Principal Component Analysis (PCA) and Factor Analysis (FA)

The functions principal_components() and factor_analysis() can be used to perform a principal component analysis (PCA) or a factor analysis (FA). They return the loadings as a data frame, and various methods and functions are available to access / display other information (see the 'Details' section).

factor_analysis(x, ...)

# S3 method for class 'data.frame'
factor_analysis(
  x,
  n = "auto",
  rotation = "oblimin",
  factor_method = "minres",
  sort = FALSE,
  threshold = NULL,
  standardize = FALSE,
  ...
)

# S3 method for class 'matrix'
factor_analysis(
  x,
  n = "auto",
  rotation = "oblimin",
  factor_method = "minres",
  n_obs = NULL,
  sort = FALSE,
  threshold = NULL,
  standardize = FALSE,
  ...
)

principal_components(x, ...)

rotated_data(x, verbose = TRUE)

# S3 method for class 'data.frame'
principal_components(
  x,
  n = "auto",
  rotation = "none",
  sparse = FALSE,
  sort = FALSE,
  threshold = NULL,
  standardize = TRUE,
  ...
)

# S3 method for class 'parameters_efa'
print_html(x, digits = 2, sort = FALSE, threshold = NULL, labels = NULL, ...)

# S3 method for class 'parameters_efa'
predict(
  object,
  newdata = NULL,
  names = NULL,
  keep_na = TRUE,
  verbose = TRUE,
  ...
)

# S3 method for class 'parameters_efa'
print(x, digits = 2, sort = FALSE, threshold = NULL, labels = NULL, ...)

# S3 method for class 'parameters_efa'
sort(x, ...)

closest_component(x)

Arguments

x

A data frame or a statistical model. For closest_component(), the output of the principal_components() function.

...

Arguments passed to or from other methods.

n

Number of components to extract. If n="all", then n is set as the number of variables minus 1 (ncol(x)-1). If n="auto" (default) or n=NULL, the number of components is selected through n_factors() resp. n_components(). Else, if n is a number, n components are extracted. If n exceeds number of variables in the data, it is automatically set to the maximum number (i.e. ncol(x)). In reduce_parameters(), can also be "max", in which case it will select all the components that are maximally pseudo-loaded (i.e., correlated) by at least one variable.

rotation

If not "none", the PCA / FA will be computed using the psych package. Possible options include "varimax", "quartimax", "promax", "oblimin", "simplimax", or "cluster" (and more). See psych::fa() for details. The default is "none" for PCA, and "oblimin" for FA.

factor_method

The factoring method to be used. Passed to the fm argument in psych::fa(). Defaults to "minres" (minimum residual). Other options include "uls", "ols", "wls", "gls", "ml", "minchi", "minrank", "old.min", and "alpha". See ?psych::fa for details.

sort

Sort the loadings.

threshold

A value between 0 and 1 indicates which (absolute) values from the loadings should be removed. An integer higher than 1 indicates the n strongest loadings to retain. Can also be "max", in which case it will only display the maximum loading per variable (the most simple structure).

standardize

A logical value indicating whether the variables should be standardized (centered and scaled) to have unit variance before the analysis (in general, such scaling is advisable). Note: This defaults to TRUE for PCA, but to FALSE for FA (because factor_analysis() computes a correlation matrix and uses that r-matrix for the factor analysis by default - therefore, standardization of the raw variables is unnecessary, and even undesirable when using cor = "poly").

n_obs

An integer or a matrix.

Integer: Number of observations in the original data set if x is a correlation matrix. Required to compute correct fit indices.
Matrix: A matrix where each cell [i, j] specifies the number of pairwise complete observations used to compute the correlation between variable i and variable j in the input x. It is crucial when x is a correlation matrix (rather than raw data), especially if that matrix was derived from a dataset containing missing values using pairwise deletion. Providing a matrix allows psych::fa() to accurately calculate statistical measures, such as chi-square fit statistics, by accounting for the varying sample sizes that contribute to each individual correlation coefficient.

verbose

Toggle warnings.

