## The Quality of a Composite Score of a Concept with Formative Indicators

The figure presents a measurement model of a complex concept with three formative indicators.

This model is very similar to the model for the sum score with two fundamental differences: The first is that the weights in this model represent the causal effects of the latent indicators ($F_{i}$) on the complex latent variable of interest ($F$). The second is that this causal relationship is normally not perfect as in case of the sum score. There remains unexplained variance which is denoted by the disturbance term $\delta$.

This model is also not the same as the model for reflective indicators. The major difference between formative and reflective indicators comes from the nature and direction of the relationships between the latent complex concepts and their indicators. With formative indicators ($F_{1}$ to $F_{k}$) the arrows go to the complex concept. This is a fundamental difference with the concepts with reflective indicators and this has a serious consequence. The consequence is that the effects of the indicators on the complex concept cannot be estimated because the dependent variable is not measured even if the correlations between the latent variables of the indicators are known. A model with latent dependent variable is, by definition, not identified.

Several solutions have been suggested for this problem (Bollen et al. 2018) but there remain many unsolved problems:

The only solution we see, if one insists to use formative indicators, is to create a model with two or more reflective indicators of the latent concept of interest and add to this model formative latent indicators and their observed variables to estimate the effects of these variables on the concept of interest.

Using these effects coefficients as the weights and the quality and the method effects of the different observed questions, one can estimate the quality of the composite score for the latent complex concept with formative indicators. The procedure in that case is exactly the same as the procedure for the complex concept which is a sum score. Given this result we will no further discuss this issue.

Aguirre-Urreta, M.I., M.Bönkkö and G.Marakas (2016) Omission of causal Indicators; Consequences and implications for measurement – A Rejoinder.

Borsboom D., G.J. Mellenbergh and j Van Heerden (2003) The theoretical status of latent variables.

Edwards J.R. (2011) The fallacy of formative measurement.

Hardin A, Marcoulides GA. (2011) A Commentary on the Use of Formative Measurement.

Lee N. and J.W. Cadogan (2013) Problems with formative and higher –order reflective variables.

This model is also not the same as the model for reflective indicators. The major difference between formative and reflective indicators comes from the nature and direction of the relationships between the latent complex concepts and their indicators. With formative indicators ($F_{1}$ to $F_{k}$) the arrows go to the complex concept. This is a fundamental difference with the concepts with reflective indicators and this has a serious consequence. The consequence is that the effects of the indicators on the complex concept cannot be estimated because the dependent variable is not measured even if the correlations between the latent variables of the indicators are known. A model with latent dependent variable is, by definition, not identified.

Several solutions have been suggested for this problem (Bollen et al. 2018) but there remain many unsolved problems:

- The effect coefficient in the model can only be estimated if one adds to the model extra dependent variables. But with the choice of these variables the causal effects will vary and so the relationships between the latent variable of interest and the composite score obtained for this variable will vary too.
- If the complex concept really exists, which is true for concepts like "Job satisfaction", the best dependent variables to use to make the estimation of the effects of the indicators possible would be two or more reflective indicators.

The only solution we see, if one insists to use formative indicators, is to create a model with two or more reflective indicators of the latent concept of interest and add to this model formative latent indicators and their observed variables to estimate the effects of these variables on the concept of interest.

Using these effects coefficients as the weights and the quality and the method effects of the different observed questions, one can estimate the quality of the composite score for the latent complex concept with formative indicators. The procedure in that case is exactly the same as the procedure for the complex concept which is a sum score. Given this result we will no further discuss this issue.

**References**Aguirre-Urreta, M.I., M.Bönkkö and G.Marakas (2016) Omission of causal Indicators; Consequences and implications for measurement – A Rejoinder.

*Measurement: Interdiciplinary Research and Percpective,*14(4):170-175.Borsboom D., G.J. Mellenbergh and j Van Heerden (2003) The theoretical status of latent variables.

*Psychological review*. 110(2):201-219.Edwards J.R. (2011) The fallacy of formative measurement.

*Organizational Research Methods*, 14:370-388.Hardin A, Marcoulides GA. (2011) A Commentary on the Use of Formative Measurement.

*Educational and Psychological Measurement*71(5):753–64.Lee N. and J.W. Cadogan (2013) Problems with formative and higher –order reflective variables.

*Journal of Business Research*66:242-247.