fedirt_2PL {FedIRT}

Federated 2PL estimate function

Description

This function implements a federated learning approach to estimate the parameters of the 2PL IRT model. It allows for collaborative estimation across multiple datasets, while maintaining the privacy of each individual data source. The federated 2PL model is particularly useful in contexts where data sharing might be limited due to privacy concerns or logistical constraints.

Usage

fedirt_2PL(J, logL_entry, g_logL_entry)

Arguments

J

An integer indicating the number of items in the IRT model across all sites. This number should be consistent for all response matrices provided.

logL_entry

A function that calculates the sum of log-likelihoods for the response matrices across all sites. This function is crucial for evaluating the fit of the model at each iteration.

g_logL_entry

A function that computes the aggregated gradient of the log-likelihood across all participating entities.

Details

The algorithm leverages federated learning techniques to estimate shared item parameters and individual ability levels without requiring the raw data to be combined into a single dataset. The estimation procedure is composed of several steps, including initialization, local computations at each data source, communication of summary statistics to a central server, and global parameter updates. This cycle is repeated until convergence criteria are met and the global parameters stabilize.

Regarding the input parameters, 'J' is the number of items across all sites, which should be consistent and known in advance. The 'logL_entry' parameter should be a function that computes the log-likelihood of the observed responses given the current model parameters. Likewise, 'g_logL_entry' is expected to be a function that computes the gradient of the log-likelihood with respect to the model parameters to inform the optimization process during parameter estimation.

Value

A list containing the following components from the federated 2PL model estimation:

• par: Numeric vector of model's fitted parameters including item discrimination (a) and item difficulty (b) parameters.

• value: The optimization objective function's value at the found solution, typically the log-likelihood.

• counts: Named integer vector with counts of function evaluations and gradient evaluations during optimization.

• convergence: Integer code indicating the optimization's convergence status (0 indicates successful convergence).

• message: Message from optimizer about optimization process, NULL if no message is available.

• loglik: The calculated log-likelihood of the fitted model, identical to the 'value' element when the objective function is log-likelihood.

• a: Numeric vector of estimated item discrimination parameters.

• b: Numeric vector of estimated item difficulty parameters.

• person: List containing person-related estimates with elements:

  • a: Vector of discrimination parameters (same as top-level 'a').

  • b: Vector of difficulty parameters (same as top-level 'b').

  • ability: List of numeric vectors with person abilities per site.

  • site: Numeric vector of abilities or locations specific to each site.

  • person: List of numeric vectors of person abilities minus site ability.