What are the main differences between components analysis and factor analysis? How might you decide to use one of these or the other in a research study? What differences might you expect in the results? Define the term as it applies to factor analysis. What is the major difference between orthogonal and oblique rotation? What are the advantages and disadvantages of each? Why would a researcher ever want to use oblique rotation in their research study?

Components analysis and factor analysis are both statistical techniques used in the field of multivariate analysis to explore the underlying structure of a set of variables. While they share similarities, there are key differences between the two methods.

Components analysis, also known as principal components analysis (PCA), aims to identify a smaller set of uncorrelated variables called principal components that explain the maximum amount of variance in the original dataset. In this technique, the original variables are transformed into a new set of variables, known as the principal components, which are linear combinations of the original variables. The goal is to reduce the dimensionality of the data while retaining as much information as possible.

On the other hand, factor analysis seeks to identify a smaller set of latent factors that underlie the observed variables. These factors are not directly observable but are inferred from the patterns of covariance among the observed variables. Factor analysis assumes that each observed variable is influenced by one or more latent factors, and the goal is to estimate the loadings of each observed variable on the latent factors.

The decision to use either components analysis or factor analysis in a research study depends on several factors. First, the nature of the research question should be considered. Components analysis is more appropriate when the goal is to summarize and reduce the dimensionality of the dataset, without necessarily trying to understand the underlying structure. It is often used in exploratory data analysis or for data reduction purposes. Factor analysis, on the other hand, is more suitable when the focus is on understanding the underlying latent factors that explain the covariance among the observed variables. It is commonly used in psychometrics, social sciences, and other fields where the goal is to identify the underlying constructs that affect the measured variables.

Another consideration is the data characteristics. Components analysis assumes that the variables are continuous and that the relationships between them are linear. Factor analysis is more flexible and can handle various types of data, including categorical variables and non-linear relationships. If the data violate the assumptions of components analysis, factor analysis may be a more appropriate choice.

The results of components analysis and factor analysis can differ in several ways. Components analysis provides a set of uncorrelated components, each accounting for a certain proportion of the total variance. These components are ordered based on the amount of variance they explain, with the first component explaining the largest proportion of variance, followed by the second component, and so on. The interpretation of these components can be challenging since they are not necessarily meaningful or interpretable in terms of underlying constructs. In contrast, factor analysis yields latent factors that are theoretically interpretable. Each factor represents a distinct latent construct, and the loadings of observed variables can be used to understand the relationships between the constructs and the variables.