Formative and reflective models to determine latent construct. Anna Simonetto
Abstract In numerous contexts, experts have to handle ordinal data and many methods have been proposed to treat this type of data. All possible approaches can be grouped into two main strands: formative and reflective models. The area of application is crucial in choosing which of two approaches to develop. We present an brief overview of the proposed models, trying to define the peculiarities of each aspect and their essential characteristics which must be verified during the analysis.
1 Introduction to formative and reflective approaches Facing a research in social sphere, it happens also for all other research areas, you have to analyze many variables (often ordinal data) with the aim to identify one (or more) underlying latent variables or simply to represent concisely the phenomenon under investigation. In literature, there are numerous proposed solutions, that vary according to the reference context and the objective of the analysis. By focusing on the adopted measurement model, we can divide thems into two main areas: formative and reflective models. Already in 1982, Fornell and Bookstein argued that "constructs such as ‘personality’ […] are typically viewed as underlying factors that give rise to something that is observed. Their indicators tend to be realized, then as reflective. On the other hand, when constructs are conceived as explanatory combinations of indicators (such as ‘population change’ or ‘marketing mix’) that are determined by a combination of variables, their indicators should be formative”. The most popular approach is the reflective model: the construct is the cause of the observed measures, so a variation in the construct leads to a variation in all its measures [1]. Constructs are phenomena that exist independently of awareness or interpretation by the researcher, even if they are not observable [6]. The observed variables are effect indicators. The classic example is the intelligence: it is the “intelligence level” of a 1
Anna Simonetto, Department of Quantitative Methods, University of Brescia;
[email protected].
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Anna Simonetto
person which determines her/his answers to a questionnaire designed to assess this aspect, not vice versa. However, there may be situations in which the observed measures are the causes of the theoretical latent variable of interest [4]. In these models, called formative model, the phenomenon is defined by, or is a function of, the observed variables: changes in the indicators determine variations in the latent variable (index), while changes in the index does not necessarily imply variations in its casual indicators. The observed variables are called causal or composite indicators and the model could be called index model [8]. Consider the socio-economic status (SES): it can be determined by several factors, such as salary, home ownership, educational qualifications and occupational prestige. If you buy a new house, you probably increase your SES, but you not necessarily increase also your qualifications or your salary. To establish the direction of the causality, we can check four conditions [4]: cause and effect must be distinct entities, there must be an association between cause and effect and a temporal precedence between the cause and its consequence, lastly we need to eliminate every other possible reason that can explain the assumed relationship between cause and effect. Figure 1: Type of models: a) direct reflective; b) indirect reflective; c) direct formative; d) indirect formative.
Obviously there are many formalizations of these models, we will present only the “pure versions”: all the relations of the model are the same type [4]. The direct reflective model (Figure 1-a) specifies the direct effects of a construct to its measure: = + . represents the i-th observed variable, is the expected effect of the j-th latent on and is the measurement error associated with . We will variable consider: and as standardized variables, uncorrelated with , , =0 = 0. When the effects of the construct latent on its for i ≠ l, and finally measures are mediated by one or more latent variables ∗ we have an indirect reflective model (Figure 1-b). It is composed of two systems of equations: ∗ ∗ = + ∗ and = + , ∗ where is the l-th "mediator" latent variable, indicates the effect of on ∗ and ∗ is the disturbance term. Figure 2-a represents the direct formative model, in which composite indicators are related causes of the phenomenon of interest (index): =∑ + . represents the i-th observed variable, expresses the contribution of the q-th observed variable to the j-th construct and is the measurement error associated with . We consider and as standardized variables, , = 0 for ∀ and = 0. Sometimes the variables have no direct impact on , but rather on one or more ∗ , which in turn influence . In this case, we have an indirect formative model (Figure 2-b), expressed by the following systems of equations: ∗ ∗ =∑ + ∗ and = ∑! + , ∗ where indicates the effect of on .
