Bind Recovery of Sparse Factor Structures by Signal Cancellation

Blind factor recovery follows from the principle that the signal of variables exclusive to a factor can be combined in a contrast (weighted sum) that cancels their factor contributions, leaving only a compound of the variables’ unique variances. Successful contrasts, uncorrelated with any remaining variable, become the signature of factors with at least two unique indicator variables. Pairwise signal cancellation, usually incomplete for variables affected by different factors, nevertheless succeeds for variables with proportional loadings on two factors, which places three cancelling clusters in the plane of two factors. This is recognized by successful cancellation among variable triplets representing the three clusters. The Signal Cancellation Recovery of Factors (SCRoF) algorithm implements these principles, only requiring that each factor has at least two unique indicators, not even requiring having pre-estimated the number of factors. Alternate sparse factor solutions are obtained through a two significance-threshold strategy. The individually estimated factor loadings and factor correlations of each potential solution are globally optimized for maximum likelihood, yielding a χ2 indication of compatibility with observed data. SCRoF is illustrated with synthetic data from a complex six-factor structure. Actual data then document that SCRoF can even benefit confirmatory factor analysis when the initial model appears inadequate.