| Type: | Package |
| Title: | Data Analysis and Parameter Estimation Using Item Response Theory |
| Version: | 0.0.2 |
| Date: | 2019-10-22 |
| Author: | Xiao Luo [aut, cre] |
| Maintainer: | Xiao Luo <xluo1986@gmail.com> |
| Description: | Parameter estimation, computation of probability, information, and (log-)likelihood, and visualization of item/test characteristic curves and item/test information functions for three uni-dimensional item response theory models: the 3-parameter-logistic model, generalized partial credit model, and graded response model. The full documentation and tutorials are at https://github.com/xluo11/Rirt. |
| License: | GPL (≥ 3) |
| Depends: | R (≥ 3.6.0) |
| URL: | https://github.com/xluo11/Rirt |
| BugReports: | https://github.com/xluo11/Rirt/issues |
| LinkingTo: | Rcpp |
| Imports: | ggplot2, Rcpp, reshape2, stats |
| Suggests: | testthat |
| RoxygenNote: | 6.1.1 |
| Encoding: | UTF-8 |
| NeedsCompilation: | yes |
| Packaged: | 2019-10-23 03:29:09 UTC; Luo.Xiao |
| Repository: | CRAN |
| Date/Publication: | 2019-10-23 14:00:02 UTC |
Estimation of the 3PL model
Description
Estimate the 3PL model using the joint or marginal maximum likelihood estimation methods
model_3pl_eap scores response vectors using the EAP method
model_3pl_map scores response vectors using the MAP method
model_3pl_jmle estimates the item and ability parameters
using the joint maximum likelihood estimation (JMLE) method
model_3pl_mmle estimates the item parameters using the
marginal maximum likelihood estimation (MMLE) method
Usage
model_3pl_eap(u, a, b, c, D = 1.702, priors = c(0, 1),
bounds_t = c(-4, 4))
model_3pl_map(u, a, b, c, D = 1.702, priors = c(0, 1),
bounds_t = c(-4, 4), iter = 30, conv = 0.001)
model_3pl_dv_Pt(t, a, b, c, D)
model_3pl_dv_Pa(t, a, b, c, D)
model_3pl_dv_Pb(t, a, b, c, D)
model_3pl_dv_Pc(t, a, b, c, D)
model_3pl_dv_jmle(pdv_fn, u, t, a, b, c, D)
model_3pl_jmle(u, t = NA, a = NA, b = NA, c = NA, D = 1.702,
iter = 100, conv = 0.001, nr_iter = 10, scale = c(0, 1),
bounds_t = c(-4, 4), bounds_a = c(0.01, 2.5), bounds_b = c(-4, 4),
bounds_c = c(0, 0.4), priors = list(t = c(0, 1)), decay = 1,
verbose = FALSE, true_params = NULL)
model_3pl_dv_mmle(pdv_fn, u, quad, a, b, c, D)
model_3pl_mmle(u, t = NA, a = NA, b = NA, c = NA, D = 1.702,
iter = 100, conv = 0.001, nr_iter = 10, bounds_t = c(-4, 4),
bounds_a = c(0.01, 2.5), bounds_b = c(-4, 4), bounds_c = c(0, 0.4),
priors = list(t = c(0, 1)), decay = 1, quad = "11",
score_fn = c("eap", "map"), verbose = FALSE, true_params = NULL)
model_3pl_fitplot(u, t, a, b, c, D = 1.702, index = NULL,
intervals = seq(-3, 3, 0.5))
Arguments
u |
observed response matrix, 2d matrix |
a |
discrimination parameters, 1d vector (fixed value) or NA (freely estimate) |
b |
difficulty parameters, 1d vector (fixed value) or NA (freely estimate) |
c |
pseudo-guessing parameters, 1d vector (fixed value) or NA (freely estimate) |
D |
the scaling constant, 1.702 by default |
priors |
prior distributions, a list |
bounds_t |
the bounds of ability parameters |
iter |
the maximum iterations, default=100 |
conv |
the convergence criterion |
t |
ability parameters, 1d vector (fixed value) or NA (freely estimate) |
pdv_fn |
the function to compute derivatives of P w.r.t the estimating parameters |
nr_iter |
the maximum newton-raphson iterations, default=10 |
scale |
the mean and SD of the theta scale, default= |
bounds_a |
the bounds of discrimination parameters |
bounds_b |
the bounds of difficulty parameters |
bounds_c |
the bounds of guessing parameters |
decay |
decay rate, default=1 |
verbose |
TRUE to print details for debugging |
true_params |
a list of true parameters for evaluating the parameter recovery |
quad |
the number of quadrature points |
