R Tutorial Series: Exploratory Factor Analysis

Exploratory factor analysis (EFA) is a common technique in the social sciences for explaining the variance between several measured variables as a smaller set of latent variables. EFA is often used to consolidate survey data by revealing the groupings (factors) that underly individual questions. This will be the context for demonstration in this tutorial.

Tutorial Files

Before we begin, you may want to download the dataset (.csv) used in this tutorial. Be sure to right-click and save the file to your R working directory. This dataset contains a hypothetical sample of 300 responses on 6 items from a survey of college students' favorite subject matter. The items range in value from 1 to 5, which represent a scale from Strongly Dislike to Strongly Like. Our 6 items asked students to rate their liking of different college subject matter areas, including biology (BIO), geology (GEO), chemistry (CHEM), algebra (ALG), calculus (CALC), and statistics (STAT). This is where our tutorial ends, because all students rated all of these content areas as Strongly Dislike, thereby rendering insufficient variance for conducting EFA (just kidding).

Beginning Steps

To begin, we need to read our datasets into R and store their contents in variables.

1. > #read the dataset into R variable using the read.csv(file) function
2. > data <- read.csv("dataset_EFA.csv")

First 10 rows of the dataset

Psych Package

Next, we need to install and load the psych package, which I prefer to use when conducting EFA. In this tutorial, we will make use of the package's fa() function.

1. > #install the package
2. > install.packages("psych")
3. > #load the package
4. > library(psych)

Number of Factors

For this tutorial, we will assume that the appropriate number of factors has already been determined to be 2, such as through eigenvalues, scree tests, and a priori considerations. Most often, you will want to test solutions above and below the determined amount to ensure the optimal number of factors was selected.

Factor Solution

To derive the factor solution, we will use the fa() function from the psych package, which receives the following primary arguments.

• r: the correlation matrix
• nfactors: number of factors to be extracted (default = 1)
• rotate: one of several matrix rotation methods, such as "varimax" or "oblimin"
• fm: one of several factoring methods, such as "pa" (principal axis) or "ml" (maximum likelihood)

Note that several rotation and factoring methods are available when conducting EFA. Rotation methods can be described as orthogonal, which do not allow the resulting factors to be correlated, and oblique, which do allow the resulting factors to be correlated. Factoring methods can be described as common, which are used when the goal is to better describe data, and component, which are used when the goal is to reduce the amount of data. The fa() function is used for common factoring. For component analysis, see princomp(). The best methods will vary by circumstance and it is therefore recommended that you seek professional council in determining the optimal parameters for your future EFAs.
In this tutorial, we will use oblique rotation (rotate = "oblimin"), which recognizes that there is likely to be some correlation between students' latent subject matter preference factors in the real world. We will use principal axis factoring (fm = "pa"), because we are most interested in identifying the underlying constructs in the data.

1. > #calculate the correlation matrix
2. > corMat <- cor(data)
3. > #display the correlation matrix
4. > corMat

The correlation matrix

1. > #use fa() to conduct an oblique principal-axis exploratory factor analysis
2. > #save the solution to an R variable
3. > solution <- fa(r = corMat, nfactors = 2, rotate = "oblimin", fm = "pa")
4. > #display the solution output
5. > solution

Complete solution output

By looking at our factor loadings, we can begin to assess our factor solution. We can see that BIO, GEO, and CHEM all have high factor loadings around 0.8 on the first factor (PA1). Therefore, we might call this factor Science and consider it representative of a student's interest in science subject matter. Similarly, ALG, CALC, and STAT load highly on the second factor (PA2), which we might call Math. Note that STAT has a much lower loading on PA2 than ALG or CALC and that it has a slight loading on factor PA1. This suggests that statistics is less related to the concept of Math than algebra and calculus. Just below the loadings table, we can see that each factor accounted for around 30% of the variance in responses, leading to a factor solution that accounted for 66% of the total variance in students' subject matter preference. Lastly, notice that our factors are correlated at 0.21 and recall that our choice of oblique rotation allowed for the recognition of this relationship.
Of course, there are many other considerations to be made in developing and assessing an EFA that will not be presented here. The intent with this tutorial was simply to demonstrate the basic execution of EFA in R. For a detailed and digestible overview of EFA, I recommend the Factor Analysis chapter of Multivariate Data Analysis by Hair, Black, Babin, and Anderson.

