Nothing
R/module2_estimate_grm.R
In xxIRT: Item Response Theory and Computer-Based Testing in R
#' Estimate Graded Response Model
#' @description Estimate the GRM using the maximum likelihood estimation
#' @name estimate_grm
NULL
#' @rdname estimate_grm
#' @description \code{model_grm_eap_scoring} scores response vectors using the EAP method
#' @param prior the prior distribution
#' @examples
#' with(model_grm_gendata(10, 50, 3), cbind(true=t, est=model_grm_eap_scoring(u, a, b)$t))
#' @importFrom stats dnorm
#' @export
model_grm_eap_scoring <- function(u, a, b, D=1.702, prior=c(0, 1), bound=c(-3, 3)){
quad <- hermite_gauss('11')
quad$w <- quad$w * exp(quad$t^2) * dnorm(quad$t, prior[1], prior[2])
n_p <- dim(u)[1]
n_i <- dim(u)[2]
n_q <- length(quad$t)
p <- model_grm_prob(quad$t, a, b, D)
ix <- model_polytomous_3dindex(u)
lh <- array(NA, c(n_p, n_i, n_q))
for(q in 1:n_q)
lh[,,q] <- array(p[q,,][ix[,-1]], c(n_p, n_i))
lh <- apply(lh, c(1, 3), prod, na.rm=T)
t <- ((lh / (lh %*% quad$w)[,1]) %*% (quad$w * quad$t))[,1]
t_sd <- ((lh / (lh %*% quad$w)[,1] * outer(t, quad$t, '-')^2) %*% quad$w)[,1]
t_sd <- sqrt(t_sd)
list(t=t, sd=t_sd)
}
#' @rdname estimate_grm
#' @description \code{model_grm_map_scoring} scores response vectors using the MAP method
#' @examples
#' with(model_grm_gendata(10, 50, 3), cbind(true=t, est=model_grm_map_scoring(u, a, b)$t))
#' @export
model_grm_map_scoring <- function(u, a, b, D=1.702, prior=NULL, bound=c(-3, 3), nr_iter=30, nr_conv=1e-3){
ix <- model_polytomous_3dindex(u)
t <- rnorm(dim(u)[1], 0, .01)
t_free <- rep(T, length(t))
for(m in 1:nr_iter){
dv <- model_grm_dv_jmle(ix, model_grm_dv_Pt(t, a, b, D))
dv$dv1 <- rowSums(dv$dv1, na.rm=T)
dv$dv2 <- rowSums(dv$dv2, na.rm=T)
if(!is.null(prior)){
dv$dv1 <- dv$dv1 - (t - prior[1]) / prior[2]^2
dv$dv2 <- dv$dv2 - 1 / prior[2]^2
}
nr <- estimate_nr_iteration(t, t_free, dv, 1.0, 1.0, bound)
t <- nr$param
if(max(abs(nr$h)) < nr_conv) break
}
list(t=t)
}
#' @rdname estimate_grm
#' @keywords internal
model_grm_dv_Pt <- function(t, a, b, D){
n_c <- ncol(b) + 1
p <- model_grm_prob(t, a, b, D, raw=T)
dv1 <- aperm(aperm(p*(1-p), c(2,1,3)) * D * a, c(2,1,3))
dv1 <- dv1[,,1:n_c] - dv1[,,-1]
dv2 <- aperm(aperm(p*(1-p)*(1-2*p), c(2,1,3)) * (D * a)^2, c(2,1,3))
dv2 <- dv2[,,1:n_c] - dv2[,,-1]
p <- p[,,1:n_c] - p[,,-1]
list(p=p, dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @keywords internal
model_grm_dv_Pa <- function(t, a, b, D){
n_c <- ncol(b) + 1
p <- model_grm_prob(t, a, b, D, raw=T)
term0 <- D * outer(t, cbind(0, b, 0), '-')
dv1 <- p * (1-p) * term0
dv1 <- dv1[,,1:n_c] - dv1[,,-1]
dv2 <- p * (1-p) * (1-2*p) * term0^2
dv2 <- dv2[,,1:n_c] - dv2[,,-1]
p <- p[,,1:n_c] - p[,,-1]
list(p=p, dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @keywords internal
model_grm_dv_Pb <- function(t, a, b, D){
n_p <- length(t)
n_i <- nrow(b)
n_c <- ncol(b) + 1
p <- model_grm_prob(t, a, b, D, raw=T)
dv1 <- dv2 <- array(0, c(n_p, n_i, n_c, n_c-1))
for(k in 1:(n_c-1)){
term0<- t(t(p[,,k+1]*(1-p[,,k+1])) * (-D * a))
dv1[,,k,k] <- -1 * term0
dv1[,,k+1,k] <- term0
term1<- t(t(p[,,k+1]*(1-p[,,k+1])*(1-2*p[,,k+1])) * (D*a)^2)
dv2[,,k,k] <- -1 * term1
dv2[,,k+1,k] <- term1
}
p <- p[,,1:n_c] - p[,,-1]
list(p=p, dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @param ix the 3d indices
#' @param dvp the derivatives of P
#' @keywords internal
model_grm_dv_jmle <- function(ix, dvp){
n_p <- max(ix[,1])
n_i <- max(ix[,2])
dv1 <- array(with(dvp, dv1[ix]/p[ix]), c(n_p, n_i))
dv2 <- array(with(dvp, dv2[ix]/p[ix]), c(n_p, n_i)) - dv1^2
list(dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @description \code{model_grm_estimate_jmle} estimates the parameters using the
#' joint maximum likelihood estimation (JMLE) method
#' @param u the observed response matrix, 2d matrix
#' @param t ability parameters, 1d vector (fixed value) or NA (freely estimate)
#' @param a discrimination parameters, 1d vector (fixed value) or NA (freely estimate)
#' @param b difficulty parameters, 2d matrix (fixed value) or NA (freely estimate)
