Screen data for exploratory factor analysis

Computes a set of diagnostics describing how suitable a correlation matrix (or raw data) is for exploratory factor analysis: the Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy overall and per variable, Bartlett's test of sphericity, the determinant and condition number of the correlation matrix, and the squared multiple correlation (SMC) of each variable with all the others.

Usage

efa_screen(
  x,
  N = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra"),
  mcd_alpha = 0.5,
  outlier_cutoff = 0.975,
  seed = NULL
)

Source

Bartlett, M. S. (1951). The effect of standardization on a Chi-square approximation in factor analysis. Biometrika, 38, 337-344.

Belsley, D. A., Kuh, E. & Welsch, R. E. (1980). Regression diagnostics: Identifying influential data and sources of collinearity. Wiley.

Croux, C. & Haesbroeck, G. (1999). Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. Journal of Multivariate Analysis, 71, 161-190.

Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). Sage.

Henze, N. & Zirkler, B. (1990). A class of invariant consistent tests for multivariate normality. Communications in Statistics - Theory and Methods, 19, 3595-3617.

Kaiser, H. F. (1970). A second generation little jiffy. Psychometrika, 35, 401-415.

Kaiser, H. F. & Rice, J. (1974). Little jiffy, mark IV. Educational and Psychological Measurement, 34, 111-117.

Mardia, K. V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519-530.

Mardia, K. V. (1974). Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhya B, 36, 115-128.

Pison, G., Van Aelst, S. & Willems, G. (2002). Small sample corrections for LTS and MCD. Metrika, 55, 111-123.

Rhemtulla, M., Brosseau-Liard, P. E. & Savalei, V. (2012). When can categorical variables be treated as continuous? A comparison of robust continuous and categorical SEM estimation methods under suboptimal conditions. Psychological Methods, 17, 354-373.

Rousseeuw, P. J. & Van Driessen, K. (1999). A fast algorithm for the minimum covariance determinant estimator. Technometrics, 41, 212-223.

Arguments

x: data.frame or matrix. Data frame or matrix of raw data, or a matrix of correlations.
N: numeric. The number of observations. Only needs to be specified when a correlation matrix is supplied; it is required for Bartlett's test of sphericity and is taken from the data when raw data are supplied. Default is NA.
use: character. Passed to stats::cor() if raw data are supplied. Default is "pairwise.complete.obs".
cor_method: character. Correlation computed from raw data: "pearson", "spearman", or "kendall" (passed to stats::cor()), or "poly" / "tetra" for polychoric / tetrachoric correlations of ordinal / binary data (a two-step estimator with no empty-cell continuity correction). Default is "pearson".
mcd_alpha: numeric. The proportion of observations covered by the minimum covariance determinant (MCD) subset used for the robust outlier diagnostics, in [0.5, 1]. The default, 0.5, gives the most robust (highest-breakdown) estimate; larger values trade robustness for efficiency. Only used when raw data are supplied.
outlier_cutoff: numeric. The probability defining the chi-square cutoff for flagging a multivariate outlier: an observation is flagged when its squared robust distance exceeds qchisq(outlier_cutoff, p) for p variables. Default is 0.975. Only used when raw data are supplied.
seed: integer. Optional seed for the random subsets drawn by the MCD algorithm, making the outlier diagnostics reproducible; the caller's random-number-generator state is left unchanged. Default is NULL. Only used when raw data are supplied.

Value

An object of class efa_screen, a list containing:

