| Title: | Item Response Theory and Computer-Based Testing |
|---|---|
| Description: | A suite of psychometric analysis tools for research and operation, including: (1) computation of probability, information, and likelihood for the 3PL, GPCM, and GRM; (2) parameter estimation using joint or marginal likelihood estimation method; (3) simulation of computerized adaptive testing using built-in or customized algorithms; (4) assembly and simulation of multistage testing. The full documentation and tutorials are at <https://github.com/xluo11/xxIRT>. |
| Authors: | Xiao Luo [aut, cre] |
| Maintainer: | Xiao Luo <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 2.1.2 |
| Built: | 2025-02-12 04:56:22 UTC |
| Source: | https://github.com/xluo11/xxirt |
ata initiates an ATA model
ata_obj_relative adds a relative objective to the model
ata_obj_absolute adds an absolute objective to the model
ata_constraint adds a constraint to the model
ata_item_use limits the minimum and maximum usage for items
ata_item_enemy adds an enemy-item constraint to the model
ata_item_fixedvalue forces an item to be selected or not selected
ata_solve solves the MIP model
ata(pool, num_form = 1, len = NULL, max_use = NULL, ...) ## S3 method for class 'ata' print(x, ...) ## S3 method for class 'ata' plot(x, ...) ata_obj_relative(x, coef, mode = c("max", "min"), tol = NULL, negative = FALSE, forms = NULL, collapse = FALSE, internal_index = FALSE, ...) ata_obj_absolute(x, coef, target, equal_tol = FALSE, tol_up = NULL, tol_down = NULL, forms = NULL, collapse = FALSE, internal_index = FALSE, ...) ata_constraint(x, coef, min = NA, max = NA, level = NULL, forms = NULL, collapse = FALSE, internal_index = FALSE) ata_item_use(x, min = NA, max = NA, items = NULL) ata_item_enemy(x, items) ata_item_fixedvalue(x, items, min = NA, max = NA, forms) ata_solve(x, solver = c("lpsolve", "glpk"), as.list = TRUE, details = TRUE, time_limit = 10, message = FALSE, ...)ata(pool, num_form = 1, len = NULL, max_use = NULL, ...) ## S3 method for class 'ata' print(x, ...) ## S3 method for class 'ata' plot(x, ...) ata_obj_relative(x, coef, mode = c("max", "min"), tol = NULL, negative = FALSE, forms = NULL, collapse = FALSE, internal_index = FALSE, ...) ata_obj_absolute(x, coef, target, equal_tol = FALSE, tol_up = NULL, tol_down = NULL, forms = NULL, collapse = FALSE, internal_index = FALSE, ...) ata_constraint(x, coef, min = NA, max = NA, level = NULL, forms = NULL, collapse = FALSE, internal_index = FALSE) ata_item_use(x, min = NA, max = NA, items = NULL) ata_item_enemy(x, items) ata_item_fixedvalue(x, items, min = NA, max = NA, forms) ata_solve(x, solver = c("lpsolve", "glpk"), as.list = TRUE, details = TRUE, time_limit = 10, message = FALSE, ...)
pool |
item pool, a data.frame |
num_form |
number of forms to be assembled |
len |
test length of each form |
max_use |
maximum use of each item |
... |
options, e.g. group, common_items, overlap_items |
x |
an ATA object |
coef |
coefficients of the objective function |
mode |
optimization mode: 'max' for maximization and 'min' for minimization |
tol |
the tolerance paraemter |
negative |
|
forms |
forms where objectives are added. |
collapse |
|
internal_index |
|
target |
the target values of the objective function |
equal_tol |
|
tol_up |
the range of upward tolerance |
tol_down |
the range of downward tolerance |
min |
the lower bound of the constraint |
max |
the upper bound of the constraint |
level |
the level of a categorical variable to be constrained |
items |
a vector of item indices, |
solver |
use 'lpsolve' for lp_solve 5.5 or 'glpk' for GLPK |
as.list |
|
details |
|
time_limit |
the time limit in seconds passed along to solvers |
message |
|
The ATA model stores the definition of a MIP model. ata_solve
converts the model definition to a real MIP object and attempts to solve it.
ata_obj_relative:
when mode='max', maximize (y-tol), subject to y <= sum(x) <= y+tol;
when mode='min', minimize (y+tol), subject to y-tol <= sum(x) <= y.
When negative is TRUE, y < 0, tol > 0.
coef can be a numeric vector that has the same length with the pool or forms,
or a variable name in the pool, or a numeric vector of theta points.
When tol is NULL, it is optimized; when FALSE, ignored;
when a number, fixed; when a range, constrained with lower and upper bounds.
ata_obj_absolute minimizes y0+y1 subject to t-y0 <= sum(x) <= t+y1.
