Hello Mohammad,

the number of indicators determines the identification of the involved parameters of a measurement model (latent variance, loading, and error variance). Remember that each model is an equation linking an observed piece of information (the variance and covariances among indicators) to unobserved coefficients (e.g., loadings). As in any other equation system, the unknown parameters have to be identified by having enough information.

In an isolated 2-indicator model (remember the term "isolated"), you have 3 observed pieces (2 variances and the covariance of both indicators) but 4 requested coefficients: 1 latent variance, 2 error variances, and 1 loading (the other has to be fixed). Hence, the isolated model is not identified, that is you cannot estimate the coefficients.

In an isolated 3-indicator model, the system of equations is "just-identified" and you have the same number of information pieces and estimated parameters (Count by yourself, can you do it?). With 4 indicators, the model is overidentified, that is coefficients emerge in more than two equations. This allows the model to be tested, as the differential equations have to converge (like two witnesses of a crime allow validation of one witnesses' claims).

Now comes the twist. All of this is restricted to the "isolation" term. If the latent variable in your unidentified 2-indicator model correlates with other model variables, covariances between the 2-indicators and these other variables will add to the count of information pieces, rendering the model identified. Hence it depends on the strength and significance whether the model is identified or "empirically unidentified". The empirical identification due to the context can go so far that even a single-indicator model may be identified, as Hayduk (2018) points out.

If a measurement model is unidentified, you can circumvent the identification problem by fixing parameters to theoretically sound values.

The most salient target of such fixation is the error variance. In an extreme of a single indicator, the loading is fixed to 1 per default to set the scale of the latent variable. The error variance is critical because its strength determines how similar the indicator will be to the latent variable. The usual approach is to fix the error to zero. This is the worst possible choice because 1) it equates the indicator with the latent variable and 2) presumes that the indicator is measured without error. Every non-zero value is theoretically more appropriate. Below I listed a lot of papers in which it is recommended to fix/set the error variance to a value that expresses the researcher's theoretical assumption of

1) an appropriate/plausible error variance (most often thought of in % of the observed variance) and

2) hence the *conceptual distance* between the indicator and the latent variable. By #2, the researcher chooses and determines how similar the indicator should be from the latent variable that the researcher thinks of when specifying his/her model.

The usual counter-argument is "but you don't KNOW the true error variance". Yes, that's true but this argument is just a manifestation of the false impression spread in the behavioural science, that you can delegate critical issues to the data and some mystical statistical machine gives you the right answers. As I argue again and again on RG, there is no free lunch: If you want to get something valuable out of a causal model there is to duck-and-cover from inputting assumptions. These concern the causal structure but also the relationship between the latent variable and the measures. And yes, these assumptions may be nonsensical but they are founded and transparent for any reader to disagree. A routine zero-fixation scheme is not founded. It represents the duck-and-cover mentality.

Your mentioning of the usual recommendations to merge variables or to eliminate variables are an additional sign of this mentality and are bad practice. They stem from the intrusion of practices in pure prediction research to the realm of causal models. If your theoretical model consists of certain latent variables that represent truly existing and causally functioning attributes you have to incorporate them in the model when they are critical. Otherwise, you will get a bias up to nonsensical estimates.

Without a doubt, I would always prefer a one-indicator model consisting of a crystal clear indicator with an appropriate error variance (or a range of tested-through candidates) that is embedded in a rigid causal structure and tested in a rigid manner (with the chi-square test instead of fit indices) to the usual SEM having a handful of heterogeneous indicators which lead to misfit and even if not, cloud the meaning of the latent variables.

Having said all of this, be aware that this is not mainstream and reviewers will probably disagree!

Below find a bunch of papers in this regard.

