# Hypothesis Templates

This vignette includes some templates for hypothesis objects that can be altered for your specific use case.

The templates currently include:

• h_1PLvs2PL: Testing a Rasch against a 2PL model
• h_DIF2PL: Testing for DIF in a 2PL model
• h_PCMvsGPCM: Testing a PCM against a GPCM model
• h_2PL_basic: Testing one item parameter
• h_3PL_basic: Testing three item parameters at once
• h_multi_basic: Comparing two item parameters
• h_multi_basic2: Testing a specific item parameter

The templates are described in some more detail in the respective sections. We are open for your suggestions to include further templates.

## h_1PLvs2PL

We test the null hypothesis that the parameters adhere to the Rasch model against the alternative hypothesis that the parameters are better described by a 2PL model.

h_1PLvs2PL <- list(res = function(altpars, nullpars = NULL) {

n.items <- length(altpars[[1]])

re <- list(n.items = n.items, itemtype = "2PL",
Amat = c(1, 0, -1, 0, rep(0, (n.items - 1) *
2)) |>
(function(x) rep(x, n.items - 2))() |>
c(1, 0, -1, 0) |>
matrix(ncol = n.items * 2, byrow = TRUE),
cvec = 0, model = mirt::mirt.model(paste("F = 1-",
n.items, "
CONSTRAIN = (1-",
n.items, ", a1)")))
return(re)
}, unres = function(altpars) {

re <- list(parsets = altpars, model = 1, itemtype = "2PL",
longpars = pars.long(pars = altpars, itemtype = "2PL"))

return(re)
}, maximizeL = function(hyp) {
# Hypothesis-specific algorithm to find the
# maximum likelihood restricted parameter set

# returns the density for each response
# pattern under the model parameters pars

patterns <- as.matrix(expand.grid(lapply(seq_len(length(pars$a)), function(x) c(0, 1)))) pre <- c() for (i in seq_len(nrow(patterns))) { pre[i] <- funs$g(patterns[i, ], pars)
}

return(pre)
}

maxl <- function(x, pars, pre, funs) {
# calculates the likelihood of parameters
# x given model 'pars'

patterns <- as.matrix(expand.grid(lapply(seq_len(length(pars$a)), function(x) c(0, 1)))) x <- list(a = rep(x[1], length(pars$a)), d = x[2:length(x)])

res <- c()
for (i in seq_len(nrow(patterns))) {
px <- pre[i]
qx <- funs$g(patterns[i, ], x) res[i] <- { px * log(qx) } } re <- -sum(res) } resmod <- hyp$resmod
unresmod <- hyp$unresmod pars <- unresmod$parsets

