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