sparse

Whether to compute sparse PCA (SPCA, using sparsepca::spca()). SPCA attempts to find sparse loadings (with few nonzero values), which improves interpretability and avoids overfitting. Can be TRUE or "robust" (see sparsepca::robspca()).

digits

Argument for print(), indicates the number of digits (rounding) to be used.

labels

Argument for print(), character vector of same length as columns in x. If provided, adds an additional column with the labels.

object

An object of class parameters_pca, parameters_efa or psych_efa.

newdata

An optional data frame in which to look for variables with which to predict. If omitted, the fitted values are used.

names

Optional character vector to name columns of the returned data frame.

keep_na

Logical, if TRUE, predictions also return observations with missing values from the original data, hence the number of rows of predicted data and original data is equal.

Value

A data frame of loadings. For factor_analysis(), this data frame is also of class parameters_efa(). Objects from principal_components() are of class parameters_pca().

Details

Methods and Utilities

n_components() and n_factors() automatically estimates the optimal number of dimensions to retain.
performance::check_factorstructure() checks the suitability of the data for factor analysis using the sphericity (see performance::check_sphericity_bartlett()) and the KMO (see performance::check_kmo()) measure.
performance::check_itemscale() computes various measures of internal consistencies applied to the (sub)scales (i.e., components) extracted from the PCA.
Running summary() returns information related to each component/factor, such as the explained variance and the Eivenvalues.
Running get_scores() computes scores for each subscale.
factor_scores() extracts the factor scores from objects returned by psych::fa(), factor_analysis(), or psych::omega().
Running closest_component() will return a numeric vector with the assigned component index for each column from the original data frame.
Running rotated_data() will return the rotated data, including missing values, so it matches the original data frame.
performance::item_omega() is a convenient wrapper around psych::omega(), which provides some additional methods to work seamlessly within the easystats framework.
performance::check_normality() checks residuals from objects returned by psych::fa(), factor_analysis(), performance::item_omega(), or psych::omega() for normality.
performance::model_performance() returns fit-indices for objects returned by psych::fa(), factor_analysis(), or psych::omega().
Running plot() visually displays the loadings (that requires the see-package to work).

Complexity

Complexity represents the number of latent components needed to account for the observed variables. Whereas a perfect simple structure solution has a complexity of 1 in that each item would only load on one factor, a solution with evenly distributed items has a complexity greater than 1 (Hofman, 1978; Pettersson and Turkheimer, 2010).

Uniqueness

Uniqueness represents the variance that is 'unique' to the variable and not shared with other variables. It is equal to 1 - communality (variance that is shared with other variables). A uniqueness of 0.20 suggests that 20% or that variable's variance is not shared with other variables in the overall factor model. The greater 'uniqueness' the lower the relevance of the variable in the factor model.

MSA

MSA represents the Kaiser-Meyer-Olkin Measure of Sampling Adequacy (Kaiser and Rice, 1974) for each item. It indicates whether there is enough data for each factor give reliable results for the PCA. The value should be > 0.6, and desirable values are > 0.8 (Tabachnick and Fidell, 2013).

PCA or FA?

There is a simplified rule of thumb that may help do decide whether to run a factor analysis or a principal component analysis:

Run factor analysis if you assume or wish to test a theoretical model of latent factors causing observed variables.
Run principal component analysis If you want to simply reduce your correlated observed variables to a smaller set of important independent composite variables.

(Source: CrossValidated)

Computing Item Scores

Use get_scores() to compute scores for the "subscales" represented by the extracted principal components or factors. get_scores() takes the results from principal_components() or factor_analysis() and extracts the variables for each component found by the PCA. Then, for each of these "subscales", raw means are calculated (which equals adding up the single items and dividing by the number of items). This results in a sum score for each component from the PCA, which is on the same scale as the original, single items that were used to compute the PCA. One can also use predict() to back-predict scores for each component, to which one can provide newdata or a vector of names for the components.

Explained Variance and Eingenvalues

Use summary() to get the Eigenvalues and the explained variance for each extracted component. The eigenvectors and eigenvalues represent the "core" of a PCA: The eigenvectors (the principal components) determine the directions of the new feature space, and the eigenvalues determine their magnitude. In other words, the eigenvalues explain the variance of the data along the new feature axes.