When level is NA, it is assumed that the constraint is on
a quantitative item property; otherwise, a categorical item property.
coef can be a variable name, a constant, or a numeric vector that has
the same size as the pool.
ata_solve takes control options in ....
For lpsolve, see lpSolveAPI::lp.control.options.
For glpk, see glpkAPI::glpkConstants
Once the model is solved, additional data are added to the model.
status shows the status of the solution, optimum
the optimal value of the objective fucntion found in the solution,
obj_vars the values of two critical variables in the objective
function, result the assembly results in a binary matrix, and
items the assembled items
## Not run: ## generate a pool of 100 items n_items <- 100 pool <- with(model_3pl_gendata(1, nitems), data.frame(id=1:n_items, a=a, b=b, c=c)) pool$content <- sample(1:3, n_items, replace=TRUE) pool$time <- round(rlnorm(n_items, log(60), .2)) pool$group <- sort(sample(1:round(n_items/3), n_items, replace=TRUE)) ## ex. 1: four 10-item forms, maximize b parameter x <- ata(pool, 4, len=10, max_use=1) x <- ata_obj_relative(x, "b", "max") x <- ata_solve(x, timeout=5) data.frame(form=1:4, b=sapply(x$items, function(x) mean(x$b))) ## ex. 2: four 10-item forms, minimize b parameter x <- ata(pool, 4, len=10, max_use=1) x <- ata_obj_relative(x, "b", "min", negative=TRUE) x <- ata_solve(x, as.list=FALSE, timeout=5) with(x$items, aggregate(b, by=list(form=form), mean)) ## ex. 3: two 10-item forms, mean(b)=0, sd(b)=1 ## content = (3, 3, 4), avg. time = 58--62 seconds constr <- data.frame(name='content',level=1:3, min=c(3,3,4), max=c(3,3,4), stringsAsFactors=F) constr <- rbind(constr, c('time', NA, 58*10, 62*10)) x <- ata(pool, 2, len=10, max_use=1) x <- ata_obj_absolute(x, pool$b, 0*10) x <- ata_obj_absolute(x, (pool$b-0)^2, 1*10) for(i in 1:nrow(constr)) x <- with(constr, ata_constraint(x, name[i], min[i], max[i], level=level[i])) x <- ata_solve(x, timeout=5) sapply(x$items, function(x) c(mean=mean(x$b), sd=sd(x$b))) ## ex. 4: two 10-item forms, max TIF over (-1, 1), consider item sets x <- ata(pool, 2, len=10, max_use=1, group="group") x <- ata_obj_relative(x, seq(-1, 1, .5), 'max') x <- ata_solve(x, timeout=5) plot(x) ## End(Not run)## Not run: ## generate a pool of 100 items n_items <- 100 pool <- with(model_3pl_gendata(1, nitems), data.frame(id=1:n_items, a=a, b=b, c=c)) pool$content <- sample(1:3, n_items, replace=TRUE) pool$time <- round(rlnorm(n_items, log(60), .2)) pool$group <- sort(sample(1:round(n_items/3), n_items, replace=TRUE)) ## ex. 1: four 10-item forms, maximize b parameter x <- ata(pool, 4, len=10, max_use=1) x <- ata_obj_relative(x, "b", "max") x <- ata_solve(x, timeout=5) data.frame(form=1:4, b=sapply(x$items, function(x) mean(x$b))) ## ex. 2: four 10-item forms, minimize b parameter x <- ata(pool, 4, len=10, max_use=1) x <- ata_obj_relative(x, "b", "min", negative=TRUE) x <- ata_solve(x, as.list=FALSE, timeout=5) with(x$items, aggregate(b, by=list(form=form), mean)) ## ex. 3: two 10-item forms, mean(b)=0, sd(b)=1 ## content = (3, 3, 4), avg. time = 58--62 seconds constr <- data.frame(name='content',level=1:3, min=c(3,3,4), max=c(3,3,4), stringsAsFactors=F) constr <- rbind(constr, c('time', NA, 58*10, 62*10)) x <- ata(pool, 2, len=10, max_use=1) x <- ata_obj_absolute(x, pool$b, 0*10) x <- ata_obj_absolute(x, (pool$b-0)^2, 1*10) for(i in 1:nrow(constr)) x <- with(constr, ata_constraint(x, name[i], min[i], max[i], level=level[i])) x <- ata_solve(x, timeout=5) sapply(x$items, function(x) c(mean=mean(x$b), sd=sd(x$b))) ## ex. 4: two 10-item forms, max TIF over (-1, 1), consider item sets x <- ata(pool, 2, len=10, max_use=1, group="group") x <- ata_obj_relative(x, seq(-1, 1, .5), 'max') x <- ata_solve(x, timeout=5) plot(x) ## End(Not run)
cat_sim runs a simulation of CAT. Use theta in options to set the starting
value of theta estimate.
