Abstract
Psychological research often relies on Exploratory Factor Analysis (EFA). As the outcome of the analysis highly depends on the chosen settings, there is a strong need for guidelines in this context. Therefore, we want to examine the recent methodological developments as well as the current practice in psychological research. We reviewed ten years of studies containing EFAs and contrasted them with new methodological options. We focused on four major issues: an adequate sample size, the extraction method, the rotation method and the factor retention criterion determining the number of factors. Finally, we present modified recommendations based on these reviewed empirical studies and practical considerations.
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Notes
Regularization means that an additional term is added to an objective function to solve an otherwise not solvable problem. Here instead of estimating several unique variances which can be infeasible when the sample size is too small, a so-called regularization parameter is selected that adjusts the initial estimates of the unique variances.
The anti-image can be pictured as the negative of the image of a matrix. The image covariance matrix contains the variation of each variable that can be explained by the other variables (partial covariance coefficients), the respective anti-image consists of the negatives which can be described as the unique components. For more detail, have a look at Kaiser (1976) or detailed EFA textbooks as the anti-image correlation matrix is a commonly used tool to evaluate whether an EFA is applicable to the data (see also Measuring Sampling Adequacy (MSA), Kaiser 1970).
The RMSE is defined as the root of the MSE which is the averaged squared distance between parameters and its estimates. In this case, the differences between the given eigenvalues and the eigenvalues obtained of the simulated data sets of the specific k-factor population are computed.
They varied the number of factors (one to five), the number of response categories (two to 20), used correlated and uncorrelated solutions and sample sizes between 200 and 1000.
It only requires the number of items and the sample size, so it can be applied without knowing much about the structure of the data – for example when evaluating published results.
The so-called complexity function is the objective function which is minimized with regard to specific constraints to achieve a particular rotation of the pattern matrix. We recommend the article of Browne (2001) explaining the link between constraints and rotation criterion in more detail.
In common ML estimation an objective function (that is derived from the log-likelihood) is minimized. Here a so-called penalty term is added to this function. It penalizes a high number of parameters (in this case loadings, especially cross-loadings). The more parameters are estimated to be non-zero, the higher this term gets and it “becomes harder” to achieve a minimum, so in turn adding this penalty yielding more small (or even zero-) loadings (depending on the type of penalty). You can read about penalizing the likelihood in the EFA estimation process in more detail in Jin, Moustaki and Yang-Wallentin (2018).
Problem of rotation indeterminacy (see introduction section)
The optimization process is done with respect to different constraints, but apart from that equivalent for all rotation methods. Therefore, theoretical considerations must be taken into account to make a reasonable decision (are cross-loadings consistent with theoretical assumptions, etc.).
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Goretzko, D., Pham, T.T.H. & Bühner, M. Exploratory factor analysis: Current use, methodological developments and recommendations for good practice. Curr Psychol 40, 3510–3521 (2021). https://doi.org/10.1007/s12144-019-00300-2
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DOI: https://doi.org/10.1007/s12144-019-00300-2