score_fn |
the scoring function: 'eap' or 'map' |
index |
the indices of items being plotted |
intervals |
intervals on the x-axis |
Value
model_3pl_eap returns theta estimates and standard errors in a list
model_3pl_map returns theta estimates in a list
model_3pl_jmle returns estimated t, a, b, c parameters in a list
model_3pl_mmle returns estimated t, a, b, c parameters in a list
model_3pl_fitplot returns a ggplot object
Examples
with(model_3pl_gendata(10, 40),
cbind(true=t, est=model_3pl_eap(u, a, b, c)$t))
with(model_3pl_gendata(10, 40),
cbind(true=t, est=model_3pl_map(u, a, b, c)$t))
# generate data
x <- model_3pl_gendata(2000, 40)
# free estimation, 40 iterations
y <- model_3pl_jmle(x$u, true_params=x, iter=40, verbose=TRUE)
# fix c-parameters, 40 iterations
y <- model_3pl_jmle(x$u, c=0, true_params=x, iter=40)
# generate data
x <- model_3pl_gendata(2000, 40)
# free estimation, 40 iterations
y <- model_3pl_mmle(x$u, true_params=x, iter=40, verbose=TRUE)
with(model_3pl_gendata(1000, 20),
model_3pl_fitplot(u, t, a, b, c, index=c(1, 3, 5)))
Estimation of the Generalizaed Partial Credit Model
Description
Estimate the GPCM using the joint or marginal maximum likelihood estimation
model_gpcm_eap scores response vectors using the EAP method
model_gpcm_map scores response vectors using the MAP method
model_gpcm_jmle estimates the parameters using the
joint maximum likelihood estimation (JMLE) method
model_gpcm_mmle estimates the parameters using the
marginal maximum likelihood estimation (MMLE) method
Usage
model_gpcm_eap(u, a, b, d, D = 1.702, priors = c(0, 1),
bounds_t = c(-4, 4))
model_gpcm_map(u, a, b, d, D = 1.702, priors = c(0, 1),
bounds_t = c(-4, 4), iter = 30, conv = 0.001)
model_gpcm_dv_Pt(t, a, b, d, D)
model_gpcm_dv_Pa(t, a, b, d, D)
model_gpcm_dv_Pb(t, a, b, d, D)
model_gpcm_dv_Pd(t, a, b, d, D)
model_gpcm_dv_jmle(u_ix, dvp)
model_gpcm_jmle(u, t = NA, a = NA, b = NA, d = NA, D = 1.702,
iter = 100, nr_iter = 10, conv = 0.001, scale = c(0, 1),
bounds_t = c(-4, 4), bounds_a = c(0.01, 2.5), bounds_b = c(-4, 4),
bounds_d = c(-4, 4), priors = list(t = c(0, 1)), decay = 1,
verbose = FALSE, true_params = NULL)
model_gpcm_dv_mmle(u_ix, quad, pdv)
model_gpcm_mmle(u, t = NA, a = NA, b = NA, d = NA, D = 1.702,
iter = 100, nr_iter = 10, conv = 0.001, bounds_t = c(-4, 4),
bounds_a = c(0.01, 2.5), bounds_b = c(-4, 4), bounds_d = c(-4, 4),
priors = list(t = c(0, 1)), decay = 1, quad_degree = "11",
score_fn = c("eap", "map"), verbose = FALSE, true_params = NULL)
model_gpcm_fitplot(u, t, a, b, d, D = 1.702, d0 = NULL, index = NULL,
intervals = seq(-3, 3, 0.5))
Arguments
u |
the observed response matrix, 2d matrix |
a |
discrimination parameters, 1d vector (fixed value) or NA (freely estimate) |
b |
difficulty parameters, 1d vector (fixed value) or NA (freely estimate) |
d |
category parameters, 2d matrix (fixed value) or NA (freely estimate) |
D |
the scaling constant, 1.702 by default |
priors |
a list of prior distributions |
bounds_t |
bounds of ability parameters |
iter |
the maximum iterations |
conv |
the convergence criterion of the -2 log-likelihood |
t |
ability parameters, 1d vector (fixed value) or NA (freely estimate) |
u_ix |
the 3d indices |
dvp |
the derivatives of P |
nr_iter |
the maximum iterations of newton-raphson |
scale |
the scale of theta parameters |
bounds_a |
bounds of discrimination parameters |
bounds_b |
bounds of location parameters |
bounds_d |
bounds of category parameters |
decay |
decay rate |
verbose |
TRUE to print debuggin information |
true_params |
a list of true parameters for evaluating the estimation accuracy |
quad_degree |
the number of quadrature points |
score_fn |
the scoring method: 'eap' or 'map' |
d0 |
insert an initial category value |