cat_estimate_mle is the maximum likelihood estimation rule. Use
map_len to apply MAP to the first K items and use map_prior to set the
prior for MAP.
cat_estimate_eap is the expected a posteriori estimation rule,
using eap_mean and eap_sd option parameters as the prior
cat_estimate_hybrid is a hybrid estimation rule, which uses MLE for
mixed responses and EAP for all 1's or 0's responses
cat_stop_default is a three-way stopping rule. When stop_se
is set in the options, it uses the standard error stopping rule. When
stop_mi is set in the options, it uses the minimum information stopping rule. When
stop_cut is set in the options, it uses the confidence interval (set by ci_width)
stopping rule.
cat_select_maxinfo is the maximum information selection rule. Use group
(a numeric vector) to group items belonging to the same set. Use info_random to implement
the random-esque item exposure control method.
cat_select_ccat is the constrained CAT selection rule. Use
ccat_var to set the content variable in the pool. Use ccat_perc to set
the desired content distribution, with the name of each element being the content code
and tue value of each element being the percentage. Use ccat_random to add randomness
to initial item selections.
cat_select_shadow is the shadow-test selection rule. Use shadow_id
to group item sets. Use constraints to set constraints. Constraints should be in a data.frame
with four columns: var (variable name), level (variable level, NA for quantitative variable),
min (lower bound), and max (upper bound).
cat_stop_projection is the projection-based stopping rule. Use
projection_method to choose the projection method ('info' or 'diff'). Use
stop_cut to set the cut score. Use constraints to set the constraints.
Constraints should be a data.frame with columns: var (variable name),
level (variable level, NA for quantitative varialbe), min (lower bound), max (upper bound)
cat_sim(true, pool, ...) cat_estimate_mle(len, theta, stats, admin, pool, opts) cat_estimate_eap(len, theta, stats, admin, pool, opts) cat_estimate_hybrid(len, theta, stats, admin, pool, opts) cat_stop_default(len, theta, stats, admin, pool, opts) cat_select_maxinfo(len, theta, stats, admin, pool, opts) cat_select_ccat(len, theta, stats, admin, pool, opts) cat_select_shadow(len, theta, stats, admin, pool, opts) ## S3 method for class 'cat' print(x, ...) ## S3 method for class 'cat' plot(x, ...) cat_stop_projection(len, theta, stats, admin, pool, opts)cat_sim(true, pool, ...) cat_estimate_mle(len, theta, stats, admin, pool, opts) cat_estimate_eap(len, theta, stats, admin, pool, opts) cat_estimate_hybrid(len, theta, stats, admin, pool, opts) cat_stop_default(len, theta, stats, admin, pool, opts) cat_select_maxinfo(len, theta, stats, admin, pool, opts) cat_select_ccat(len, theta, stats, admin, pool, opts) cat_select_shadow(len, theta, stats, admin, pool, opts) ## S3 method for class 'cat' print(x, ...) ## S3 method for class 'cat' plot(x, ...) cat_stop_projection(len, theta, stats, admin, pool, opts)
true |
the true theta |
pool |
the item pool (data.frame) |
... |
option/control parameters |
len |
the current test length |
theta |
the current theta estimate |
stats |
a matrix of responses, theta estimate, information and std error |
admin |
a data frame of administered items |
opts |
a list of option/control parameters |
x |
a |
... takes a variety of option/control parameters for the simulations from users.
min and max are mandatory for setting limits on the test length. User-defined
selection, estimation, and stopping rules are also passed to the simulator via options.
To write a new rule, the function siganiture must be: function(len, theta, stats, admin, pool, opts).
See built-in rules for examples.