#' @param D the scaling constant, 1.702 by default
#' @param iter the maximum iterations
#' @param conv the convergence criterion for the -2 log-likelihood
#' @param nr_iter the maximum iterations of newton-raphson
#' @param nr_conv the convegence criterion of newton-raphson
#' @param scale the scale of theta parameters
#' @param bounds_t bounds of ability parameters
#' @param bounds_a bounds of discrimination parameters
#' @param bounds_b bounds of location parameters
#' @param priors a list of prior distributions
#' @param decay decay rate
#' @param debug TRUE to print debuggin information
#' @param true_params a list of true parameters for evaluating the estimation accuracy
#' @examples
#' \dontrun{
#' # generate data
#' x <- model_grm_gendata(1000, 40, 3)
#' # free calibration
#' y <- model_grm_estimate_jmle(x$u, true_params=x)
#' # no priors
#' y <- model_grm_estimate_jmle(x$u, priors=NULL, true_params=x)
#' }
#' @importFrom stats cor
#' @importFrom reshape2 melt
#' @import ggplot2
#' @export
model_grm_estimate_jmle <- function(u, t=NA, a=NA, b=NA, D=1.702, iter=100, nr_iter=10, conv=1e-0, nr_conv=1e-3, scale=c(0, 1), bounds_t=c(-4, 4), bounds_a=c(.01, 2), bounds_b=c(-4, 4), priors=list(t=c(0, 1), a=c(-.1, .2), b=c(0, 1)), decay=1, debug=FALSE, true_params=NULL){
# configuration
h_max <- 1.0
tracking <- list(fit=rep(NA, iter), t=rep(NA, iter), a=rep(NA, iter), b=rep(NA, iter))
# initial values
n_p <- dim(u)[1]
n_i <- dim(u)[2]
n_c <- max(u) + 1
u_ix <- model_polytomous_3dindex(u)
if(length(t) == 1) t <- rep(t, n_p)
t[t_free <- is.na(t)] <- rnorm(sum(is.na(t)), 0, .01)
if(length(a) == 1) a <- rep(a, n_i)
a[a_free <- is.na(a)] <- rlnorm(sum(is.na(a)), -.1, .01)
if(length(b) == 1) b <- array(b, c(n_i, n_c-1))
b[b_free <- is.na(b)] <- rnorm(sum(is.na(b)), 0, .1)
b <- t(apply(b, 1, sort))
# estimate parameters
for (k in 1:iter){
# t parameters
if(any(t_free)){
for(j in 1:nr_iter){
dv_t <- model_grm_dv_jmle(u_ix, model_grm_dv_Pt(t, a, b, D))
dv_t$dv1 <- rowSums(dv_t$dv1, na.rm=T)
dv_t$dv2 <- rowSums(dv_t$dv2, na.rm=T)
if(!is.null(priors$t)){
dv_t$dv1 <- dv_t$dv1 - (t - priors$t[1]) / priors$t[2]^2
dv_t$dv2 <- dv_t$dv2 - 1 / priors$t[2]^2
}
nr_t <- estimate_nr_iteration(t, t_free, dv_t, h_max, decay, bounds_t)
t <- nr_t$param
if(max(abs(nr_t$h)) < nr_conv) break
}
# rescale theta
if(!is.null(scale)) t <- (t - mean(t)) / sd(t) * scale[2] + scale[1]
}
# b parameters
if(any(b_free)){
for(j in 1:nr_iter){
dv_b <- model_grm_dv_Pb(t, a, b, D)
dv_bh <- array(0, c(n_i, n_c-1))
for(m in 1:(n_c-1)){
dv <- model_grm_dv_jmle(u_ix, with(dv_b, list(p=p, dv1=dv1[,,,m], dv2=dv2[,,,m])))
dv$dv1 <- colSums(dv$dv1, na.rm=T)
dv$dv2 <- colSums(dv$dv2, na.rm=T)
if(!is.null(priors$b)){
dv$dv1 <- dv$dv1 - (b[,m] - priors$b[1]) / priors$b[2]^2
dv$dv2 <- dv$dv2 - 1 / priors$b[2]^2
}
nr <- estimate_nr_iteration(b[,m], b_free[,m], dv, h_max, decay, bounds_b)
b[,m] <- nr$param
dv_bh[,m] <- nr$h
}
b <- t(apply(b, 1, sort))
if(max(abs(dv_bh)) < nr_conv) break
}
}
# a parameters
if(any(a_free)){
for(j in 1:nr_iter){
dv_a <- model_grm_dv_jmle(u_ix, model_grm_dv_Pa(t, a, b, D))
dv_a$dv1 <- colSums(dv_a$dv1, na.rm=T)
dv_a$dv2 <- colSums(dv_a$dv2, na.rm=T)
if(!is.null(priors$a)){
dv_a$dv1 <- dv_a$dv1 - 1/a * (1 + (log(a)-priors$a[1])/priors$a[2]^2)
dv_a$dv2 <- dv_a$dv2 - 1/a^2 * (1/priors$a[2]^2 - (1 + (log(a)-priors$a[1])/priors$a[2]^2))
}
nr_a <- estimate_nr_iteration(a, a_free, dv_a, h_max, decay, bounds_a)
a <- nr_a$param
if(max(abs(nr_a$h)) < nr_conv) break
}
}
decay <- decay * decay
# model fit
loglh <- -2 * sum(model_grm_lh(u, t, a, b, D, log=T), na.rm=T)
if(debug) cat('iter #', k, ': -2 log-likelihood = ', round(loglh, 2), '\n', sep='')
if(k > 1 && tracking$fit[k-1] - loglh < conv) break
tracking$fit[k] <- loglh
if(any(t_free)) tracking$t[k] <- mean(abs(nr_t$h[t_free]))
if(any(a_free)) tracking$a[k] <- mean(abs(nr_a$h[a_free]))
if(any(b_free)) tracking$b[k] <- mean(abs(dv_bh[b_free]))
}
# debugging
if(debug){
xx <- with(tracking, data.frame(iteration=1:iter, fit=fit, t=t, a=a, b=b))[1:k,]
