probability {catSurv} | R Documentation |

Calculates the probability of specific responses or the left-cumulative probability of responses to `item`

conditioned on a respondent's ability (*θ*).

probability(catObj, theta, item)

`catObj` |
An object of class |

`theta` |
A numeric or an integer indicating the value for |

`item` |
An integer indicating the index of the question item |

For the `ltm`

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

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

where *θ_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|θ_j)=c_i+(1-c_i)\frac{\exp(a_i + b_i θ_j)}{1+\exp(a_i + b_i θ_j)}*

where *θ_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 | θ_j) = (exp(α_ik - β_i θ_ij))/(1 + exp(α_ik - β_i θ_ij))*

where *θ_j* is respondent *j* 's position on the latent scale of interest, *α_ik* the *k*-th element of item *i* 's difficulty parameter,
*β_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|θ_j)=\frac{\exp(∑_{t=1}^k α_{i} [θ_j - (β_i - τ_{it})])}
{∑_{r=1}^{K_i}\exp(∑_{t=1}^{r} α_{i} [θ_j - (β_i - τ_{it}) )}*

where *θ_j* is respondent *j* 's position on the latent scale of interest, *α_i* is the discrimination parameter for item *i*,
*β_i* is the difficulty parameter for item *i*, and *τ_{it}* is the category *t* threshold parameter for item *i*, with *k = 1,...,K_i* response options
for item *i*. For identification purposes *τ_{i0} = 0* and *∑_{t=1}^1 α_{i} [θ_j - (β_i - τ_{it})] = 0*. Note that when fitting the model,
the *β_i* and *τ_{it}* are not distinct, but rather, the difficulty parameters are *β_{it}* = *β_{i}* - *τ_{it}*.

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.

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.

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

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.

## 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)

[Package *catSurv* version 1.4.0 Index]