**3. Measurement and structural models**

In medicine and health sciences, it is common to use one of the four types of SEM that exist in the literature. Next, each one is briefly described.

**Path analysis models:** This method of SEM only includes observed variables, similar that multiple regression models (MRM), but it has the advantage that a variable can be a dependent variable and an independent variable at the same time, in addition, there can be several dependent variables, and the indirect and direct effects can be measured (**Figure 2**). In **Figure 2** (below), the Socioeconomic Status (SES) and Disease, both exogenous variables, represent direct effects on the endogenous variable obesity.

**Confirmatory Factor Analysis models (CFA):** The CFA, like the measurement model, analyzes the relation among latent and observed variables, emphasizing that the theoretical factorial structure predetermined by the researcher is confirmed by the data; that is, it must be predetermined to which factor the observed variables will be loaded and the CFA will be useful to confirm or not the default structure.

**Structural Regression Model:** This method of SEM is a regression model between latent variables. The idea consists on to combine the techniques of CFA and MRM, further include the measurement errors.

The causal relationships between latent variables are represented by the directional arrows according to the hypothetical model. Typically, model fit indices are examined first, followed by hypothesis tests. The latent growth curve model is a statistical technique of longitudinal analysis that estimates or explains the growth over a period of time.

In this chapter, we will only address the Structural Regression Model.

#### **3.1 Measurement model**

This part of the path diagram is necessary to analyze all the items or observed variables that are "loaded" in the latent variable, their variances, and errors, as well as the relation between the observed variables.

The measurement model quantifies linkages among the latent variables and observed variables that characterize the hypothetical model.

The latent variables are representations of the concepts of interest. Previously the concept is selected, Bollen [4] recommends for the measurement process: (1) Determine its meaning, (2) Represent it with latent variables, (3) Form measures, and (4) Establish the relation among latent variables and measures variables.

The measurement model analyzes the relation between the measure and latent variables. The latent variable is the representation of a concept. This relation can be described or represented by an equation or in a path diagram (**Figure 3**).

The CFA is a method for evaluating a measurement model. Klein [1] mentioning Bollen suggests applying some rules to ensure the identification of the measurement model re specifying it as a CFA. When it comes to a CFA, a factor must have at least three observed variables, when there are two or more factors or latent variables each factor must have at least two observed variables.

#### **Figure 2.**

*The path diagram above represents the indirect effect between FAT and LV diastolic dysfunction, and the path diagram below shows the direct effects of socioeconomic status and obesity, and between disease and obesity [3].*

*The Basics of Structural Equations in Medicine and Health Sciences DOI: http://dx.doi.org/10.5772/intechopen.104957*

#### **Figure 3.**

*A path diagram of the CFA model on Matsuda index with 11 observed variables: Percentage of FAT (FAT), body mass index (BMI), abdominal circumference (AC), arginine (ARG), glycine (GLY), leucine (LEU), phenylalanine (PHE), valine (VAL), liver ultrasound (USG), alanine aminotransferase (ALT), aspartate aminotransferase (AST); and 3 latent variables: Amino acids (AA1), fatty liver, and obesity [5].*

### **3.2 Structural regression model**

The path diagram of structural regression (SR) includes the set of latent variables and their relationships. Unlike the measurement model (CFA) where all the factors or latent variables are exogenous and can be assumed to covary or have a dependency, the causal effects between latent variables are described only in the SR. Causal inference in latent variable modeling is more laborious than measurement model analysis. In SR models the effects between latent variables can also be direct or indirect. Similarly, the structural component can also be recursive or non-recursive. A recursive SR is a model in which causation is directed in one single direction, while a nonrecursive structural model has causality going in both directions on some variables.

#### **3.3 Identification of SR model**

Identification of the SR model is analogous to the identification of the measurement model. However, before validating the SR, the measurement model needs to be identified (i.e., valid) and then evaluate the fully SEM model. The only valid identification of the CFA does not guarantee the identification of the SR.

Therefore, the analysis of a fully SEM must include the variances and covariances between the factors or latent variables A fully SR model is identified by [4]: (1) In the first, the researcher must analyze the measurement model as a CFA, that is, ignore in the analysis the relations among the latent variables of the SR model. After reformulating the model, discover if the model is identified. If identification is obtained, apply it to the second step; (2) in the second step, you must analyze the equation or equations that contain the relation among the latent variables of the SR model must be analyzed and

#### **Figure 4.**

*SEM to analyze relationships between adiposity, inflammatory responses, LV diastolic dysfunction. Fasting plasma glucose (FPG); high-density lipoprotein (HDL); homeostasis model of insulin resistance (HOMA); high sensitivity C-reactive protein (hsCRP); peritoneum fat area (Peri fat); retroperitoneum fat area (retro fat); subcutaneous fat area (sub fat); triglyceride (TG). The latent variable FAT directly influences the inflammation variable and indirectly on the observed variable LV diastolic [6].*

then determine if the SR model is identified, assuming that the latent variables are observed variables. If in step 1 it is proved that the measurement parameters are identified and in step 2 that the parameters of the SR are also identified, both conditions are sufficient to fully identify the SR model. **Figure 4** shows a path diagram of a complete SEM, which includes 9 observed variables and two latent variables. The objective is to analyze the relationship among central obesity (FAT), systemic inflammation (Inflammation), and left ventricular diastolic dysfunction (LV diastolic). This figure does not show the variances or the disturbances or errors.
