Calculates the probability of specific responses or the left-cumulative probability of responses to item conditioned on a respondent's ability (\(\theta\)).
probability(catObj, theta, item)An object of class Cat
A numeric or an integer indicating the value for \(\theta_j\)
An integer indicating the index of the question item
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.
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 \(\beta_{it}\) = \(\beta_{i}\) - \(\tau_{it}\).
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.
# NOT RUN {
## 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)
# }
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