DMAR (version 1.0.0)

cfa_k: Multiple-Factor Confirmatory Factor Analysis Model

Description

Fits a confirmatory factor analysis model with one or more factors, where each factor is specified by naming its indicator variables and the measurement structure is specified by describing what is constrained (equal_loading, equal_intercept, equal_error) rather than by more technical terms. The function then reports which classical measurement structure the description implies (congeneric, essentially tau-equivalent, tau-equivalent, essentially parallel, or parallel), the parameter estimates with confidence intervals, fit information, and, per factor, coefficient omega, the average variance extracted (AVE), and coefficient H, each with a delta method standard error and confidence interval computed by lavaan from defined parameters (no additional packages are involved).

Usage

cfa_k(
  data = NULL,
  factors,
  S = NULL,
  N = NULL,
  M = NULL,
  equal_loading = FALSE,
  equal_intercept = FALSE,
  equal_error = FALSE,
  correlated_factors = TRUE,
  meanstructure = NULL,
  estimator = "ML",
  missing = "listwise",
  ordered = NULL,
  se = "standard",
  conf_level = 0.95,
  output = c("verbose", "measurement", "summary", "standardized", "fit"),
  ...
)

Value

For output = "verbose" (default) and output = "measurement", a data.frame (classes dmar_cfa_k, dmar_tbl) with columns syntax, term, estimate, se, z_value, p_value, ci_lower, ci_upper. The "model" attribute is a named character vector giving, per factor, the implied classical structure; the printed header displays it. For output = "summary", the lavaan summary object; for "standardized", the standardized solution; for "fit", the lavaan fit object.

Arguments

data

A raw data matrix or data frame, rows are respondents and columns include the items named in factors. Supply exactly one of data or S.

factors

Named list. Each element names a factor and gives the character vector of its indicator columns (two or more per factor; three or more when only one factor is specified). Each item loads on exactly one factor (simple structure).

S

A symmetric covariance matrix of the items, with dimnames naming the items; N is then required. Supply exactly one of data or S.

N

Total sample size. Required with S; ignored (inferred from the rows) with data.

M

Optional named numeric vector of item means, used with a covariance matrix to model the mean structure (required there when equal_intercept is used). Ignored for raw data.

equal_loading

Logical, or a named logical vector with one element per factor. TRUE constrains the loadings within a factor to a single value (across factors nothing is equated). Defaults to FALSE.

equal_intercept

Logical, or a named logical vector with one element per factor. TRUE constrains the item intercepts within a factor to a single value; this requires the mean structure (raw data, or M with a covariance matrix). Defaults to FALSE.

equal_error

Logical, or a named logical vector with one element per factor. TRUE constrains the error variances within a factor to a single value. Defaults to FALSE.

correlated_factors

Logical. If TRUE (default) the factors covary freely; because each factor variance is fixed to 1, the phi terms for factor pairs are the latent correlations. If FALSE the factor covariances are fixed to zero.

meanstructure

Logical or NULL. NULL (default) models the mean structure exactly when it is needed: when any equal_intercept is TRUE or when M is supplied. Set TRUE to model intercepts regardless (raw data or M required), or FALSE to suppress them (an error if equal_intercept is used).

estimator

Character; estimator passed to lavaan. Must be one of "ML" (default; maximum likelihood, fully efficient under multivariate normality), "MLR" (robust maximum likelihood: maximum likelihood estimates with standard errors and test statistic corrected for nonnormality; Satorra & Bentler, 1994), "WLS" (the asymptotic distribution free estimator of Browne, 1984; raw data and a large sample required), "WLSMV" (diagonally weighted least squares with mean- and variance-adjusted test statistic; Muthén, 1984; Muthén, du Toit, & Spisic, 1997; the standard choice for ordered categorical items, and what ordered switches to), or "GLS" (generalized least squares; Browne, 1974). With a robust estimator the reported fit indices are the robust versions.

missing

Character; missing-data handling passed to lavaan when raw data are supplied. Common values are "listwise" (default, listwise deletion) and "ml" (full information maximum likelihood). Ignored with S. With ordered items, "ml"/"fiml" are not available; use "pairwise" or "listwise".