xx <- melt(xx, id.vars='iteration')
xx <- xx[!is.na(xx$value), ]
g <- ggplot(xx, aes_string(x="iteration", y="value", color="variable")) +
geom_line() + facet_wrap(~variable, scales="free") + guides(color=F) +
xlab('Iterations') + ylab('') + theme_bw()
print(g)
}
# compare with true parameters
if(!is.null(true_params)){
xx <- rbind(data.frame(true=true_params$t, est=t, params='t'),
data.frame(true=true_params$a, est=a, params='a'))
for(i in 1:(n_c-1))
xx <- rbind(xx, data.frame(true=true_params$b[,i], est=b[,i], params=paste('b',i,sep='')))
g <- ggplot(xx, aes_string(x="true", y="est", color="params")) +
geom_point(alpha=.3) + geom_smooth(method='gam', se=F) +
facet_wrap(~params, nrow=1, scales='free') + guides(color=F) +
xlab('True Parameters') + ylab('Est. Parameters') + theme_bw()
print(g)
if(any(t_free)) cat('t: corr = ', round(cor(t, true_params$t), 3), ', rmse = ', round(rmse(t, true_params$t), 3),'\n', sep='')
if(any(a_free)) cat('a: corr = ', round(cor(a, true_params$a), 3), ', rmse = ', round(rmse(a, true_params$a), 3),'\n', sep='')
for(i in 1:(n_c-1)) if(any(b_free[,i])) cat('b_', i, ': corr = ', round(cor(b[,i], true_params$b[,i]), 3), ', rmse = ', round(rmse(b[,i], true_params$b[,i]), 3),'\n', sep='')
}
list(t=t, a=a, b=b)
}
#' @rdname estimate_grm
#' @keywords internal
model_grm_dv_mmle <- function(u_ix, quad, pdv){
n_p <- max(u_ix[,1])
n_i <- max(u_ix[,2])
n_q <- length(quad$t)
p0 <- array(NA, c(n_p, n_i, n_q))
for(q in 1:n_q)
p0[,,q] <- array(pdv$p[q,,][u_ix[,-1]], c(n_p, n_i))
p1 <- apply(p0, c(1, 3), prod, na.rm=T)
p2 <- (p1 %*% quad$w)[,1]
dv1 <- dv2 <- array(0, c(n_p, n_i))
dv_common <- t(t(p1 / p2) * quad$w)
for(q in 1:n_q)
dv1 <- dv1 + dv_common[,q] / p0[,,q] * array(pdv$dv1[q,,][u_ix[,-1]], c(n_p, n_i))
for(q in 1:n_q)
dv2 <- dv2 + dv_common[,q] / p0[,,q] * (array(pdv$dv2[q,,][u_ix[,-1]], c(n_p, n_i)) - array(pdv$dv1[q,,][u_ix[,-1]], c(n_p, n_i)) * dv1)
list(dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @description \code{model_grm_estimate_mmle} estimates the parameters using the
#' marginal maximum likelihood estimation (MMLE) method
#' @param quad_degree the number of quadrature points
#' @param scoring the scoring method: 'eap' or 'map'
#' @examples
#' \dontrun{
#' # generate data
#' x <- model_grm_gendata(1000, 40, 3)
#' # free estimation
#' y <- model_grm_estimate_mmle(x$u, true_params=x)
#' # no priors
#' y <- model_grm_estimate_mmle(x$u, priors=NULL, true_params=x)
#' }
#' @importFrom stats cor
#' @importFrom reshape2 melt
#' @import ggplot2
#' @export
model_grm_estimate_mmle <- function(u, t=NA, a=NA, b=NA, d=NA, D=1.702, iter=100, nr_iter=10, conv=1e-0, nr_conv=1e-3, bounds_t=c(-4, 4), bounds_a=c(.01, 2), bounds_b=c(-4, 4), bounds_d=c(-4, 4), priors=list(t=c(0, 1), a=c(-.1, .2), b=c(0, 1)), decay=1, quad_degree='11', scoring=c('eap', 'map'), debug=FALSE, true_params=NULL){
# configuration
h_max <- 1.0
if(is.null(priors$t)) priors$t <- c(0, 1)
quad <- hermite_gauss(quad_degree)
quad$w <- quad$w * exp(quad$t^2) * dnorm(quad$t, priors$t[1], priors$t[2])
tracking <- list(fit=rep(NA, iter), t=rep(NA, iter), a=rep(NA, iter), b=rep(NA, iter), d=rep(NA, iter))
# initial values
n_p <- dim(u)[1]
n_i <- dim(u)[2]
n_c <- max(u) + 1
u_ix <- model_polytomous_3dindex(u)
if(length(t) == 1) t <- rep(t, n_p)
t[t_free <- is.na(t)] <- rnorm(sum(is.na(t)), 0, .01)
if(length(a) == 1) a <- rep(a, n_i)
a[a_free <- is.na(a)] <- rlnorm(sum(is.na(a)), -.1, .01)
if(length(b) == 1) b <- array(b, c(n_i, n_c-1))
b[b_free <- is.na(b)] <- rnorm(sum(is.na(b)), 0, .1)
b <- t(apply(b, 1, sort))
# estimate parameters
for (k in 1:iter){
# b parameters
if(any(b_free)){
for(j in 1:nr_iter){
dv_b <- model_grm_dv_Pb(t, a, b, D)
dv_bh <- array(0, c(n_i, n_c-1))
for(m in 2:n_c-1){
dv <- model_grm_dv_mmle(u_ix, quad, with(dv_b, list(p=p, dv1=dv1[,,,m], dv2=dv2[,,,m])))
dv$dv1 <- colSums(dv$dv1, na.rm=T)
dv$dv2 <- colSums(dv$dv2, na.rm=T)
if(!is.null(priors$b)){
dv$dv1 <- dv$dv1 - (b[,m] - priors$b[1]) / priors$b[2]^2
dv$dv2 <- dv$dv2 - 1 / priors$b[2]^2
}
