The fundamental premise of Structural Equation Modeling (SEM) is that a market researcher "can test whether certain variables are interrelated through a set of linear relationships by examining the variances and covariances of the variables" (StatSoft, 2011) This is perhaps one of the clearest statements about SEM, If you understand the terms used in the sentence:
- Variable - (Noun) According to Merriam-Webster: "1). An element or factor that is liable to vary or change; 2) A quantity that during a calculation is assumed to vary or be capable of varying in value."
- Linear Relationship - According to Investopedia: In the simplest of terms, "the relationship between a variable and a constant that can be expressed in a graphic in which a constant and a variable are connected by a straight line." An example would be the cost of sailboats that increases in a linear fashion as one moves up the line to larger and larger vessels as measured by square footage.
- Variance - According to the Business Dictionary: "1) The difference between an expected result and the actual result; 2) In statistics, the arithmetic mean of the squares of the deviation of all values in a set of numbers from their arithmetic mean. Variance and its square root (the standard deviation) are of fundamental importance as a measure of dispersion."
- Variable Covariance - According to Merriam-Webster: "In statistics and probability theory, covariance is a measure of how much two variables change together."
The SEM Is Based on Structure Which Is Based on Mathematics
This first step in the SEM process is basically one of the market researcher stating -- or drawing, through the use of a path diagram -- the way that she/he believes the variables are inter-related.
It may help to think about the effect of additive and multiplicative transformations. For example, if a list of numbers is multiplied by a constant K, the mean and the standard deviation are also multiplied by the absolute value of K. It's automatic. With numbers, it looks like this:
For numbers 1,2,& 3: The mean is 2, and the standard deviation is 1. Say K = 4. Multiplying 1, 2, & 3 by K results in 4, 8, & 12. For 4, 8, & 12, the mean is 8, and the standard deviation is 4. The variance is 16. Remember, " variance is a measure of how far each value in the dataset is from the mean." Hence, the standard deviation squared.
Since you know that the two sets of numbers are related, and you know what the variance is, you can indirectly test the hypothesis that one set of numbers is related to the other set of numbers by comparing the variance of the variables.
The information on Structural Equation Modeling below is based on content from the book by R. H. Hoyle (ed.) 1995. Structural Equation Modeling. SAGE Publications, Inc. Thousand Oaks, CA courtesy of Google Books, and also on the gracious interpretation of complex writing about SEM by Ricka Stoelting, formerly of San Francisco State University.
In the model specification step, the model is defined in terms of its parameters. Two kinds of parameters are considered: Fixed parameters and free parameters.
Parameters Designated Fixed or Free
Identifying which parameters are fixed and which parameters are free is critical to the integrity and application of the SEM model. The fixed or free designations determine how the components of the model will be compared. The model components are 1) The hypothesized diagram, 2) the sample population variance, and 3) the covariance matrix. Each of these components is important for testing the fit of the model.
The market researcher determines which parameters are designated free and which parameters are designated fixed. The choices made by the market researcher are a reflection of the a priori hypothesis means that the "from the former" in Latin, so it refers to the hypothesis made before the research or experiment has taken place. So an a priori hypothesis is the best guess about the relationships to be explored through the SEM process.
The market researcher makes a best guess about which pathways will be important in the relational structure. The market researcher surmises which parameters will play a part in the sample variance (which is observable) and in the covariance matrix. In other words, where does the market researcher expect the relationships to occur?
A fixed parameter is generally established at zero. Zero means that there is not a relationship between the variables. Because the model is based on paths, the fixed parameters will have paths that have numerical labels. An exception, of course, occurs if a value of zero has been assigned to a path. No path is drawn in the SEM diagram for a path with a value of zero.
A market researcher expects the free parameters to have values other than zero. The free parameters are estimated from the data that is observable. In the SEM diagram, the paths of the free parameters are marked with asterisks.