Principal Component Analysis vs Factor Analysis vs regression

In summary, the conversation discusses the differences between principal component analysis (PCA) and factor analysis (FA) in terms of their results and their relationship to multiple linear regression. Both PCA and FA produce factors, which can be seen as independent and normalized random variables that explain the variability of the dataset. However, the exact differences between PCA and FA are not clear. The comparison with multiple linear regression also raises questions about calculating the slope of simple linear regression from PCA factor loadings. Further research and helpful comments are needed to fully understand and compare these methods.
  • #1
lalbatros
1,256
2
I just gave a try to a statistical excel add-in and found PCA and FA quite interresting.
However, I don't see where are the differences between these two analysis, except for the layout of the results.
Additionaly, I see the link with multiple regression, but I don't see the link precisely enough.

I see that both PCA and FA produce factors.
I understand factors as independent and normalized random variables from which the variability of the dataset can be reproduced.
I also appreciate the ranking of the factors according to the their contribution to the global variance.
But where are the difference between PCA and FA?

The comparison with multiple linear regression left me uneasy too.
I first tought I would be able to calculate the slope of simple linear regression from the PCA factor loadings.
This works rather well when dispersion is small, but there are large differences when dispersion is large.

Digging more into the details,
- I observed small differences in the "factor loading" from PCA or FA
- I had difficulty to reproduce all the details by SVD decomposition of the correlation matrix, but numbers are close

Could some of you provide me with helpful comments.
Web links to well-written descriptions and/or comparisons of the methods would be highly appreciated too.
 
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  • #3


Principal Component Analysis (PCA) and Factor Analysis (FA) are both multivariate statistical techniques used for data reduction and dimensionality reduction. However, they have different underlying principles and objectives.

PCA is a data-driven technique that aims to identify the underlying structure or patterns in a dataset by finding the linear combinations of variables that explain the most variance in the data. These linear combinations are called principal components, and they are orthogonal (uncorrelated) to each other. PCA is commonly used for dimensionality reduction, data visualization, and identifying important variables in a dataset.

On the other hand, FA is a theory-driven technique that aims to identify the underlying latent factors that explain the correlations among a set of observed variables. These factors are not directly observable, but they can be inferred from the observed variables. FA assumes that the observed variables are influenced by a smaller number of underlying factors, and it tries to identify the structure of these factors. FA is commonly used for data reduction, hypothesis testing, and developing latent constructs in social sciences.

Both PCA and FA produce factors, but the main difference is in the underlying assumptions and objectives. PCA is used to identify the most important variables in a dataset, while FA is used to identify the latent factors that explain the correlations among variables. Additionally, PCA assumes that the observed variables are independent, while FA allows for correlations among the observed variables.

Regression, on the other hand, is a predictive modeling technique that aims to identify the relationship between a dependent variable and one or more independent variables. Unlike PCA and FA, regression does not reduce the dimensionality of the data, but it uses all the variables in the model to predict the outcome variable. Regression can also be used to identify the most important predictors in a dataset, but it does not consider the correlations among the variables.

In terms of the link between PCA and FA, both techniques use a similar mathematical approach called singular value decomposition (SVD) to extract the factors. However, in FA, the factors are determined based on the correlations among the observed variables, while in PCA, the factors are determined based on the variance in the data.

In summary, PCA and FA are both useful techniques for data reduction and dimensionality reduction, but they have different underlying principles and objectives. Regression, on the other hand, is a predictive modeling technique that aims to identify the relationship between variables. It is important to choose the appropriate technique based on the objectives of the analysis and the underlying assumptions of the data.
 

1. What is the main difference between Principal Component Analysis (PCA) and Factor Analysis?

PCA is a statistical method used for dimensionality reduction, while Factor Analysis is a tool for identifying underlying latent variables in a dataset.

2. How are PCA and Factor Analysis used in data analysis?

PCA is often used for visualization and data compression, while Factor Analysis is commonly used for exploratory factor analysis and hypothesis testing.

3. Can PCA and Factor Analysis be used together?

Yes, they can be used together to gain a deeper understanding of the relationships between variables in a dataset. For example, PCA can be used to reduce the dimensionality of the data, followed by Factor Analysis to identify underlying factors.

4. How does regression differ from PCA and Factor Analysis?

Regression is a predictive modeling technique that aims to find the relationship between a dependent variable and one or more independent variables. PCA and Factor Analysis, on the other hand, are unsupervised learning methods that do not require a dependent variable.

5. When should I use PCA, Factor Analysis, or regression in my analysis?

PCA and Factor Analysis are useful for exploratory data analysis and understanding the underlying structure of a dataset. Regression, on the other hand, is better suited for making predictions and identifying relationships between variables. The choice of method will depend on the objectives of the analysis and the type of data being analyzed.

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