Complete EFA Example

To see a complete example of how EFA data can be organized using the psych package in R, please download the EFA example (.txt) file. For the code used in this tutorial, download the EFA Example (.R) file.

References

Revelle, W. (2011). psych: Procedures for Personality and Psychological Research. http://personality-project.org/r/psych.manual.pdf

R Tutorial Series: Applying the Reshape Package to Organize ANOVA Data

As demonstrated in the preceding ANOVA tutorials, data organization is central to conducting ANOVA in R. In standard ANOVA, we used the tapply() function to generate a table for a single summary function. In repeated measures ANOVA, we used separate datasets for our omnibus ANOVA and follow-up comparisons. This tutorial will demonstrate how the reshape package can be used to simplify the ANOVA data organization process in R.

Tutorial Files

Before we begin, you may want to download the between group and repeated measures datasets (.csv) used in this tutorial. Be sure to right-click and save the files to your R working directory. The between groups dataset contains a hypothetical sample of 30 cases separated into three groups (a, b, and c). The repeated measures dataset contains a hypothetical sample of 10 cases across three measurements (a, b, and c). In both cases, the values are represented on a scale that ranges from 1 to 5.

Beginning Steps

To begin, we need to read our datasets into R and store their contents in variables.

1. > #read the datasets into R variables using the read.csv(file) function
2. > dataBetween <- read.csv("dataset_ANOVA_reshape_1.csv")
3. > dataRepeated <- read.csv("dataset_ANOVA_reshape_2.csv")

Reshape Package

Next, we need to install and load the reshape package. In this tutorial, we will make use of the package's cast() and melt() functions.

1. > #install the package
2. > install.packages("reshape")
3. > #load the package
4. > library(reshape)

Using cast() to Derive ANOVA Descriptives

The cast() function can be used to easily derive summary statistics for a between groups ANOVA dataset. The cast() function receives the following primary arguments.

• data: the dataset
• formula: in our case, a one-sided formula indicating the grouping variable
• fun.aggregate: a function or vector of functions for deriving summary statistics, such as mean, var, or sd

1. > #display the raw between groups data
2. > dataBetween

The raw between groups data

1. > #cast the between groups data using cast(data, formula, fun.aggregate) to get the group means
2. > cast(dataBetween, formula = ~group, fun.aggregate = mean)

The casted data with means

Note that the fun.aggregate argument can also receive a vector of summary statistics functions. This will yield all of the requested descriptives via a single cast() function.

1. > #cast the between groups data using cast(data, formula, fun.aggregate) to get the group means, variances, and standard deviations
2. > cast(dataBetween, formula = ~group, fun.aggregate = c(mean, var, sd))

The casted data with descriptives

Using melt() to Prepare Repeated Measures Data for Pairwise Comparisons

The melt() function can be used to morph a repeated measures ANOVA dataset prior to conducting pairwise comparisons. The melt() function receives the following primary arguments.

• data: the dataset
• id.vars: the id variable or a vector of values that can be used as ids
• measure.vars: a vector containing the variables to be melted
• variable_name: the name of the column containing the melted variables

1. > #display the repeated measures data
2. > dataRepeated

The raw repeated measures data

1. > #melt the repeated measures data using melt(data, id.vars, measure.vars, variable_name) to organize it for pairwise comparisons
2. > melt(dataRepeated, id.vars = "case", measure.vars = c("valueA", "valueB", "valueC"), variable_name = "abcValues")

The melted repeated measures data

Note that the data are now prepared to be used in the pairwise.t.test() function. See the One-Way ANOVA with Pairwise Comparisons tutorial for details on using the pairwise.t.test() function.

Complete ANOVA Reshape Example

To see a complete example of how ANOVA data can be organized using the reshape package in R, please download the ANOVA reshape example (.txt) file.