Solved – In CFA, does it matter which factor loading is set to 1

confirmatory-factorstructural-equation-modeling

I'd been previously taught that, aside from the fact that fixing a loading to 1 means you won't get a significance test on that loading, it was totally arbitrary which loading got fixed to 1.

However, a noted authority on SEM (Jeremy Miles) makes an interesting comment here that

It doesn't make any difference empirically which is fixed – it
rescales the loadings. Sometimes it makes theoretical sense to choose
one of the variables to have its loading fixed to one – this is the
variable with the closest conceptual relationship to the latent
variable of interest.

Would anyone care to explain why it can make theoretical sense to fix the variable with the closest conceptual relationship to the latent variable to 1? Why does this make sense "sometimes" and not always?

Best Answer

To add (and then to digress a bit...): selecting a particular marker variable over another can be a reasonable thing to do if one is known to be a high-consensus "gold-standard" indicator of your latent variable of interest (Little, 2013). Imagine you have three tests $x_1$, $x_2$, and $x_3$, that attempt to assess Latent Variable $X$. Perhaps $x_1$ and $x_2$ are cheap/quick/easy to administer assessments--they are convenient to the researcher, and capture some amount of signal in $X$, but are known to not be the most reliable/valid assessment tools. $x_3$, meanwhile, is perhaps a longer assessment that's been put through a more rigorous development process, and though perhaps not quite as convenient to use, is a more reliable/valid indicator of $X$.

In this contrived hypothetical, it would make good sense to use $x_3$ as your marker variable for scale-setting, as opposed to $x_1$ (which would be the default marker variable selected by many SEM software options) or $x_2$. As most texts will aptly point out, however, the choice of marker variable won't make a lick of difference for your indices of model fit, so why does the selection matter? In the context of estimating a measurement model for $X$, the answer is that by anchoring to $x_3$, you will get a more accurate estimate of the variance of $X$ (this is mentioned in the Steiger, 2002, paper that Jeremy Miles references), because $x_3$ ostensibly has a much higher factor loading and contains less unique/error variance than either $x_1$ and/or $x_2$.

Cue digression

In many applications of CFA/SEM, the selection of a particular marker variable impacts more than just the estimate of a latent variance, in ways that others (and I) think is deeply problematic. Stated simply: your choice of marker variable will impact the estimate and standard error (and therefore p-values) of structural associations with other latent variables. This may be acceptable (or even desirable) in a case like my above hypothetical--when the gold-standard indicator is clear--but in many cases, no such consensus of which indicator is best exists, and you can get different patterns of results depending on which you select.

Here is a reproducible example showing how the selection of marker variable impacts estimation/testing, using the HolzingerSwineford1939 data set and the lavaan package:

Fit the same model (predicting latent textual from latent visual), and vary the selected marker variables for each ($x_1$/$x_4$, $x_2$/$x_5$, $x_3$/$x_6$):

model.x1 <- '
visual  =~ 1*x1 + x2 + x3
textual =~ 1*x4 + x5 + x6
textual ~ visual
'

model.x2 <- '
visual  =~ NA*x1 + 1*x2 + x3
textual =~ NA*x4 + 1*x5 + x6
textual ~ visual
'

model.x3 <- '
visual  =~ NA*x1 + x2 + 1*x3
textual =~ NA*x4 + x5 + 1*x6
textual ~ visual
'

The fit of each model is identical (marker variable selection doesn't impact it), $\chi^2$(8) = 24.361, p = .002. But the estimated slope and statistical test of regressing textual on visual does change:

$b$ = 0.503, z = 5.235
$b$ = 1.000, z = 4.745
$b$ = 0.658, z = 5.386

In absence of a clear gold-standard indicator of either textual or visual, this presents (or rather, and in my opinion, it should present) a fairly large problem to those wishing to use a marker variable approach to scale setting. For my part, though it has yet to be studied formally from this perspective, I see the arbitrary and (potentially) flexible selection of marker variables as a "researcher degree of freedom" (John et al., 2012; Simmons et al., 2011) ripe for exploitation in the measurement/structural modelling context (e.g., Flake & Fried, 2019).

