Principal component and factor analysis

Principal component analysis and factor analysis are some of the most confusing techniques in statistics, perhaps because they are very similar in so many ways. On the surface, these two methods appear to be the same, yet there is a huge difference between them that can significantly affect how you use them for data analysis. Below are some of the most notable similarities:

  • They are both data reduction methods, meaning, they allow you to determine the variance in smaller sets of variables
  • They are both run in a statistics program using a similar procedure and the resulting output looks relatively the same
  • They both use the same steps to select the number of components or factors; extraction, interpretation, and rotation

However, despite the techniques displaying all these resemblances, there is a substantial difference between them – principal component analysis measures the linear combination of variables while factor analysis measures a latent or hidden variable. Let’s look into each of them in detail.

Principal component analysis

Principal component analysis is a statistical method used when there are lots of variables to deal with. It minimizes the dimensions of the data being analyzed to help you understand the data better and plot it with a lesser dimension than the initial data. As the name suggests, principal component analysis enables researchers to study principal components in data. So, what exactly are principal components, you may ask? They are simply linearly uncorrelated vectors that have a variance within data.

When should you use principal component analysis?

  • When you want to decrease the number of variables but can’t identify the exact variables that you want to remove
  • When you want to ascertain that the variables of the data you are analyzing do not depend on each other
  • When you want to make your independent variables more interpretable

Properties of principal component analysis

Principal components are obtained by solving a specific optimization problem, and therefore, they have some naturally built in properties that data scientists find desirable. They include the following:

  • The variances of each component are given by the eigenvalues and so is the proportion of the combined variance of the initial variables
  • Component values may be computed. This helps display the value of each component in every observation
  • Component loadings that illustrate the relationship between each variable and each component may also be computed

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Factor analysis

Factor analysis is the process of taking a large set of data and shrinking it into a smaller, more understandable and more manageable set. It helps researchers find hidden patterns of data, see how these patterns overlap and find out what characteristics are exhibited in multiple patterns. Due to this capability, factor analysis has become an essential tool in studying complex data sets involving socioeconomic status, psychological studies, and other intricate concepts.

A factor can be defined as a set of variables that have the same response patterns. There are two types of factors and hence two types of factor analyses:

  • Exploratory factor analysis: Performed when one does not know what structure the data being observed takes or how many dimensions are present in a given set of variables.
  • Confirmatory factor analysis: Carried out to verify the structure of data as well as the number of dimensions in a particular set of variables.

Latent variable

Factor analysis is said to be a measurement model of latent or hidden variable. A latent variable is generally a variable that cannot be observed or measured directly. For instance, an individual’s level of anxiety, openness, or neurosis are all hidden variables. Even though you cannot see these variables because they are usually not considered part of the data obtained from an experiment, they can affect the results obtained from the experiment. Latent variables are also referred to as:

  • Factors
  • Hypothetical constructs
  • Hypothetical variables
  • True scores
  • Unobserved variables
  • Unmeasured variables

Some of the statistical modeling techniques used to determine latent variables include:

  • Factor analysis
  • Expectation maximization algorithms
  • Latent semantic analysis
  • Hidden Markov models
  • Structural equation modeling
  • Principal component analysis

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