How do you choose the optimal number of factors for latent variable discovery?

1 What is factor analysis?

Factor analysis is a statistical method that reduces a large number of observed variables into a smaller number of unobserved variables, called factors. Factors are assumed to be linear combinations of the observed variables, with some error or noise. For example, if you have data on students' scores on different subjects, you can use factor analysis to find out how many factors, such as intelligence, motivation, or interest, influence their performance. Factor analysis can also help you visualize your data by plotting the factors on a graph, where each point represents an observation and each axis represents a factor.

2 How to perform factor analysis?

To perform factor analysis, you need to have a data set that meets some assumptions, such as having enough observations, having continuous or ordinal variables, and having linear relationships and multivariate normality among the variables. You also need to choose a method of factor extraction and a method of factor rotation. Factor extraction is the process of finding the initial factors that account for the most variance in the data. Factor rotation is the process of adjusting the factors to make them more interpretable and meaningful. There are different types of factor extraction and rotation methods, such as principal component analysis, maximum likelihood, varimax, and oblimin, that have different advantages and disadvantages depending on your data and research questions.

3 How to choose the number of factors?

When deciding how many factors to retain for factor analysis, there is no definite answer. Common methods and criteria include the eigenvalue rule, the scree plot, and the parallel analysis. The eigenvalue rule suggests keeping factors with an eigenvalue greater than 1, as this indicates that the factor explains more variance than a single variable. The scree plot shows the eigenvalues of the factors in descending order and identifies the point where the curve changes dramatically. The parallel analysis compares the eigenvalues of your data with those of a random data set and keeps only those with an eigenvalue higher than the corresponding random eigenvalue. However, each of these methods may overestimate or underestimate the number of factors, be subjective or ambiguous, or be computationally intensive.

4 How to evaluate the quality of your factors?

After selecting the number of factors, you need to evaluate how well they represent your data and research questions. To do this, you can look at the factor loadings, which show the strength of each variable's relation to each factor. This can help you interpret the meaning and direction of each factor, as well as assess the reliability and validity of your factors. Additionally, you can measure the communality, which is the proportion of variance in each variable that is explained by the factors. This can help you identify outliers or problematic variables with low or high communality values. Lastly, you can measure the variance explained, which is the percentage of variance in the data that is explained by the factors. This can help evaluate the overall fit and adequacy of your factor model, as well as compare different models with different numbers of factors or extraction and rotation methods.

5 How to report and present your factors?

The last step in factor analysis is to report and present your findings in an understandable way. Your report should include the purpose and rationale of your factor analysis, including the research questions and hypotheses. Additionally, you should describe and summarize your data set, including the number of observations, variables, and missing values. You should also explain the assumptions and methods used for your factor analysis, such as the extraction and rotation methods, the number of factors, and the criteria for choosing them. Furthermore, your report should include the results and interpretation of your factor analysis, such as the factor loadings, communality, variance explained, and meaning of each factor. Last but not least, you should discuss the limitations and implications of your factor analysis, such as potential sources of error or bias, generalizability and applicability of your factors, and suggestions for future research or practice.