What's the alternative, then? One approach is to fix the latent variances to 1 (often accompanied by fixing the latent means, when modelling mean structures, to 0), effectively standardizing the latent variable, and allowing a factor loading for each indicator to be estimated:

model.ff <- '
visual  =~ NA*x1 + x2 + x3
textual =~ NA*x4 + x5 + x6
visual ~~ 1*visual
textual ~~ 1*textual
textual ~ visual
'

Model fit is the still the same, but like before, we get a different estimated slope/statistical test based on our choice of scale-setting: $b$ = 0.519, z = 5.668. Why is this approach potentially preferable? First, researchers are often more interested in the factor loadings of their indicators than in the estimate of their latent variances. Second--and perhaps what carries the argument in most cases--though standardized scalings for latent variables are still arbitrary in a sense, it's an approach that is at least scientifically normative (i.e., we standardize variables all the time and no one seems to lose their heads).

There's a third option for scale-setting though, for those wanting a solution that is non-arbitrary while still providing estimates of all factor loadings: effects coding (Little et al., 2006). Here, we constrain all loadings to average 1 (and if we were modelling a mean structure, we would constrain all item intercepts to equal 0):

model.ec <- '
visual  =~ NA*x1 + a*x1 + b*x2 + c*x3
textual =~ NA*x4 + d*x4 + e*x5 + f*x6
a+b+c ==3
d+e+f ==3
textual ~ visual
'

Model fit is the same, we get estimated loadings for all indicators and estimated latent variances for both factors, and a slightly different estimate/test of the latent slope: $b$ = 0.674, z = 6.159. The main perk of effects coding is that it puts the latent variable back on the original metric/scale of all of its specified indicators, and so if you estimated its latent mean, it would be the same mean (on the same scale) that you would get by calculating a crude average of the--just with a smaller latent variance because you've removed error/unique variance--lending itself to a more intuitive interpretation.

tl;dr: Selecting a particular marker variable is worth while when your indicators are on different scales and you want your LV on the scale of one of them in particular (as per Noah's good answer), and/or when one of your indicators is a gold-standard indicator. The latter case, however, is often unclear, and given its impact on estimates/tests of structural parameters, I think it's a somewhat questionable thing to do without strong evidence.

References

Flake, J. K., & Fried, E. I. (2019, January 17). Measurement Schmeasurement: Questionable Measurement Practices and How to Avoid Them. https://doi.org/10.31234/osf.io/hs7wm

John, L. K., Loewenstein, G., & Prelec, D. (2012). Measuring the prevalence of questionable research practices with incentives for truth telling. Psychological Science, 23, 524-532.

Little, T. D. (2013). Longitudinal Structural Equation Modeling. New York, NY: Guilford Press.

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, 59-72.

Simmons, J. P., Nelson, L. D., & Simonsohn, U. (2011). False-positive psychology: Undisclosed flexibility in data collection and analysis allows presenting anything as significant. Psychological science, 22(11), 1359-1366.

Steiger, J. H. (2002). When constraints interact: A caution about reference variables, identification constraints, and scale dependencies in structural equation modeling. Psychological Methods, 7, 210-227.

Related Solutions

Weights – Why Set Weights to 1 in Confirmatory Factor Analysis?

Because it then allows you to use the relationship between the latent variable and the observed variable to determine the variance of the latent variable. For example, consider the regression of Y on X. If I am allowed to change the variance of X, say, by multiplying it by a constant, then I can change the regression coefficient arbitrarily. If instead I fix the value of the regression coefficient, then this determines the variance of X.
By convention, and to make it easier to compare the coefficients to each other.
In that case, the latent variable simply becomes reversed. For example, suppose our latent variable is math ability, our observed variable is the number of errors on a test, and we fix the regression coefficient to 1. Then our latent variable will become "difficulty with math" instead of math ability, and the coefficients for any other observed variables will change accordingly.
If the observed variable and the latent variable are both standardized (i.e., standard deviation equal to 1), then the regression coefficient is equal to the covariance.
It is fixing spatial -> visperc to 1 that permits estimation of the variance of spatial (see answer to (1) above). Likewise, fixing verbal -> paragrap permits estimation of the variance of verbal. A model with only one of these constraints would not be identifiable.
Because the differences between the unstandardized and standardized coefficients depend not only on the variance of verbal, but also on the variances of paragrap, sentence and wordmean. For example, the standardized coefficient for wordmean equals the unstandardized coefficient multiplied by $\frac{SD_{verbal}}{SD_{wordmean}}$, or $2.234 \times \frac{\sqrt{9.682}}{\sqrt{(2.234^2 \times 9.682) + 19.925}} = 0.841$.

