Automated Test Assembly (ATA)

Description

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

Usage

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, ...)

Arguments

Details

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

Examples

## 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)

Helper functions of ATA

Description

miscellaneous helper functions of ATA

ata_append_constraints appends constraint definitions to the model

ata_form_index converts input forms into actual form indices in the model

ata_obj_coef processes input coefficients of the objective functions

ata_solve_lpsolve solves the the MIP model using lp_solve

ata_solve_glpk solves the the MIP model using GLPK

Usage

ata_append_constraints(x, mat, dir, rhs)

ata_form_index(x, forms, collapse, internal_index)

ata_obj_coef(x, coef, compensate)

ata_solve_lpsolve(x, time_limit, message, ...)

ata_solve_glpk(x, time_limit, message, ...)

Arguments

Simulation of Computerized Adaptive Testing (CAT)

Description

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)

Usage

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)

Arguments

Details

... 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.

Value

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

Examples

## 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's alpha

Description

cronbach_alpha computes Cronbach's alpha internal consistency reliability

Usage

cronbach_alpha(responses)

Arguments

Examples

cronbach_alpha(model_3pl_gendata(1000, 20)$u)

Estimate 3-parameter-logistic model

Description

Estimate the 3PL model using the maximum likelihood estimation

model_3pl_eap_scoring scores response vectors using the EAP method

model_3pl_map_scoring scores response vectors using the MAP method

model_3pl_dv_jmle calculates the first and second derivatives for the joint maximum likelihood estimation

model_3pl_estimate_jmle estimates the parameters using the joint maximum likelihood estimation (JMLE) method

model_3pl_dv_mmle calculates the first and second derivatives for the marginal maximum likelihood estimation

model_3pl_estimate_mmle estimates the parameters using the marginal maximum likelihood estimation (MMLE) method

Usage

model_3pl_eap_scoring(u, a, b, c, D = 1.702, prior = c(0, 1),
  bound = c(-3, 3))

model_3pl_map_scoring(u, a, b, c, D = 1.702, prior = c(0, 1),
  bound = c(-3, 3), nr_iter = 30, nr_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(dv, u)

model_3pl_estimate_jmle(u, t = NA, a = NA, b = NA, c = NA,
  D = 1.702, iter = 100, conv = 1, nr_iter = 10, nr_conv = 0.001,
  scale = c(0, 1), bounds_t = c(-3, 3), bounds_a = c(0.01, 2),
  bounds_b = c(-3, 3), bounds_c = c(0, 0.25), priors = list(t = c(0,
  1), a = c(-0.1, 0.2), b = c(0, 1), c = c(4, 20)), decay = 1,
  debug = FALSE, true_params = NULL)

model_3pl_dv_mmle(pdv_fn, u, quad, a, b, c, D)

model_3pl_estimate_mmle(u, t = NA, a = NA, b = NA, c = NA,
  D = 1.702, iter = 100, conv = 1, nr_iter = 10, nr_conv = 0.001,
  bounds_t = c(-3, 3), bounds_a = c(0.01, 2), bounds_b = c(-3, 3),
  bounds_c = c(0, 0.25), priors = list(t = c(0, 1), a = c(-0.1, 0.2), b
  = c(0, 1), c = c(4, 20)), decay = 1, quad_degree = "11",
  scoring = c("eap", "map"), debug = FALSE, true_params = NULL)

model_3pl_fitplot(u, t, a, b, c, D = 1.702, index = NULL,
  intervals = seq(-3, 3, 0.5), show_points = TRUE)

Arguments

Examples

with(model_3pl_gendata(10, 40), cbind(true=t, est=model_3pl_eap_scoring(u, a, b, c)$t))
with(model_3pl_gendata(10, 40), cbind(true=t, est=model_3pl_map_scoring(u, a, b, c)$t))
## Not run: 
# generate data
x <- model_3pl_gendata(2000, 40)
# free estimation
y <- model_3pl_estimate_jmle(x$u, true_params=x)
# fix c-parameters
y <- model_3pl_estimate_jmle(x$u, c=0, true_params=x)
# no priors
y <- model_3pl_estimate_jmle(x$u, priors=NULL, iter=30, debug=T)

## End(Not run)
## Not run: 
# generate data
x <- model_3pl_gendata(2000, 40)
# free estimation
y <- model_3pl_estimate_mmle(x$u, true_params=x)
# fix c-parameters
y <- model_3pl_estimate_mmle(x$u, c=0, true_params=x)
# no priors
y <- model_3pl_estimate_mmle(x$u, priors=NULL, iter=30, debug=T)

