Principal Component Analysis (PCA) and Factor Analysis (FA)
Source:R/factor_analysis.R, R/principal_components.R, R/utils_pca_efa.R
principal_components.RdThe 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).
Usage
factor_analysis(
x,
n = "auto",
rotation = "none",
sort = FALSE,
threshold = NULL,
standardize = TRUE,
cor = NULL,
...
)
principal_components(
x,
n = "auto",
rotation = "none",
sparse = FALSE,
sort = FALSE,
threshold = NULL,
standardize = TRUE,
...
)
rotated_data(pca_results, verbose = TRUE)
# 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(pca_results)Arguments
- x
A data frame or a statistical model.
- n
Number of components to extract. If
n="all", thennis set as the number of variables minus 1 (ncol(x)-1). Ifn="auto"(default) orn=NULL, the number of components is selected throughn_factors()resp.n_components(). Else, ifnis a number,ncomponents are extracted. Ifnexceeds number of variables in the data, it is automatically set to the maximum number (i.e.ncol(x)). Inreduce_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). Seepsych::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).
- cor
An optional correlation matrix that can be used (note that the data must still be passed as the first argument). If
NULL, will compute it by runningcor()on the passed data.- ...
Arguments passed to or from other methods.
- 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 beTRUEor"robust"(seesparsepca::robspca()).- pca_results
The output of the
principal_components()function.- verbose
Toggle warnings.
- object
An object of class
parameters_pcaorparameters_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.- digits
Argument for
print(), indicates the number of digits (rounding) to be used.- labels
Argument for
print(), character vector of same length as columns inx. If provided, adds an additional column with the labels.
Details
Methods and Utilities
n_components()andn_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 (seeperformance::check_sphericity_bartlett()) and the KMO (seeperformance::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.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.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. get_scores() takes the results from
principal_components() 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)
#> # Loadings from Factor Analysis (no rotation)
#>
#> Variable | MR1 | MR2 | MR3 | MR4 | MR5 | MR6 | Complexity | Uniqueness
#> ----------------------------------------------------------------------------
#> mpg | -0.92 | | | 0.22 | | | 1.16 | 0.08
#> cyl | 0.96 | | | | | | 1.14 | 0.01
#> disp | 0.95 | | | 0.20 | | | 1.13 | 0.03
#> hp | 0.86 | -0.35 | | | | | 1.38 | 0.12
#> drat | -0.72 | -0.40 | 0.29 | | | | 1.96 | 0.24
#> wt | 0.89 | 0.38 | 0.24 | | | | 1.53 | 5.00e-03
#> qsec | -0.53 | 0.76 | | | | | 1.81 | 0.14
#>
#> The 6 latent factors accounted for 90.99% of the total variance of the original data (MR1 = 71.60%, MR2 = 14.55%, MR3 = 2.91%, MR4 = 1.35%, MR5 = 0.45%, MR6 = 0.12%).
#>
factor_analysis(mtcars[, 1:7], n = 2, rotation = "oblimin", 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, 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 | 0.782 | 0.080
#> Variance Explained (Cumulative) | 0.782 | 0.861
#> Variance Explained (Proportion) | 0.908 | 0.092
predict(efa, verbose = FALSE)
#> MR1 MR2
#> 1 -0.41953471 -0.35544083
#> 2 -0.41953471 -0.35544083
#> 3 -0.93451791 0.05188420
#> 4 -0.02247401 -1.06704187
#> 5 0.80423811 -0.57723722
#> 6 0.05132529 -1.40382795
#> 7 1.17796287 0.82565287
#> 8 -0.97981546 -0.77961043
#> 9 -0.87835186 0.02703875
#> 10 -0.31368972 -0.16476124
#> 11 -0.28032337 -0.23570818
#> 12 0.73053450 -0.34851886
#> 13 0.70908470 -0.30291012
#> 14 0.75913424 -0.40933052
#> 15 1.40523363 -0.78714835
#> 16 1.40381792 -0.48501471
#> 17 1.26324243 0.25588537
#> 18 -1.39062136 0.13411535
#> 19 -1.62307787 0.15402661
#> 20 -1.48149201 0.28280884
#> 21 -0.82463188 -0.03789222
#> 22 0.79425443 -1.37121293
#> 23 0.67357950 -1.14361360
#> 24 1.04878537 1.07073183
#> 25 0.89062471 -0.71650907
#> 26 -1.26846941 -0.12530276
#> 27 -1.14242028 0.42262235
#> 28 -1.03993655 0.89398689
#> 29 0.94311149 1.87551864
#> 30 -0.08324543 0.96674422
#> 31 1.32372015 3.27012728
#> 32 -0.87651282 0.43537850
# \donttest{
# Automated number of components
factor_analysis(mtcars[, 1:4], n = "auto")
#> # Loadings from Factor Analysis (no 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 accounted for 83.55% of the total variance of the original data.
#>
# }