Formative and reflective models to determine latent construct
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2 Index or latent variable measurement As we have seen, when we have more observed variables, to extract the latent construct (reflective approach) or the index (formative approach) we proceed with the measurement phase [2]. One of the first checks to do concerns the internal consistency: indicators positively associated with the same concept must be positively correlated to each other. For effects indicators, consider two indicators belonging to the same construct (Figure 1-a). Since the variables are standardized, represents the with , so we have that " , # = have a correlation of # . If both the positive association with , then even " , # is certainly positive. For the formative model, since indicators are determined exogenously, we don’t know in advance the correlation between and # , or in general between any pair of composite indicators. Observed variables referring to the same concept may have a positive or negative correlation, or they could be uncorrelated. The internal consistency checking could lead to erroneous conclusions. One solution would be to check the correlation between indicators and a variable that is external to the process but we feel strongly connected with the latent phenomenon under investigation [6]. With regard to the optimum correlation of indicators, we know that in the reflective model " $ , $# = # : a high correlation means that almost one of the indicators is strongly correlated with . If the correlation is low, the reliability of the found measure is reduced, as almost one of the indicators doesn’t have high correlation with the latent variable under study. The formative model does not explain the correlation between indicators referring to the same phenomenon, so we can only observe that it is difficult to distinguish the impact that each observed variable has on the construct if composite indicators are strongly correlated with each other. Another important factor is the selection of aspects (and the choice of the indicators that represent them) related to the latent phenomenon under investigation. Obviously, we must try to identify all its possible facets and analyze each of them, but it is not easy to understand the impact of the addition or removal of an indicator with respect to the global definition of the construct. For reflective models, if we remove an indicator, the correlation of the remaining with the latent variable and the correlation between the remaining indicators do not change. So, if we maintain a sufficient number of indicators (it obviously depends on the specific considered case), the obtained measure still sufficiently represents the original different facets of the latent construct (probably the reliability of this measure decreases). For this approach, we can talk about interchangeability of effects indicators. In case of formative models, to remove an indicator has more stringent and direct consequences on the estimate of , since it is precisely a linear combination of all the composite indicators. To exclude an indicator means to exclude an aspect of the phenomenon, so composite indicators are not interchangeable. To determine which indicator refer to a particular construct, often analysts follow this empirical rule: correlation between indicators referred to the same must be greater than correlation between indicators referring to different . For reflective model, consider three indicators: and # are effects of and % is an effect of # . With all the hypotheses previously described, " , # = " , % = # and " , # . Following the rule of thumb cited above, we have %# & # , where & # = to check if " , # > " , % , or equivalently if # > %# & # . If the two latent constructs are not orthogonal (so & # ≠ 0), there may be particular pattern of values for
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, %# and & # (for example # = 0.5, : # = 0.8, & # = 0.8) for which this inequality is not met, although really and # are effects of and % is an effect of # . For composite indicators, the discussion has even less value because the model does not place restrictions on the correlation between indicators. Another fundamental aspect to is the presence of measurement errors. In reflective models, the measurement error ( ) is taken into account at the individual indicator effetc ( ). In formative models, the indicators ( ) have no error term, for which the variance of the errors is represented only by the disturbance term ( ), that is uncorrelated with . So the true variance in scores is greater than the variance of the observed values, while in reflective models occurs the opposite. One element that strongly differentiates the two approaches concerns the identifiability of the model. The formative model, if considered individually, is not identified. In order to estimate the model, it must be inserted into a larger model that incorporates the effects for the latent construct [1]. The reflective model is identified if it has at least three effect indicators. A final brief consideration regards the use of specific estimation models for these two approaches. The reflective models are widely discussed in the literature, and because of their wide dissemination, they also have a strong methodological apparatus that allows a more conscious use by analysts. It is more difficult to approach the formative models. Among the various models that allow to deal the analysis following both of these approaches there is the family of the Structural Equation Model (SEM), in particular with Partial Least Squares (PLS) estimation [3]. With ordinal data, another family of models widely used are the Rasch models. In this regard, however, the traditional software to estimate Rasch model are often not able to distinguish between latent variables that are causes or effects of their indicators [7], since both approach are basically of associative nature. As example of this confusion, often analysts interpret index variables (formative) as measure (reflective). It is important to never forget that a correctly estimated Rasch model does not provide the flow of causal direction and therefore it does not provide unambiguous evidence in interpreting latent variable.
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