## End(Not run)
with(model_3pl_gendata(1000, 20), model_3pl_fitplot(u, t, a, b, c, index=c(1, 3, 5)))

Estimate Generalizaed Partial Credit Model

Description

Estimate the GPCM using the maximum likelihood estimation

model_gpcm_eap_scoring scores response vectors using the EAP method

model_gpcm_map_scoring scores response vectors using maximum a posteriori

model_gpcm_estimate_jmle estimates the parameters using the joint maximum likelihood estimation (JMLE) method

model_gpcm_estimate_mmle estimates the parameters using the marginal maximum likelihood estimation (MMLE) method

Usage

model_gpcm_eap_scoring(u, a, b, d, D = 1.702, prior = c(0, 1),
  bound = c(-3, 3))

model_gpcm_map_scoring(u, a, b, d, D = 1.702, prior = NULL,
  bound = c(-3, 3), nr_iter = 30, nr_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(ix, dvp)

model_gpcm_estimate_jmle(u, t = NA, a = NA, b = NA, d = NA,
  D = 1.702, iter = 100, nr_iter = 10, conv = 1, nr_conv = 0.001,
  scale = c(0, 1), bounds_t = c(-4, 4), bounds_a = c(0.01, 2),
  bounds_b = c(-4, 4), bounds_d = c(-4, 4), priors = list(t = c(0,
  1), a = c(-0.1, 0.2), b = c(0, 1), d = c(0, 1)), decay = 1,
  debug = FALSE, true_params = NULL)

model_gpcm_dv_mmle(u_ix, quad, pdv)

model_gpcm_estimate_mmle(u, t = NA, a = NA, b = NA, d = NA,
  D = 1.702, iter = 100, nr_iter = 10, conv = 1, nr_conv = 0.001,
  bounds_t = c(-4, 4), bounds_a = c(0.01, 2), bounds_b = c(-4, 4),
  bounds_d = c(-4, 4), priors = list(t = c(0, 1), a = c(-0.1, 0.2), b =
  c(0, 1), d = c(0, 1)), decay = 1, quad_degree = "11",
  scoring = c("eap", "map"), debug = FALSE, true_params = NULL)

model_gpcm_fitplot(u, t, a, b, d, D = 1.702, insert_d0 = NULL,
  index = NULL, intervals = seq(-3, 3, 0.5), show_points = TRUE)

Arguments

Examples

with(model_gpcm_gendata(10, 40, 3), cbind(true=t, est=model_gpcm_eap_scoring(u, a, b, d)$t))
with(model_gpcm_gendata(10, 40, 3), cbind(true=t, est=model_gpcm_map_scoring(u, a, b, d)$t))
## Not run: 
# generate data
x <- model_gpcm_gendata(1000, 40, 3)
# free calibration
y <- model_gpcm_estimate_jmle(x$u, true_params=x)
# no priors
y <- model_gpcm_estimate_jmle(x$u, priors=NULL, true_params=x)

## End(Not run)
## Not run: 
# generate data
x <- model_gpcm_gendata(1000, 40, 3)
# free estimation
y <- model_gpcm_estimate_mmle(x$u, true_params=x)
# no priors
y <- model_gpcm_estimate_mmle(x$u, priors=NULL, true_params=x)

## End(Not run)
with(model_gpcm_gendata(1000, 20, 3), model_gpcm_fitplot(u, t, a, b, d, index=c(1, 3, 5)))

Estimate Graded Response Model

Description

Estimate the GRM using the maximum likelihood estimation

model_grm_eap_scoring scores response vectors using the EAP method

model_grm_map_scoring scores response vectors using the MAP method

model_grm_estimate_jmle estimates the parameters using the joint maximum likelihood estimation (JMLE) method

model_grm_estimate_mmle estimates the parameters using the marginal maximum likelihood estimation (MMLE) method

Usage

model_grm_eap_scoring(u, a, b, D = 1.702, prior = c(0, 1),
  bound = c(-3, 3))

model_grm_map_scoring(u, a, b, D = 1.702, prior = NULL, bound = c(-3,
  3), nr_iter = 30, nr_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(ix, dvp)

model_grm_estimate_jmle(u, t = NA, a = NA, b = NA, D = 1.702,
  iter = 100, nr_iter = 10, conv = 1, nr_conv = 0.001,
  scale = c(0, 1), bounds_t = c(-4, 4), bounds_a = c(0.01, 2),
  bounds_b = c(-4, 4), priors = list(t = c(0, 1), a = c(-0.1, 0.2), b =
  c(0, 1)), decay = 1, debug = FALSE, true_params = NULL)

model_grm_dv_mmle(u_ix, quad, pdv)

model_grm_estimate_mmle(u, t = NA, a = NA, b = NA, d = NA,
  D = 1.702, iter = 100, nr_iter = 10, conv = 1, nr_conv = 0.001,
  bounds_t = c(-4, 4), bounds_a = c(0.01, 2), bounds_b = c(-4, 4),
  bounds_d = c(-4, 4), priors = list(t = c(0, 1), a = c(-0.1, 0.2), b =
  c(0, 1)), decay = 1, quad_degree = "11", scoring = c("eap", "map"),
  debug = FALSE, true_params = NULL)

model_grm_fitplot(u, t, a, b, D = 1.702, index = NULL,
  intervals = seq(-3, 3, 0.5), show_points = TRUE)