References

Kaiser, H.F. and Rice. J. (1974). Little jiffy, mark iv. Educational and Psychological Measurement, 34(1):111–117
Hofmann, R. (1978). Complexity and simplicity as objective indices descriptive of factor solutions. Multivariate Behavioral Research, 13:2, 247-250, doi:10.1207/s15327906mbr1302_9
Pettersson, E., & Turkheimer, E. (2010). Item selection, evaluation, and simple structure in personality data. Journal of research in personality, 44(4), 407-420, doi:10.1016/j.jrp.2010.03.002
Tabachnick, B. G., and Fidell, L. S. (2013). Using multivariate statistics (6th ed.). Boston: Pearson Education.

Examples

library(parameters)

# \donttest{
# Principal Component Analysis (PCA) -------------------
principal_components(mtcars[, 1:7], n = "all", threshold = 0.2)
#> # Loadings from Principal Component Analysis (no rotation)
#> 
#> Variable |   PC1 |   PC2 |  PC3 |   PC4 |  PC5 |   PC6 | Complexity
#> -------------------------------------------------------------------
#> mpg      | -0.93 |       |      | -0.30 |      |       |       1.30
#> cyl      |  0.96 |       |      |       |      | -0.21 |       1.18
#> disp     |  0.95 |       |      | -0.23 |      |       |       1.16
#> hp       |  0.87 |  0.36 |      |       | 0.30 |       |       1.64
#> drat     | -0.75 |  0.48 | 0.44 |       |      |       |       2.47
#> wt       |  0.88 | -0.35 | 0.26 |       |      |       |       1.54
#> qsec     | -0.54 | -0.81 |      |       |      |       |       1.96
#> 
#> The 6 principal components accounted for 99.30% of the total variance of the original data (PC1 = 72.66%, PC2 = 16.52%, PC3 = 4.93%, PC4 = 2.26%, PC5 = 1.85%, PC6 = 1.08%).

# Automated number of components
principal_components(mtcars[, 1:4], n = "auto")
#> # Loadings from Principal Component Analysis (no rotation)
#> 
#> Variable |   PC1 | Complexity
#> -----------------------------
#> mpg      | -0.93 |       1.00
#> cyl      |  0.96 |       1.00
#> disp     |  0.95 |       1.00
#> hp       |  0.91 |       1.00
#> 
#> The unique principal component accounted for 87.55% of the total variance of the original data.

# labels can be useful if variable names are not self-explanatory
print(
  principal_components(mtcars[, 1:4], n = "auto"),
  labels = c(
    "Miles/(US) gallon",
    "Number of cylinders",
    "Displacement (cu.in.)",
    "Gross horsepower"
  )
)
#> # Loadings from Principal Component Analysis (no rotation)
#> 
#> Variable | Label                 |   PC1 | Complexity
#> -----------------------------------------------------
#> mpg      | Miles/(US) gallon     | -0.93 |       1.00
#> cyl      | Number of cylinders   |  0.96 |       1.00
#> disp     | Displacement (cu.in.) |  0.95 |       1.00
#> hp       | Gross horsepower      |  0.91 |       1.00

