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In relation to the following figure

Figure 9.3 Kline 2016

Kline (2016) writes on p194:

The numerals (1) in Figure 9.3(b) that appear next to paths from the factors to one of their indicators are scaling constants, or unit loading identification (ULI) constraints. The specifications that

$A \rightarrow X_1 = 1.0$ and $B \rightarrow X_4 = 1.0$

scale the factors in a metric related to that of the explained (common) variance of the corresponding indicator, or reference (marker) variable.

I understand from this answer that the reason for having the reference variable is so we can determine the variance of the latent variable. However, it wasn't clear to me why this goal was not also relevant to exploratory factor analysis. Why don't we set $A \rightarrow X_1 = 1.0$ in the EFA model?

Kline, R. B. (2016). Principles and practice of structural equation modeling. Guilford Press.

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2 Answers 2

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There are multiple ways to set the scaling of latent variables:

  1. Reference/marker variables: Fixing the loading of one item per factor to a value of one--the default of most CFA/SEM software.
  2. Fixed-factor: Fixing the variance(s) of factor(s) to one (in essence, standardizing it/them).
  3. Effects-coding: A lesser-known approach (see Little et al., 2006, for an overview). Loadings are fixed to average one, and intercepts are fixed to average zero; this gives the latent variable(s) the same ("non-arbitrary", in Little's words) scaling as their indicators. It's described as a useful strategy when comparing latent means is the primary goal of an analysis.

Anyways, while reference/marker variable scale-setting is the default for most software, it prevents you from estimating all the factor loadings (as some are fixed). And since the focus of EFA is generally to estimate exploratory factor loadings, the latent variables are standardized instead (and this is why EFA software usually provides standardized factor loadings).

References

Little, T. D., Slegers, D. W., & Card, N. A. (2006). A non-arbitrary method of identifying and scaling latent variables in SEM and MACS models. Structural Equation Modeling, 13(1), 59-72.

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First, kudos for this question...it demonstrates a critical approach to EFA that I always try to pass on to all of my SEM students.

In brief, we don’t set a factor loading to 1 in EFA because we set the latent factor variances to one (as mentioned in jsakaluk answer).  And, the result of either approach (anchoring the latent factor to one of its items or standardizing the latent factor) is the same:  it provides a single solution for the parameter estimates (out of an infinite choice of possible solutions).

Just to clarify, there are infinitely many solutions, but they are all variations of the same solution that would result in a a comparable variance/covariance matrix.  (One could say the solutions are all “proportional” to each other.)

Hope this helps clarify the situation.

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