probability function

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

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)

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

Returns

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.

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

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

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 $\beta_{it}$ = $\beta_{i}$ - $\tau_{it}$.

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.

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)

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

Author(s)

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