cat_sim returns a cat object
an estimation rule should return a theta estimate
a stopping rule should return a boolean: TRUE to stop the CAT, FALSE to continue
a selection rule should return a list of (a) the selected item and (b) the updated pool
## Not run: ## generate a 100-item pool num_items <- 100 pool <- with(model_3pl_gendata(1, num_items), data.frame(a=a, b=b, c=c)) pool$set_id <- sample(1:30, num_items, replace=TRUE) pool$content <- sample(1:3, num_items, replace=TRUE) pool$time <- round(rlnorm(num_items, mean=4.1, sd=.2)) ## MLE, EAP, and hybrid estimation rule cat_sim(1.0, pool, min=10, max=20, estimate_rule=cat_estimate_mle) cat_sim(1.0, pool, min=10, max=20, estimate_rule=cat_estimate_eap) cat_sim(1.0, pool, min=10, max=20, estimate_rule=cat_estimate_hybrid) ## SE, MI, and CI stopping rule cat_sim(1.0, pool, min=10, max=20, stop_se=.3) cat_sim(1.0, pool, min=10, max=20, stop_mi=.6) cat_sim(1.0, pool, min=10, max=20, stop_cut=0) cat_sim(1.0, pool, min=10, max=20, stop_cut=0, ci_width=2.58) ## maximum information selection with item sets cat_sim(1.0, pool, min=10, max=20, group="set_id")$admin ## maximum information with item exposure control cat_sim(1.0, pool, min=10, max=20, info_random=5)$admin ## Constrained-CAT selection rule with and without initial randomness cat_sim(1.0, pool, min=10, max=20, select_rule=cat_select_ccat, ccat_var="content", ccat_perc=c("1"=.2, "2"=.3, "3"=.5)) cat_sim(1.0, pool, min=10, max=20, select_rule=cat_select_ccat, ccat_random=5, ccat_var="content", ccat_perc=c("1"=.2, "2"=.3, "3"=.5)) ## Shadow-test selection rule cons <- data.frame(var='content', level=1:3, min=c(3,3,4), max=c(3,3,4)) cons <- rbind(cons, data.frame(var='time', level=NA, min=55*10, max=65*10)) cat_sim(1.0, pool, min=10, max=10, select_rule=cat_select_shadow, constraints=cons) ## Projection-based stopping rule cons <- data.frame(var='content', level=1:3, min=5, max=15) cons <- rbind(cons, data.frame(var='time', level=NA, min=60*20, max=60*40)) cat_sim(1.0, pool, min=20, max=40, select_rule=cat_select_shadow, stop_rule=cat_stop_projection, projection_method="diff", stop_cut=0, constraints=cons) ## End(Not run)## Not run: ## generate a 100-item pool num_items <- 100 pool <- with(model_3pl_gendata(1, num_items), data.frame(a=a, b=b, c=c)) pool$set_id <- sample(1:30, num_items, replace=TRUE) pool$content <- sample(1:3, num_items, replace=TRUE) pool$time <- round(rlnorm(num_items, mean=4.1, sd=.2)) ## MLE, EAP, and hybrid estimation rule cat_sim(1.0, pool, min=10, max=20, estimate_rule=cat_estimate_mle) cat_sim(1.0, pool, min=10, max=20, estimate_rule=cat_estimate_eap) cat_sim(1.0, pool, min=10, max=20, estimate_rule=cat_estimate_hybrid) ## SE, MI, and CI stopping rule cat_sim(1.0, pool, min=10, max=20, stop_se=.3) cat_sim(1.0, pool, min=10, max=20, stop_mi=.6) cat_sim(1.0, pool, min=10, max=20, stop_cut=0) cat_sim(1.0, pool, min=10, max=20, stop_cut=0, ci_width=2.58) ## maximum information selection with item sets cat_sim(1.0, pool, min=10, max=20, group="set_id")$admin ## maximum information with item exposure control cat_sim(1.0, pool, min=10, max=20, info_random=5)$admin ## Constrained-CAT selection rule with and without initial randomness cat_sim(1.0, pool, min=10, max=20, select_rule=cat_select_ccat, ccat_var="content", ccat_perc=c("1"=.2, "2"=.3, "3"=.5)) cat_sim(1.0, pool, min=10, max=20, select_rule=cat_select_ccat, ccat_random=5, ccat_var="content", ccat_perc=c("1"=.2, "2"=.3, "3"=.5)) ## Shadow-test selection rule cons <- data.frame(var='content', level=1:3, min=c(3,3,4), max=c(3,3,4)) cons <- rbind(cons, data.frame(var='time', level=NA, min=55*10, max=65*10)) cat_sim(1.0, pool, min=10, max=10, select_rule=cat_select_shadow, constraints=cons) ## Projection-based stopping rule cons <- data.frame(var='content', level=1:3, min=5, max=15) cons <- rbind(cons, data.frame(var='time', level=NA, min=60*20, max=60*40)) cat_sim(1.0, pool, min=20, max=40, select_rule=cat_select_shadow, stop_rule=cat_stop_projection, projection_method="diff", stop_cut=0, constraints=cons) ## End(Not run)
cronbach_alpha computes Cronbach's alpha internal consistency reliability
