Screen data for exploratory factor analysis

Checks whether your data are suitable for exploratory factor analysis (EFA). From a correlation matrix or raw data, it reports the Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy, Bartlett's test of sphericity, the condition index of the correlation matrix, and each variable's squared multiple correlation (SMC). When you supply raw data, it also reports each variable's variance and percentage of missing values, category counts for categorical variables, Mardia's tests of multivariate normality, and multivariate outliers.

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 = 0.75,
  outlier_cutoff = 0.975,
  seed = lifecycle::deprecated()
)

Source

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

Belsley, D. A. (1991). A guide to using the collinearity diagnostics. Computer Science in Economics and Management, 4, 33-50.

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.

Hubert, M., Debruyne, M. & Rousseeuw, P. J. (2018). Minimum covariance determinant and extensions. WIREs Computational Statistics, 10, e1421.

Hubert, M., Rousseeuw, P. J. & Verdonck, T. (2012). A deterministic algorithm for robust location and scatter. Journal of Computational and Graphical Statistics, 21, 618-637.

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.

Leys, C., Delacre, M., Mora, Y. L., Lakens, D. & Ley, C. (2019). How to classify, detect, and manage univariate and multivariate outliers, with emphasis on pre-registration. International Review of Social Psychology, 32, 5.

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.

Zijlstra, W. P., van der Ark, L. A. & Sijtsma, K. (2007). Outlier detection in test and questionnaire data. Multivariate Behavioral Research, 42, 531-555.

Arguments

x: data.frame or matrix. Raw data, or a correlation matrix. Needs at least three variables, none of which is a perfect linear combination of the others.
N: numeric. The number of observations. Set this only when you supply a correlation matrix; it is needed for Bartlett's test of sphericity and is taken from the data automatically when you supply raw data. Default is NA.
use: character. How to handle missing values in raw data. For cor_method = "pearson", "spearman", or "kendall" this is passed to stats::cor(). For "poly" or "tetra" the same rule is applied to the raw data before the correlations are estimated; "all.obs" and "everything" then stop with an error on any missing value, instead of returning NA correlations. Default is "pairwise.complete.obs".
cor_method: character. How to compute correlations from raw data: "pearson", "spearman", or "kendall" (via stats::cor()), or "poly" / "tetra" for polychoric / tetrachoric correlations of ordinal or binary data. Spearman and Kendall matrices are screened as they are, without conversion to the Pearson scale. Default is "pearson".
mcd_alpha: numeric. The proportion of cases used to build the robust outlier estimate, between 0.5 and 1. 0.5 resists the most outliers (up to half the cases). The default, 0.75, resists up to a quarter but is more efficient (Hubert et al., 2018), and Leys et al. (2019) recommend it. Used only with raw data in which all variables are continuous.
outlier_cutoff: numeric. The probability used to set the cutoff for flagging a multivariate outlier, between 0.5 and 0.9999. Default is 0.975. Used only with raw data in which all variables are continuous.
seed: Accepted and ignored. The outlier detection is deterministic, so it needs no seed.

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 was too small for the correction. NULL when N is unavailable.
determinant: The determinant of the correlation matrix.
condition: The condition number of the correlation matrix (largest eigenvalue over smallest). Its square root is the condition index.
collinear: The names of the variables involved in a near linear dependency (see Details). NULL when no condition index is above 30.
smc: The per-variable squared multiple correlations.
per_item: A data frame with one row per variable (row names are the variable names): variance, missing (percentage), smc, kmo_i, and flags (any sparse/empty-category issues). NULL when a correlation matrix is supplied instead of raw data.
normality: A list with mardia (skewness skewness, skewness_df, skewness_p, kurtosis kurtosis and kurtosis_p, and the underlying b1p/b2p) and n_complete (the number of complete cases used). NULL without raw data, or a note ($message) that tells why the tests were skipped.
outliers: A list with distances (each complete case's distance, named by its row number), cutoff (the flagging threshold, on the same scale as distances), flagged (the row numbers above cutoff), center and cov (the location and scatter used), method, fallback_reason, and n_complete. method is "mcd", "classical" (the fallback for continuous data; fallback_reason tells why), or "ranking" (categorical data: classical distances, cutoff is NA, and flagged is empty). NULL without raw data, or a note ($message) that tells why no distances could be computed.
categories: A named list with the response-category counts for each categorical variable (in category order); NA for a variable treated as continuous. NULL without raw data.
note: A note ($message) that tells why the raw-data diagnostics (per_item, normality, outliers, categories) are missing, when a correlation matrix is supplied instead of raw data. NULL when raw data are supplied.
settings: The settings used: N, n_obs (rows in the raw data supplied, NA for a correlation-matrix input), use, cor_method, mcd_alpha, and outlier_cutoff.

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; see efa_kmo(). It shows how much common variance your variables share. Higher values are better; a common rule of thumb treats values below .50 as unacceptable.

Bartlett

Bartlett's (1951) test of sphericity: the likelihood-ratio test of whether the correlation matrix is an identity matrix, i.e., whether your variables correlate with each other at all; see efa_bartlett(). At usual sample sizes the test is almost always significant, so a significant result is weak evidence of suitability, but a non-significant result is a strong warning. The test needs the sample size N; without it, $bartlett is NULL. If N is too small relative to the number of variables, the statistic is NA.

Determinant

The determinant of $R$, returned but not used for a verdict. It falls as you add variables, also when they are not collinear, so a fixed cut-off on it (such as the 0.00001 often quoted from Field, 2018) mostly reflects the number of variables.