kmo: A list with the overall KMO (KMO) and the per-variable KMO (KMO_i).
bartlett: A list with Bartlett's chi-square statistic (chisq), its p_value, and its degrees of freedom (df); chisq and p_value are NA when N is too small for the Bartlett correction. NULL when N is unavailable.
determinant: The determinant of the correlation matrix.
condition: The condition number of the correlation matrix (largest over smallest eigenvalue).
smc: The per-variable squared multiple correlations.
per_item: A data frame with one row per variable (row names are the variable names) holding the per-item diagnostics: variance (over the available values), missing (percentage of missing values), smc, kmo_i, and flags (a comma-separated list of any sparse/empty-category issues, the empty string "" for a categorical variable with none, and NA for a variable treated as continuous). NULL when a correlation matrix is supplied.
normality: Multivariate-normality diagnostics computed from the complete cases of the raw data: a list with mardia (Mardia's multivariate skewness statistic skewness, its degrees of freedom skewness_df and p-value skewness_p, a small_sample flag recording whether the small-sample skewness correction was applied, the standardised kurtosis statistic and its p-value kurtosis_p, and the underlying coefficients b1p and b2p), hz (the Henze-Zirkler statistic and its p_value), and n_complete (the number of complete cases used). When the complete-case covariance is singular the tests are skipped and a classed note (of class efa_screen_no_mvn) is returned instead, alongside a warning. NULL when a correlation matrix is supplied.
outliers: Multivariate-outlier diagnostics from the complete cases of the raw data: a list with distances (each complete case's robust Mahalanobis distance, named by its row number in the supplied data), cutoff (the flagging threshold on the distance scale, sqrt(qchisq(outlier_cutoff, p)), directly comparable to distances), flagged (the row numbers, in the supplied data, whose robust distance exceeds cutoff), center and cov (the robust location and scatter underlying the distances), method ("mcd" for the robust estimate, or "classical" when the fallback was used), and n_complete (the number of complete cases). When neither a robust nor a classical covariance can be formed a classed note (of class efa_screen_no_outliers) is returned instead. NULL when a correlation matrix is supplied.
categories: A named list with, for each variable treated as categorical, the response-category counts (in category order); NA for a variable treated as continuous. NULL when a correlation matrix is supplied.
note: A classed note explaining that the raw-data diagnostics need raw data; NULL when raw data are supplied.
settings: A list of the settings used, including n_obs, the number of rows in the supplied raw data (NA for a correlation-matrix input).

Details

The diagnostics are computed from the analysis correlation matrix $R$:

KMO: The Kaiser-Meyer-Olkin measure of sampling adequacy (Kaiser, 1970; Kaiser & Rice, 1974), overall and for each variable. Larger values (a rough floor of .50) indicate greater suitability for factor analysis; see efa_kmo().
Bartlett: Bartlett's (1951) test of sphericity, the likelihood-ratio test of the hypothesis that $R$ is an identity matrix, with $df = p(p - 1)/2$ for $p$ variables. It requires the sample size N; when N is unavailable (a correlation matrix supplied without N) the test is skipped with a warning, $bartlett is NULL, and the remaining diagnostics are still returned. See efa_bartlett().
Determinant: The determinant of $R$. A value near zero signals extreme multicollinearity or a (near-)singular matrix; as a rough guide, a determinant below about 0.00001 is commonly taken as a sign of multicollinearity (Field, 2018).
Condition number: The ratio of the largest to the smallest eigenvalue of $R$. Large values indicate an ill-conditioned matrix with near-linear dependencies among the variables. Its square root is the condition index: an index above 30 is a conventional sign of strong multicollinearity, and 10 to 30 of moderate multicollinearity (Belsley, Kuh & Welsch, 1980).
SMC: The squared multiple correlation of each variable with all the others, $1 - 1/(R^{-1})_{ii}$. A low SMC flags a variable that shares little variance with the rest of the set.
Variance and missing data: The variance of each variable (over its available values) and the percentage of missing values. These, and the category tabulation below, are computed column by column from the supplied data using every non-missing value, and so do not depend on use, which governs only the correlation matrix. Under a listwise use ("complete.obs" / "na.or.complete") the correlation matrix and N are based on the complete cases, while the missingness is reported over every supplied row so that it explains why N is smaller; the number of supplied rows is recorded in settings$n_obs. Available only from raw data.
Categories: For each variable with fewer than ten distinct values (treated as categorical), the response-category counts, flagging a sparse category (fewer than five responses) and, for integer-coded variables, an empty interior category (an unused category between the smallest and largest observed value). A variable with ten or more distinct values is treated as continuous and is not tabulated. As a rough guide, items with fewer than five response categories are better analysed with an ordinal estimator (polychoric correlations with categorical least squares) than with normal-theory maximum likelihood (Rhemtulla et al., 2012). Available only from raw data.
Multivariate normality: Two tests of multivariate normality computed from the complete cases of the raw data: Mardia's (1970) multivariate skewness and kurtosis, and the Henze-Zirkler (1990) omnibus test. A small p-value indicates a departure from multivariate normality, a reason to prefer robust or ordinal estimation over normal-theory maximum likelihood. Available only from raw data, and skipped with a note if the complete-case covariance is singular.
Outliers: Multivariate outliers flagged by their robust Mahalanobis distance. A high-breakdown robust location and scatter are estimated from the complete cases with the fast minimum covariance determinant (MCD) algorithm (Rousseeuw & Van Driessen, 1999), using a subset covering a proportion mcd_alpha of the observations; an observation whose squared robust distance exceeds qchisq(outlier_cutoff, p) is flagged. With too few complete cases ($n \le 2p$) or collinear variables the robust covariance is undefined, so the classical Mahalanobis distance is used instead with a warning; if even that covariance is singular the diagnostic is skipped with a note. Available only from raw data.