cronbach_alpha(responses)cronbach_alpha(responses)
responses |
the oberved responses, 2d matrix |
cronbach_alpha(model_3pl_gendata(1000, 20)$u)cronbach_alpha(model_3pl_gendata(1000, 20)$u)
Calculate the distribution of expected raw scores
expected_raw_score_dist(t, a, b, c)expected_raw_score_dist(t, a, b, c)
t |
the ability parameters, 1d vector |
a |
the item discrimination parameters, 1d vector |
b |
the item difficulty parameters, 1d vector |
c |
the item guessing parameters, 1d vector |
Frequency counts of a vector
freq(x, values = NULL, rounding = NULL)freq(x, values = NULL, rounding = NULL)
x |
a numeric or character vector |
values |
valid values, |
rounding |
round percentage to n-th decimal places |
Routine functions for the 3PL model
model_3pl_prob(t, a, b, c, D = 1.702) model_3pl_info(t, a, b, c, D = 1.702) model_3pl_lh(u, t, a, b, c, D = 1.702, log = FALSE) model_3pl_rescale(t, a, b, c, param = c("t", "b"), mean = 0, sd = 1) model_3pl_gendata(n_p, n_i, t = NULL, a = NULL, b = NULL, c = NULL, D = 1.702, t_dist = c(0, 1), a_dist = c(-0.1, 0.2), b_dist = c(0, 0.7), c_dist = c(5, 46), missing = NULL) model_3pl_plot(a, b, c, D = 1.702, type = c("prob", "info"), total = FALSE, xaxis = seq(-4, 4, 0.1)) model_3pl_plot_loglh(u, a, b, c, D = 1.702, xaxis = seq(-4, 4, 0.1), show_mle = FALSE)model_3pl_prob(t, a, b, c, D = 1.702) model_3pl_info(t, a, b, c, D = 1.702) model_3pl_lh(u, t, a, b, c, D = 1.702, log = FALSE) model_3pl_rescale(t, a, b, c, param = c("t", "b"), mean = 0, sd = 1) model_3pl_gendata(n_p, n_i, t = NULL, a = NULL, b = NULL, c = NULL, D = 1.702, t_dist = c(0, 1), a_dist = c(-0.1, 0.2), b_dist = c(0, 0.7), c_dist = c(5, 46), missing = NULL) model_3pl_plot(a, b, c, D = 1.702, type = c("prob", "info"), total = FALSE, xaxis = seq(-4, 4, 0.1)) model_3pl_plot_loglh(u, a, b, c, D = 1.702, xaxis = seq(-4, 4, 0.1), show_mle = FALSE)
t |
ability parameters, 1d vector |
a |
discrimination parameters, 1d vector |
b |
difficulty parameters, 1d vector |
c |
guessing parameters, 1d vector |
D |
the scaling constant, 1.702 by default |
u |
observed responses, 2d matrix |
log |
True to return log-likelihood |
param |
the parameter of the new scale: 't' or 'b' |
mean |
the mean of the new scale |
sd |
the standard deviation of the new scale |
n_p |
the number of people to be generated |
n_i |
the number of items to be generated |
t_dist |
parameters of the normal distribution used to generate t-parameters |
a_dist |
parameters of the lognormal distribution used to generate a-parameters |
b_dist |
parameters of the normal distribution used to generate b-parameters |
c_dist |
parameters of the beta distribution used to generate c-parameters |
missing |
the proportion or number of missing responses |
type |
the type of plot: 'prob' for item characteristic curve (ICC) and 'info' for item information function curve (IIFC) |
total |
TRUE to sum values over items |
xaxis |
the values of x-axis |
show_mle |
TRUE to print maximum likelihood estimates |