index |
the indices of items being plotted |
intervals |
intervals on the x-axis |
Value
model_gpcm_eap returns theta estimates and standard errors in a list
model_gpcm_map returns theta estimates in a list
model_gpcm_jmle returns estimated t, a, b, d parameters in a list
model_gpcm_mmle returns estimated t, a, b, d parameters in a list
model_gpcm_fitplot returns a ggplot object
Examples
with(model_gpcm_gendata(10, 40, 3),
cbind(true=t, est=model_gpcm_eap(u, a, b, d)$t))
with(model_gpcm_gendata(10, 40, 3),
cbind(true=t, est=model_gpcm_map(u, a, b, d)$t))
# generate data
x <- model_gpcm_gendata(1000, 40, 3)
# free calibration, 40 iterations
y <- model_gpcm_jmle(x$u, true_params=x, iter=40, verbose=TRUE)
# generate data
x <- model_gpcm_gendata(1000, 40, 3)
# free estimation, 40 iterations
y <- model_gpcm_mmle(x$u, true_params=x, iter=40, verbose=TRUE)
with(model_gpcm_gendata(1000, 20, 3),
model_gpcm_fitplot(u, t, a, b, d, index=c(1, 3, 5)))
Estimation of the Graded Response Model
Description
Estimate the GRM using the joint or marginal maximum likelihood estimation
model_grm_eap scores response vectors using the EAP method
model_grm_map scores response vectors using the MAP method
model_grm_jmle estimates the parameters using the
joint maximum likelihood estimation (JMLE) method
model_grm_mmle estimates the parameters using the
marginal maximum likelihood estimation (MMLE) method
Usage
model_grm_eap(u, a, b, D = 1.702, priors = c(0, 1), bounds_t = c(-4,
4))
model_grm_map(u, a, b, D = 1.702, priors = c(0, 1), bounds_t = c(-4,
4), iter = 30, conv = 0.001)
model_grm_dv_Pt(t, a, b, D)
model_grm_dv_Pa(t, a, b, D)
model_grm_dv_Pb(t, a, b, D)
model_grm_dv_jmle(u_ix, dvp)
model_grm_jmle(u, t = NA, a = NA, b = NA, D = 1.702, iter = 100,
nr_iter = 10, conv = 0.001, scale = c(0, 1), bounds_t = c(-4, 4),
bounds_a = c(0.01, 2.5), bounds_b = c(-4, 4), priors = list(t =
c(0, 1)), decay = 1, verbose = FALSE, true_params = NULL)
model_grm_dv_mmle(u_ix, quad, pdv)
model_grm_mmle(u, t = NA, a = NA, b = NA, d = NA, D = 1.702,
iter = 100, nr_iter = 10, conv = 0.001, bounds_t = c(-4, 4),
bounds_a = c(0.01, 2.5), bounds_b = c(-4, 4), priors = list(t =
c(0, 1)), decay = 1, quad_degree = "11", score_fn = c("eap",
"map"), verbose = FALSE, true_params = NULL)
model_grm_fitplot(u, t, a, b, D = 1.702, index = NULL,
intervals = seq(-3, 3, 0.5))
Arguments
u |
the observed response matrix, 2d matrix |
a |
discrimination parameters, 1d vector (fixed value) or NA (freely estimate) |
b |
difficulty parameters, 2d matrix (fixed value) or NA (freely estimate) |
D |
the scaling constant, 1.702 by default |
priors |
a list of prior distributions |
bounds_t |
bounds of ability parameters |
iter |
the maximum iterations |
conv |
the convergence criterion for the -2 log-likelihood |
t |
ability parameters, 1d vector (fixed value) or NA (freely estimate) |
u_ix |
the 3d indices |
dvp |
the derivatives of P |
nr_iter |
the maximum newton-raphson iterations, default=10 |
scale |
the scale of theta parameters |
bounds_a |
bounds of discrimination parameters |
bounds_b |
bounds of location parameters |
decay |
decay rate |
verbose |
TRUE to print debuggin information |
true_params |
a list of true parameters for evaluating the estimation accuracy |
quad_degree |
the number of quadrature points |
score_fn |
the scoring method: 'eap' or 'map' |
index |
the indices of items being plotted |
intervals |
intervals on the x-axis |
Value
model_grm_eap returns theta estimates and standard errors in a list
model_grm_map returns theta estimates in a list
model_grm_jmle returns estimated t, a, b parameters in a list
model_grm_mmle returns estimated t, a, b parameters in a list
model_grm_fitplot returns a ggplot object
