#' Nonlinear regression DIF models estimation.
#'
#' @aliases estimNLR
#'
#' @description Estimates parameters of nonlinear regression models for DIF
#' detection using either nonlinear least squares or maximum likelihood method
#' with various algorithms.
#'
#' @param y numeric: binary vector of responses.
#' @param match numeric: vector of matching criterion.
#' @param group numeric: binary vector of group membership. \code{"0"} for
#' reference group, \code{"1"} for focal group.
#' @param formula formula: specification of the model. Can be obtained by the
#' \code{formulaNLR()} function.
#' @param method character: method used to estimate parameters. The options are
#' \code{"nls"} for nonlinear least squares (default), \code{"mle"} for
#' maximum likelihood method, and \code{"irls"} for maximum likelihood estimation
#' with iteratively reweighted least squares. See \strong{Details}.
#' @param lower numeric: lower bounds for parameters of model specified in \code{formula}.
#' @param upper numeric: upper bounds for parameters of model specified in \code{formula}.
#' @param start numeric: initial parameters. Can be obtained by the
#' \code{startNLR()} function.
#'
#' @usage estimNLR(y, match, group, formula, method, lower, upper, start)
#'
#' @details
#' Function offers either non-linear least squares estimation via
#' \code{\link[stats]{nls}} function, maximum likelihood method with
#' \code{"L-BFGS-B"} method via \code{\link[stats]{optim}} function,
#' or maximum likelihood method with iteratively reweighted least
#' squares via \code{\link[stats]{glm}} function.
#'
#' @author
#' Adela Hladka (nee Drabinova) \cr
#' Institute of Computer Science of the Czech Academy of Sciences \cr
#' Faculty of Mathematics and Physics, Charles University \cr
#' \email{hladka@@cs.cas.cz} \cr
#'
#' Patricia Martinkova \cr
#' Institute of Computer Science of the Czech Academy of Sciences \cr
#' \email{martinkova@@cs.cas.cz} \cr
#'
#' @references
#' Hladka, A. (2021). Statistical models for detection of differential item
#' functioning. Dissertation thesis.
#' Faculty of Mathematics and Physics, Charles University.
#'
#' @examples
#' # loading datadata(GMAT)
#' y <- GMAT[, 1] # item 1
#' match <- scale(rowSums(GMAT[, 1:20])) # standardized total score
#' group <- GMAT[, "group"] # group membership variable
#'
#' # formula for 3PL model with the same guessing for both groups
#' M <- formulaNLR(model = "3PLcg", type = "both")
#'
#' # starting values for 3PL model with the same guessing for item 1
#' start <- startNLR(GMAT[, 1:20], group, model = "3PLcg", parameterization = "classic")
#' start <- start[[1]][M$M1$parameters]
#'
#' # nonlinear least squares
#' (fit_nls <- estimNLR(
#' y = y, match = match, group = group,
#' formula = M$M1$formula, method = "nls",
#' lower = M$M1$lower, upper = M$M1$upper, start = start
#' ))
#'
#' coef(fit_nls)
#' logLik(fit_nls)
#' vcov(fit_nls)
#' vcov(fit_nls, sandwich = TRUE)
#' fitted(fit_nls)
#' residuals(fit_nls)
#'
#' # maximum likelihood method
#' (fit_mle <- estimNLR(
#' y = y, match = match, group = group,
#' formula = M$M1$formula, method = "mle",
#' lower = M$M1$lower, upper = M$M1$upper, start = start
#' ))
#'
#' coef(fit_mle)
#' logLik(fit_mle)
#' vcov(fit_mle)
#' fitted(fit_mle)
#' residuals(fit_mle)
#'
#' # iteratively reweighted least squares for 2PL model
#' M <- formulaNLR(model = "2PL", parameterization = "logistic")
#' (fit_irls <- estimNLR(
#' y = y, match = match, group = group,
#' formula = M$M1$formula, method = "irls"
#' ))
#'
#' coef(fit_irls)
#' logLik(fit_irls)
#' vcov(fit_irls)
#' fitted(fit_irls)
#' residuals(fit_irls)
#' @keywords DIF
#' @export
estimNLR <- function(y, match, group, formula, method, lower, upper, start) {
M <- switch(method,
nls = tryCatch(
nls(
formula = formula,
data = data.frame(y = y, x = match, g = group),
algorithm = "port",
start = start,
lower = lower,
upper = upper
),
error = function(e) {},
finally = ""
),
mle = tryCatch(
.MLE_estimation(
formula = formula,
data = data.frame(y = y, x = match, g = group),
par = start,
lower = lower,
upper = upper
),
error = function(e) {},
finally = ""
),
irls = tryCatch(
glm(
formula = formula,
family = binomial(),
data = data.frame(y = y, x = match, g = group)
),
error = function(e) {},
finally = ""
)
)
if (!is.null(M)) {
class(M) <- c("estimNLR", class(M))
}
return(M)
}
#' @rdname estimNLR
#' @export
logLik.estimNLR <- function(object, ...) {
val <- switch(class(object)[2],
"nls" = {
res <- object$m$resid()
N <- length(res)
w <- rep(1, N)
zw <- w == 0
N <- sum(!zw)
- N * (log(2 * pi) + 1 - log(N) - sum(log(w + zw))/N + log(sum(res^2)))/2
},
"mle" = -object$value,
"glm" = object$rank - object$aic/2
)
attr(val, "df") <- length(coef(object))
class(val) <- "logLik"
val
}
#' @rdname estimNLR
#' @export
coef.estimNLR <- function(object, ...) {
switch(class(object)[2],
"nls" = object$m$getPars(),
"mle" = object$par,
"glm" = object$coefficients
)
}
#' @rdname estimNLR
#' @export
fitted.estimNLR <- function(object, ...) {
val <- switch(class(object)[2],
"nls" = as.vector(object$m$fitted()),
"mle" = object$fitted,
"glm" = object$fitted.values
)
lab <- "Fitted values"
attr(val, "label") <- lab
val
}
#' @rdname estimNLR
#' @export
residuals.estimNLR <- function(object, ...) {
val <- switch(class(object)[2],
"nls" = as.vector(object$m$resid()),
"mle" = object$data$y - object$fitted,
"glm" = object$residuals
)
lab <- "Residuals"
attr(val, "label") <- lab
val
}
#' @rdname estimNLR
#' @param x an object of \code{"estimNLR"} class.
