get.beta: Calculate $\beta$

Description

The function is able to calculate the $\beta$ index for all items after fitting CDM or directly.

Usage

get.beta(Y = NULL, Q = NULL, CDM.obj = NULL, model = "GDINA")

Value

An object of class matrix, which consisted of $\beta$ index for each item and each possible attribute mastery pattern.

Arguments

Y: A required $N$ × $I$ matrix or data.frame consisting of the responses of N individuals to $N$ × $I$ items. Missing values need to be coded as NA.
Q: A required binary $I$ × $K$ matrix containing the attributes not required or required master the items. The ith row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which attributes are required (coded by 1) to master item $i$.
CDM.obj: An object of class CDM.obj. Can be NULL, but when it is not NULL, it enables rapid validation of the Q-matrix without the need for parameter estimation. @seealso CDM.
model: Type of model to be fitted; can be "GDINA", "LCDM", "DINA", "DINO", "ACDM", "LLM", or "rRUM". Default = "GDINA".

Author

Haijiang Qin <Haijiang133@outlook.com>

Details

For item $i$ with the q-vector of the $c$-th ($c = 1, 2, ..., 2^{K}$) type, the $\beta$ index is computed as follows:

$$ \beta_{ic} = \sum_{l=1}^{2^K} \left| \frac{r_{li}}{n_l} P_{ic}(\boldsymbol{\alpha}_{l}) - \left(1 - \frac{r_{li}}{n_l}\right) \left[1 - P_{ic}(\boldsymbol{\alpha}_{l})\right] \right| = \sum_{l=1}^{2^K} \left| \frac{r_{li}}{n_l} - \left[1 - P_{ic}(\boldsymbol{\alpha}_{l}) \right] \right| $$

In the formula, $r_{li}$ represents the number of examinees in attribute mastery pattern $\boldsymbol{\alpha}_{l}$ who correctly answered item $i$, while $n_l$ is the total number of examinees in attribute mastery pattern $\boldsymbol{\alpha}_{l}$. $P_{ic}(\boldsymbol{\alpha}_{l})$ denotes the probability that an examinee in attribute mastery pattern $\boldsymbol{\alpha}_{l}$ answers item $i$ correctly when the q-vector for item $i$ is of the $c$-th type. In fact, $\frac{r_{li}}{n_l}$ is the observed probability that an examinee in attribute mastery pattern $\boldsymbol{\alpha}_{l}$ answers item $i$ correctly, and $\beta_{jc}$ represents the difference between the actual proportion of correct answers for item $i$ in each attribute mastery pattern and the expected probability of answering the item incorrectly in that state. Therefore, to some extent, $\beta_{jc}$ can be considered as a measure of discriminability.

References

Li, J., & Chen, P. (2024). A new Q-matrix validation method based on signal detection theory. British Journal of Mathematical and Statistical Psychology, 00, 1–33. DOI: 10.1111/bmsp.12371

Examples

Run this code

library(Qval)

set.seed(123)

## generate Q-matrix and data
K <- 3
I <- 20
example.Q <- sim.Q(K, I)
IQ <- list(
  P0 = runif(I, 0.0, 0.2),
  P1 = runif(I, 0.8, 1.0)
)

model <- "DINA"
example.data <- sim.data(Q = example.Q, N = 500, IQ = IQ, model = model, distribute = "horder")

## calculate beta directly
beta <-get.beta(Y = example.data$dat, Q = example.Q, model = model)
print(beta)

## calculate beta after fitting CDM
example.CDM.obj <- CDM(example.data$dat, example.Q, model=model)
beta <-get.beta(CDM.obj = example.CDM.obj)
print(beta)

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get.beta: Calculate \(\beta\)