The formative-reflective measurement dichotomy

I have been asked several times in the past about the formative-reflective measurement dichotomy, and whether formative measurement should be used at all. Recently there seems to be an emerging belief shared among various methodological researchers that formative measurement should not be used, under any circumstances. My view on the issue is not as extreme, and is summarized through the following text, adapted from the article listed below (whose full text is linked).

Kock, N., & Mayfield, M. (2015). PLS-based SEM algorithms: The good neighbor assumption, collinearity, and nonlinearity. Information Management and Business Review, 7(2), 113-130.

The formative-reflective measurement dichotomy is intimately related to a characteristic shared by the PLS-based SEM algorithms discussed here. These algorithms generate approximations of factors via exact linear combinations of indicators, without explicitly modeling measurement error. Recently new PLS-based SEM algorithms have been proposed that explicitly model measurement error. These new algorithms suggest that formative and reflective latent variables may be conceptually the same, but at the ends of a reliability scale, where reliability can be measured through various coefficients (e.g., Dijkstra's consistent PLS reliability, and the Cronbach’s alpha coefficient).

That is, a properly designed formative latent variable would typically have a lower reliability than a properly designed reflective latent variable. Nevertheless, both reliabilities would have to satisfy the same criterion – be above a certain threshold (e.g., .7). While reflective latent variables can achieve high reliabilities with few indicators (e.g., 3), formative latent variables require more indicators (e.g., 10). This mathematical property is in fact consistent with formative measurement theory, where many different facets of the same construct should be measured so that the corresponding formative latent variable can be seen as a complete depiction of the underlying formative construct.

Future research opportunities stem from the above discussion, leading to important methodological questions. What is the best measure of reliability to be used? It is possible that the composite reliability coefficient is a better choice than the Cronbach’s alpha coefficient, under certain circumstances. Will the new PLS-based SEM algorithms that explicitly model measurement error (i.e., factor-based PLS algorithms, a.k.a. PLSF algorithms) obviate the need for the classic composite-based algorithms, or will the new algorithms have a more limited scope of applicability? Will formative measurement be re-conceptualized as being at the low end of a reliability scale that also includes reflective measurement, providing a unified view of what could be seen as an artificial dichotomy? These and other related methodological questions give a glimpse of the exciting future of PLS-based SEM.

WarpPLS’ treatment of formative latent variables: PLS regression is more conservative and stable

For a more detailed discussion of the issues addressed in this post, please see the  publication below.

Kock, N., & Mayfield, M. (2015). PLS-based SEM algorithms: The good neighbor assumption, collinearity, and nonlinearity. Information Management and Business Review, 7(2), 113-130.

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WarpPLS implements a number of composite-based and factor-based algorithms. One of the composite-based algorithms is often referred to as Wold’s original “PLS regression” algorithm to calculate indicator weights, for both formative and reflective variables. PLS regression was developed by Wold, and is slightly different from the modified versions often referred to as modes A and B, which are the ones normally used in other publicly available PLS-based structural equation modeling software. These modified versions implement an underlying algorithmic assumption that Lohmöller called the "good neighbor" assumption, whereby weights are influenced by inner model links.

Generally speaking, the PLS regression algorithm generates coefficients that are more stable and robust – i.e., reliable for hypothesis testing. It also tends to minimize collinearity. On the other hand, it may be lead to a higher demand for computational power in some cases, which may be the reason why modified versions have been implemented. Lohmöller discusses multiple algorithm versions, with some characteristics placing them within broad types called “modes” – see Lohmöller (1989), the PLS "bible", for more details. Personal computers were not that powerful in the 1980s.

Moreover, the type of nonlinear treatment employed by WarpPLS is difficult to perform with Lohmöller’s underlying algorithm (the "good neighbor" assumption), whereby the outer model is influenced by the inner model. The problem is that with Lohmöller’s algorithm, as a model changes, the weights and loadings also change, even if the latent variables do not change. That is, with Lohmöller’s algorithm, two models with the same latent variables but different structures (i.e., links among latent variables) will have different weights and loadings.

The weights of formative latent variables will be essentially the same in WarpPLS as they would be if the variables were defined as reflective. That is, they will be obtained by an iterative algorithm that stops when two conditions are met: (a) the weights between indicators and latent variable are standardized partial regression coefficients calculated with the indicators as independent variables and the latent variable as the dependent variable; and (b) the regression equation expressing the latent variable as a combination of the indicators has an error term of zero.

So why should the user define a latent variable as formative or reflective? The reason are the interpretations of the outputs generated by the software. When a latent variable is formative, both the P values for the weights and the variance inflation factors for the indicators should be generally low; ideally below 0.05 and 2.5, respectively.

True formative variables are fundamentally different from true reflective variables; there are cases that can be seen as “in between” formative and reflective. True formative and reflective variables behave differently, whether the software treats them differently or not. For example, with true formative variables you would expect indicators to be significantly associated with the scores of their respective latent variable; which is indicated by low P values for their weights. However, you would not normally expect the indicators to be redundant; which is indicated by low variance inflation factors for the indicators.

The way formative variables are treated in Lohmöller’s approach leads to unstable weights, with the signs of weights frequently changing in the resample set. See Temme et al. (2006) for a discussion on this phenomenon. Lohmöller’s approach also leads to “lateral” collinearity; or collinearity between predictor and criteria latent variables. This “stealth” type of collinearity often leads to inflated path coefficients for links involving formative latent variables.

