# 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. ```r 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,...) ``` ## 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. ## Returns 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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1007/BF02294457") Stukel, T. A. (1988). Generalized logistic models. **Journal of the American Statistical Association, 83**(402), 426-431. tools:::Rd_expr_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 ```r ############################################################################# # 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 ## End(Not run) ```

latent.regression.em.raschtype function