All the best,

--Holger

Duncan, O. D., Haller, A. O., & Portes, A. P. (1968). Peer influences on aspirations: A reinterpretation. American Journal of Sociology, 74(2), 119-137. doi:10.1086/224615

Alwin, D. F., Jackson, D. J., & Jackson, E. F. (1980). Measurement models for response errors in surveys: Issues and applications. Sociological Methodology, 11(1980), 68-119. doi:10.2307/270860

Hayduk, L. A., & Littvay, L. (2012). Should researchers use single indicators, best indicators, or multiple indicators. BMC Medical Research Methodology, 12(159), 1-17. doi:10.1186/1471-2288-12-159

Hayduk, L. a., Stratkotter, R. F., & Rovers, M. W. (1997). Sexual orientation and the willingness of catholic seminary students to conform to church teachings. Journal for the Scientific Study of Religion, 36(3), 455-467.

Hayduk, L. (2018). Review essay on rex b. Kline’s principles and practice of structural equation modeling: Encouraging a fifth edition. Canadian Studies in Population, 45(3-4), 154-187.

Oberski, D. L., & Satorra, A. (2013). Measurement error models with uncertainty about the error variance. Structural Equation Modeling: A Multidisciplinary Journal, 20(3), 409-428.

Onyskiw, J. E., & Hayduk, L. a. (2001). Processes underlying children's adjustment in families characterized by physical aggression. Family Relations, 50(4), 376-385. doi:10.1111/j.1741-3729.2001.00376.x

Hello Mohammad,

the number of indicators determines the identification of the involved parameters of a measurement model (latent variance, loading, and error variance). Remember that each model is an equation linking an observed piece of information (the variance and covariances among indicators) to unobserved coefficients (e.g., loadings). As in any other equation system, the unknown parameters have to be identified by having enough information.

In an isolated 2-indicator model (remember the term "isolated"), you have 3 observed pieces (2 variances and the covariance of both indicators) but 4 requested coefficients: 1 latent variance, 2 error variances, and 1 loading (the other has to be fixed). Hence, the isolated model is not identified, that is you cannot estimate the coefficients.

In an isolated 3-indicator model, the system of equations is "just-identified" and you have the same number of information pieces and estimated parameters (Count by yourself, can you do it?). With 4 indicators, the model is overidentified, that is coefficients emerge in more than two equations. This allows the model to be tested, as the differential equations have to converge (like two witnesses of a crime allow validation of one witnesses' claims).

Now comes the twist. All of this is restricted to the "isolation" term. If the latent variable in your unidentified 2-indicator model correlates with other model variables, covariances between the 2-indicators and these other variables will add to the count of information pieces, rendering the model identified. Hence it depends on the strength and significance whether the model is identified or "empirically unidentified". The empirical identification due to the context can go so far that even a single-indicator model may be identified, as Hayduk (2018) points out.

If a measurement model is unidentified, you can circumvent the identification problem by fixing parameters to theoretically sound values.

The most salient target of such fixation is the error variance. In an extreme of a single indicator, the loading is fixed to 1 per default to set the scale of the latent variable. The error variance is critical because its strength determines how similar the indicator will be to the latent variable. The usual approach is to fix the error to zero. This is the worst possible choice because 1) it equates the indicator with the latent variable and 2) presumes that the indicator is measured without error. Every non-zero value is theoretically more appropriate. Below I listed a lot of papers in which it is recommended to fix/set the error variance to a value that expresses the researcher's theoretical assumption of

1) an appropriate/plausible error variance (most often thought of in % of the observed variance) and

2) hence the *conceptual distance* between the indicator and the latent variable. By #2, the researcher chooses and determines how similar the indicator should be from the latent variable that the researcher thinks of when specifying his/her model.

The usual counter-argument is "but you don't KNOW the true error variance". Yes, that's true but this argument is just a manifestation of the false impression spread in the behavioural science, that you can delegate critical issues to the data and some mystical statistical machine gives you the right answers. As I argue again and again on RG, there is no free lunch: If you want to get something valuable out of a causal model there is to duck-and-cover from inputting assumptions. These concern the causal structure but also the relationship between the latent variable and the measures. And yes, these assumptions may be nonsensical but they are founded and transparent for any reader to disagree. A routine zero-fixation scheme is not founded. It represents the duck-and-cover mentality.