# Sparse PCA
principal_components(mtcars[, 1:7], n = 4, sparse = TRUE)
#> # Loadings from Principal Component Analysis (no rotation)
#> 
#> Variable |   PC1 |   PC2 |   PC3 |   PC4 | Complexity
#> -----------------------------------------------------
#> mpg      | -0.92 |  0.03 | -0.11 | -0.31 |       1.27
#> cyl      |  1.00 |  0.07 | -0.07 | -0.05 |       1.03
#> disp     |  0.96 | -0.06 |  0.08 | -0.23 |       1.14
#> hp       |  0.74 |  0.32 |  0.07 |  0.00 |       1.38
#> drat     | -0.68 |  0.46 |  0.47 | -0.03 |       2.62
#> wt       |  1.03 | -0.32 |  0.24 | -0.03 |       1.31
#> qsec     | -0.49 | -0.85 |  0.17 |  0.00 |       1.69
#> 
#> The 4 principal components accounted for 96.42% of the total variance of the original data (PC1 = 72.75%, PC2 = 16.53%, PC3 = 4.91%, PC4 = 2.24%).
principal_components(mtcars[, 1:7], n = 4, sparse = "robust")
#> # Loadings from Principal Component Analysis (no rotation)
#> 
#> Variable |   PC1 |   PC2 |   PC3 |   PC4 | Complexity
#> -----------------------------------------------------
#> mpg      | -0.92 |  0.03 | -0.11 | -0.31 |       1.27
#> cyl      |  1.00 |  0.07 | -0.07 | -0.05 |       1.03
#> disp     |  0.96 | -0.06 |  0.08 | -0.23 |       1.14
#> hp       |  0.74 |  0.32 |  0.07 |  0.00 |       1.38
#> drat     | -0.68 |  0.46 |  0.47 | -0.03 |       2.62
#> wt       |  1.03 | -0.32 |  0.24 | -0.03 |       1.31
#> qsec     | -0.49 | -0.85 |  0.17 |  0.00 |       1.69
#> 
#> The 4 principal components accounted for 96.42% of the total variance of the original data (PC1 = 72.75%, PC2 = 16.53%, PC3 = 4.91%, PC4 = 2.24%).

# Rotated PCA
principal_components(mtcars[, 1:7],
  n = 2, rotation = "oblimin",
  threshold = "max", sort = TRUE
)
#> # Rotated loadings from Principal Component Analysis (oblimin-rotation)
#> 
#> Variable |   TC1 |   TC2 | Complexity | Uniqueness |  MSA
#> ---------------------------------------------------------
#> wt       |  0.98 |       |       1.03 |       0.10 | 0.77
#> drat     | -0.95 |       |       1.19 |       0.21 | 0.85
#> disp     |  0.89 |       |       1.07 |       0.08 | 0.85
#> mpg      | -0.87 |       |       1.07 |       0.13 | 0.87
#> cyl      |  0.78 |       |       1.38 |       0.08 | 0.87
#> qsec     |       | -0.98 |       1.00 |       0.06 | 0.61
#> hp       |       |  0.61 |       1.97 |       0.10 | 0.90
#> 
#> The 2 principal components (oblimin rotation) accounted for 89.18% of the total variance of the original data (TC1 = 63.90%, TC2 = 25.28%).
principal_components(mtcars[, 1:7], n = 2, threshold = 2, sort = TRUE)
#> # Loadings from Principal Component Analysis (no rotation)
#> 
#> Variable |  PC1 |   PC2 | Complexity
#> ------------------------------------
#> cyl      | 0.96 |       |       1.02
#> disp     | 0.95 |       |       1.02
#> mpg      |      |       |       1.02
#> wt       |      |       |       1.30
#> hp       |      |       |       1.33
#> drat     |      |  0.48 |       1.71
#> qsec     |      | -0.81 |       1.75
#> 
#> The 2 principal components accounted for 89.18% of the total variance of the original data (PC1 = 72.66%, PC2 = 16.52%).

