Calculate PVAF

The function is able to calculate the proportion of variance accounted for ($PVAF$) for all items after fitting CDM or directly.

get.PVAF(
  Y = NULL,
  Q = NULL,
  att.str = NULL,
  CDM.obj = NULL,
  mono.constraint = FALSE,
  model = "GDINA"
)

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 $i$.
att.str: Specify attribute structures. NULL, by default, means there is no structure. Attribute structure needs be specified as a list - which will be internally handled by att.structure function. See examples. It can also be a matrix giving all permissible attribute profiles.
CDM.obj: An object of class CDM.obj. Can be NULL, but when it is not NULL, it enables rapid verification of the Q-matrix without the need for parameter estimation.
mono.constraint: Logical indicating whether monotonicity constraints should be fulfilled in estimation. Default = FALSE.
model: Type of model to be fitted; can be "GDINA", "LCDM", "DINA", "DINO", "ACDM", "LLM", or "rRUM". Default = "GDINA".

Value

An object of class matrix, which consisted of $PVAF$ for each item and each possible q-vector.

Details

The intrinsic essence of the GDI index (as denoted by $\zeta_{2}$) is the weighted variance of all $2^{K\ast}$ attribute mastery patterns' probabilities of correctly responding to item $i$, which can be computed as: $$ \zeta^2 = \sum_{l=1}^{2^K} \pi_{l}{(P(X_{pi}=1|\boldsymbol{\alpha}_{l}) - \bar{P}_{i})}^2 $$ where $\pi_{l}$ represents the prior probability of mastery pattern $l$; $\bar{P}_{i}=\sum_{k=1}^{2^K}\pi_{l}P(X_{pi}=1|\boldsymbol{\alpha}_{l})$ is the weighted average of the correct response probabilities across all attribute mastery patterns. When the q-vector is correctly specified, the calculated $\zeta^2$ should be maximized, indicating the maximum discrimination of the item.

Theoretically, $\zeta^{2}$ is larger when $\boldsymbol{q}_{i}$ is either specified correctly or over-specified, unlike when $\boldsymbol{q}_{i}$ is under-specified, and that when $\boldsymbol{q}_{i}$ is over-specified, $\zeta^{2}$ is larger than but close to the value of $\boldsymbol{q}_{i}$ when specified correctly. The value of $\zeta^{2}$ continues to increase slightly as the number of over-specified attributes increases, until $\boldsymbol{q}_{i}$ becomes $\boldsymbol{q}_{i1:K}$ ($\boldsymbol{q}_{i1:K}$ = [11...1]). Thus, $\zeta^{2} / \zeta_{max}^{2}$ is computed to indicate the proportion of variance accounted for by $\boldsymbol{q}_{i}$, called the $PVAF$.

References

de la Torre, J., & Chiu, C. Y. (2016). A General Method of Empirical Q-matrix Validation. Psychometrika, 81(2), 253-273. DOI: 10.1007/s11336-015-9467-8.

Author

Haijiang Qin <Haijiang133@outlook.com>

Examples