Formative variables don't "become reflective", or vice-versa, if one or another algorithm is used. This is a common misconception among users of PLS-based SEM software.

References

Kock, N., & Mayfield, M. (2015). PLS-based SEM algorithms: The good neighbor assumption, collinearity, and nonlinearity. Information Management and Business Review, 7(2), 113-130.

Lohmöller, J.-B. (1989). Latent variable path modeling with partial least squares. Heidelberg, Germany: Physica-Verlag.

Temme, D., Kreis, H., & Hildebrandt, L. (2006). PLS path modeling – A software review. Berlin, Germany: Institute of Marketing, Humboldt University Berlin.

Use formative latent variables with caution

One should use formative latent variables (LVs) with caution in structural equation modeling analyses using WarpPLS. It is not uncommon to see formative LVs being created simply by casually aggregating indicators, without much concern about the indicators being actually facets of the same construct. See this post for more details.

It is also important to stress that formative LVs are better assessed when included as part of a model. This is preferable to analyzing formative LVs individually; that is, as “models” that include one single LV. The loadings and cross-loadings table takes into consideration both formative and reflective LVs in its calculation, and may suggest that some indicators do not “belong” to a formative LV.

Also, certain model parameters may become unstable due to collinearity. High collinearity among indicators is to be expected in reflective LV measurement, but not in formative LV measurement. In the context of formative LV assessment, collinearity may be reflected in unstable weights, where unexpected P values (usually statistically non-significant) are associated with weights.

In formative LVs, indicators are expected to measure different facets of the LV, not the same thing. If two (or more) indicators are collinear in a formative LV, it may be a good idea to collapse them into one indicator. This can be done by defining second order LVs (a two-step, somewhat complex procedure), averaging the indicators, or simply eliminating one of the indicators from the analysis.

Second order latent variables in WarpPLS: YouTube videos by Jaime León

The blog post below refers to a procedure employed with earlier versions of WarpPLS. For a more recent, and less time-consuming, approach see the video linked immediately below. The video shows how to create and use second (and higher) order latent variables with WarpPLS.

http://youtu.be/bkO6YoRK8Zg

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The YouTube videos below have been created by WarpPLS user and blog commenter Jaime León. They illustrate how steps 1 and 2, described in this post, can be implemented in WarpPLS. The goal of those steps is to use second order latent variables (LVs) in an SEM analysis. Latent variable (LV) scores are generated, saved, and then used in a subsequent SEM analysis.

Step 1: YouTube video 1.

Step 2: YouTube video 2.

In the first video Jaime includes only LVs in the model, without any links among them, and then runs the SEM analysis. This generates the LV scores for the LVs, which Jaime then saves into a .txt file. The LV scores generated are then combined with indicators from the original dataset.

Note that Jaime does not set the LVs in the first video as formative before generating the scores. That is okay if the LVs are reflective; that is, if the indicators of the LVs are highly correlated. (In reflective LVs the loadings are expected to be all high, ideally greater than .7, and significant.) If not, then the LVs should be set as formative.

Also, note that Jaime combined the LV scores in standardized format with indicator data from the original dataset, which were not standardized. That is fine because WarpPLS always standardizes the raw data before proceeding to an SEM analysis. Standardized data, when used as input, will not be affected by standardization (since they are already standardized).

In the second video Jaime creates a model with new LVs, some of which include the previously generated LV scores as indicators. These are frequently referred to as second order LVs. (Although sometimes the original LVs, shown in the first video, are the ones called second order LVs.) Jaime then builds a model by creating several direct links among the LVs.

Cool example, with a Bob Marley song in the background; thanks Jaime!

Reflective and formative latent variable measurement in WarpPLS

A reflective latent variable is one in which all the indicators are expected to be highly correlated with the latent variable score. For example, the answers to certain question-statements by a group of people, measured on a 1 to 7 scale (1=strongly disagree; 7 strongly agree) and answered after a meal, are expected to be highly correlated with the latent variable “satisfaction with a meal”. The question-statements are: “I am satisfied with this meal”, and “After this meal, I feel good”. Therefore, the latent variable “satisfaction with a meal”, can be said to be reflectively measured through two indicators. Those indicators store answers to the two question-statements. This latent variable could be represented in a model graph as “Satisf”, and the indicators as “Satisf1” and “Satisf2”.

A formative latent variable is one in which the indicators are expected to measure certain attributes of the latent variable, but the indicators are not expected to be highly correlated with the latent variable score, because they (i.e., the indicators) are not expected to be highly correlated with one another. For example, let us assume that the latent variable “Satisf” (“satisfaction with a meal”) is now measured using the two following question-statements: “I am satisfied with the main course” and “I am satisfied with the dessert”. Here, the meal comprises the main course, say, filet mignon; and a dessert, a fruit salad. Both main course and dessert make up the meal (i.e., they are part of the same meal) but their satistisfaction indicators are not expected to be highly correlated with each other. The reason is that some people may like the main course very much, and not like the dessert. Conversely, other people may be vegetarians and hate the main course, but may like the dessert very much.

If the indicators are not expected to be highly correlated with one anoother, they cannot be expected to be highly correlated with their latent variable’s score. So here is a general rule of thumb that can be used to decide if a latent variable is reflectively or formatively measured. If the indicators are expected to be highly correlated, then the measurement model should be set as reflective in WarpPLS. If the indicators are not expected to be highly correlated, even though they clearly refer to the same latent variable, then the measurement model should be set as formative.