How to Measure an Unobserved Construct in SEM Model

With an unobservable construct, we are trying to use indicators to measure the concept.With any unobservable analysis, you will rarely be able to say you are capturing the “true” or actual score without some degree of error. Thus, an indicator is a function of the “true” score plus error:

Construct Indicator =True or Actual Score of Concept + Error in Measuring the Concept Or, put in mathematical form: X = A + e

With this formula, the construct indicator (X) is the measured item which is going to cap- ture the “true” value of the concept (A) while also accounting for any error in the meas- urement (e). You will often see unobservable constructs using a single measured indicator to measure the concept. This can be problematic for a number of reasons. With a single measure, you are making the claim that a single item can capture the “true” value of an unobservable construct. Referring back to our formula of X = A + e, you cannot deter- mine the error of a single indicator measure.You have two unknowns (A and e) with only one known (X). The formula is unsolvable. Thus, with a single indicator, the error is ignored, and the formula is converted to X = A. Obviously, this is a big leap of faith that a single observed indicator can perfectly measure an unobservable construct. For this reason, you will often see unobservable constructs measured with multiple indicators, which have a higher likelihood of capturing the “true” or actual score of the unobservable. Additionally, multiple indicators allow us enough information to determine the error of each indicator.Thus, we are using more indicators to capture a potentially complex concept and at the same time accounting for error in the measurement. Let’s look at how single and multiple indicators of an unobservable construct are represented in SEM.

Single Indicators—one observed item/ indicator is used to capture the entire unobserv- able construct. In the representation shown in Figure 1.7, the unobserved construct of “A” is going to represent the “true” or actual score of the unobserved concept. The single indicator of “X1” is a measure of the concept and “e1” represents the error in measuring the unobserved construct in the single indicator. Again, if you use a single item to measure an unobservable construct, you cannot account for error.You are saying this indi- cator is a perfect measure and has no unexplained variance.

Multiple Indicators—more than one observed item is used to capture the construct. In the example shown in Figure 1.8, the unobserv- able construct “A” is being captured or measured by three indicators (X1–X3). The rationale is that complex constructs cannot be fully captured with- out multiple indicators. As well, multiple indicators aid the researcher in understanding the reliability and validity of the captured unobserved variable. Lastly, you can determine the measurement error of each indicator, giving you a better understanding of whether your indicators/items are capturing the unobservable concept.

Measurement    Model    vs.    Structural

Model—the measurement model in SEM is where the researcher is going to assess the validity of the indicators for each construct. After showing the validity of the measurement model, the researcher can proceed to the structural model.The structural model is concerned with the influence and significance between constructs. The term “full structural model” means that the measurement and structural relationships of each construct are included in the model testing.

Figure 1.9

Parameters—the term “parameter” indicates the size and nature of the relationship between two objects in a model. Parameters can be fixed to a constant or can be estimated freely from the data. A parameter estimate will take place on the measurement level with indicators and error terms as well as on the structural level between constructs.

Source: Thakkar, J.J. (2020). “Procedural Steps in Structural Equation Modelling”. In: Structural Equation Modelling. Studies in Systems, Decision and Control, vol 285. Springer, Singapore.