Examples

# From a correlation matrix (supply N for Bartlett's test of sphericity)
efa_screen(test_models$baseline$cormat, N = 500)
#> 
#> ── Sampling adequacy and sphericity ────────────────────────────────────────────
#> 
#> ✔ The overall KMO value for your data is marvellous (Overall KMO = 0.916).
#> These data are probably suitable for factor analysis.
#> 
#> ✔ The Bartlett's test of sphericity was significant at an alpha level of .05.
#> These data are probably suitable for factor analysis.
#> 𝜒²(153) = 2173.28, p < .001
#> 
#> ── Multicollinearity ───────────────────────────────────────────────────────────
#> 
#> ✔ Determinant: 0.0121. No concern (a value near 0 signals multicollinearity).
#> ✔ Condition number: 11.680. No concern (large values signal near-collinear variables).
#> 
#> ── Per-variable diagnostics ────────────────────────────────────────────────────
#> 
#>       MSA   SMC
#> V1  0.900 0.309
#> V2  0.914 0.250
#> V3  0.924 0.260
#> V4  0.932 0.315
#> V5  0.923 0.269
#> V6  0.891 0.305
#> V7  0.928 0.304
#> V8  0.919 0.288
#> V9  0.916 0.282
#> V10 0.892 0.303
#> V11 0.928 0.278
#> V12 0.908 0.352
#> V13 0.922 0.322
#> V14 0.905 0.283
#> V15 0.924 0.308
#> V16 0.934 0.283
#> V17 0.907 0.313
#> V18 0.923 0.299
#> 
#> ── Recommendations ─────────────────────────────────────────────────────────────
#> 
#> ✔ The data appear suitable for factor analysis.
#> ℹ Per-item variance, missing-data, category, normality, and outlier diagnostics
#>   require raw data; only a correlation matrix was supplied.

# From raw data (N is taken from the data)
efa_screen(GRiPS_raw)
#> 
#> ── Sampling adequacy and sphericity ────────────────────────────────────────────
#> 
#> ✔ The overall KMO value for your data is marvellous (Overall KMO = 0.955).
#> These data are probably suitable for factor analysis.
#> 
#> ✔ The Bartlett's test of sphericity was significant at an alpha level of .05.
#> These data are probably suitable for factor analysis.
#> 𝜒²(28) = 5054.06, p < .001
#> 
#> ── Multicollinearity ───────────────────────────────────────────────────────────
#> 
#> ✔ Determinant: 0.00188. No concern (a value near 0 signals multicollinearity).
#> ✔ Condition number: 24.340. No concern (large values signal near-collinear variables).
#> 
#> ── Per-variable diagnostics ────────────────────────────────────────────────────
#> 
#>           variance missing   SMC   MSA  flags
#> fun          1.427       0 0.609 0.955       
#> friends      1.309       0 0.685 0.950       
#> enjoy        1.154       0 0.715 0.947 sparse
#> hurt         1.136       0 0.557 0.968 sparse
#> part         1.271       0 0.636 0.955       
#> commonly     1.099       0 0.653 0.958       
#> chances      1.389       0 0.592 0.961       
#> attracted    1.259       0 0.679 0.950 sparse
#> 
#> ── Multivariate normality ──────────────────────────────────────────────────────
#> 
#> ✖ Mardia's skewness: 𝜒²(120) = 910.1, p < .001.
#> ✖ Mardia's kurtosis: z = 49.32, p < .001.
#> ✖ Henze-Zirkler: HZ = 11.65, p < .001.
#> These data depart from multivariate normality.
#> 
#> ── Outliers ────────────────────────────────────────────────────────────────────
#> 
#> ℹ 217 of 810 observations were flagged as multivariate outliers (robust distance > 4.19).
#> 
#> ── Recommendations ─────────────────────────────────────────────────────────────
#> 
#> ! These data depart from multivariate normality; normal-theory standard errors
#>   and fit statistics may be biased - prefer robust (sandwich) or bootstrapped
#>   standard errors.
#> ! Bartlett's test is significant, but it assumes multivariate normality and
#>   grows more sensitive as N increases; because these data are non-normal, treat
#>   it as uninformative here and rely on the KMO.
#> ! 3 variables have a sparse response category (< 5 responses): enjoy, hurt, and
#>   attracted; a low-frequency category can destabilise polychoric estimates -
#>   consider collapsing it into an adjacent category.
#> ! 217 observations were flagged as potential multivariate outliers; inspect
#>   them (see `$outliers$flagged`) before down-weighting or excluding.