.simple pseudoN.latent
  ## X1 -0.7929    0.0243 0.0315 -25.1818 0 0.0000 0.4044     2000       1247.306
  ## x   0.5025    0.0188 0.0241  20.8273 0 0.5093 0.3936     2000       1247.306
  ## y   0.2149    0.0209 0.0266   8.0850 0 0.1960 0.3831     2000       1247.306
  ##
  ## Residual Variance=0.9338
  ## Explained Variance=0.5487
  ## Total Variance=1.4825
  ##                 R2=0.3701

#############################################################################
# EXAMPLE 3: Measurement error in dependent variable
#############################################################################

set.seed(8766)
N <- 4000       # number of persons
X <- stats::rnorm(N)           # independent variable
Z <- stats::rnorm(N)           # independent variable
y <- .45 * X + .25 * Z + stats::rnorm(N)   # dependent variable true score
sig.e <- stats::runif( N, .5, .6 )       # measurement error standard deviation
yast <- y + stats::rnorm( N, sd=sig.e ) # dependent variable measured with error

#****
# Model 1: Estimation with latent.regression.em.raschtype using
#          individual likelihood
# define theta grid for evaluation of density
theta.list <- mean(yast) + stats::sd(yast) * seq( - 5, 5, length=21)
# compute individual likelihood
f.yi.qk <- stats::dnorm( outer( yast, theta.list, "-" ) / sig.e )
f.yi.qk <- f.yi.qk / rowSums(f.yi.qk)
# define predictor matrix
X1 <- as.matrix(data.frame( "intercept"=1, "X"=X, "Z"=Z ))

# latent regression model
res <- sirt::latent.regression.em.raschtype( f.yi.qk=f.yi.qk,
                    X=X1, theta.list=theta.list)
  ##   Regression Parameters
  ##
  ##                est se.simple     se       t      p   beta    fmi N.simple pseudoN.latent
  ##   intercept 0.0112    0.0157 0.0180  0.6225 0.5336 0.0000 0.2345     4000       3061.998
  ##   X         0.4275    0.0157 0.0180 23.7926 0.0000 0.3868 0.2350     4000       3061.998
  ##   Z         0.2314    0.0156 0.0178 12.9868 0.0000 0.2111 0.2349     4000       3061.998
  ##
  ##   Residual Variance=0.9877
  ##   Explained Variance=0.2343
  ##   Total Variance=1.222
  ##                   R2=0.1917

#****
# Model 2: Estimation with latent.regression.em.normal
res2 <- sirt::latent.regression.em.normal( y=yast, sig.e=sig.e, X=X1)
  ##   Regression Parameters
  ##
  ##                est se.simple     se       t      p   beta    fmi N.simple pseudoN.latent
  ##   intercept 0.0112    0.0157 0.0180  0.6225 0.5336 0.0000 0.2345     4000       3062.041
  ##   X         0.4275    0.0157 0.0180 23.7927 0.0000 0.3868 0.2350     4000       3062.041
  ##   Z         0.2314    0.0156 0.0178 12.9870 0.0000 0.2111 0.2349     4000       3062.041
  ##
  ##   Residual Variance=0.9877
  ##   Explained Variance=0.2343
  ##   Total Variance=1.222
  ##                   R2=0.1917

  ## -> Results between Model 1 and Model 2 are identical because they use
  ##    the same input.

#***
# Model 3: Regression model based on true scores y
mod3 <- stats::lm( y ~ X + Z )
summary(mod3)
  ##   Coefficients:
  ##               Estimate Std. Error t value Pr(>|t|)
  ##   (Intercept)  0.02364    0.01569   1.506    0.132
  ##   X            0.42401    0.01570  27.016   <2e-16 ***
  ##   Z            0.23804    0.01556  15.294   <2e-16 ***
  ##   Residual standard error: 0.9925 on 3997 degrees of freedom
  ##   Multiple R-squared:  0.1923,    Adjusted R-squared:  0.1919
  ##   F-statistic: 475.9 on 2 and 3997 DF,  p-value: < 2.2e-16

#***
# Model 4: Regression model based on observed scores yast
mod4 <- stats::lm( yast ~ X + Z )
summary(mod4)
  ##   Coefficients:
  ##               Estimate Std. Error t value Pr(>|t|)
  ##   (Intercept)  0.01101    0.01797   0.613     0.54
  ##   X            0.42716    0.01797  23.764   <2e-16 ***
  ##   Z            0.23174    0.01783  13.001   <2e-16 ***
  ##   Residual standard error: 1.137 on 3997 degrees of freedom
  ##   Multiple R-squared:  0.1535,    Adjusted R-squared:  0.1531
  ##   F-statistic: 362.4 on 2 and 3997 DF,  p-value: < 2.2e-16
}