Examples
with(model_grm_gendata(10, 50, 3),
cbind(true=t, est=model_grm_eap(u, a, b)$t))
with(model_grm_gendata(10, 50, 3),
cbind(true=t, est=model_grm_map(u, a, b)$t))
# generate data
x <- model_grm_gendata(1000, 40, 3)
# free calibration, 40 iterations
y <- model_grm_jmle(x$u, true_params=x, iter=40, verbose=TRUE)
# generate data
x <- model_grm_gendata(1000, 40, 3)
# free estimation, 40 iterations
y <- model_grm_mmle(x$u, true_params=x, iter=40, verbose=TRUE)
with(model_grm_gendata(1000, 20, 3),
model_grm_fitplot(u, t, a, b, index=c(1, 3, 5)))
Estimation of the Mixed Format Model
Description
Estimate the mixed format model
Usage
model_mixed_eap(u, items, D = 1.702, priors = c(0, 1),
bounds_t = c(-4, 4))
model_mixed_map(u, items, D = 1.702, priors = c(0, 1),
bounds_t = c(-4, 4), iter = 30, conv = 0.001)
Arguments
u |
the response data, 2d marix |
items |
a list of 3pl, gpcm, grm items |
D |
the scaling constant |
priors |
the prior distribution |
bounds_t |
the lower- and upper-bound of the parameter |
iter |
the maximum number of newton-raphson iterations |
conv |
the convergence criterion |
Value
model_mixed_eap returns a list of point estimates and
standard error of the ability parameters
model_mixed_map returns a list of point estimates of the ability parameters
Examples
x <- model_mixed_gendata(200, 30, 5, 5, 3)
y <- model_mixed_eap(x$u, x$items)
c('corr'=cor(x$t, y$t), 'rmse'=rmse(x$t, y$t))
x <- model_mixed_gendata(200, 30, 5, 5, 3)
y <- model_mixed_map(x$u, x$items)
c('corr'=cor(x$t, y$t), 'rmse'=rmse(x$t, y$t))
Helper Functions
Description
model_polytomous_3dindex creates indices extracting 3D stats
model_polytomous_3dresponse converts responses from 2D to 3D
hermite_gauss stores pre-computed hermite gaussian
quadratures points and weights
nr_iteration updates the parameters using the Newton-Raphson method
Usage
model_polytomous_3dindex(u)
model_polytomous_3dresponse(u)
hermite_gauss(degree = c("20", "11", "7"))
nr_iteration(param, free, dv, h_max, lr, bound)
estimate_3pl_debug(tracking, k)
estimate_3pl_eval(true_params, t, a, b, c, t_free, a_free, b_free, c_free)
estimate_gpcm_debug(tracking, k)
estimate_gpcm_eval(true_params, n_c, t, a, b, d, t_free, a_free, b_free,
d_free)
estimate_grm_debug(tracking, k)
estimate_grm_eval(true_params, n_c, t, a, b, t_free, a_free, b_free)
Arguments
u |
the observed response, 2d matrix, values start from 0 |
degree |
the degree of hermite-gauss quadrature: '20', '11', '7' |
param |
the parameter being estimated |
free |
TRUE to free parameters, otherwise fix parameters |
dv |
the first and second derivatives |
h_max |
the maximum value of h |
lr |
the learning rate |
bound |
the lower and upper bounds of the parameter |
tracking |
estimation tracking information |
k |
the number of iterations in estimation |
true_params |
a list of true parameters |
t |
estimated ability parameters |
a |
estimated discrimination parameters |
b |
estimated difficulty parameters |
c |
estimated guessing parameters |
t_free |
TRUE to estimate ability parameters, otherwise fix |
a_free |
TRUE to estimate discrimination parameters, otherwise fix |
b_free |
TRUE to estimate difficulty parameters, otherwise fix |
c_free |
TRUE to estimate guessing parameters, otherwise fix |
3-parameter-logistic model
Description
Common computations and operations for the 3PL model
Usage
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, scale = 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), t_bounds = c(-3, 3), a_bounds = c(0.01,
2.5), b_bounds = c(-3, 3), c_bounds = c(0, 0.5), 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),
verbose = FALSE)
Arguments
t |
ability parameters, 1d vector |
a |
discrimination parameters, 1d vector |
b |
difficulty parameters, 1d vector |