nr <- estimate_nr_iteration(b[,m], b_free[,m], dv, h_max, decay, bounds_b)
b[,m] <- nr$param
dv_bh[,m] <- nr$h
}
b <- t(apply(b, 1, sort))
if(max(abs(dv_bh)) < nr_conv) break
}
}
# a parameters
if(any(a_free)){
for(j in 1:nr_iter){
dv_a <- model_grm_dv_mmle(u_ix, quad, model_grm_dv_Pa(quad$t, a, b, D))
dv_a$dv1 <- colSums(dv_a$dv1, na.rm=T)
dv_a$dv2 <- colSums(dv_a$dv2, na.rm=T)
if(!is.null(priors$a)){
dv_a$dv1 <- dv_a$dv1 - 1/a * (1 + (log(a)-priors$a[1])/priors$a[2]^2)
dv_a$dv2 <- dv_a$dv2 - 1/a^2 * (1/priors$a[2]^2 - (1 + (log(a)-priors$a[1])/priors$a[2]^2))
}
nr_a <- estimate_nr_iteration(a, a_free, dv_a, h_max, decay, bounds_a)
a <- nr_a$param
if(max(abs(nr_a$h)) < nr_conv) break
}
}
# scoring
if(any(t_free))
t[t_free] <- switch(match.arg(scoring, scoring), 'eap'=model_grm_eap_scoring, 'map'=model_grm_map_scoring)(u, a, b, D, prior=priors$t, bound=bounds_t)$t[t_free]
decay <- decay * decay
# model fit
loglh <- -2 * sum(model_grm_lh(u, t, a, b, D, log=T), na.rm=T)
if(debug) cat('iter #', k, ': -2 log-likelihood = ', round(loglh, 2), '\n', sep='')
if(k > 1 && tracking$fit[k-1] - loglh < conv) break
tracking$fit[k] <- loglh
if(any(a_free)) tracking$a[k] <- mean(abs(nr_a$h[a_free]))
if(any(b_free)) tracking$d[k] <- mean(abs(dv_bh[b_free]))
}
# debugging
if(debug){
xx <- with(tracking, data.frame(iteration=1:iter, fit=fit, t=t, a=a, b=b))[1:k, ]
xx <- melt(xx, id.vars='iteration')
xx <- xx[!is.na(xx$value), ]
g <- ggplot(xx, aes_string(x="iteration", y="value", color="variable")) +
geom_line() + facet_wrap(~variable, scales="free") + guides(color=F) +
xlab('Iterations') + ylab('') + theme_bw()
print(g)
}
# compare with true parameters
if(!is.null(true_params)){
xx <- rbind(data.frame(true=true_params$t, est=t, params='t'),
data.frame(true=true_params$a, est=a, params='a'))
for(i in 2:n_c-1)
xx <- rbind(xx, data.frame(true=true_params$b[,i], est=b[,i], params=paste('b',i,sep='')))
g <- ggplot(xx, aes_string(x="true", y="est", color="params")) +
geom_point(alpha=.3) + geom_smooth(method='gam', se=F) +
facet_wrap(~params, nrow=1, scales='free') + guides(color=F) +
xlab('True Parameters') + ylab('Est. Parameters') + theme_bw()
print(g)
if(any(t_free)) cat('t: corr = ', round(cor(t, true_params$t), 3), ', rmse = ', round(rmse(t, true_params$t), 3),'\n', sep='')
if(any(a_free)) cat('a: corr = ', round(cor(a, true_params$a), 3), ', rmse = ', round(rmse(a, true_params$a), 3),'\n', sep='')
for(i in 2:n_c-1) if(any(b_free[,i])) cat('b_', i, ': corr = ', round(cor(b[,i], true_params$b[,i]), 3), ', rmse = ', round(rmse(b[,i], true_params$b[,i]), 3),'\n', sep='')
}
list(t=t, a=a, b=b)
}
#' @rdname estimate_grm
#' @param index the indices of items being plotted
#' @param intervals intervals on the x-axis
#' @param show_points TRUE to show points
#' @examples
#' with(model_grm_gendata(1000, 20, 3), model_grm_fitplot(u, t, a, b, index=c(1, 3, 5)))
#' @importFrom reshape2 melt
#' @import ggplot2
#' @export
model_grm_fitplot <- function(u, t, a, b, D=1.702, index=NULL, intervals=seq(-3, 3, .5), show_points=TRUE){
if(is.null(index)) index <- seq(b)
groups <- cut(t, intervals, labels=(intervals[-length(intervals)] + intervals[-1]) / 2)
obs <- aggregate(u, by=list(intervals=groups), mean, na.rm=TRUE)[, c(1, index+1)]
obs <- melt(obs, id.vars='intervals', variable.name='items')
obs[, 'type'] <- 'Observed'
p <- model_grm_prob(t, a, b, D)
p <- apply(p, 1:2, function(x) sum(x * (seq(x)-1), na.rm=T))
exp <- aggregate(p, by=list(intervals=groups), mean, na.rm=TRUE)[, c(1, index+1)]
exp <- melt(exp, id.vars='intervals', variable.name='items')
exp[, 'type'] <- 'Expected'
data <- rbind(obs, exp)
data$intervals <- as.numeric(levels(data$intervals)[data$intervals])
levels(data$items) <- gsub('V', 'Item ', levels(data$items))
g <- ggplot(data, aes_string('intervals', 'value', color='type', group='type')) +
geom_line() + facet_wrap(~items) + xlab(expression(theta)) + ylab('Probability') +
scale_color_discrete(guide=guide_legend("")) + theme_bw()
if(show_points) g <- g + geom_point(fill='white', pch=1)
g
}