Your mentioning of the usual recommendations to merge variables or to eliminate variables are an additional sign of this mentality and are bad practice. They stem from the intrusion of practices in pure prediction research to the realm of causal models. If your theoretical model consists of certain latent variables that represent truly existing and causally functioning attributes you have to incorporate them in the model when they are critical. Otherwise, you will get a bias up to nonsensical estimates.

Without a doubt, I would always prefer a one-indicator model consisting of a crystal clear indicator with an appropriate error variance (or a range of tested-through candidates) that is embedded in a rigid causal structure and tested in a rigid manner (with the chi-square test instead of fit indices) to the usual SEM having a handful of heterogeneous indicators which lead to misfit and even if not, cloud the meaning of the latent variables.

Having said all of this, be aware that this is not mainstream and reviewers will probably disagree!

Below find a bunch of papers in this regard.

All the best,

--Holger

Duncan, O. D., Haller, A. O., & Portes, A. P. (1968). Peer influences on aspirations: A reinterpretation. American Journal of Sociology, 74(2), 119-137. doi:10.1086/224615

Alwin, D. F., Jackson, D. J., & Jackson, E. F. (1980). Measurement models for response errors in surveys: Issues and applications. Sociological Methodology, 11(1980), 68-119. doi:10.2307/270860

Hayduk, L. A., & Littvay, L. (2012). Should researchers use single indicators, best indicators, or multiple indicators. BMC Medical Research Methodology, 12(159), 1-17. doi:10.1186/1471-2288-12-159

Hayduk, L. a., Stratkotter, R. F., & Rovers, M. W. (1997). Sexual orientation and the willingness of catholic seminary students to conform to church teachings. Journal for the Scientific Study of Religion, 36(3), 455-467.

Hayduk, L. (2018). Review essay on rex b. Kline’s principles and practice of structural equation modeling: Encouraging a fifth edition. Canadian Studies in Population, 45(3-4), 154-187.

Oberski, D. L., & Satorra, A. (2013). Measurement error models with uncertainty about the error variance. Structural Equation Modeling: A Multidisciplinary Journal, 20(3), 409-428.

Onyskiw, J. E., & Hayduk, L. a. (2001). Processes underlying children's adjustment in families characterized by physical aggression. Family Relations, 50(4), 376-385. doi:10.1111/j.1741-3729.2001.00376.x

The answer from Holger Steinmetz is clear and to the point. One additional thought: sometimes, if you are not comfortable constraining the error variance to a specific %, you might plausibly identify the model by constraining the two indicators to be "parallel measures," with equal loadings and equal (but unknown) error variance, to be estimated from the data.

I would not say that the loadings for the two-item model are "not identified." Instead, I would point out they are each forced to be the sq-root of the single correlation.

Hello David,

yes that's true in an overall identified model. In the two-indicator model, however, there are 2 additional parameters (2 error variances) (while it requires that you would estimate both loadings and fix the latent variance to a number, often 1)--and these are not all identified.

All the best,

--Holger

In your case, I advise you to use PLS instead of confirmatory factor analysis with AMOS. Because with PLS it is acceptable to use two items for a single construct. This is not the case with AMOS.

Lyes Rahmani perhaps, you should read the post in a thread before you post. Two indicators are identified when the latent variable correlate with other variables in the model.

Best,

Holger

Dear Mohammad Masukujjaman,

What to do for the lower items in a construct in SEM analysis using AMOS? May I please ask you to give a clarification of what the lower items in a constuct in SEM analysis using AMOS are? Please give some information as to how the construct in sEM analysis using AMOS? Your answer will enable us to understand the subject matter and to participate by making the necessary contributions in the discussion.

Best Regards

Desalegn Abraha

Dear Holger Steinmetz

really appreciate your simple explanation on the amount of information one has and one needs.

Best

Marcel