pca <- principal_components(mtcars[, 1:5], n = 2, rotation = "varimax")
pca # Print loadings
#> # Rotated loadings from Principal Component Analysis (varimax-rotation)
#> 
#> Variable |   RC1 |   RC2 | Complexity | Uniqueness |  MSA
#> ---------------------------------------------------------
#> mpg      | -0.77 |  0.53 |       1.77 |       0.14 | 0.92
#> cyl      |  0.81 | -0.52 |       1.70 |       0.08 | 0.84
#> disp     |  0.77 | -0.56 |       1.82 |       0.10 | 0.88
#> hp       |  0.95 | -0.16 |       1.05 |       0.06 | 0.81
#> drat     | -0.27 |  0.95 |       1.16 |       0.03 | 0.80
#> 
#> The 2 principal components (varimax rotation) accounted for 92.00% of the total variance of the original data (RC1 = 56.46%, RC2 = 35.54%).
summary(pca) # Print information about the factors
#> # (Explained) Variance of Components
#> 
#> Parameter                       |   RC1 |   RC2
#> -----------------------------------------------
#> Eigenvalues                     | 4.038 | 0.562
#> Variance Explained              | 0.565 | 0.355
#> Variance Explained (Cumulative) | 0.565 | 0.920
#> Variance Explained (Proportion) | 0.614 | 0.386
predict(pca, names = c("Component1", "Component2")) # Back-predict scores
#>     Component1  Component2
#> 1  -0.23906186  0.38058456
#> 2  -0.23906186  0.38058456
#> 3  -0.87523403  0.30496365
#> 4  -0.83256507 -1.21853890
#> 5   0.37716377 -0.82007976
#> 6  -1.10127278 -1.86114104
#> 7   1.23453249 -0.26702364
#> 8  -1.30024506 -0.23007562
#> 9  -0.74129623  0.41446358
#> 10 -0.02461777  0.47612740
#> 11  0.02534454  0.45386195
#> 12  0.29853804 -0.88051969
#> 13  0.26641941 -0.86620618
#> 14  0.34136287 -0.89960437
#> 15  0.93247714 -1.25088059
#> 16  1.06844670 -1.03682781
#> 17  1.22799380 -0.41588998
#> 18 -1.30762202  0.71198099
#> 19 -0.60701035  2.14073148
#> 20 -1.25685030  0.99325379
#> 21 -0.90720118  0.02588588
#> 22 -0.15711335 -1.72688554
#> 23  0.18137850 -1.00150788
#> 24  1.72116683  0.68171810
#> 25  0.36060813 -0.98161661
#> 26 -1.12513535  0.63056351
#> 27 -0.46730967  1.39310467
#> 28 -1.05428733  0.43877634
#> 29  2.24906373  1.75900256
#> 30  0.13210722  0.33766027
#> 31  2.24244855  1.06973229
#> 32 -0.42316749  0.86380202

# which variables from the original data belong to which extracted component?
closest_component(pca)
#>  mpg  cyl disp   hp drat 
#>    1    1    1    1    2 
# }

# Factor Analysis (FA) ------------------------

factor_analysis(mtcars[, 1:7], n = "all", threshold = 0.2, rotation = "Promax")
#> # Rotated loadings from Factor Analysis (Promax-rotation)
#> 
#> Variable |   MR1 |   MR3 |   MR2 |  MR6 |  MR4 |  MR5 | Complexity | Uniqueness
#> -------------------------------------------------------------------------------
#> mpg      | -0.56 |  0.28 |       |      |      |      |       1.94 |       0.08
#> cyl      |       | -0.58 | -0.34 |      |      | 0.24 |       2.28 |       0.01
#> disp     |  0.48 | -0.37 |       |      | 0.28 |      |       2.91 |       0.03
#> hp       |       |       | -0.43 | 0.45 |      |      |       2.17 |       0.12
#> drat     |       |  1.01 |       |      |      |      |       1.08 |       0.24
#> wt       |  1.07 |       |  0.20 |      |      |      |       1.08 |   5.00e-03
#> qsec     |       |       |  1.01 |      |      |      |       1.04 |       0.14
#> 
#> The 6 latent factors (Promax rotation) accounted for 90.99% of the total variance of the original data (MR1 = 31.21%, MR3 = 28.13%, MR2 = 23.60%, MR6 = 5.81%, MR4 = 1.49%, MR5 = 0.75%).
factor_analysis(mtcars[, 1:7], n = 2, threshold = "max", sort = TRUE)
#> # Rotated loadings from Factor Analysis (oblimin-rotation)
#> 
#> Variable |   MR1 |  MR2 | Complexity | Uniqueness
#> -------------------------------------------------
#> wt       |  1.00 |      |       1.07 |       0.10
#> disp     |  0.92 |      |       1.02 |       0.09
#> mpg      | -0.88 |      |       1.02 |       0.15
#> drat     | -0.84 |      |       1.14 |       0.39
#> cyl      |  0.82 |      |       1.23 |       0.08
#> hp       |  0.60 |      |       1.94 |       0.18
#> qsec     |       | 1.00 |       1.00 |   4.75e-03
#> 
#> The 2 latent factors (oblimin rotation) accounted for 85.83% of the total variance of the original data (MR1 = 63.85%, MR2 = 21.99%).
factor_analysis(mtcars[, 1:7], n = 2, rotation = "none", threshold = 2, sort = TRUE)
#> # Loadings from Factor Analysis (no rotation)
#> 
#> Variable |  MR1 |  MR2 | Complexity | Uniqueness
#> ------------------------------------------------
#> cyl      | 0.96 |      |       1.01 |       0.08
#> disp     | 0.95 |      |       1.03 |       0.09
#> mpg      |      |      |       1.03 |       0.15
#> wt       |      | 0.36 |       1.33 |       0.10
#> hp       |      |      |       1.23 |       0.18
#> drat     |      |      |       1.49 |       0.39
#> qsec     |      | 0.83 |       1.74 |   4.75e-03
#> 
#> The 2 latent factors accounted for 85.83% of the total variance of the original data (MR1 = 70.67%, MR2 = 15.16%).

