probability {catSurv}

Probability of Responses to a Question Item or the Left-Cumulative Probability of Responses

Description

Calculates the probability of specific responses or the left-cumulative probability of responses to item conditioned on a respondent's ability (\theta).

Usage

probability(catObj, theta, item)

Arguments

catObj

An object of class Cat

theta

A numeric or an integer indicating the value for \theta_j

item

An integer indicating the index of the question item

Details

For the ltm model, the probability of non-zero response for respondent j on item i is

Pr(y_{ij}=1|\theta_j)=\frac{\exp(a_i + b_i \theta_j)}{1+\exp(a_i + b_i \theta_j)}

where \theta_j is respondent j 's position on the latent scale of interest, a_i is item i 's discrimination parameter, and b_i is item i 's difficulty parameter.

For the tpm model, the probability of non-zero response for respondent j on item i is

Pr(y_{ij}=1|\theta_j)=c_i+(1-c_i)\frac{\exp(a_i + b_i \theta_j)}{1+\exp(a_i + b_i \theta_j)}

where \theta_j is respondent j 's position on the latent scale of interest, a_i is item i 's discrimination parameter, b_i is item i 's difficulty parameter, and c_i is item i 's guessing parameter.

For the grm model, the probability of a response in category k or lower for respondent j on item i is

Pr(y_{ij} < k|\theta_j)=\frac{\exp(\alpha_{ik} - \beta_i \theta_{ij})}{1+\exp(\alpha_{ik} - \beta_i \theta_{ij})}

where \theta_j is respondent j 's position on the latent scale of interest, \alpha_ik the k-th element of item i 's difficulty parameter, \beta_i is discrimination parameter vector for item i. Notice the inequality on the left side and the absence of guessing parameters.

For the gpcm model, the probability of a response in category k for respondent j on item i is

Pr(y_{ij} = k|\theta_j)=\frac{\exp(\sum_{t=1}^k \alpha_{i} [\theta_j - (\beta_i - \tau_{it})])} {\sum_{r=1}^{K_i}\exp(\sum_{t=1}^{r} \alpha_{i} [\theta_j - (\beta_i - \tau_{it}) )}

where \theta_j is respondent j 's position on the latent scale of interest, \alpha_i is the discrimination parameter for item i, \beta_i is the difficulty parameter for item i, and \tau_{it} is the category t threshold parameter for item i, with k = 1,...,K_i response options for item i. For identification purposes \tau_{i0} = 0 and \sum_{t=1}^1 \alpha_{i} [\theta_j - (\beta_i - \tau_{it})] = 0. Note that when fitting the model, the \beta_i and \tau_{it} are not distinct, but rather, the difficulty parameters are \beta_{it} = \beta_{i} - \tau_{it}.

Value

When the model slot of the catObj is "ltm", the function probability returns a numeric vector of length one representing the probability of observing a non-zero response.

When the model slot of the catObj is "tpm", the function probability returns a numeric vector of length one representing the probability of observing a non-zero response.

When the model slot of the catObj is "grm", the function probability returns a numeric vector of length k+1, where k is the number of possible responses. The first element will always be zero and the (k+1)th element will always be one. The middle elements are the cumulative probability of observing response k or lower.

When the model slot of the catObj is "gpcm", the function probability returns a numeric vector of length k, where k is the number of possible responses. Each number represents the probability of observing response k.

Note

This function is to allow users to access the internal functions of the package. During item selection, all calculations are done in compiled C++ code.

Author(s)

Haley Acevedo, Ryden Butler, Josh W. Cutler, Matt Malis, Jacob M. Montgomery, Tom Wilkinson, Erin Rossiter, Min Hee Seo, Alex Weil

References

Baker, Frank B. and Seock-Ho Kim. 2004. Item Response Theory: Parameter Estimation Techniques. New York: Marcel Dekker.

Choi, Seung W. and Richard J. Swartz. 2009. “Comparison of CAT Item Selection Criteria for Polytomous Items." Applied Psychological Measurement 33(6):419-440.

Muraki, Eiji. 1992. “A generalized partial credit model: Application of an EM algorithm." ETS Research Report Series 1992(1):1-30.

van der Linden, Wim J. 1998. “Bayesian Item Selection Criteria for Adaptive Testing." Psychometrika 63(2):201-216.

See Also

Cat-class

Examples

## Loading ltm Cat object
## Probability for Cat object of the ltm model
data(ltm_cat)
probability(ltm_cat, theta = 1, item = 1)

## Loading tpm Cat object
## Probability for Cat object of the tpm model
probability(tpm_cat, theta = 1, item = 1)

## Loading grm Cat object
## Probability for Cat object of the grm model
probability(grm_cat, theta = 1, item = 1)

## Loading gpcm Cat object
## Probability for Cat object of the gpcm model
probability(gpcm_cat, theta = -3, item = 2)