Factor Analysis in R: Data interpretation Made Easy!

Factor analysis in R is a statistical technique that simplifies data interpretation by reducing the initial variables into a smaller number of factors. You must be thinking – what are factors? Well, the definition of factors states that they are a representation of the ‘latent variables’ that underlie the original variables.

Thus, factor analysis represents dataset variables y1, y2,… yp as a linear combination of latent variables called factors, denoted by f1, f2,…fm where m<p. However, the factors cannot be observed or measured, and thus, their existence is hypothetical. 

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Introducing the factor analysis model

Before we discuss the details of factor analysis in R, let us get introduced to the basic idea of the factor analysis model. The factor analysis model, as stated in the previous section, is a linear combination of random, hypothetical, and latent variables called factors (f1, f2,…fm). Since the factors are theoretical, they may not exist. Thus, for the variables in the observation vectors of a sample, the factor analysis model is defined as:

a1−α1=Ɣ11l1+Ɣ12l2+…+Ɣ1mlm+δ1a1−α1=Ɣ11l1+Ɣ12l2+…+Ɣ1mlm+δ1

a2−α2=Ɣ21l1+Ɣ22l2+…+Ɣ2mlm+δ2a2−α2=Ɣ21l1+Ɣ22l2+…+Ɣ2mlm+δ2

⋮

ap−αp=Ɣp1l1+Ɣp2l2+…+Ɣpmlm+δp

In the above model, α denotes the mean vector, Ɣ represents the factor loading that represents the relationship between the pth observed variable and the mth latent factor, and δ indicates the random error to show that there is no exact relationship between the factors.

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An example of factor analysis in R

In this section, we will look at an example to understand factor analysis in R. For the study, we will use two R packages – ‘psych’ and ‘GPArotation’. We will begin by describing the dataset and then move on to selecting the number of factors for analysis. Finally, we will perform factor analysis by using the fa() function of the ‘psych’ package.

So, here is a step-by-step example of factor analysis in R:

1. The dataset

To succinctly understand the factor analysis method, we shall use an example to elucidate on the model. Let us consider a dataset consisting of 13 diverse variables that a prospective consumer considers while investing in a property. The variables are:

• Rent/Cost/Lease amount
• Location
• Size 
• Amenities
• Distance from landmarks
• Availability of essential services
• Locality
• Maintenance cost
• Resale value
• Prospect as per requirement
• Demography
• Resale value
• Reliability of the builder

We will apply the factor analysis method on a dataset that contains 14 different variables that customers usually consider while purchasing a car. The variables are:

• Price
• Safety
• Exterior looks
• Space and comfort
• Technology
• Fuel type
• Fuel efficiency
• After-sales service
• Resale value
• Colour
• Maintenance
• Test drive
• Product reviews
• Testimonials

Read: Best Datasets for Machine Learning Projects

2. Data importing

In this step, the CSV format dataset is read to R to store as the variable. A window opens for choosing the CSV file, and the ‘header’ option ensures that the first row in the file is taken to be the header.

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3. Installing Packages

 The install.packages() function is called for installing the ‘psyche’ and ‘GPArotation’ packages to carry out further analysis.

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4. Number of factors

In this step, the number of factors to be selected for analysis is evaluated through methods like ‘Parallel Analysis’ and ‘eigenvalue’, and a scree plot is generated. In this example, the ‘psych’ package’s ‘fa.parallel’ function performs Parallel Analysis. The data frame and the factor method (‘minres’) are specified. Subsequently, the maximum number of considerable factors and a scree plot are generated.

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5. Factor Analysis

The fa() function of ‘psyche’ package runs the factor analysis with a supply of the following arguments:

• r – Covariance matrix or the raw data
• nfactors – Number of factors to be extracted
• rotate – Oblique rotation (rotate = “oblimin”) is used in this example
• fm – It is the factor extraction technique. In this example, Ordinary Least Squared, or Minres (fm = “minres”) has been used. Three factors are considered first:
• The output generated indicates the loadings and factors. Now, the loadings, usually used as single loading, would chart the relative propensity needed to consider these factors as essentially viable. For instance, if the loading was represented as thus:

Loadings	MR2	MR1	MR3
Rent/Cost/Lease amount	0.793	0.648	0.432
Location	0.890	0.634	0.421
Size	0.567	0.432	0.678
Amenities	0.348	0.454	0.678
Distance from Landmark	0.666	0.783	-0.142
Availability of essential services	0.787	0.132	-0.231
Locality	-0.876	-0.467	0.589
Maintenance cost	0.322	0.356	0.389
Resale value	0.479	0.245	0.167
Prospect as per requirement	0.424	0.327	0.675
Demography	0.121	0.278	0.437
Resale value	-0.298	0.798	0.398
Reliability of the builder	0.646	0.785	-0.478

• Now, considering loadings above 0.3 are established as cut-off and the highest reading on each factor is taken into account. You can even consider negative values if they represent the highest loading.
• This model is further replicated under four factors in a simple structure, however with single loading as displayed above. 

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6. Adequacy Test

In this step, the model is validated by examining the output of factor analysis:

Here depending on the final outcome, the values are judged on the basis of parameters like RMSR value, RSMEA value, and finally the Tucker-Lewis Index. These values determine the general validity and sustainability of the model.

The last step pertains to the theoretical aspect of the analysis. The factors could then be summarized according to the value of the loadings obtained. In the current context, such factors could be:

1. Location 
2. Locality
3. Rent/Cost/Lease amount
4. Availability of essential services

Also Read: Data Manipulation in R: What is, Variables, Using dplyr package

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Conclusion

In this article, we discussed the basic idea of factor analysis in R through the factor analysis model. We further illustrated the concept with the help of a real-world example where the number of variables related to car purchasing was reduced to some common factors. Now, go ahead and try it out!

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