c |
guessing parameters, 1d vector |
D |
the scaling constant, default=1.702 |
u |
observed responses, 2d matrix |
log |
True to return log-likelihood |
scale |
the scale, 't' for theta or 'b' for b-parameters |
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 |
t_bounds |
bounds of the ability parameters |
a_bounds |
bounds of the discrimination parameters |
b_bounds |
bounds of the difficulty parameters |
c_bounds |
bounds of the guessing parameters |
missing |
the proportion or number of missing responses |
... |
additional arguments |
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 |
verbose |
TRUE to print rough maximum likelihood estimates |
Value
model_3pl_prob returns the resulting probabilities in a matrix
model_3pl_info returns the resulting information in a matrix
model_3pl_lh returns the resulting likelihood in a matrix
model_3pl_rescale returns t, a, b, c parameters on the new scale
model_3pl_gendata returns the generated response matrix and parameters in a list
model_3pl_plot returns a ggplot object
model_3pl_plot_loglh returns a ggplot object
Examples
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))
Generalized Partial Credit Model
Description
Common computations and operatoins for the GPCM
Usage
model_gpcm_prob(t, a, b, d, D = 1.702, d0 = NULL)
model_gpcm_info(t, a, b, d, D = 1.702, d0 = NULL)
model_gpcm_lh(u, t, a, b, d, D = 1.702, 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), d_dist = c(0, 1),
t_bounds = c(-3, 3), a_bounds = c(0.01, 2.5), b_bounds = c(-3, 3),
d_bounds = c(-3, 3), missing = NULL, ...)
model_gpcm_rescale(t, a, b, d, scale = c("t", "b"), mean = 0, sd = 1)
model_gpcm_plot(a, b, d, D = 1.702, d0 = NULL, type = c("prob",
"info"), item_level = FALSE, total = FALSE, xaxis = seq(-6, 6,
0.1))
model_gpcm_plot_loglh(u, a, b, d, D = 1.702, d0 = NULL,
xaxis = seq(-6, 6, 0.1), verbose = FALSE)
Arguments
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, default=1.702 |
d0 |
insert an initial category value |
u |
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 |
d_dist |
parameters of the normal distribution used to generate d-parameters |
t_bounds |
the bounds of the ability parameters |
a_bounds |
the bounds of the discrimination parameters |
b_bounds |
the bounds of the difficulty parameters |
d_bounds |
the bounds of the category parameters |
missing |
the proportion or number of missing responses |
... |
additional arguments |
scale |
the scale, 't' for theta or 'b' for b-parameters |
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 |
item_level |
TRUE to add item level data |
total |
TRUE to sum values over items |
xaxis |
the values of x-axis |
verbose |
TRUE to print rough maximum likelihood values |
Details
Use NA to represent unused category.
Value
model_gpcm_prob returns the resulting probabilities in a 3d array
model_gpcm_info returns the resulting information in a 3d array
model_gpcm_lh returns the resulting likelihood in a matrix
model_gpcm_gendata returns the generated response matrix and parameters
model_gpcm_rescale returns t, a, b, d parameters on the new scale
model_gpcm_plot returns a ggplot object
model_gpcm_plot_loglh returns a ggplot object
Examples
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, 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, 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, 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', item_level=TRUE)
with(model_gpcm_gendata(5, 50, 3), model_gpcm_plot_loglh(u, a, b, d))
Graded Response Model
Description
Common computations and operations for the GRM
Usage
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), t_bounds = c(-3, 3), a_bounds = c(0.01, 2.5),
b_bounds = c(-3, 3), missing = NULL, ...)