funs <- load.functions(unresmod$itemtype) startval <- c(mean(pars$a), as.numeric(pars$d)) maxlpre <- maxlpreload(pars, funs) optpar <- stats::optim(startval, function(x) { maxl(x, pars, maxlpre, funs) }, method = "BFGS") re <- pars re$a <- rep(optpar$par[1], length(pars$a))
re$d <- optpar$par[2:length(optpar$par)] return(re) }) ## h_DIF2PL We test for differential item functioning in the 2PL model. Our alternative hypothesis is that certain item parameters differ in two known participant groups. Which item parameters differ can be determined in the altpars object, see the “demo” vignette for an example. h_DIF2PL <- list(res = function(altpars, nullpars = NULL) { n.items <- length(altpars[[1]][[1]]) reA <- altpars[[1]] reB <- altpars[[2]] hyp_a <- which(reA$a != reB$a) hyp_d <- which(reA$d != reB$d) Amat <- matrix(0, nrow = length(c(hyp_a, hyp_d)), ncol = n.items * 2) i <- 1 for (j in hyp_a) { Amat[i, j * 2 - 1] <- 1 i <- i + 1 } for (j in hyp_d) { Amat[i, j * 2] <- 1 i <- i + 1 } Amat <- cbind(Amat, -Amat) delcols <- (colSums(Amat) == 0) & (1:(n.items * 2 * 2)) > 2 * n.items relpars <- colSums(Amat[, 1:(2 * n.items)]) == 1 Amat <- Amat[, !delcols] re <- list(n.items = n.items, itemtype = "2PL", Amat = Amat, cvec = 0, model = mirt::mirt.model(paste("F = 1-", n.items, " CONSTRAINB = (1-", n.items, ", d), (1-", n.items, ", a1)")), multigroup = TRUE, delcols = delcols, relpars = relpars) return(re) }, unres = function(altpars) { n.items <- length(altpars[[1]][[1]]) reA <- altpars[[1]] reB <- altpars[[2]] reA$itemtype <- reB$itemtype <- "2PL" reA$longpars <- pars.long(pars = reA, itemtype = "2PL")
reB$longpars <- pars.long(pars = reB, itemtype = "2PL") constrain_a <- which(reA$a == reB$a) constrain_d <- which(reA$d == reB$d) hyp_a <- which(reA$a != reB$a) hyp_d <- which(reA$d != reB$d) Amat <- matrix(0, nrow = length(c(hyp_a, hyp_d)), ncol = n.items * 2) i <- 1 for (j in hyp_a) { Amat[i, j * 2 - 1] <- 1 i <- i + 1 } for (j in hyp_d) { Amat[i, j * 2] <- 1 i <- i + 1 } Amat <- cbind(Amat, -Amat) delcols <- (colSums(Amat) == 0) & (1:(n.items * 2 * 2)) > 2 * n.items longpars <- c(reA$longpars, reB$longpars)[!delcols] re <- list(parsets = list(reA, reB), model = mirt::mirt.model(paste("F = 1-", n.items, " CONSTRAINB = (", paste(constrain_d, collapse = ","), ", d), (", paste(constrain_a, collapse = ","), ", a1)")), longpars = longpars, multigroup = TRUE, itemtype = "2PL", delcols = delcols) return(re) }, maximizeL = function(hyp) { # Hypothesis-specific algorithm to find the # maximum likelihood restricted parameter set # L Optimizer maxl <- function(x, pars1, pars2, i) { px1 <- function(th) { funs$f(th, pars1$a[i], pars1$d[i], 1)
}
px2 <- function(th) {
funs$f(th, pars2$a[i], pars2$d[i], 1) } qx <- function(th) { funs$f(th, x[1], x[2], 1)
}
kl <- function(th) {
px1(th) * log(qx(th)) + (1 - px1(th)) *
log(1 - qx(th)) + px2(th) * log(qx(th)) +
(1 - px2(th)) * log((1 - qx(th)))
}
re <- -spatstat.random::gauss.hermite(kl, order = 20)
}

resmod <- hyp$resmod unresmod <- hyp$unresmod

pars <- unresmod$parsets pars1 <- pars[[1]] pars2 <- pars[[2]] funs <- load.functions(pars1$itemtype)
re <- pars1

for (i in seq_len(length(pars1$a))) { startval <- c(re$a[i], re$d[i]) optpar <- stats::optim(startval, function(x) { maxl(x, pars1, pars2, i) }, method = "BFGS") re$a[i] <- optpar$par[1] re$d[i] <- optpar$par[2] } return(re) }) ## h_PCMvsGPCM We test the null hypothesis that the parameters adhere to the partial credit model against the alternative hypothesis that the parameters are better described by the generalized partial credit model. h_PCMvsGPCM <- list(res = function(altpars, nullpars = NULL) { n.items <- length(altpars[[1]]) nkat <- ncol(altpars$d)

re <- list(n.items = n.items, itemtype = "gpcm",
Amat = c(1, rep(0, nkat - 1), -1, rep(0, nkat -
1), rep(0, (n.items - 1) * nkat)) |>
(function(x) rep(x, n.items - 2))() |>
c(1, rep(0, nkat - 1), -1, rep(0, nkat -
1)) |>
matrix(ncol = n.items * nkat, byrow = TRUE),
cvec = 0, model = mirt::mirt.model(paste("F = 1-",
n.items, "
CONSTRAIN = (1-",
n.items, ", a1)")))
return(re)
}, unres = function(altpars) {

re <- list(parsets = altpars, model = 1, itemtype = "gpcm",
longpars = pars.long(pars = altpars, itemtype = "gpcm"))

return(re)
}, maximizeL = function(hyp) {
# Hypothesis-specific algorithm to find the
# maximum likelihood restricted parameter set

