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Factor Analysis

Correlated Errors in Confirmatory Factor Analysis

by Jeff Meyer 3 Comments

Latent constructs, such as liberalism or conservatism, are theoretical and cannot be measured directly.

But we can use a set of questions on a scale, called indicators, to represent the construct together by combining them into a latent factor.

Often prior research has determined which indicators represent the latent construct. Prudent researchers will run a confirmatory factor analysis (CFA) to ensure the same indicators work in their sample.

[Read more…] about Correlated Errors in Confirmatory Factor Analysis

Tagged With: Confirmatory Factor Analysis, error term, Factor Analysis, latent variable, Model Fit

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Life After Exploratory Factor Analysis: Estimating Internal Consistency

by guest contributer 2 Comments

by Christos Giannoulis, PhD

After you are done with the odyssey of exploratory factor analysis (aka a reliable and valid instrument)…you may find yourself at the beginning of a journey rather than the ending.

The process of performing exploratory factor analysis usually seeks to answer whether a given set of items form a coherent factor (or often several factors). If you decide on the number and type of factors, the next step is to evaluate how well those factors are measured.

[Read more…] about Life After Exploratory Factor Analysis: Estimating Internal Consistency

Tagged With: Coefficient alpha, Cronbach's alpha, Exploratory Factor Analysis, Factor Analysis, latent variable, reliability, scale reliability

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Member Training: Confirmatory Factor Analysis

by guest contributer Leave a Comment

There are two main types of factor analysis: exploratory and confirmatory. Exploratory factor analysis (EFA) is data driven, such that the collected data determines the resulting factors. Confirmatory factor analysis (CFA) is used to test factors that have been developed a priori.

Think of CFA as a process for testing what you already think you know.

CFA is an integral part of structural equation modeling (SEM) and path analysis. The hypothesized factors should always be validated with CFA in a measurement model prior to incorporating them into a path or structural model. Because… garbage in, garbage out.

CFA is also a useful tool in checking the reliability of a measurement tool with a new population of subjects, or to further refine an instrument which is already in use.

Elaine will provide an overview of CFA. She will also [Read more…] about Member Training: Confirmatory Factor Analysis

Tagged With: a priori, Confirmatory Factor Analysis, Factor Analysis, reliability, Structural Equation Modeling

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The Fundamental Difference Between Principal Component Analysis and Factor Analysis

by Karen Grace-Martin 25 Comments

One of the many confusing issues in statistics is the confusion between Principal Component Analysis (PCA) and Factor Analysis (FA).

They are very similar in many ways, so it’s not hard to see why they’re so often confused. They appear to be different varieties of the same analysis rather than two different methods. Yet there is a fundamental difference between them that has huge effects on how to use them.

(Like donkeys and zebras. They seem to differ only by color until you try to ride one).

Both are data reduction techniques—they allow you to capture the variance in variables in a smaller set.

Both are usually run in stat software using the same procedure, and the output looks pretty much the same.

The steps you take to run them are the same—extraction, interpretation, rotation, choosing the number of factors or components.

Despite all these similarities, there is a fundamental difference between them: PCA is a linear combination of variables; Factor Analysis is a measurement model of a latent variable.

Principal Component Analysis

PCA’s approach to data reduction is to create one or more index variables from a larger set of measured variables. It does this using a linear combination (basically a weighted average) of a set of variables. The created index variables are called components.

The whole point of the PCA is to figure out how to do this in an optimal way: the optimal number of components, the optimal choice of measured variables for each component, and the optimal weights.

The picture below shows what a PCA is doing to combine 4 measured (Y) variables into a single component, C. You can see from the direction of the arrows that the Y variables contribute to the component variable. The weights allow this combination to emphasize some Y variables more than others.

This model can be set up as a simple equation:

C = w1(Y1) + w2(Y2) + w3(Y3) + w4(Y4)

Factor Analysis

A Factor Analysis approaches data reduction in a fundamentally different way. It is a model of the measurement of a latent variable. This latent variable cannot be directly measured with a single variable (think: intelligence, social anxiety, soil health).  Instead, it is seen through the relationships it causes in a set of Y variables.

For example, we may not be able to directly measure social anxiety. But we can measure whether social anxiety is high or low with a set of variables like “I am uncomfortable in large groups” and “I get nervous talking with strangers.” People with high social anxiety will give similar high responses to these variables because of their high social anxiety. Likewise, people with low social anxiety will give similar low responses to these variables because of their low social anxiety.

The measurement model for a simple, one-factor model looks like the diagram below. It’s counter intuitive, but F, the latent Factor, is causing the responses on the four measured Y variables. So the arrows go in the opposite direction from PCA. Just like in PCA, the relationships between F and each Y are weighted, and the factor analysis is figuring out the optimal weights.

In this model we have is a set of error terms. These are designated by the u’s. This is the variance in each Y that is unexplained by the factor.

You can literally interpret this model as a set of regression equations:

Y1 = b1*F + u1
Y2 = b2*F + u2
Y3 = b3*F + u3
Y4 = b4*F + u4

As you can probably guess, this fundamental difference has many, many implications. These are important to understand if you’re ever deciding which approach to use in a specific situation.

Tagged With: data manipulation, Factor Analysis, latent variable, principal component analysis, Statistical analysis

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  • How To Calculate an Index Score from a Factor Analysis
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Structural Equation Modeling: What is a Latent Variable?

by guest contributer 6 Comments

By Manolo Romero Escobar

What is a latent variable?

“The many, as we say, are seen but not known, and the ideas are known but not seen” (Plato, The Republic)

My favourite image to explain the relationship between latent and observed variables comes from the “Myth of the Cave” from Plato’s The Republic.  In this myth a group of people are constrained to face a wall.  The only things they see are shadows of objects that pass in front of a fire [Read more…] about Structural Equation Modeling: What is a Latent Variable?

Tagged With: Factor Analysis, latent variable, SEM, Structural Equation Modeling

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How To Calculate an Index Score from a Factor Analysis

by Karen Grace-Martin 13 Comments

One common reason for running Principal Component Analysis (PCA) or Factor Analysis (FA) is variable reduction.

In other words, you may start with a 10-item scale meant to measure something like Anxiety, which is difficult to accurately measure with a single question.

You could use all 10 items as individual variables in an analysis–perhaps as predictors in a regression model.

But you’d end up with a mess.

Not only would you have trouble interpreting all those coefficients, but you’re likely to have multicollinearity problems.

And most importantly, you’re not interested in the effect of each of those individual 10 items on your [Read more…] about How To Calculate an Index Score from a Factor Analysis

Tagged With: Factor Analysis, Factor Score, index variable, PCA, principal component analysis

Related Posts

  • Four Common Misconceptions in Exploratory Factor Analysis
  • In Factor Analysis, How Do We Decide Whether to Have Rotated or Unrotated Factors?
  • Can We Use PCA for Reducing Both Predictors and Response Variables?
  • The Fundamental Difference Between Principal Component Analysis and Factor Analysis

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