Arguments

Examples

with(model_grm_gendata(10, 50, 3), cbind(true=t, est=model_grm_eap_scoring(u, a, b)$t))
with(model_grm_gendata(10, 50, 3), cbind(true=t, est=model_grm_map_scoring(u, a, b)$t))
## Not run: 
# 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)

## End(Not run)
## Not run: 
# 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)

## End(Not run)
with(model_grm_gendata(1000, 20, 3), model_grm_fitplot(u, t, a, b, index=c(1, 3, 5)))

Helper functions of Model Estimation

Description

miscellaneous helper functions for estimating IRT models

estimate_nr_iteration updates the parameters using the newton-raphson method

Usage

estimate_nr_iteration(param, free, dv, h_max, lr, bound)

model_polytomous_3dindex(u)

model_polytomous_3dresponse(u)

Arguments

#' Distribution of Expected Raw Scores

Description

Calculate the distribution of expected raw scores

Usage

expected_raw_score_dist(t, a, b, c)

Arguments

Frequency Counts

Description

Frequency counts of a vector

Usage

freq(x, values = NULL, rounding = NULL)

Arguments

Hermite-Gauss Quadrature

Description

Pre-computed hermite gaussian quadratures points and weights

Usage

hermite_gauss(degree = c("20", "11", "7"))

Arguments

3-parameter-logistic model

Description

Routine functions 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, 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)

Arguments

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, show_mle=TRUE))

Generalized Partial Credit Model

Description

Routine functions for the GPCM

Usage

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)

Arguments

Details

Use NA to represent unused category.

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, 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))

Graded Response Model

Description

Routine functions 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), 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)

Arguments

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', by_item=TRUE))
with(model_grm_gendata(5, 50, 3), model_grm_plot_loglh(u, a, b))

Computerized Multistage Testing (MST)

Description

mst creates a multistage (MST) object for assembly

mst_route adds/removes a route to/from the MST

mst_get_indices maps the input indices to the actual indices

mst_obj adds objective functions to the MST

mst_constraint adds constraints to the MST

mst_stage_length sets length limits on stages

mst_rdp anchors the routing decision point (rdp) between adjacent modules

mst_module_mininfo sets the minimum information for modules

mst_assemble assembles the mst

mst_get_items extracts items from the assembly results

Usage

mst(pool, design, num_panel, method = c("topdown", "bottomup"),
  len = NULL, max_use = NULL, group = NULL, ...)

mst_route(x, route, op = c("+", "-"))

mst_get_indices(x, indices)

mst_obj(x, theta, indices = NULL, target = NULL, ...)

mst_constraint(x, coef, min = NA, max = NA, level = NULL,
  indices = NULL)

mst_stage_length(x, stages, min = NA, max = NA)

mst_rdp(x, theta, indices, tol = 0)

mst_module_info(x, thetas, min, max, indices)

mst_assemble(x, ...)

## S3 method for class 'mst'
print(x, ...)

## S3 method for class 'mst'
plot(x, ...)

mst_get_items(x, panel_ix = NULL, stage_ix = NULL, module_ix = NULL,
  route_ix = NULL)

Arguments

Details

There are two methods for designing a MST. The bottom-up approach adds objectives and constraints on individual modules, whereas the topdown approach adds objectives and constraints directly on routes.

plot.mst draws module information functions when byroute=FALSE and route information functions when byroute=TRUE

Examples

## Not run: 
## generate item pool
num_item <- 300
pool <- with(model_3pl_gendata(1, num_item), data.frame(a=a, b=b, c=c))
pool$id <- 1:num_item
pool$content <- sample(1:3, num_item, replace=TRUE)
pool$time <- round(rlnorm(num_item, 4, .3))
pool$group <- sort(sample(1:round(num_item/3), num_item, replace=TRUE))