# returns the density for each response
# pattern under the model parameters pars

n.items <- length(pars$a) n.kat <- max(ncol(pars$d), 2)
patterns <- as.matrix(expand.grid(lapply(1:n.items,
function(x) 0:(n.kat - 1))))

pre <- c()
for (i in seq_len(nrow(patterns))) {
pre[i] <- funs$g(patterns[i, ], pars) } return(pre) } maxl <- function(x, pars, pre) { # calculates the likelihood of parameters # x given model 'pars' n.items <- length(pars$a)
n.kat <- max(ncol(pars$d), 2) patterns <- as.matrix(expand.grid(lapply(1:n.items, function(x) 0:(n.kat - 1)))) x <- list(a = rep(x[1], n.items), d = matrix(c(rep(0, n.items), x[2:length(x)]), ncol = ncol(pars$d)))

res <- c()
for (i in seq_len(nrow(patterns))) {
px <- pre[i]
qx <- funs$g(patterns[i, ], x) res[i] <- { px * log(qx) } } re <- -sum(res) } resmod <- hyp$resmod
unresmod <- hyp$unresmod pars <- unresmod$parsets
funs <- load.functions(unresmod$itemtype) n.kat <- max(ncol(pars$d), 2)
n.items <- length(pars$a) startval <- c(mean(pars$a), as.numeric(pars$d[, 2:n.kat])) maxlpre <- maxlpreload(pars) optpar <- stats::optim(startval, function(x) { maxl(x, pars, maxlpre) }, method = "BFGS") re <- pars re$a <- rep(optpar$par[1], n.items) re$d <- matrix(c(rep(0, n.items), optpar$par[2:length(optpar$par)]),
ncol = ncol(pars$d)) return(re) }) ## h_2PL_basic The null hypothesis states that the first item has difficulty 0. The underlying model is the 2PL model. h_2PL_basic <- list(res = function(altpars, nullpars = NULL) { n.items <- length(altpars[[1]]) re <- list(n.items = n.items, itemtype = "2PL", Amat = c(0, 1, rep(0, (n.items - 1) * 2)) |> (function(x) matrix(x, ncol = n.items * 2, byrow = TRUE))(), cvec = 0, model = mirt::mirt.model(paste("F = 1-", n.items, " FIXED = (1, d) START = (1,d,0)"))) return(re) }, unres = function(altpars) { re <- list(parsets = altpars, model = 1, itemtype = "2PL", longpars = pars.long(pars = altpars, itemtype = "2PL")) return(re) }, maximizeL = function(hyp) { # Hypothesis-specific algorithm to find the # maximum likelihood restricted parameter set maxlpreload <- function(pars) { # returns the density for each response # pattern under the model parameters pars patterns <- as.matrix(expand.grid(lapply(seq_len(length(pars$a)),
function(x) c(0, 1))))

pre <- c()
for (i in seq_len(nrow(patterns))) {
pre[i] <- funs$g(patterns[i, ], pars) } return(pre) } maxl <- function(x, pars, pre) { # calculates the likelihood of parameters # x given model 'pars' patterns <- as.matrix(expand.grid(lapply(seq_len(length(pars$a)),
function(x) c(0, 1))))

x <- list(a = c(x, pars$a[2:length(pars$a)]),
d = c(0, pars$d[2:length(pars$d)]))

res <- c()
for (i in seq_len(nrow(patterns))) {
px <- pre[i]
qx <- funs$g(patterns[i, ], x) res[i] <- { px * log(qx) } } re <- -sum(res) } resmod <- hyp$resmod
unresmod <- hyp$unresmod pars <- unresmod$parsets
funs <- load.functions(unresmod$itemtype) startval <- pars$a[1]

optpar <- stats::optim(startval, function(x) {
maxl(x, pars, maxlpre)
}, method = "BFGS")
re <- pars
re$a <- c(optpar$par[1], pars$a[2:length(pars$a)])
re$d <- c(0, pars$d[2:length(pars$d)]) return(re) }) ## h_3PL_basic We test the null hypothesis that the first item parameters are (1,0,.2) for the a,d, and g parameters respectively. The underlying model is the 3PL model. h_3PL_basic <- list(res = function(altpars, nullpars = NULL) { n.items <- length(altpars[[2]]) re <- list(n.items = n.items, itemtype = "3PL", Amat = c(1, 0, 0, rep(0, (n.items - 1) * 3), 0, 1, 0, rep(0, (n.items - 1) * 3), 0, 0, 1, rep(0, (n.items - 1) * 3)) |> matrix(ncol = n.items * 3, byrow = TRUE), cvec = c(1, 0, 0.2), model = mirt::mirt.model(paste("F = 1-", n.items, " FIXED = (1, d), (1,a1), (1,g) START = (1,d,0),(1,a1,1),(1,g,.2)"))) return(re) }, unres = function(altpars) { n.items <- length(altpars[[2]]) re <- list(parsets = altpars, model = 1, itemtype = "3PL", longpars = pars.long(pars = altpars, itemtype = "3PL")) return(re) }, maximizeL = function(hyp) { # Hypothesis-specific algorithm to find the # maximum likelihood restricted parameter set maxlpreload <- function(pars) { # returns the density for each response # pattern under the model parameters pars patterns <- as.matrix(expand.grid(lapply(seq_len(length(pars$a)),
function(x) c(0, 1))))

