Hello Myo,

You ask a good question, which points out discrepancies among scholarly journals in reporting of analytic results (it's not limited in any sense to just the SEM family!).

Here are my suggestions for inferring whether a model consists solely of manifest/observed variables (and, therefore, is a path model--which is still a legitimate member of the SEM family):

1. Are summary statistics given for all variables in the model? If "yes," it is likely that all variables are manifest/observed (and not latent).

2. Do all "downstream" variables (those that are pointed to by antecedent variables) have error terms associated with them? If "yes," it is likely that these variables are manifest/observed.

3. Does each variable that is proposed to influence/relate to another variable link to just one other variable? If "yes," then such variables are likely manifest/observed.

As to reporting, there are some good guides as to what researchers should furnish when presenting SEM results. Here are a couple:

1. https://steinhardtapps.es.its.nyu.edu/create/courses/3311/reading/7-Reporting_SEM_and_CFA__Schreiber__Stage__King__Nora__Barlow_.pdf

2. Article Reporting Results From Structural Equation Modeling Analyses...

Good luck with your work!

Good information David

Myo,

there is an easy way to spot the difference: Take a look onto the degrees of freedom (at least, when they are reported). Roughly, if the DF are around (+/- a few) the number of variables in the model, then it is a path analysis; if they are substantially higher, then it's a latent model.

The more detailed version of this is also very informative to spot non-reported tinkering (e.g., estimating error covariances but not reporting them). The basis is to know how DF are reported. The simple formula is

DF = S - P

where S is the number of measured variances and covariances of the variables (the input) and P is the number of estimated parameters (i.e. variances and covariances of the exogeneous model variables, error variables of the dependent/endogeneous variables, error covariances, and effects).

You can get S by either simply counting the entries of the empirical covariance matrix or by an equally easy formula that is

S = (q * (q + 1))/2,

where q = number of variables. For instance, if the model consists of 10 variables, the covariance matrix contains 10 * 11 / 2 = 55 elements, that is 10 variances and 45 covariances.

You can get P by simply counting the parameter (and don't forget the hidden ones, referring to the variances and error variables).

Here's an example for the case of a path model: We assume a model with two exo variables (X1, X2), two mediators (M1, M2), and one dependent variable Y. X1 and X2 influence both mediators, and both mediators have both an effect on Y. This is a very typical model. S in this case is 5*6/2 = 15 elements of the covariance matrix. P contains

a) 2 exo variances

b) 1 exo covariance

c) 3 error variables (for M1, M2, and Y)

d) 6 effects

that is 12 estimated parameters. Hence, this model has 15-12 = 3 DF.

Now let's turn this model in a full latent variables model. We keep the structural part identical but now each of the 5 variable is latent, and let's assume, each is measured by 3 manifest indicators.

The number of observed variables, hence is 15 (5 x 3), and S = 15*16/2 = 120.

P is now (again) 12 estimated parameters on the latent level (this does not change). Added, however, are all the parameters of the measurement models:

a) 10 factor loadings (in each measurement model, one loading is fixed for scaling)

b) 15 error variances

in total, 12 + 10 + 15 = 37 estimated parameters

The DF hence are 120 - 37 = 83

Quite a huge difference. If you do that and the reported DF are only 82 or 81 or so, then you know there is something problematic (e.g., tinkering) as the authors estimated something (consuming 1 or 2 DF) which is not contained in the path diagram.

I have made routine of calculating DF's for reported models and I can tell there is often something problematic (besides your reported lack of clarity or intentional erroneous depiction of manifest variables with ellipses).

HTH,

Holger