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#' Estimate Graded Response Model
#' @description Estimate the GRM using the maximum likelihood estimation
#' @name estimate_grm
NULL
#' @rdname estimate_grm
#' @description \code{model_grm_eap_scoring} scores response vectors using the EAP method
#' @param prior the prior distribution
#' @examples
#' with(model_grm_gendata(10, 50, 3), cbind(true=t, est=model_grm_eap_scoring(u, a, b)$t))
#' @importFrom stats dnorm
#' @export
model_grm_eap_scoring <- function(u, a, b, D=1.702, prior=c(0, 1), bound=c(-3, 3)){
quad <- hermite_gauss('11')
quad$w <- quad$w * exp(quad$t^2) * dnorm(quad$t, prior[1], prior[2])
n_p <- dim(u)[1]
n_i <- dim(u)[2]
n_q <- length(quad$t)
p <- model_grm_prob(quad$t, a, b, D)
ix <- model_polytomous_3dindex(u)
lh <- array(NA, c(n_p, n_i, n_q))
for(q in 1:n_q)
lh[,,q] <- array(p[q,,][ix[,-1]], c(n_p, n_i))
lh <- apply(lh, c(1, 3), prod, na.rm=T)
t <- ((lh / (lh %*% quad$w)[,1]) %*% (quad$w * quad$t))[,1]
t_sd <- ((lh / (lh %*% quad$w)[,1] * outer(t, quad$t, '-')^2) %*% quad$w)[,1]
t_sd <- sqrt(t_sd)
list(t=t, sd=t_sd)
}
#' @rdname estimate_grm
#' @description \code{model_grm_map_scoring} scores response vectors using the MAP method
#' @examples
#' with(model_grm_gendata(10, 50, 3), cbind(true=t, est=model_grm_map_scoring(u, a, b)$t))
#' @export
model_grm_map_scoring <- function(u, a, b, D=1.702, prior=NULL, bound=c(-3, 3), nr_iter=30, nr_conv=1e-3){
ix <- model_polytomous_3dindex(u)
t <- rnorm(dim(u)[1], 0, .01)
t_free <- rep(T, length(t))
for(m in 1:nr_iter){
dv <- model_grm_dv_jmle(ix, model_grm_dv_Pt(t, a, b, D))
dv$dv1 <- rowSums(dv$dv1, na.rm=T)
dv$dv2 <- rowSums(dv$dv2, na.rm=T)
if(!is.null(prior)){
dv$dv1 <- dv$dv1 - (t - prior[1]) / prior[2]^2
dv$dv2 <- dv$dv2 - 1 / prior[2]^2
}
nr <- estimate_nr_iteration(t, t_free, dv, 1.0, 1.0, bound)
t <- nr$param
if(max(abs(nr$h)) < nr_conv) break
}
list(t=t)
}
#' @rdname estimate_grm
#' @keywords internal
model_grm_dv_Pt <- function(t, a, b, D){
n_c <- ncol(b) + 1
p <- model_grm_prob(t, a, b, D, raw=T)
dv1 <- aperm(aperm(p*(1-p), c(2,1,3)) * D * a, c(2,1,3))
dv1 <- dv1[,,1:n_c] - dv1[,,-1]
dv2 <- aperm(aperm(p*(1-p)*(1-2*p), c(2,1,3)) * (D * a)^2, c(2,1,3))
dv2 <- dv2[,,1:n_c] - dv2[,,-1]
p <- p[,,1:n_c] - p[,,-1]
list(p=p, dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @keywords internal
model_grm_dv_Pa <- function(t, a, b, D){
n_c <- ncol(b) + 1
p <- model_grm_prob(t, a, b, D, raw=T)
term0 <- D * outer(t, cbind(0, b, 0), '-')
dv1 <- p * (1-p) * term0
dv1 <- dv1[,,1:n_c] - dv1[,,-1]
dv2 <- p * (1-p) * (1-2*p) * term0^2
dv2 <- dv2[,,1:n_c] - dv2[,,-1]
p <- p[,,1:n_c] - p[,,-1]
list(p=p, dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @keywords internal
model_grm_dv_Pb <- function(t, a, b, D){
n_p <- length(t)
n_i <- nrow(b)
n_c <- ncol(b) + 1
p <- model_grm_prob(t, a, b, D, raw=T)
dv1 <- dv2 <- array(0, c(n_p, n_i, n_c, n_c-1))
for(k in 1:(n_c-1)){
term0<- t(t(p[,,k+1]*(1-p[,,k+1])) * (-D * a))
dv1[,,k,k] <- -1 * term0
dv1[,,k+1,k] <- term0
term1<- t(t(p[,,k+1]*(1-p[,,k+1])*(1-2*p[,,k+1])) * (D*a)^2)
dv2[,,k,k] <- -1 * term1
dv2[,,k+1,k] <- term1
}
p <- p[,,1:n_c] - p[,,-1]
list(p=p, dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @param ix the 3d indices
#' @param dvp the derivatives of P
#' @keywords internal
model_grm_dv_jmle <- function(ix, dvp){
n_p <- max(ix[,1])
n_i <- max(ix[,2])
dv1 <- array(with(dvp, dv1[ix]/p[ix]), c(n_p, n_i))
dv2 <- array(with(dvp, dv2[ix]/p[ix]), c(n_p, n_i)) - dv1^2
list(dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @description \code{model_grm_estimate_jmle} estimates the parameters using the
#' joint maximum likelihood estimation (JMLE) method
#' @param u the observed response matrix, 2d matrix
#' @param t ability parameters, 1d vector (fixed value) or NA (freely estimate)
#' @param a discrimination parameters, 1d vector (fixed value) or NA (freely estimate)
#' @param b difficulty parameters, 2d matrix (fixed value) or NA (freely estimate)
#' @param D the scaling constant, 1.702 by default
