SMT: Sequential Chi Square Model Tests, RMSEA lower bound, and AIC

In EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools

Sequential Chi Square Model Tests, RMSEA lower bound, and AIC

Description

Sequential Chi Square Model Tests (SMT) are a factor retention method where multiple EFAs with increasing numbers of factors are fitted and the number of factors for which the Chi Square value first becomes non-significant is taken as the suggested number of factors. Preacher, Zhang, Kim, & Mels (2013) suggested a similar approach with the lower bound of the 90% confidence interval of the Root Mean Square Error of Approximation (RMSEA; Browne & Cudeck, 1992; Steiger & Lind, 1980), and with the Akaike Information Criterion (AIC). For the RMSEA, the number of factors for which this lower bound first falls below .05 is the suggested number of factors to retain. For the AIC, it is the number of factors where the AIC is lowest.

Usage

SMT(
  x,
  N = NA,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall")
)

Arguments

x	data.frame or matrix. Dataframe or matrix of raw data or matrix with correlations.
N	numeric. The number of observations. Needs only be specified if a correlation matrix is used.
use	character. Passed to stats::cor if raw data is given as input. Default is "pairwise.complete.obs".
cor_method	character. Passed to stats::cor. Default is "pearson".

Details

As a first step in the procedure, a maximum number of factors to extract is determined for which the model is still over-identified (df > 0).

Then, EFAs with increasing numbers of factors from 1 to the maximum number are fitted with maximum likelihood estimation.

For the SMT, first the significance of the chi square value for a model with 0 factors is determined. If this value is not significant, 0 factors are suggested to retain. If it is significant, a model with 1 factor is estimated and the significance of its chi square value is determined, and so on, until a non-significant result is obtained. The suggested number of factors is the number of factors for the model where the chi square value first becomes non-significant.

Regarding the RMSEA, the suggested number of factors is the number of factors for the model where the lower bound of the 90% confidence interval of the RMSEA first falls below the .05 threshold.

Regarding the AIC, the suggested number of factors is the number of factors for the model with the lowest AIC.

In comparison with other prominent factor retention criteria, SMT performed well at determining the number of factors to extract in EFA (Auerswald & Moshagen, 2019). The RMSEA lower bound also performed well at determining the true number of factors, while the AIC performed well at determining the most generalizable model (Preacher, Zhang, Kim, & Mels, 2013).

The SMT function can also be called together with other factor retention criteria in the N_FACTORS function.

Value

A list of class SMT containing

nfac_chi	The number of factors to retain according to the significance of the chi square value.
nfac_RMSEA	The number of factors to retain according to the RMSEA lower bound
nfac_AIC	The number of factors to retain according to the AIC
p_null	The p-value for the null model (zero factors)
ps_chi	The p-values for EFA models with increasing numbers of factors, starting with 1 factor
RMSEA_LB_null	The lower bounds of the 90% confidence interval for the RMSEA for the null model (zero factors).
RMSEA_LBs	The lower bounds of the 90% confidence interval for the RMSEA for EFA models with increasing numbers of factors, starting with 1 factor
AIC_null	The AICs for the null model (zero factors)
AICs	The AICs for EFA models with increasing numbers of factors, starting with 1 factor

Source

Auerswald, M., & Moshagen, M. (2019). How to determine the number of factors to retain in exploratory factor analysis: A comparison of extraction methods under realistic conditions. Psychological Methods, 24(4), 468–491. https://doi.org/10.1037/met0000200

Browne, M.W., & Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods and Research, 21, 230–258.

Preacher, K. J., Zhang G., Kim, C., & Mels, G. (2013). Choosing the Optimal Number of Factors in Exploratory Factor Analysis: A Model Selection Perspective, Multivariate Behavioral Research, 48(1), 28-56, doi:10.108/00273171.2012.710386

Steiger, J. H., & Lind, J. C. (1980, May). Statistically based tests for the number of common factors. Paper presented at the annual meeting of the Psychometric Society, Iowa City, IA.

See Also

Other factor retention criteria: CD, EKC, HULL, KGC, PARALLEL

N_FACTORS as a wrapper function for this and all the above-mentioned factor retention criteria.

Examples

SMT_base <- SMT(test_models$baseline$cormat, N = 500)
SMT_base

EFAtools documentation built on Jan. 6, 2023, 5:16 p.m.