pre <- c()
for (i in seq_len(nrow(patterns))) {
pre[i] <- funs$g(patterns[i, ], pars) } return(pre) } maxl <- function(x, pars, pre) { # calculates the likelihood of parameters # x given model 'pars' patterns <- as.matrix(expand.grid(lapply(seq_len(length(pars$a)),
function(x) c(0, 1))))

x <- list(a = c(x, pars$a[2:length(pars$a)]),
d = c(0, pars$d[2:length(pars$d)]))

res <- c()
for (i in seq_len(nrow(patterns))) {
px <- pre[i]
qx <- funs$g(patterns[i, ], x) res[i] <- { px * log(qx) } } re <- -sum(res) } resmod <- hyp$resmod
unresmod <- hyp$unresmod pars <- unresmod$parsets
funs <- load.functions(unresmod$itemtype) startval <- pars$a[1]

optpar <- stats::optim(startval, function(x) {
maxl(x, pars, maxlpre)
}, method = "BFGS")
re <- pars
re$a <- c(optpar$par[1], pars$a[2:length(pars$a)])
re$d <- c(0, pars$d[2:length(pars\$d)])

return(re)
})

## h_multi_basic

We test the null hypothesis that the difficulty of the first two item parameters is equal. The underlying model is the two-dimensional 2PL model. Only the sampling-based approach is available for now.

h_multi_basic <- list(res = function(altpars, nullpars = NULL) {
n.items <- length(altpars[[2]])

re <- list(n.items = n.items, itemtype = "2PL",
Amat = c(0, 0, 1, 0, 0, -1, rep(0, (n.items -
3) * 3 + 2)) |>
matrix(ncol = n.items * 3 - 1, byrow = TRUE),
cvec = 0, model = mirt::mirt.model(paste("F1 = 1-",
n.items, "
F2 = 1-",
n.items - 1, "
CONSTRAIN = (1-2, d")))
return(re)
}, unres = function(altpars) {
n.items <- length(altpars[[2]])

re <- list(parsets = altpars, model = mirt::mirt.model(paste("F1 = 1-",
n.items, "
F2 = 1-",
n.items - 1, "")), itemtype = "2PL", longpars = pars.long(pars = altpars,
itemtype = "2PL"))

return(re)
}, maximizeL = function(hyp) {
# Hypothesis-specific algorithm to find the
# maximum likelihood restricted parameter set

# not written yet, only sampling-based
# available for now
})

## h_multi_basic2

We test the null hypothesis that the difficulty of the first item is equal to 2. The underlying model is the two-dimensional 2PL model. Only the sampling-based approach is available for now.

h_multi_basic2 <- list(res = function(altpars, nullpars = NULL) {
n.items <- length(altpars[[2]])

re <- list(n.items = n.items, itemtype = "2PL",
Amat = c(0, 0, 1, rep(0, (n.items - 2) * 3 +
2)) |>
matrix(ncol = n.items * 3 - 1, byrow = TRUE),
cvec = 2, model = mirt::mirt.model(paste("F1 = 1-",
n.items, "
F2 = 1-",
n.items - 1, "
FIXED = (1, d)
START = (1,d,2)")))
return(re)
}, unres = function(altpars) {
n.items <- length(altpars[[2]])

re <- list(parsets = altpars, model = mirt::mirt.model(paste("F1 = 1-",
n.items, "
F2 = 1-",
n.items - 1, "")), itemtype = "2PL", longpars = pars.long(pars = altpars,
itemtype = "2PL"))

return(re)
}, maximizeL = function(hyp) {
# Hypothesis-specific algorithm to find the
# maximum likelihood restricted parameter set

# not written yet, only sampling-based
# available for now
})