I have read the paper "Testing Measurement Invariance of Composites Using Partial Least Squares". Also, I have consulted other related papers (Henseler et al. 2016) and (Rigdon, 2012).

Regarding the issue related to the composites and factors, what I understood from the paper "Testing measurement invariance of composites using partial least squares", is that all constructs in PLS are composites, and these composites can be built on either formative or reflective measurement models. Also, in the empirical example section, the authors used the MICOM model invaraince test with formative and reflective constructs.

**On page 20 and 21:**

"

*We use the corporate reputation model (Schwaiger, 2004), as shown with its latent variables in*

Figure 5, to provide a MICOM example with empirical data......... While the exogenous latent variables

(i.e., QUAL, PERF, CSOR, ATTR) represent composites that build on a formative measurement model

(Mode A), the endogenous latent variables (i.e., LIKE, COMP, CUSL) are composites with a reflective

measurement model (Mode A);"

Figure 5, to provide a MICOM example with empirical data......... While the exogenous latent variables

(i.e., QUAL, PERF, CSOR, ATTR) represent composites that build on a formative measurement model

(Mode A), the endogenous latent variables (i.e., LIKE, COMP, CUSL) are composites with a reflective

measurement model (Mode A)

However, what I understood from Henseler et al 2016 paper "PLS path modeling in new technology research: updated guidelins", is that PLS can contain two types of constructs, namely, factors and composites.

**On page 4:**

"

*PLS path models can contain two different forms of construct measurement: factor*

models or composite models (see Rigdon, 2012, for a nice comparison of both types

of measurement models). The factor model hypothesizes that the variance of a set of

indicators can be perfectly explained by the existence of one unobserved variable (the

common factor) and individual random error. It is the standard model of behavioral

research. In Figure 1, the exogenous construct ξ and the endogenous construct η are

modeled as factors. In contrast, composites are formed as linear combinations of their

respective indicators. The composite model does not impose any restrictions on the

covariances between indicators of the same construct, i.e. it relaxes the assumption that

all the covariation between a block of indicators is explained by a common factor.

The composites serve as proxies for the scientific concept under investigation

(Ketterlinus et al., 1989; Rigdon, 2012; Maraun and Halpin, 2008; Tenenhaus, 2008)[1].

The fact that composite models are less restrictive than factor models makes it likely

that they have a higher overall model fit (Landis et al., 2000).".

models or composite models (see Rigdon, 2012, for a nice comparison of both types

of measurement models). The factor model hypothesizes that the variance of a set of

indicators can be perfectly explained by the existence of one unobserved variable (the

common factor) and individual random error. It is the standard model of behavioral

research. In Figure 1, the exogenous construct ξ and the endogenous construct η are

modeled as factors. In contrast, composites are formed as linear combinations of their

respective indicators. The composite model does not impose any restrictions on the

covariances between indicators of the same construct, i.e. it relaxes the assumption that

all the covariation between a block of indicators is explained by a common factor.

The composites serve as proxies for the scientific concept under investigation

(Ketterlinus et al., 1989; Rigdon, 2012; Maraun and Halpin, 2008; Tenenhaus, 2008)[1].

The fact that composite models are less restrictive than factor models makes it likely

that they have a higher overall model fit (Landis et al., 2000).

**On page 6:**

"

*In some PLS path modeling software (e.g. SmartPLS and PLS-Graph), the depicted*

direction of arrows in the measurement model does not indicate whether a factor or

composite model is estimated, but whether correlation weights (Mode A, represented by

arrows pointing from a construct to its indicators) or regression weights (Mode B,

represented by arrows pointing from indicators to their construct) shall be used to create

the proxy. In both cases PLS will estimate a composite model. Indicator weights estimated

by Mode B are consistent (Dijkstra, 2010) whereas indicators weights estimated by

Mode A are not, but the latter excel in out-of-sample prediction (Rigdon, 2012)."

direction of arrows in the measurement model does not indicate whether a factor or

composite model is estimated, but whether correlation weights (Mode A, represented by

arrows pointing from a construct to its indicators) or regression weights (Mode B,

represented by arrows pointing from indicators to their construct) shall be used to create

the proxy. In both cases PLS will estimate a composite model. Indicator weights estimated

by Mode B are consistent (Dijkstra, 2010) whereas indicators weights estimated by

Mode A are not, but the latter excel in out-of-sample prediction (Rigdon, 2012)

In addition, what I understood is that MICOM is suitable for composites, and there are other model invariance tests for factor models.

**On page 13:**

"

*There is a plethora of papers discussing how to assess the measurement invariance of factor*

models (see e.g. French and Finch, 2006), there is only one approach for assessing the

measurement invariance of composite models (Henseler et al., forthcoming)."

models (see e.g. French and Finch, 2006), there is only one approach for assessing the

measurement invariance of composite models (Henseler et al., forthcoming).

From the above, the distinction between reflective constructs and factor models is not clear to me.

1. What is the real difference between reflective constructs and factor models? What would be a reflective composite and what would be a factor? And how the difference is transferred to the PLS context in terms of model specification?

2. Can I use the MICOM for both reflective and formative constructs? if not, how can I test for measurement model invariance test for reflective constructs. I could not find a suitable model invariance test for reflective constructs that is available in SmartPLS 3. The program provides the permutation p values for the difference between the two groups in terms of, for example, AVE and composite reliability. Should I use these for reflective constructs? Isn't similar to Ringle et al. (2011) approach when they tested for measurement invariance for their model's reflective constructs.

Thanks,

Henseler, J., Hubona, G., & Ray, P. A. (2016). Using PLS path modeling in new technology research: updated guidelines. Industrial Management & Data Systems, 116(1), 2-20

Henseler, J., Ringle, C.M. and Sarstedt, M. (forthcoming), “Testing measurement invariance of

composites using partial least squares”, International Marketing Review (in print).

Rigdon, E.E. (2012), "Rethinking partial least squares path modeling: in praise of simple

methods", Long Range Planning, Vol. 45 Nos 5/6, pp. 341-358.

Ringle, C. M., Sarstedt, M. and Zimmermann, L. (2011), "Customer Satisfaction with Commercial

Airlines: The Role of Perceived Safety and Purpose of Travel", Journal of Marketing Theory and

Practice, Vol. 19 No. 4, pp. 459-472.