Knowledge Tracing Machines (KTMs) behave exactly like the Additive Factor Model (AFM) and Performance Factor Analysis (PFA) when specific features and weights are used. This equivalence occurs if the features are skills (with wins and fails at the skill level), the Q-matrix between items and skills is known, the weights are encoded as $$(\beta_1, \dots, \beta_s, \lambda_1, \dots, \lambda_s, \delta_1, \dots, \delta_s)$$, and the features are encoded as $$(q_{j1}, \dots, q_{js}, W_{i1}, \dots, W_{is}, F_{i1}, \dots, F_{is})$$, where $$W$$ and $$F$$ are counters for successful and unsuccessful attempts.

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Knowledge Tracing Machines (KTMs) generalize several existing models, including Item Response Theory (IRT), the Additive Factor Model (AFM), Performance Factor Analysis (PFA), and Multidimensional Item Response Theory (MIRT). The specific model a KTM simplifies to depends on the chosen features (such as users, items, or skills) and the dimensionality of the embeddings ($$d$$).

Relation to existing models (Knowledge Tracing Machines: Factorization Machines for Knowledge Tracing)

When the embedding dimension is $$d = 0$$ (without embeddings), and the features consist only of users and items represented as one-hot encodings, the Knowledge Tracing Machine (KTM) expression becomes:

$$ \log \frac{p(x)}{1 - p(x)} = \mu + w_i + w_{n+j} = \theta_i - d_i $$

Under these conditions, the KTM simplifies to the Rasch Model, which is the 1-Parameter Logistic (1-PL) Item Response Theory (IRT) model.

Relation of Knowledge Tracing Machines to Item Response Theory (IRT)

Relation of Knowledge Tracing Machines to AFM and PFA

Knowledge Tracing Machines (KTMs) act like Multidimensional Item Response Theory (MIRT) when the embedding dimension is strictly greater than zero ($$d > 0$$).

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