Condition index

The square root of the condition number (the ratio of the largest to the smallest eigenvalue of $R$): the collinearity diagnostic of Belsley, Kuh & Welsch (1980). An index of 10 or less is rarely of interest; an index above 30 flags a near linear dependency (Belsley, 1991). The report grades such an index as moderate (30 to 100), strong (100 to 300), or very strong (above 300), and names the variables involved: those whose variance-decomposition proportions, summed over all indices above 30, are above .5 (Belsley, Kuh & Welsch, 1980). These values come from regression on data that are not centred, but a correlation matrix is centred, so use them as a guide and not as a test.

SMC

The squared multiple correlation of each variable with all the others. A low value flags a variable that has little in common with the rest of your set.

Variance and missing data

For raw data: each variable's variance (over its available values) and percentage of missing values over all rows you supplied. Factor columns are recoded to integer levels first, so variance reflects those codes. With use = "complete.obs" or "na.or.complete", N is the number of complete rows; otherwise it is the number of rows.

Categories

For raw data: for each variable with fewer than ten distinct values (treated as categorical), the count of responses in each category. A category with fewer than five responses is flagged as sparse, and an unused category between the smallest and largest response is flagged as empty. As a rule of thumb, items with fewer than five response categories are better analysed with cor_method = "poly" or "tetra" than with Pearson correlations (Rhemtulla et al., 2012).

Multivariate normality

For raw data, using only complete cases: Mardia's (1970) tests of multivariate skewness and kurtosis, with the small-sample corrections of Mardia (1974). A small p-value suggests your data depart from a multivariate normal distribution, a reason to prefer robust standard errors over normal-theory ones. In a very small sample the kurtosis statistic is NA. With cor_method = "poly" or "tetra" the tests are skipped: they test the observed codes, but a polychoric analysis assumes normality only of the underlying continuous variables.

Outliers

For raw data, using only complete cases: each case's distance from the centre of your data.

If all variables are continuous, the distance is a robust one: it uses the reweighted minimum covariance determinant (MCD; Rousseeuw & Van Driessen, 1999), found with the deterministic DetMCD algorithm (Hubert, Rousseeuw & Verdonck, 2012) and corrected for consistency (Croux & Haesbroeck, 1999) and small samples (Pison et al., 2002). A case is flagged when its squared distance is above the outlier_cutoff quantile of the chi-square distribution. When there are too few complete cases or the variables are nearly collinear, the classical (non-robust) distance is used instead, and $outliers$fallback_reason tells why.
If any variable is categorical (fewer than ten distinct values, e.g. a rating-scale item), no case is flagged, and the report names the cases with the largest classical distances. The MCD theory needs continuous data (Hubert, Debruyne & Rousseeuw, 2018): with tied answers, many cases can share a hyperplane, which makes the MCD degenerate. Also, the distances of rating-scale data do not follow the chi-square distribution, so a chi-square cutoff flags many clean cases when items are skewed (Zijlstra, van der Ark & Sijtsma, 2007). To screen for careless responding, use indices made for that purpose.

Distance-based screening does not detect straight-lining (the same answer to all items).

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 ────────────────────────────────────────────
#> 
#> ✔ Overall KMO = 0.916: marvellous (Kaiser & Rice, 1974).
#> ✔ Bartlett's test of sphericity: χ²(153) = 2173.28, p < .001 (significant).
#> 
#> ── Multicollinearity ───────────────────────────────────────────────────────────
#> 
#> ✔ Condition index: 3.418. An index of 10 or less is rarely of interest
#> (Belsley, 1991).
#> 
#> ── 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 ────────────────────────────────────────────
#> 
#> ✔ Overall KMO = 0.955: marvellous (Kaiser & Rice, 1974).
#> ✔ Bartlett's test of sphericity: χ²(28) = 5054.06, p < .001 (significant).
#> 
#> ── Multicollinearity ───────────────────────────────────────────────────────────
#> 
#> ✔ Condition index: 4.934. An index of 10 or less is rarely of interest
#> (Belsley, 1991).
#> 
#> ── Per-variable diagnostics ────────────────────────────────────────────────────
#> 
#>           variance   SMC   MSA  flags
#> fun          1.427 0.609 0.955       
#> friends      1.309 0.685 0.950       
#> enjoy        1.154 0.715 0.947 sparse
#> hurt         1.136 0.557 0.968 sparse
#> part         1.271 0.636 0.955       
#> commonly     1.099 0.653 0.958       
#> chances      1.389 0.592 0.961       
#> attracted    1.259 0.679 0.950 sparse
#> 
#> ── Multivariate normality ──────────────────────────────────────────────────────
#> 
#> ✖ Mardia's skewness: χ²(120) = 914.23, p < .001.
#> ✖ Mardia's kurtosis: z = 50.44, p < .001.
#> These data depart from multivariate normality: both tests reject it.
#> 
#> ── Outliers ────────────────────────────────────────────────────────────────────
#> 
#> ℹ No case is flagged: some variables have fewer than ten distinct values
#> (categorical), and a chi-square cutoff is not valid for their Mahalanobis
#> distances. The largest distances are in rows 561, 238, 622, 549, and 186.
#> 
#> ── Recommendations ─────────────────────────────────────────────────────────────
#> 
#> ! These data depart from multivariate normality; normal-theory standard errors
#>   and fit statistics may be biased - prefer robust (sandwich) or bootstrapped
#>   standard errors.
#> ! 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.