with(model_3pl_gendata(10, 5), model_3pl_prob(t, a, b, c)) with(model_3pl_gendata(10, 5), model_3pl_info(t, a, b, c)) with(model_3pl_gendata(10, 5), model_3pl_lh(u, t, a, b, c)) model_3pl_gendata(10, 5) model_3pl_gendata(10, 5, a=1, c=0, missing=.1) with(model_3pl_gendata(10, 5), model_3pl_plot(a, b, c, type="prob")) with(model_3pl_gendata(10, 5), model_3pl_plot(a, b, c, type="info", total=TRUE)) with(model_3pl_gendata(5, 50), model_3pl_plot_loglh(u, a, b, c, show_mle=TRUE))with(model_3pl_gendata(10, 5), model_3pl_prob(t, a, b, c)) with(model_3pl_gendata(10, 5), model_3pl_info(t, a, b, c)) with(model_3pl_gendata(10, 5), model_3pl_lh(u, t, a, b, c)) model_3pl_gendata(10, 5) model_3pl_gendata(10, 5, a=1, c=0, missing=.1) with(model_3pl_gendata(10, 5), model_3pl_plot(a, b, c, type="prob")) with(model_3pl_gendata(10, 5), model_3pl_plot(a, b, c, type="info", total=TRUE)) with(model_3pl_gendata(5, 50), model_3pl_plot_loglh(u, a, b, c, show_mle=TRUE))
Routine functions for the GPCM
model_gpcm_prob(t, a, b, d, D = 1.702, insert_d0 = NULL) model_gpcm_info(t, a, b, d, D = 1.702, insert_d0 = NULL) model_gpcm_lh(u, t, a, b, d, D = 1.702, insert_d0 = NULL, log = FALSE) model_gpcm_gendata(n_p, n_i, n_c, t = NULL, a = NULL, b = NULL, d = NULL, D = 1.702, sort_d = FALSE, t_dist = c(0, 1), a_dist = c(-0.1, 0.2), b_dist = c(0, 0.8), missing = NULL) model_gpcm_rescale(t, a, b, d, param = c("t", "b"), mean = 0, sd = 1) model_gpcm_plot(a, b, d, D = 1.702, insert_d0 = NULL, type = c("prob", "info"), by_item = FALSE, total = FALSE, xaxis = seq(-6, 6, 0.1)) model_gpcm_plot_loglh(u, a, b, d, D = 1.702, insert_d0 = NULL, xaxis = seq(-6, 6, 0.1), show_mle = FALSE)model_gpcm_prob(t, a, b, d, D = 1.702, insert_d0 = NULL) model_gpcm_info(t, a, b, d, D = 1.702, insert_d0 = NULL) model_gpcm_lh(u, t, a, b, d, D = 1.702, insert_d0 = NULL, log = FALSE) model_gpcm_gendata(n_p, n_i, n_c, t = NULL, a = NULL, b = NULL, d = NULL, D = 1.702, sort_d = FALSE, t_dist = c(0, 1), a_dist = c(-0.1, 0.2), b_dist = c(0, 0.8), missing = NULL) model_gpcm_rescale(t, a, b, d, param = c("t", "b"), mean = 0, sd = 1) model_gpcm_plot(a, b, d, D = 1.702, insert_d0 = NULL, type = c("prob", "info"), by_item = FALSE, total = FALSE, xaxis = seq(-6, 6, 0.1)) model_gpcm_plot_loglh(u, a, b, d, D = 1.702, insert_d0 = NULL, xaxis = seq(-6, 6, 0.1), show_mle = FALSE)
t |
ability parameters, 1d vector |
a |
discrimination parameters, 1d vector |
b |
item location parameters, 1d vector |
d |
item category parameters, 2d vector |
D |
the scaling constant, 1.702 by default |
insert_d0 |
insert an initial category value |
u |
the observed scores (starting from 0), 2d matrix |
log |
TRUE to return log-likelihood |
n_p |
the number of people to be generated |
n_i |
the number of items to be generated |
n_c |
the number of score categories |
sort_d |
|
t_dist |
parameters of the normal distribution used to generate t-parameters |
a_dist |
parameters of the lognormal distribution parameters of a-parameters |
b_dist |
parameters of the normal distribution used to generate b-parameters |
missing |
the proportion or number of missing responses |
param |
the parameter of the new scale: 't' or 'b' |
mean |
the mean of the new scale |
sd |
the standard deviation of the new scale |
type |
the type of plot, prob for ICC and info for IIFC |
by_item |
TRUE to combine categories |
total |
TRUE to sum values over items |
xaxis |
the values of x-axis |
show_mle |
TRUE to print maximum likelihood values |
Use NA to represent unused category.
with(model_gpcm_gendata(10, 5, 3), model_gpcm_prob(t, a, b, d)) with(model_gpcm_gendata(10, 5, 3), model_gpcm_info(t, a, b, d)) with(model_gpcm_gendata(10, 5, 3), model_gpcm_lh(u, t, a, b, d)) model_gpcm_gendata(10, 5, 3) model_gpcm_gendata(10, 5, 3, missing=.1) # Figure 1 in Muraki, 1992 (APM) b <- matrix(c(-2,0,2,-.5,0,2,-.5,0,2), nrow=3, byrow=TRUE) model_gpcm_plot(a=c(1,1,.7), b=rowMeans(b), d=rowMeans(b)-b, D=1.0, insert_d0=0) # Figure 2 in Muraki, 1992 (APM) b <- matrix(c(.5,0,NA,0,0,0), nrow=2, byrow=TRUE) model_gpcm_plot(a=.7, b=rowMeans(b, na.rm=TRUE), d=rowMeans(b, na.rm=TRUE)-b, D=1.0, insert_d0=0) # Figure 3 in Muraki, 1992 (APM) b <- matrix(c(1.759,-1.643,3.970,-2.764), nrow=2, byrow=TRUE) model_gpcm_plot(a=c(.778,.946), b=rowMeans(b), d=rowMeans(b)-b, D=1.0, insert_d0=0) # Figure 1 in Muraki, 1993 (APM) b <- matrix(c(0,-2,4,0,-2,2,0,-2,0,0,-2,-2,0,-2,-4), nrow=5, byrow=TRUE) model_gpcm_plot(a=1, b=rowMeans(b), d=rowMeans(b)-b, D=1.0) # Figure 2 in Muraki, 1993 (APM) b <- matrix(c(0,-2,4,0,-2,2,0,-2,0,0,-2,-2,0,-2,-4), nrow=5, byrow=TRUE) model_gpcm_plot(a=1, b=rowMeans(b), d=rowMeans(b)-b, D=1.0, type='info', by_item=TRUE) with(model_gpcm_gendata(5, 50, 3), model_gpcm_plot_loglh(u, a, b, d))with(model_gpcm_gendata(10, 5, 3), model_gpcm_prob(t, a, b, d)) with(model_gpcm_gendata(10, 5, 3), model_gpcm_info(t, a, b, d)) with(model_gpcm_gendata(10, 5, 3), model_gpcm_lh(u, t, a, b, d)) model_gpcm_gendata(10, 5, 3) model_gpcm_gendata(10, 5, 3, missing=.1) # Figure 1 in Muraki, 1992 (APM) b <- matrix(c(-2,0,2,-.5,0,2,-.5,0,2), nrow=3, byrow=TRUE) model_gpcm_plot(a=c(1,1,.7), b=rowMeans(b), d=rowMeans(b)-b, D=1.0, insert_d0=0) # Figure 2 in Muraki, 1992 (APM) b <- matrix(c(.5,0,NA,0,0,0), nrow=2, byrow=TRUE) model_gpcm_plot(a=.7, b=rowMeans(b, na.rm=TRUE), d=rowMeans(b, na.rm=TRUE)-b, D=1.0, insert_d0=0) # Figure 3 in Muraki, 1992 (APM) b <- matrix(c(1.759,-1.643,3.970,-2.764), nrow=2, byrow=TRUE) model_gpcm_plot(a=c(.778,.946), b=rowMeans(b), d=rowMeans(b)-b, D=1.0, insert_d0=0) # Figure 1 in Muraki, 1993 (APM) b <- matrix(c(0,-2,4,0,-2,2,0,-2,0,0,-2,-2,0,-2,-4), nrow=5, byrow=TRUE) model_gpcm_plot(a=1, b=rowMeans(b), d=rowMeans(b)-b, D=1.0) # Figure 2 in Muraki, 1993 (APM) b <- matrix(c(0,-2,4,0,-2,2,0,-2,0,0,-2,-2,0,-2,-4), nrow=5, byrow=TRUE) model_gpcm_plot(a=1, b=rowMeans(b), d=rowMeans(b)-b, D=1.0, type='info', by_item=TRUE) with(model_gpcm_gendata(5, 50, 3), model_gpcm_plot_loglh(u, a, b, d))
Routine functions for the GRM
model_grm_prob(t, a, b, D = 1.702, raw = FALSE) model_grm_info(t, a, b, D = 1.702) model_grm_lh(u, t, a, b, D = 1.702, log = FALSE) model_grm_gendata(n_p, n_i, n_c, t = NULL, a = NULL, b = NULL, D = 1.702, t_dist = c(0, 1), a_dist = c(-0.1, 0.2), b_dist = c(0, 0.8), missing = NULL) model_grm_rescale(t, a, b, param = c("t", "b"), mean = 0, sd = 1) model_grm_plot(a, b, D = 1.702, type = c("prob", "info"), by_item = FALSE, total = FALSE, xaxis = seq(-6, 6, 0.1), raw = FALSE) model_grm_plot_loglh(u, a, b, D = 1.702, xaxis = seq(-6, 6, 0.1), show_mle = FALSE)model_grm_prob(t, a, b, D = 1.702, raw = FALSE) model_grm_info(t, a, b, D = 1.702) model_grm_lh(u, t, a, b, D = 1.702, log = FALSE) model_grm_gendata(n_p, n_i, n_c, t = NULL, a = NULL, b = NULL, D = 1.702, t_dist = c(0, 1), a_dist = c(-0.1, 0.2), b_dist = c(0, 0.8), missing = NULL) model_grm_rescale(t, a, b, param = c("t", "b"), mean = 0, sd = 1) model_grm_plot(a, b, D = 1.702, type = c("prob", "info"), by_item = FALSE, total = FALSE, xaxis = seq(-6, 6, 0.1), raw = FALSE) model_grm_plot_loglh(u, a, b, D = 1.702, xaxis = seq(-6, 6, 0.1), show_mle = FALSE)
t |
ability parameters, 1d vector |
a |
discrimination parameters, 1d vector |
b |
item location parameters, 2d matrix |
D |
the scaling constant, 1.702 by default |
raw |
TRUE to return P* |
u |
the observed scores (starting from 0), 2d matrix |
log |
TRUE to return log-likelihood |
n_p |
the number of people to be generated |
n_i |
the number of items to be generated |
n_c |
the number of score categories |
t_dist |
parameters of the normal distribution used to generate t-parameters |
a_dist |
parameters of the lognormal distribution used to generate a-parameters |
b_dist |
parameters of the normal distribution used to generate b-parameters |
missing |
the proportion or number of missing responses |
param |
the parameter of the new scale: 't' or 'b' |
mean |
the mean of the new scale |
sd |
the standard deviation of the new scale |
type |
the type of plot, prob for ICC and info for IIFC |
