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.
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 class 'latent.regression'
summary(object,...)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 |
An optional matrix which contains the individual likelihood.
This matrix is produced by |
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 |
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 |
a |
Item discriminations (optional). |
c |
Guessing parameter (lower asymptotes) (optional). |
d |
One minus slipping parameter (upper asymptotes) (optional). |
alpha1 |
Upper tail parameter α_1 in the generalized logistic item response model. Default is 0. |
alpha2 |
Lower tail parameter α_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 |
... |
Further arguments to be passed |
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.
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 |
Using the defaults in a, c, d,
alpha1 and alpha2 corresponds to the
Rasch model.
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
See also plausible.value.imputation.raschtype
for plausible value imputation of generalized logistic
item type models.
#############################################################################
# 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$b )
## Not run:
#***
# Model 2: Latent regression with generalized link function
# estimate alpha parameters for link function
mod2 <- sirt::rasch.mml2( dat[,items], est.alpha=TRUE)
# use model estimated likelihood for latent regression model
lm2 <- sirt::latent.regression.em.raschtype( f.yi.qk=mod2$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
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