efa <- factor_analysis(mtcars[, 1:5], n = 2)
summary(efa)
#> # (Explained) Variance of Components
#> 
#> Parameter                       |   MR1 |   MR2
#> -----------------------------------------------
#> Eigenvalues                     | 3.908 | 0.398
#> Variance Explained              | 0.531 | 0.331
#> Variance Explained (Cumulative) | 0.531 | 0.861
#> Variance Explained (Proportion) | 0.616 | 0.384
#> 
#> # Factor Correlations
#> 
#> Factor |    MR1 |    MR2
#> ------------------------
#> MR1    |  1.000 | -0.647
#> MR2    | -0.647 |  1.000
predict(efa, verbose = FALSE)
#>           MR1         MR2
#> 1   0.2845528 -0.54283674
#> 2   0.2845528 -0.54283674
#> 3   0.9024152 -0.77336128
#> 4  -0.3183886 -0.57046758
#> 5  -0.9461598  0.39041871
#> 6  -0.4955643 -0.68125903
#> 7  -0.8538301  1.43513473
#> 8   0.6806493 -1.24168944
#> 9   0.8412617 -0.73810505
#> 10  0.2449176 -0.35370543
#> 11  0.1907012 -0.36178666
#> 12 -0.8034782  0.44546524
#> 13 -0.7686248  0.45066032
#> 14 -0.8499494  0.43853847
#> 15 -1.5827127  0.79657207
#> 16 -1.4851863  0.95143952
#> 17 -1.1160586  1.21381686
#> 18  1.3609673 -1.12141178
#> 19  1.5876686 -1.31016252
#> 20  1.4944469 -1.12240438
#> 21  0.7696658 -0.72565105
#> 22 -1.1894582 -0.02829063
#> 23 -1.0026052 -0.01404053
#> 24 -0.6533523  1.45113448
#> 25 -1.0723893  0.39243905
#> 26  1.1625846 -1.15083452
#> 27  1.2175257 -0.75985383
#> 28  1.2704330 -0.42860113
#> 29 -0.2969720  1.77639903
#> 30  0.3866779  0.42813430
#> 31 -0.2138043  2.82273115
#> 32  0.9695135 -0.52558562

# \donttest{
# Automated number of components
factor_analysis(mtcars[, 1:4], n = "auto")
#> Warning: convergence not obtained in GPFoblq. 1000 iterations used.
#> # Rotated loadings from Factor Analysis (oblimin-rotation)
#> 
#> Variable |   MR1 | Complexity | Uniqueness
#> ------------------------------------------
#> mpg      | -0.90 |       1.00 |       0.19
#> cyl      |  0.96 |       1.00 |       0.08
#> disp     |  0.93 |       1.00 |       0.13
#> hp       |  0.86 |       1.00 |       0.26
#> 
#> The unique latent factor (oblimin rotation) accounted for 83.55% of the total variance of the original data.
# }