probability {catSurv} R Documentation

## 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.

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)

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