ordered

Ordered categorical items: NULL (none, the default), TRUE (every item), or a character vector of item names. Requires raw data. Each factor must be all ordered or all continuous. Declaring ordered items switches the estimator to "WLSMV" (with a message, unless a categorical estimator was requested), fits thresholds in place of intercepts, and reports each ordered factor's omega on the categorical sum score metric via the Green and Yang (2009) computation; see Details.

se

Standard error type passed to lavaan; see cfa. Common values are "standard" (default), "robust.sem" (with estimator = "MLR"), and "none" (point estimates only; fastest).

conf_level

Confidence level for the parameter confidence intervals, including the delta method intervals for omega, AVE, and H. Defaults to 0.95. The RMSEA interval is a separate convention (see Details).

output

Format of the returned object:

"verbose"

(default) Parameter estimates with confidence intervals, the per-factor defined parameters (loading_sum, error_sum, omega, ave, H), and fit information.

"measurement"

The measurement-property rows only: per factor omega, ave, and H (with delta method standard errors and confidence intervals), the latent correlation phi for every factor pair (with its confidence interval), and, for raw data, the heterotrait-monotrait ratio htmt for every factor pair via htmt.

"summary"

The raw summary() output from lavaan (not a data frame).

"standardized"

The standardized parameter estimates from lavaan::standardizedSolution().

"fit"

The raw lavaan fit object. Escape hatch for direct lavaan access, including likelihood ratio tests between two cfa_k() fits via lavaan::lavTestLRT().

...

Additional arguments forwarded to lavaan (e.g., group, cluster, bootstrap).

Author

Ken Kelley [email protected]

Details

With ordered items, the model is fit by WLSMV to polychoric correlations with thresholds. A sum score of ordered items lives on the metric of the observed categories, not on the latent response metric of the polychoric loadings, so for a factor whose items are ordered the reported omega is the Green and Yang (2009) categorical sum score omega computed from the same fit; the substitution is announced in a message and recorded in the omega_metric attribute, and the delta method interval columns are NA for those rows because the delta method interval describes the latent response metric. The categorical sum score omega is the coefficient Kelley and Pornprasertmanit (2016) call categorical omega. AVE and coefficient H concern the latent response correlations themselves and are reported unchanged on that metric. For a bootstrap confidence interval on a categorical omega, use reliability_omega_categorical on the factor's items.

cfa_1 and cfa_2 are convenience wrappers around this function for the one and two factor cases: cfa_1() takes a vector of items and fits one factor over them, and cfa_2() takes the items of each of two factors. Both forward every argument here, so their results are this function's results, with the factors named f1 (and f2).

Describing the model instead of naming it. The classical measurement structures are nested patterns of within-factor equality constraints (Lord & Novick, 1968; Graham, 2006):

congeneric

loadings, intercepts, and error variances all free.

essentially tau-equivalent

equal loadings; intercepts and error variances free.

tau-equivalent

equal loadings and equal intercepts; error variances free.

essentially parallel

equal loadings and equal error variances; intercepts free.

parallel

equal loadings, equal intercepts, and equal error variances.

The caller states the constraints; the function reports the implied name, per factor, in the printed header and in the "model" attribute of the returned table. The distinction between tau-equivalent and essentially tau-equivalent (and between parallel and essentially parallel) lives entirely in the mean structure: the covariance structure of the two members of each pair is identical, so without intercepts in the model only the "essentially" form can be claimed. That is why equal_intercept requires raw data or M: covariances alone cannot speak to it. A constraint pattern outside the classical list (for example equal error variances with free loadings) is fit as requested and labeled descriptively, since it has no conventional name.

Identification fixes each factor variance to 1 (and each factor mean to 0 when the mean structure is modeled), so all loadings are estimated and within-factor equality constraints are meaningful. Two cfa_k() fits that differ only in descriptor settings are nested, so output = "fit" feeds lavaan::lavTestLRT() directly (with estimator = "MLR", lavaan applies the scaled difference test).