#' @param iter the maximum iterations
#' @param conv the convergence criterion for the -2 log-likelihood
#' @param nr_iter the maximum iterations of newton-raphson
#' @param nr_conv the convegence criterion of newton-raphson
#' @param scale the scale of theta parameters
#' @param bounds_t bounds of ability parameters
#' @param bounds_a bounds of discrimination parameters
#' @param bounds_b bounds of location parameters
#' @param priors a list of prior distributions
#' @param decay decay rate
#' @param debug TRUE to print debuggin information
#' @param true_params a list of true parameters for evaluating the estimation accuracy
#' @examples
#' \dontrun{
#' # generate data
#' x <- model_grm_gendata(1000, 40, 3)
#' # free calibration
#' y <- model_grm_estimate_jmle(x$u, true_params=x)
#' # no priors
#' y <- model_grm_estimate_jmle(x$u, priors=NULL, true_params=x)
#' }
#' @importFrom stats cor
#' @importFrom reshape2 melt
#' @import ggplot2
#' @export
model_grm_estimate_jmle <- function(u, t=NA, a=NA, b=NA, D=1.702, iter=100, nr_iter=10, conv=1e-0, nr_conv=1e-3, scale=c(0, 1), bounds_t=c(-4, 4), bounds_a=c(.01, 2), bounds_b=c(-4, 4), priors=list(t=c(0, 1), a=c(-.1, .2), b=c(0, 1)), decay=1, debug=FALSE, true_params=NULL){
# configuration
h_max <- 1.0
tracking <- list(fit=rep(NA, iter), t=rep(NA, iter), a=rep(NA, iter), b=rep(NA, iter))
# initial values
n_p <- dim(u)[1]
n_i <- dim(u)[2]
n_c <- max(u) + 1
u_ix <- model_polytomous_3dindex(u)
if(length(t) == 1) t <- rep(t, n_p)
t[t_free <- is.na(t)] <- rnorm(sum(is.na(t)), 0, .01)
if(length(a) == 1) a <- rep(a, n_i)
a[a_free <- is.na(a)] <- rlnorm(sum(is.na(a)), -.1, .01)
if(length(b) == 1) b <- array(b, c(n_i, n_c-1))
b[b_free <- is.na(b)] <- rnorm(sum(is.na(b)), 0, .1)
b <- t(apply(b, 1, sort))
# estimate parameters
for (k in 1:iter){
# t parameters
if(any(t_free)){
for(j in 1:nr_iter){
dv_t <- model_grm_dv_jmle(u_ix, model_grm_dv_Pt(t, a, b, D))
dv_t$dv1 <- rowSums(dv_t$dv1, na.rm=T)
dv_t$dv2 <- rowSums(dv_t$dv2, na.rm=T)
if(!is.null(priors$t)){
dv_t$dv1 <- dv_t$dv1 - (t - priors$t[1]) / priors$t[2]^2
dv_t$dv2 <- dv_t$dv2 - 1 / priors$t[2]^2
}
nr_t <- estimate_nr_iteration(t, t_free, dv_t, h_max, decay, bounds_t)
t <- nr_t$param
if(max(abs(nr_t$h)) < nr_conv) break
}
# rescale theta
if(!is.null(scale)) t <- (t - mean(t)) / sd(t) * scale[2] + scale[1]
}
# b parameters
if(any(b_free)){
for(j in 1:nr_iter){
dv_b <- model_grm_dv_Pb(t, a, b, D)
dv_bh <- array(0, c(n_i, n_c-1))
for(m in 1:(n_c-1)){
dv <- model_grm_dv_jmle(u_ix, with(dv_b, list(p=p, dv1=dv1[,,,m], dv2=dv2[,,,m])))
dv$dv1 <- colSums(dv$dv1, na.rm=T)
dv$dv2 <- colSums(dv$dv2, na.rm=T)
if(!is.null(priors$b)){
dv$dv1 <- dv$dv1 - (b[,m] - priors$b[1]) / priors$b[2]^2
dv$dv2 <- dv$dv2 - 1 / priors$b[2]^2
}
nr <- estimate_nr_iteration(b[,m], b_free[,m], dv, h_max, decay, bounds_b)
b[,m] <- nr$param
dv_bh[,m] <- nr$h
}
b <- t(apply(b, 1, sort))
if(max(abs(dv_bh)) < nr_conv) break
}
}
# a parameters
if(any(a_free)){
for(j in 1:nr_iter){
dv_a <- model_grm_dv_jmle(u_ix, model_grm_dv_Pa(t, a, b, D))
dv_a$dv1 <- colSums(dv_a$dv1, na.rm=T)
dv_a$dv2 <- colSums(dv_a$dv2, na.rm=T)
if(!is.null(priors$a)){
dv_a$dv1 <- dv_a$dv1 - 1/a * (1 + (log(a)-priors$a[1])/priors$a[2]^2)
dv_a$dv2 <- dv_a$dv2 - 1/a^2 * (1/priors$a[2]^2 - (1 + (log(a)-priors$a[1])/priors$a[2]^2))
}
nr_a <- estimate_nr_iteration(a, a_free, dv_a, h_max, decay, bounds_a)
a <- nr_a$param
if(max(abs(nr_a$h)) < nr_conv) break
}
}
decay <- decay * decay
# model fit
loglh <- -2 * sum(model_grm_lh(u, t, a, b, D, log=T), na.rm=T)
if(debug) cat('iter #', k, ': -2 log-likelihood = ', round(loglh, 2), '\n', sep='')
if(k > 1 && tracking$fit[k-1] - loglh < conv) break
tracking$fit[k] <- loglh
if(any(t_free)) tracking$t[k] <- mean(abs(nr_t$h[t_free]))
if(any(a_free)) tracking$a[k] <- mean(abs(nr_a$h[a_free]))
if(any(b_free)) tracking$b[k] <- mean(abs(dv_bh[b_free]))
}
# debugging
if(debug){
xx <- with(tracking, data.frame(iteration=1:iter, fit=fit, t=t, a=a, b=b))[1:k,]
xx <- melt(xx, id.vars='iteration')