model_grm_rescale(t, a, b, scale = c("t", "b"), mean = 0, sd = 1)
model_grm_plot(a, b, D = 1.702, type = c("prob", "info"),
item_level = 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),
verbose = FALSE)
Arguments
t |
ability parameters, 1d vector |
a |
discrimination parameters, 1d vector |
b |
item location parameters, 2d matrix |
D |
the scaling constant, default=1.702 |
raw |
TRUE to return P* |
u |
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 |
t_bounds |
the bounds of the ability parameters |
a_bounds |
the bounds of the discrimination parameters |
b_bounds |
the bounds of the difficulty parameters |
missing |
the proportion or number of missing responses |
... |
additional arguments |
scale |
the scale, 't' for theta or 'b' for b-parameters |
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 |
item_level |
TRUE to combine categories |
total |
TRUE to sum values over items |
xaxis |
the values of x-axis |
verbose |
TRUE to print rough maximum likelihood values |
Value
model_grm_prob returns the resulting probabilities in a 3d array
model_grm_info returns the resulting information in a 3d array
model_grm_lh returns the resulting likelihood in a matrix
model_grm_gendata returns the generated response data and parameters in a list
model_grm_rescale returns t, a, b parameters on the new scale
model_grm_plot returns a ggplot object
model_grm_plot_loglh returns a ggplot object
Examples
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', item_level=TRUE))
with(model_grm_gendata(5, 50, 3), model_grm_plot_loglh(u, a, b))
Mixed-format model
Description
Common computations and operations for the mixed format model
Usage
model_mixed_gendata(n_p, n_3pl = 0, n_gpcm = 0, n_grm = 0, n_c, ...)
model_mixed_prob(t, items, D = 1.702)
model_mixed_info(t, items, D = 1.702, combine = TRUE)
model_mixed_lh(u, t, items, D = 1.702, log = FALSE, combine = TRUE)
Arguments
n_p |
the number of test takers |
n_3pl |
the number of 3pl items |
n_gpcm |
the number of gpcm items |
n_grm |
the number of grm items |
n_c |
the number of score categories for polytomous items |
... |
additional arguments |
t |
ability parameters, a vector |
items |
a list of '3pl', 'gpcm', and 'grm' items |
D |
the scaling constant, default=1.702 |
combine |
|
u |
the response data, a 2d matrix |
log |
|
Value
model_mixed_gendata returns a list of generated responses, ability paramters and items
model_mixed_prob returns a list of probabilities for '3pl', 'gpcm', and 'grm' items
model_mixed_info returns a list or matrix of information
Examples
# generate 10 3pl items, 5 gpcm items and 5 grm items
model_mixed_gendata(10, n_3pl=10, n_gpcm=5, n_grm=5, n_c=3)
# generate 5 gpcm items and 5 grm items, 4 score categories each
model_mixed_gendata(10, n_gpcm=5, n_grm=5, n_c=4)
# generate 5 people and 4 items of each type
with(model_mixed_gendata(n_p=5, n_3pl=4, n_gpcm=4, n_grm=4, n_c=3),
model_mixed_prob(t, items))
# generate 10 people and 5 gpcm and 5 grm items
with(model_mixed_gendata(n_p=10, n_gpcm=4, n_grm=4, n_c=3),
model_mixed_prob(t, items))
with(model_mixed_gendata(10, 4, 4, 4, 3), model_mixed_info(t, items))
with(model_mixed_gendata(10, 0, 4, 4, 3), model_mixed_info(t, items))
with(model_mixed_gendata(10, 4, 4, 4, 3), model_mixed_lh(u, t, items))
Utility functions
Description
rmse computes the root mean squared error (RMSE)
of two numeric vectors/matrices
freq computes the frequency counts of
a numeric or character vector
cronbach_alpha computes the Cronbach's alpha
internal consistency reliability index
spearman_brown predicts the reliability when the
current test is extended to n times longer
spearman_brown_reverse computes how many times
the current test length needs to be extended in order to reach targeted
reliability
quadratic kappa computes the quadratic weighted kappa
of two numeric vectors
Usage
rmse(x1, x2)
freq(x, vals = NULL, decimal = NULL)
cronbach_alpha(u)
spearman_brown(rho, n_len)
spearman_brown_reverse(rho, target_rho)
quadratic_kappa(x1, x2)
Arguments
x1 |
a numeric vector or matrix |
x2 |
a numeric vector or matrix |
x |
a numeric or character vector |
vals |
valid values, |
decimal |
round results to n-th decimal places |
u |
oberved responses, 2d matrix |
rho |
the reliability of the current test |
n_len |
extend the test to n times longer |
target_rho |
the targeted reliability |
Value
freq returns the frequency counts and percentages in a data.frame
Examples
rmse(rnorm(10), rnorm(10))
freq(round(runif(100, 1, 5)))
cronbach_alpha(model_3pl_gendata(1000, 20)$u)
spearman_brown(.70, 2)
spearman_brown_reverse(.70, .85)
quadratic_kappa(round(runif(100, 1, 5)), round(runif(100, 1, 5)))