#' @export
print.estimNLR <- function(x, ...) {
formula <- switch(class(x)[2],
"nls" = paste(x$m$formula()[2], x$m$formula()[1], x$m$formula()[3]),
"mle" = paste(x$formula[2], x$formula[1], x$formula[3]),
"glm" = paste0(x$formula[2], " ", x$formula[1], " exp(", x$formula[3], ") / (1 + exp(", x$formula[3], "))")
)
cat(
"Nonlinear regression model \n\n",
"Model: ", formula, "\n"
)
pars <- switch(class(x)[2],
"nls" = x$m$getPars(),
"mle" = x$par,
"glm" = x$coefficients
)
alg <- switch(class(x)[2],
"nls" = "Nonlinear least squares estimation",
"mle" = "Maximum likelihood estimation using L-BFGS-B algorithm",
"glm" = "Maximum likelihood estimation using iteratively reweighted least squares algorithm"
)
cat("\nCoefficients:\n")
print(round(pars, 4))
cat("\n", alg)
}
#' @rdname estimNLR
#' @param object an object of \code{"estimNLR"} class.
#' @param sandwich logical: should be sandwich estimator used for covariance
#' matrix of parameters when using \code{method = "nls"}? Default is
#' \code{FALSE}.
#' @param ... other generic parameters for S3 methods.
#' @export
vcov.estimNLR <- function(object, sandwich = FALSE, ...) {
if (inherits(object, "nls")) {
if (sandwich) {
e <- object$m$getEnv()
y <- e$y
x <- e$x
g <- e$g
cov.object <- .sandwich.cov.nls(formula = object$m$formula(), y, x, g, par = object$m$getPars())
} else {
cov.object <- tryCatch(
{
sm <- summary(object)
sm$cov.unscaled * sm$sigma^2
},
error = function(e) NULL
)
}
} else {
if (sandwich) {
message("Sandwich estimator of covariance is available only for method = 'nls'. ")
}
if (inherits(object, "mle")) {
cov.object <- tryCatch(
{
solve(object$hessian)
},
error = function(e) NULL
)
} else {
cov.object <- tryCatch(
{
vcov(summary(object))
},
error = function(e) NULL
)
}
}
return(cov.object)
}
#' @noRd
.sandwich.cov.nls <- function(formula, y, x, group, par) {
n <- length(y)
f <- paste0("(y - ", "(", gsub("y ~ ", "", paste(deparse(formula), collapse = "")), "))^2")
f <- gsub(" ", "", f)
psi <- calculus::derivative(
f = f,
var = names(par)
)
psi.fun <- eval(
parse(
text =
paste0(
"function(",
paste(c("y", "x", "g", names(par)), collapse = ", "), ") {
return(list(", paste(as.list(psi), collapse = ", "), "))}"
)
)
)
hess <- calculus::hessian(
f = f,
var = names(par)
)
hess.fun <- eval(
parse(text = paste0(
"function(",
paste(c("y", "x", "g", names(par)), collapse = ", "), ") {
return(list(", paste(as.list(hess), collapse = ", "), "))}"
))
)
psi.val <- do.call(
cbind,
do.call(
psi.fun,
append(list(y = y, x = x, g = group), par)
)
)
# calculating matrix Sigma
mat.sigma <- t(psi.val) %*% psi.val / n
# calculating matrix Gamma
mat.gamma <- matrix(
sapply(
do.call(
hess.fun,
append(list(y = y, x = x, g = group), par)
),
mean
),
ncol = length(par), nrow = length(par)
)
# sandwich estimator of covariance matrix
cov.sandwich <- solve(mat.gamma) %*% mat.sigma %*% solve(mat.gamma) / n
rownames(cov.sandwich) <- colnames(cov.sandwich) <- names(par)
return(cov.sandwich)
}
# ---------------------------------------------------------------------------------
# Maximum likelihood estimation
# ---------------------------------------------------------------------------------
# log-likelihood function to be maximized in MLE estimation
.likelihood <- function(data, formula, par) {
param <- as.list(par)
param[[length(param) + 1]] <- data$x
names(param)[length(param)] <- "x"
param[[length(param) + 1]] <- data$g
names(param)[length(param)] <- "g"
y <- data$y
h. <- parse(text = as.character(formula)[3])
h <- eval(h., envir = param)
l <- -sum((y * log(h)) + ((1 - y) * log(1 - h)), na.rm = TRUE)
return(l)
}
# Maximum likelihood estimation
.MLE_estimation <- function(formula, data, par, lower, upper) {
m <- optim(
par = par,
fn = .likelihood,
data = data,
formula = formula,
method = "L-BFGS-B",
lower = lower,
upper = upper,
hessian = TRUE
)
h. <- parse(text = as.character(formula)[3])
param <- as.list(par)
param[[length(param) + 1]] <- data$x
names(param)[length(param)] <- "x"
param[[length(param) + 1]] <- data$g
names(param)[length(param)] <- "g"
m$fitted <- eval(h., param)
m$data <- data
m$formula <- formula
class(m) <- "mle"
return(m)
}
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