The Principal Axis Factoring (PAF) method involves the following steps:
Step 1. **Basic Principle**:
The core principle of factor analysis using Principal Axis Factoring (PAF) is expressed as:
$$\mathbf{R} = \mathbf{\Lambda} \mathbf{\Lambda}^T + \mathbf{\Phi}$$
$$R_{ii} = H_i^2 + \Phi_{ii}$$
where \(\mathbf{\Lambda}\) is the matrix of factor loadings, and \(\mathbf{\Phi}\) is the diagonal
matrix of unique variances. Here, \(H_i^2\) represents the portion of the i-th item's variance explained by the factor model.
\(\mathbf{H}^2\) reflects the amount of total variance in the variable accounted for by the factors in the model, indicating the
explanatory power of the factor model for that variable.
Step 2. **Factor Extraction by Iteratoin**:
- Initial Communalities:
Compute the initial communalities as the squared multiple correlations:
$$H_{i(t)}^2 = R_{ii(t)}$$
where \(H_{i(t)}^2\) is the communality of i-th item in the \(t\)-th iteration, and \(R_{ii(t)}\) is the i-th
diagonal element of the correlation matrix in the \(t\)-th iteration.
- Extract Factors and Update Communalities:
$$\Lambda_{ij} = \sqrt{\lambda_j} \times v_{ij}$$
$$H_{i(t+1)}^2 = \sum_j \Lambda_{ij}^2$$
$$R_{ii(t+1)} = H_{i(t+1)}^2$$
where \(\Lambda_{ij}\) represents the j-th factor loading for the i-th item, \(\lambda_j\) is the j-th
eigenvalue, \(H_{i(t+1)}^2\) is the communality of i-th item in the \(t+1\)-th iteration, and \(v_{ij}\) is
the j-th value of the i-th item in the eigen vector matrix \(\mathbf{v}\).
Step 3. **Iterative Refinement**:
- Calculate the Change between \(\mathbf{H}_{t}^2\) and \(\mathbf{H}_{t+1}^2\):
$$\Delta H_i^2 = \lvert H_{i(t+1)}^2 - H_{i(t)}^2 \lvert$$
where \(\Delta H_i^2\) represents the change in communalities between iterations \(t\) and \(t+1\).
- Convergence Criterion:
Continue iterating until the change in communalities is less than the specified criterion \(criterion\):
$$\sum_i \Delta H_i^2 < criterion$$
The iterative process is implemented using C++ code to ensure computational speed.