by_item |
TRUE to combine categories |
total |
TRUE to sum values over items |
xaxis |
the values of x-axis |
show_mle |
TRUE to print maximum likelihood values |
with(model_grm_gendata(10, 5, 3), model_grm_prob(t, a, b)) with(model_grm_gendata(10, 5, 3), model_grm_info(t, a, b)) with(model_grm_gendata(10, 5, 3), model_grm_lh(u, t, a, b)) model_grm_gendata(10, 5, 3) model_grm_gendata(10, 5, 3, missing=.1) with(model_grm_gendata(10, 5, 3), model_grm_plot(a, b, type='prob')) with(model_grm_gendata(10, 5, 3), model_grm_plot(a, b, type='info', by_item=TRUE)) with(model_grm_gendata(5, 50, 3), model_grm_plot_loglh(u, a, b))with(model_grm_gendata(10, 5, 3), model_grm_prob(t, a, b)) with(model_grm_gendata(10, 5, 3), model_grm_info(t, a, b)) with(model_grm_gendata(10, 5, 3), model_grm_lh(u, t, a, b)) model_grm_gendata(10, 5, 3) model_grm_gendata(10, 5, 3, missing=.1) with(model_grm_gendata(10, 5, 3), model_grm_plot(a, b, type='prob')) with(model_grm_gendata(10, 5, 3), model_grm_plot(a, b, type='info', by_item=TRUE)) with(model_grm_gendata(5, 50, 3), model_grm_plot_loglh(u, a, b))
mst_sim simulates a MST administration
mst_sim(x, true, rdp = NULL, ...) ## S3 method for class 'mst_sim' print(x, ...) ## S3 method for class 'mst_sim' plot(x, ...)mst_sim(x, true, rdp = NULL, ...) ## S3 method for class 'mst_sim' print(x, ...) ## S3 method for class 'mst_sim' plot(x, ...)
x |
the assembled MST |
true |
the true theta parameter (numeric) |
rdp |
routing decision points (list) |
... |
additional option/control parameters |
## Not run: ## assemble a MST nitems <- 200 pool <- with(model_3pl_gendata(1, nitems), data.frame(a=a, b=b, c=c)) pool$content <- sample(1:3, nrow(pool), replace=TRUE) x <- mst(pool, "1-2-2", 2, 'topdown', len=20, max_use=1) x <- mst_obj(x, theta=-1, indices=1) x <- mst_obj(x, theta=0, indices=2:3) x <- mst_obj(x, theta=1, indices=4) x <- mst_constraint(x, "content", 6, 6, level=1) x <- mst_constraint(x, "content", 6, 6, level=2) x <- mst_constraint(x, "content", 8, 8, level=3) x <- mst_stage_length(x, 1:2, min=5) x <- mst_assemble(x) ## ex. 1: administer the MST using fixed RDP for routing x_sim <- mst_sim(x, .5, list(stage1=0, stage2=0)) plot(x_sim) ## ex. 2: administer the MST using the max. info. for routing x_sim <- mst_sim(x, .5) plot(x_sim, ylim=c(-5, 5)) ## End(Not run)## Not run: ## assemble a MST nitems <- 200 pool <- with(model_3pl_gendata(1, nitems), data.frame(a=a, b=b, c=c)) pool$content <- sample(1:3, nrow(pool), replace=TRUE) x <- mst(pool, "1-2-2", 2, 'topdown', len=20, max_use=1) x <- mst_obj(x, theta=-1, indices=1) x <- mst_obj(x, theta=0, indices=2:3) x <- mst_obj(x, theta=1, indices=4) x <- mst_constraint(x, "content", 6, 6, level=1) x <- mst_constraint(x, "content", 6, 6, level=2) x <- mst_constraint(x, "content", 8, 8, level=3) x <- mst_stage_length(x, 1:2, min=5) x <- mst_assemble(x) ## ex. 1: administer the MST using fixed RDP for routing x_sim <- mst_sim(x, .5, list(stage1=0, stage2=0)) plot(x_sim) ## ex. 2: administer the MST using the max. info. for routing x_sim <- mst_sim(x, .5) plot(x_sim, ylim=c(-5, 5)) ## End(Not run)
Root mean squared error (RMSE) of two numeric vectors/matrices
rmse(x, y)rmse(x, y)
x |
a numeric vector/matrix |
y |
a numeric vector/matrix |
Use Spearman-brown formula to compute the predicted reliability when the test length is extened to n-fold or reversely the n-fold extension of test length in order to reach the targeted reliability
spearman_brown(n, rho) spearman_brown_reverse(rho, target)spearman_brown(n, rho) spearman_brown_reverse(rho, target)
n |
extend the test length to n-fold |
rho |
the reliability of current test |
target |
the targeted reliability |
spearman_brown(2, .70) spearman_brown_reverse(.70, .85)spearman_brown(2, .70) spearman_brown_reverse(.70, .85)