Finally, note that err_v is analogous to the error term in a regression model, e.g., $$visperc = \beta_0 + \beta_1 spatial + err\_v$$ We fix the coefficient on err_v (i.e., on the error term) to 1 so that we can estimate the error variance (i.e., the variance of err_v).

Solved – Unable to estimate standard error after freeing first indicator in SEM model – Why is it so

As Maarten points out, your problem is that you have not set the scale of the second model. True, you have more observed variances/covariances than what you need to identify your model, but you still need to provide a point of reference from which other model parameters can be calculated (Brown, 2015).

You can set the scale using one of three methods:

Marker variable: one factor loading per latent variable is fixed to 1
Fixed factor: each latent variable's variance is fixed to 1
Effects-coding: factor loadings for each latent variable are constrained to average 1

Code for each approach (using the lavaan package's HolzingerSwineford1939 dataset) is presented below. The latent variable I've created is nonsensical/poor-fitting, but it has the same number of indicators as your model, so the example will hopefully be more transferable to your situation.

library(lavaan)

#marker-variable; first factor loading fixed to 1 by default
marker.variable<-'f1=~ x1+x2+x3+x4+x5+x6'
summary(output.marker<-cfa(marker.variable, data=HolzingerSwineford1939), fit.measures=TRUE)

#fixed-factor method; manually free first factor loading/fix latent variance to 1
fixed.factor<-'f1=~ NA*x1+x2+x3+x4+x5+x6
          f1~~1*f1'
summary(output.fixed<-cfa(fixed.factor, data=HolzingerSwineford1939), fit.measures=TRUE)

#effects coding; manually free first loading/constrain loadings to average 1
effects.coding<-'f1=~ NA*x1+a*x1+b*x2+c*x3+d*x4+e*x5+f*x6
          a+b+c+d+e+f==6'
summary(output.effects<-cfa(effects.coding, data=HolzingerSwineford1939), fit.measures=TRUE)

Note that model fit is identical, regardless of which method of scale-setting that you use; the fit in all three models is $\chi^2 (df = 9) = 103.23, ~p < .001$.

Which method you should use largely depends on the nature of your data and your research goals. The marker variable method is a highly arbitrary method of scale-setting. Like Maarten stated, your latent variables will take on the units of their respective marker variables, so this approach is only informative to the extent that your marker variables are especially meaningful, or perhaps represent some "gold standard" indicator of your latent construct.

The fixed factor method, alternatively, is easy to specify, and essentially standardizes your latent variables (if you're examining mean structures, you would fix the latent means to zero as well). Since we standardize variables all the time, this is a highly intuitive and widely acceptable form of scale-setting for latent variables, though the resultant scaling is not inherently meaningful. Even so, it's probably the best method to "default" to, unless you have a strong imperative to use one of the other methods.

Effects-coding is a relative new-comer to methods of scale-setting (see Little, Slegers, & Card, 2006, for a thorough discussion). It's greatest advantage is when you are modeling latent means. When doing so, you would also constrain item intercepts to average 0. The effect of these constraints is that your latent variables will be on the exact same scale as your original items. For example, if the average of your indicators was "5", your latent mean would also be "5", though your latent variance would be smaller than you observed variance. Because the constraints on the loadings and intercepts can be more computationally demanding, especially in more complicated models, and occasionally result in convergence errors, effects-coding is probably not worth it unless you plan to examine latent means. But for the particular purpose of examining latent means, it's great.

References

Brown, T. A. (2015). Confirmatory factor analysis for applied research (2nd Edition). New York, NY: Guilford Press.

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

Best Answer

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Weights – Why Set Weights to 1 in Confirmatory Factor Analysis?

Solved – Unable to estimate standard error after freeing first indicator in SEM model – Why is it so

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