est_ability {irt}

Estimate Examinee Ability

Description

This function estimates examinee ability using different methods, including Owen's Bayesian estimation, Maximum Likelihood estimation, Maximum-a-Posteriori and Expected-a-Posteriori.

Usage

est_ability(
  resp,
  ip = NULL,
  method = c("eap", "ml", "map", "bm", "owen", "sum_score"),
  ...,
  prior_dist = c("norm", "unif", "lnorm", "gamma", "t", "cauchy"),
  prior_pars = c(0, 1),
  theta_range = c(-5, 5),
  number_of_quads = 41,
  tol = 1e-06,
  output_type = c("list", "data.frame", "tibble")
)

Arguments

resp

A Response_set-class, matrix or a data.frame object holding responses. Missing responses are excluded from the ability estimation.

ip

An Item-class, Itempool-class or a Testlet-class object. If ip is not an Itempool-class object, the function attempts to convert it. While the default is NULL, this argument is required for all methods except when method = "sum_score".

method

The method used for ability estimation. The default is "eap".

Available methods:

'sum_score'

Basic sum (raw) score of responses.

'owen'

Owen's Bayesian Ability Estimation.

This method is suitable for dichotomous IRT models (e.g., 'Rasch', '1PL', '2PL', '3PL' and '4PL'). Testlet groupings are ignored and items within testlets are treated as standalone items.

Formulas were implemented in Owen (1975) and Vale (1977). The original formulation does not include the D parameter. If D = 1, the original solution is obtained. If D = 1.7, the a parameter is multiplied by this number.

The user needs to provide prior parameters, i.e., prior_pars. These should be a numeric vector of length two, with the first component as the prior mean and the second as the prior standard deviation (not variance). For example, if the prior mean is 0.1 and the prior standard deviation is 2, set the prior parameters as prior_pars = c(0.1, 2).

'ml'

Maximum Likelihood Ability Estimation via Newton-Raphson Algorithm.

'eap'

Expected-a-Posteriori Ability Estimation. Prior information must be provided for this function. The number of quadrature points can also be specified using the argument number_of_quads.

'map' or 'bm'

Maximum-a-Posteriori Ability Estimation (or Bayes Modal estimation). Prior information must be provided for this function. Currently, only 'norm' prior distribution is available.

...

Additional arguments passed to specific methods.

prior_dist

The shape of the prior distribution. Available options are:

'norm'

Normal distribution

'unif'

Uniform distribution

't'

t distribution

'cauchy'

Cauchy distribution

The default value is 'norm'.

prior_pars

Parameters of the prior distribution. Default value is c(0, 1), where 0 is the mean and 1 is the standard deviation of the default normal prior distribution. For example, uniform prior parameter can be set as c(a, b) where a is the minimum value and b is the maximum value. For t distribution, prior parameter can be set as df to represent the degree of freedom. For Cauchy distribution, prior parameters can be set as c(location, scale).

If method is "owen", provide c(<Prior Mean>, <Prior SD>).

theta_range

The limits of the ability estimation scale. The estimation result will be bounded within this interval. Default is c(-5, 5).

number_of_quads

Number of quadratures. The default value is 41. As this number increases, the precision of the estimate will also increase.

tol

The precision level of ability estimate. The final ability estimates will be rounded to remove precision smaller than the tol value. Default is 1e-06.

output_type

A string specifying the output type of the function. Default is "list". Options include:

"list"

Function returns a list object with elements est and se.

"data.frame"

Function returns a data.frame object with columns examinee_id, est and se.

"tibble"

If the tibble package is available, the function returns a tibble object with columns examinee_id, est and se.

Value

est The estimated examinee abilities. If the response vector for a subject contains all NAs, then est will be NA to differentiate from cases where all answers are incorrect.

se The standard errors of the ability estimates. For "sum_score" method, all standard errors will be NA. For Bayesian methods (like EAP, MAP or Owen's), this value is the square root of the posterior variance.

Author(s)

Emre Gonulates

References

Owen, R. J. (1975). A Bayesian sequential procedure for quantal response in the context of adaptive mental testing. Journal of the American Statistical Association, 70(350), 351-356.

Vale, C. D., & Weiss, D. J. (1977). A Rapid Item-Search Procedure for Bayesian Adaptive Testing. Research Report 77-4. Minneapolis, MN.

Examples

ip <- generate_ip(n = 7)
resp <- sim_resp(ip, theta = rnorm(3))

### EAP estimation ###
est_ability(resp, ip)
est_ability(resp, ip, number_of_quads = 81)
# The default prior_dist is 'norm'. prior_pars = c(mean, sd)
est_ability(resp, ip, prior_pars = c(0, 3))
# prior_pars = c(min, max)
est_ability(resp, ip, prior_dist = 'unif',  prior_pars = c(-3, 3))
# prior_pars = c(df)
est_ability(resp, ip, prior_dist = 't',  prior_pars = 3)
# prior_pars = c(location, scale)
est_ability(resp, ip, prior_dist = 'cauchy',  prior_pars = c(0, 1))


### MAP estimation (Bayes Modal estimation) ###
est_ability(resp, ip, method = "map")
# The default prior_dist is 'norm'. prior_pars = c(mean, sd)
est_ability(resp, ip, method = "map", prior_pars = c(0, 2))


### Maximum Likelihood estimation ###
est_ability(resp, ip, method = 'ml')
est_ability(resp, ip, method = 'ml', tol = 1e-8)
est_ability(resp = rep(1, length(ip)), ip, method = 'ml')
est_ability(resp = rep(1, length(ip)), ip, method = 'ml',
            theta_range = c(-3, 3))

### Owen's Bayesian ability estimation ###
est_ability(resp, ip, method = 'owen')
est_ability(resp, ip, method = 'owen', prior_pars = c(0, 3))