Confirmatory Factor Analysis: How To Measure Something We Cannot Observe or Measure Directly

Many times in science we are intrigued to measure an underlying characteristic that cannot be observed or measured directly. This measure is hypothesized to exist to explain variables, such as behavior, that can be observed.

The measurable variables are called manifest variables. The unmeasurable are called latent variables.

Latent variables are often called factors, especially in the context of factor analysis.

Anytime we want to measure something in science we have to take into account that our measurements contains various kinds of error. That error can be random and/or systematic. So what we want to do in our statistical approach to the data is to isolate the true score in a variable and remove the error. This is really what we’re trying to do using latent variables for measurement.

In our approach to measure something accurately, we want to decompose our measure X (i.e. what we actually measured) into the true score (T) and the error (E):

X = T + E

One useful way of splitting X into T and E is simply to add the scores across a number of different X variables.

For instance, if we have four variables that are all measuring the same underlying concept then we could just add those up and take a summed score.

This approach has the benefit that the random error in each of those measurements will cancel out as we add items together. However, with this approach we assign equal weight to each item in the construction of the true score and that may not lead to the most accurate measurement.

Another approach is to actually estimate some kind of latent variable model.

In order to understand how we estimate latent variables, think about this approach through Factor Analysis. We have talked before about the conceptual and procedural differences between Exploratory and Confirmatory Factor Analysis.

Exploratory Factor Analysis (EFA) is conducted to discover what latent variables are behind a set of variables or measures.

In contrast, Confirmatory Factor Analysis is conducted to test theories and hypotheses about the factors or latent variables one expects to find. Therefore, the key difference between CFA and EFA is that we specify our measurement model before we’ve looked at our data based on theory.

The challenge with a latent variable is that it does not have a metric since it is an unobserved-hypothetical variable. In other words, it has no units.

Confirmatory Factor Analysis allows us to give a specific metric to the latent variable that makes sense. There are two approaches that we usually follow.

One approach is to essentially produce a standardized solution so that all variables are measured in standard deviation units. This can be done by constraining the variance of the latent variable to one. The downside of this approach is that we no longer have a non-standardized metric that could be given to this latent (unobserved) variable.

Another approach is to make a reference item from the group of items that make up the latent variable. Then we compare all the other items of this latent variable with the reference item. This reference item has a fixed loading for ease of comparison purposes. The value of the loading of this reference items is one.

The latter approach is more flexible as it can yield a standardized solution or an unstandardized solution.


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