## ex. 1: 1-2-2 MST, 2 panels, topdown
## 20 items in total and 10 items in content area 1 in each route
## maximize info. at -1 and 1 for easy and hard routes
x <- mst(pool, "1-2-2", 2, 'topdown', len=20, max_use=1)
x <- mst_obj(x, theta=-1, indices=1:2)
x <- mst_obj(x, theta=1, indices=3:4)
x <- mst_constraint(x, "content", 10, 10, level=1)
x <- mst_assemble(x, timeout=5)
plot(x, byroute=TRUE)
for(p in 1:x$num_panel)
  for(r in 1:x$num_route) {
     route <- paste(x$route[r, 1:x$num_stage], collapse='-')
     count <- sum(mst_get_items(x, panel_ix=p, route_ix=r)$content==1)
     cat('panel=', p, ', route=', route, ': ', count, ' items in content area #1\n', sep='')
  }

## ex. 2: 1-2-3 MST, 2 panels, bottomup,
## remove two routes with large theta change: 1-2-6, 1-3-4 
## 10 items in total and 4 items in content area 2 in each module
## maximize info. at -1, 0 and 1 for easy, medium, and hard modules
x <- mst(pool, "1-2-3", 2, 'bottomup', len=10, max_use=1)
x <- mst_route(x, c(1, 2, 6), "-")
x <- mst_route(x, c(1, 3, 4), "-")
x <- mst_obj(x, theta= 0, indices=c(1, 5))
x <- mst_obj(x, theta=-1, indices=c(2, 4))
x <- mst_obj(x, theta= 1, indices=c(3, 6))
x <- mst_constraint(x, "content", 4, 4, level=2)
x <- mst_assemble(x, timeout=10) 
plot(x, byroute=FALSE)
for(p in 1:x$num_panel)
  for(m in 1:x$num_module){
    count <- sum(mst_get_items(x, panel_ix=p, module_ix=m)$content==2)
    cat('panel=', p, ', module=', m, ': ', count, ' items in content area #2\n', sep='')
  }
 
## ex.3: same with ex.2 (w/o content constraints), but group-based  
x <- mst(pool, "1-2-3", 2, 'bottomup', len=12, max_use=1, group="group")
x <- mst_route(x, c(1, 2, 6), "-")
x <- mst_route(x, c(1, 3, 4), "-")
x <- mst_obj(x, theta= 0, indices=c(1, 5))
x <- mst_obj(x, theta=-1, indices=c(2, 4))
x <- mst_obj(x, theta= 1, indices=c(3, 6))
x <- mst_assemble(x, timeout=10)
plot(x, byroute=FALSE)
for(p in 1:x$num_panel)
  for(m in 1:x$num_module){
    items <- mst_get_items(x, panel_ix=p, module_ix=m)
    cat('panel=', p, ', module=', m, ': ', length(unique(items$id)), ' items from ', 
        length(unique(items$group)), ' groups\n', sep='')
  }
  
## ex.4: 2 panels of 1-2-3 top-down design
## 20 total items and 10 items in content area 3
## 6+ items in stage 1 & 2
x <- mst(pool, "1-2-3", 2, "topdown", len=20, max_use=1)
x <- mst_route(x, c(1, 2, 6), "-")
x <- mst_route(x, c(1, 3, 4), "-")
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", 10, 10, level=3)
x <- mst_stage_length(x, 1:2, min=6)
x <- mst_assemble(x, timeout=15)
head(x$items)
plot(x, byroute=FALSE)
for(p in 1:x$num_panel)
  for(s in 1:x$num_stage){
    items <- mst_get_items(x, panel_ix=p, stage_ix=s)
    cat('panel=', p, ', stage=', s, ': ', length(unique(items$id)), ' items\n', sep='')
  }

## ex.5: same with ex.4, but use RDP instead of stage length to control routing errors
x <- mst(pool, "1-2-3", 2, "topdown", len=20, max_use=1)
x <- mst_route(x, c(1, 2, 6), "-")
x <- mst_route(x, c(1, 3, 4), "-")
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", 10, 10, level=3)
x <- mst_rdp(x, 0, 2:3, .1)
x <- mst_module_mininfo(x, 0, 5, 2:3)
x <- mst_assemble(x, timeout=15)
plot(x, byroute=FALSE)

## End(Not run)

Simulation of Multistage Testing

Description

mst_sim simulates a MST administration

Usage

mst_sim(x, true, rdp = NULL, ...)

## S3 method for class 'mst_sim'
print(x, ...)

## S3 method for class 'mst_sim'
plot(x, ...)

Arguments

Examples

## 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

Description

Root mean squared error (RMSE) of two numeric vectors/matrices

Usage

rmse(x, y)

Arguments

Spearman Brown Prophecy

Description

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

Usage

spearman_brown(n, rho)

spearman_brown_reverse(rho, target)

Arguments

Examples

spearman_brown(2, .70)
spearman_brown_reverse(.70, .85)