9.3 Maximum Likelihood Estimation (MLE)
Let us recall:
Maximum Likelihood Estimation (MLE) is a statistical approach to estimate the parameter of a statistical model. In this approach, we find the parameter values that make the observed data most probable.
Maximum Likelihood Estimation (MLE) of attribute profiles in CDM is relevant to estimatimating the most likely attribute profile for an examinee based on their responses to assessment items.
The process can be summarized as follows: - The process starts with the examinee’s responses to assessment items, which are typically designed to measure a set of cognitive attributes.
MLE relies on the likelihood function, which models the probability of an examinee’s observed item responses given different potential attribute profiles.
MLE seeks to find the attribute profile that maximizes the likelihood of the observed responses. In other words, it selects the profile that makes the examinee’s responses most probable.
The likelihood of observing the response vector \(\mathbf{Y}_i\) for student \(i\) given attribute profile \(\mathbf{\alpha}_c\) is \[ L(\mathbf{\alpha}_c;\mathbf{Y}_i)=\prod_{j=1}^JP(Y_{ij}=1|\mathbf{\alpha}_c)^{Y_{ij}}[1-P(Y_{ij}=1|\mathbf{\alpha}_c)]^{1-Y_{ij}} \]
The log likelihood is \[ \log L(\mathbf{\alpha}_c;\mathbf{Y}_i)=\sum_{j=1}^J{Y_{ij}}\log P(Y_{ij}=1|\mathbf{\alpha}_c)+{(1-Y_{ij})}\log[1-P(Y_{ij}=1|\mathbf{\alpha}_c)] \]
The Maximum Likelihood Estimation (MLE) of attribute profile for student \(i\) is \(\mathbf{\alpha}_c\) that maximize \(L(\mathbf{\alpha}_c;\mathbf{Y}_i)\) or \(\log L(\mathbf{\alpha}_c;\mathbf{Y}_i)\).
MLE is a straightforward method to estimate the most likely attribute profile based on observed data, however, it often shows sensitivity to sparse data. Also, it is a completely data driven process and do not incorporates any prior information.