xx <- xx[!is.na(xx$value), ]
g <- ggplot(xx, aes_string(x="iteration", y="value", color="variable")) +
geom_line() + facet_wrap(~variable, scales="free") + guides(color=F) +
xlab('Iterations') + ylab('') + theme_bw()
print(g)
}
# compare with true parameters
if(!is.null(true_params)){
xx <- rbind(data.frame(true=true_params$t, est=t, params='t'),
data.frame(true=true_params$a, est=a, params='a'))
for(i in 1:(n_c-1))
xx <- rbind(xx, data.frame(true=true_params$b[,i], est=b[,i], params=paste('b',i,sep='')))
g <- ggplot(xx, aes_string(x="true", y="est", color="params")) +
geom_point(alpha=.3) + geom_smooth(method='gam', se=F) +
facet_wrap(~params, nrow=1, scales='free') + guides(color=F) +
xlab('True Parameters') + ylab('Est. Parameters') + theme_bw()
print(g)
if(any(t_free)) cat('t: corr = ', round(cor(t, true_params$t), 3), ', rmse = ', round(rmse(t, true_params$t), 3),'\n', sep='')
if(any(a_free)) cat('a: corr = ', round(cor(a, true_params$a), 3), ', rmse = ', round(rmse(a, true_params$a), 3),'\n', sep='')
for(i in 1:(n_c-1)) if(any(b_free[,i])) cat('b_', i, ': corr = ', round(cor(b[,i], true_params$b[,i]), 3), ', rmse = ', round(rmse(b[,i], true_params$b[,i]), 3),'\n', sep='')
}
list(t=t, a=a, b=b)
}
#' @rdname estimate_grm
#' @keywords internal
model_grm_dv_mmle <- function(u_ix, quad, pdv){
n_p <- max(u_ix[,1])
n_i <- max(u_ix[,2])
n_q <- length(quad$t)
p0 <- array(NA, c(n_p, n_i, n_q))
for(q in 1:n_q)
p0[,,q] <- array(pdv$p[q,,][u_ix[,-1]], c(n_p, n_i))
p1 <- apply(p0, c(1, 3), prod, na.rm=T)
p2 <- (p1 %*% quad$w)[,1]
dv1 <- dv2 <- array(0, c(n_p, n_i))
dv_common <- t(t(p1 / p2) * quad$w)
for(q in 1:n_q)
dv1 <- dv1 + dv_common[,q] / p0[,,q] * array(pdv$dv1[q,,][u_ix[,-1]], c(n_p, n_i))
for(q in 1:n_q)
dv2 <- dv2 + dv_common[,q] / p0[,,q] * (array(pdv$dv2[q,,][u_ix[,-1]], c(n_p, n_i)) - array(pdv$dv1[q,,][u_ix[,-1]], c(n_p, n_i)) * dv1)
list(dv1=dv1, dv2=dv2)
}
#' @rdname estimate_grm
#' @description \code{model_grm_estimate_mmle} estimates the parameters using the
#' marginal maximum likelihood estimation (MMLE) method
#' @param quad_degree the number of quadrature points
#' @param scoring the scoring method: 'eap' or 'map'
#' @examples
#' \dontrun{
#' # generate data
#' x <- model_grm_gendata(1000, 40, 3)
#' # free estimation
#' y <- model_grm_estimate_mmle(x$u, true_params=x)
#' # no priors
#' y <- model_grm_estimate_mmle(x$u, priors=NULL, true_params=x)
#' }
#' @importFrom stats cor
#' @importFrom reshape2 melt
#' @import ggplot2
#' @export
model_grm_estimate_mmle <- function(u, t=NA, a=NA, b=NA, d=NA, D=1.702, iter=100, nr_iter=10, conv=1e-0, nr_conv=1e-3, bounds_t=c(-4, 4), bounds_a=c(.01, 2), bounds_b=c(-4, 4), bounds_d=c(-4, 4), priors=list(t=c(0, 1), a=c(-.1, .2), b=c(0, 1)), decay=1, quad_degree='11', scoring=c('eap', 'map'), debug=FALSE, true_params=NULL){
# configuration
h_max <- 1.0
if(is.null(priors$t)) priors$t <- c(0, 1)
quad <- hermite_gauss(quad_degree)
quad$w <- quad$w * exp(quad$t^2) * dnorm(quad$t, priors$t[1], priors$t[2])
tracking <- list(fit=rep(NA, iter), t=rep(NA, iter), a=rep(NA, iter), b=rep(NA, iter), d=rep(NA, iter))
# initial values
n_p <- dim(u)[1]
n_i <- dim(u)[2]
n_c <- max(u) + 1
u_ix <- model_polytomous_3dindex(u)
if(length(t) == 1) t <- rep(t, n_p)
t[t_free <- is.na(t)] <- rnorm(sum(is.na(t)), 0, .01)
if(length(a) == 1) a <- rep(a, n_i)
a[a_free <- is.na(a)] <- rlnorm(sum(is.na(a)), -.1, .01)
if(length(b) == 1) b <- array(b, c(n_i, n_c-1))
b[b_free <- is.na(b)] <- rnorm(sum(is.na(b)), 0, .1)
b <- t(apply(b, 1, sort))
# estimate parameters
for (k in 1:iter){
# b parameters
if(any(b_free)){
for(j in 1:nr_iter){
dv_b <- model_grm_dv_Pb(t, a, b, D)
dv_bh <- array(0, c(n_i, n_c-1))
for(m in 2:n_c-1){
dv <- model_grm_dv_mmle(u_ix, quad, with(dv_b, list(p=p, dv1=dv1[,,,m], dv2=dv2[,,,m])))
dv$dv1 <- colSums(dv$dv1, na.rm=T)
dv$dv2 <- colSums(dv$dv2, na.rm=T)
if(!is.null(priors$b)){
dv$dv1 <- dv$dv1 - (b[,m] - priors$b[1]) / priors$b[2]^2
dv$dv2 <- dv$dv2 - 1 / priors$b[2]^2
}
nr <- estimate_nr_iteration(b[,m], b_free[,m], dv, h_max, decay, bounds_b)
b[,m] <- nr$param
dv_bh[,m] <- nr$h
}
b <- t(apply(b, 1, sort))
if(max(abs(dv_bh)) < nr_conv) break
}
}
# a parameters
if(any(a_free)){
for(j in 1:nr_iter){
dv_a <- model_grm_dv_mmle(u_ix, quad, model_grm_dv_Pa(quad$t, a, b, D))
dv_a$dv1 <- colSums(dv_a$dv1, na.rm=T)
dv_a$dv2 <- colSums(dv_a$dv2, na.rm=T)
if(!is.null(priors$a)){
dv_a$dv1 <- dv_a$dv1 - 1/a * (1 + (log(a)-priors$a[1])/priors$a[2]^2)
dv_a$dv2 <- dv_a$dv2 - 1/a^2 * (1/priors$a[2]^2 - (1 + (log(a)-priors$a[1])/priors$a[2]^2))
}
nr_a <- estimate_nr_iteration(a, a_free, dv_a, h_max, decay, bounds_a)
a <- nr_a$param
if(max(abs(nr_a$h)) < nr_conv) break
}
}
# scoring
if(any(t_free))
t[t_free] <- switch(match.arg(scoring, scoring), 'eap'=model_grm_eap_scoring, 'map'=model_grm_map_scoring)(u, a, b, D, prior=priors$t, bound=bounds_t)$t[t_free]
decay <- decay * decay
# model fit
loglh <- -2 * sum(model_grm_lh(u, t, a, b, D, log=T), na.rm=T)
if(debug) cat('iter #', k, ': -2 log-likelihood = ', round(loglh, 2), '\n', sep='')
if(k > 1 && tracking$fit[k-1] - loglh < conv) break
tracking$fit[k] <- loglh
if(any(a_free)) tracking$a[k] <- mean(abs(nr_a$h[a_free]))
if(any(b_free)) tracking$d[k] <- mean(abs(dv_bh[b_free]))
}
# debugging
if(debug){
xx <- with(tracking, data.frame(iteration=1:iter, fit=fit, t=t, a=a, b=b))[1:k, ]
xx <- melt(xx, id.vars='iteration')
xx <- xx[!is.na(xx$value), ]
g <- ggplot(xx, aes_string(x="iteration", y="value", color="variable")) +
geom_line() + facet_wrap(~variable, scales="free") + guides(color=F) +
xlab('Iterations') + ylab('') + theme_bw()
print(g)
}
# compare with true parameters
if(!is.null(true_params)){
xx <- rbind(data.frame(true=true_params$t, est=t, params='t'),
data.frame(true=true_params$a, est=a, params='a'))
for(i in 2:n_c-1)
xx <- rbind(xx, data.frame(true=true_params$b[,i], est=b[,i], params=paste('b',i,sep='')))
g <- ggplot(xx, aes_string(x="true", y="est", color="params")) +
geom_point(alpha=.3) + geom_smooth(method='gam', se=F) +
facet_wrap(~params, nrow=1, scales='free') + guides(color=F) +
xlab('True Parameters') + ylab('Est. Parameters') + theme_bw()
print(g)
if(any(t_free)) cat('t: corr = ', round(cor(t, true_params$t), 3), ', rmse = ', round(rmse(t, true_params$t), 3),'\n', sep='')
if(any(a_free)) cat('a: corr = ', round(cor(a, true_params$a), 3), ', rmse = ', round(rmse(a, true_params$a), 3),'\n', sep='')
for(i in 2:n_c-1) if(any(b_free[,i])) cat('b_', i, ': corr = ', round(cor(b[,i], true_params$b[,i]), 3), ', rmse = ', round(rmse(b[,i], true_params$b[,i]), 3),'\n', sep='')
}
list(t=t, a=a, b=b)
}
#' @rdname estimate_grm
#' @param index the indices of items being plotted
#' @param intervals intervals on the x-axis
#' @param show_points TRUE to show points
#' @examples
#' with(model_grm_gendata(1000, 20, 3), model_grm_fitplot(u, t, a, b, index=c(1, 3, 5)))
#' @importFrom reshape2 melt
#' @import ggplot2
#' @export
model_grm_fitplot <- function(u, t, a, b, D=1.702, index=NULL, intervals=seq(-3, 3, .5), show_points=TRUE){
if(is.null(index)) index <- seq(b)
groups <- cut(t, intervals, labels=(intervals[-length(intervals)] + intervals[-1]) / 2)
obs <- aggregate(u, by=list(intervals=groups), mean, na.rm=TRUE)[, c(1, index+1)]
obs <- melt(obs, id.vars='intervals', variable.name='items')
obs[, 'type'] <- 'Observed'
p <- model_grm_prob(t, a, b, D)
p <- apply(p, 1:2, function(x) sum(x * (seq(x)-1), na.rm=T))
exp <- aggregate(p, by=list(intervals=groups), mean, na.rm=TRUE)[, c(1, index+1)]
exp <- melt(exp, id.vars='intervals', variable.name='items')
exp[, 'type'] <- 'Expected'
data <- rbind(obs, exp)
data$intervals <- as.numeric(levels(data$intervals)[data$intervals])
levels(data$items) <- gsub('V', 'Item ', levels(data$items))
g <- ggplot(data, aes_string('intervals', 'value', color='type', group='type')) +
geom_line() + facet_wrap(~items) + xlab(expression(theta)) + ylab('Probability') +
scale_color_discrete(guide=guide_legend("")) + theme_bw()
if(show_points) g <- g + geom_point(fill='white', pch=1)
g
}
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