Measurement properties. For factor \(f\) with unstandardized loadings \(\lambda_j\) and error variances \(\psi_j\) (factor variance 1): coefficient omega \(= (\sum_j \lambda_j)^2 / ((\sum_j \lambda_j)^2 + \sum_j \psi_j)\) (McDonald, 1999), the reliability of the unit-weighted composite; the average variance extracted \(= J^{-1} \sum_j \lambda_j^2 / (\lambda_j^2 + \psi_j)\) (Fornell & Larcker, 1981), the mean proportion of item variance the factor accounts for; and coefficient \(H = (1 + (\sum_j \lambda_j^2/\psi_j)^{-1})^{-1}\) (Hancock & Mueller, 2001), the reliability of the optimally weighted composite, which no single item can drag below its value for any subset. All three are computed as lavaan defined parameters, so each carries a delta method standard error and a conf_level confidence interval in the same table as the model parameters.

Discriminant validity. Three complementary readings come from output = "measurement": (a) the latent correlation phi for a factor pair, with a confidence interval whose upper limit near 1 means the data cannot distinguish the two factors; (b) the Fornell and Larcker (1981) comparison, which asks whether each factor's ave exceeds the squared phi of its pairs (the factor should share more variance with its own items than with the other factor); and (c) for raw data, the model-free htmt ratio (Henseler, Ringle, & Sarstedt, 2015). The rows report the numbers and their uncertainty; the judgment is the researcher's.

Confidence interval conventions. Parameter rows (including omega, ave, H, and phi) use conf_level. The RMSEA interval follows its own convention: the rmsea_ci_level row records the level actually used (0.90, the conventional level for RMSEA, as in lavaan), and rmsea_ci_lower / rmsea_ci_upper are that interval.

Common row names under term: lambda_<factor>_<j> (loadings; lambda_<factor> when equated), psi_<factor>_<j> (error variances; psi_<factor> when equated), nu_<factor>_<j> (intercepts, when the mean structure is modeled; nu_<factor> when equated), phi_<factor> (factor variance, fixed to 1), phi_<factor1>_<factor2> (latent correlation), the per-factor defined parameters (loading_sum_<factor>, error_sum_<factor>, omega_<factor>, ave_<factor>, H_<factor>), and the fit rows chi_square, df, p_chi_square, cfi, tli, nnfi, rmsea, rmsea_ci_lower, rmsea_ci_upper, rmsea_ci_level, srmr, AIC, BIC, H0, H1.

References

Browne, M. W. (1974). Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1--24.

Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62--83.

Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49(1), 115--132.

Muthén, B., du Toit, S. H. C., & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes. Unpublished technical report.

Satorra, A., & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. von Eye & C. C. Clogg (Eds.), Latent variables analysis: Applications for developmental research (pp. 399--419). Thousand Oaks, CA: Sage.

Fornell, C., & Larcker, D. F. (1981). Evaluating structural equation models with unobservable variables and measurement error. Journal of Marketing Research, 18(1), 39--50.

Graham, J. M. (2006). Congeneric and (essentially) tau-equivalent estimates of score reliability: What they are and how to use them. Educational and Psychological Measurement, 66(6), 930--944. tools:::Rd_expr_doi("10.1177/0013164406288165")

Green, S. B., & Yang, Y. (2009). Reliability of summed item scores using structural equation modeling: An alternative to coefficient alpha. Psychometrika, 74(1), 155--167. tools:::Rd_expr_doi("10.1007/s11336-008-9099-3")

Hancock, G. R., & Mueller, R. O. (2001). Rethinking construct reliability within latent variable systems. In R. Cudeck, S. du Toit, & D. Sörbom (Eds.), Structural equation modeling: Present and future (pp. 195--216). Scientific Software International.

Henseler, J., Ringle, C. M., & Sarstedt, M. (2015). A new criterion for assessing discriminant validity in variance-based structural equation modeling. Journal of the Academy of Marketing Science, 43(1), 115--135. tools:::Rd_expr_doi("10.1007/s11747-014-0403-8")

Kelley, K., & Pornprasertmanit, S. (2016). Confidence intervals for population reliability coefficients: Evaluation of methods, recommendations, and software for composite measures. Psychological Methods, 21, 69--92. tools:::Rd_expr_doi("10.1037/a0040086")

Lord, F. M., & Novick, M. R. (1968). Statistical theories of mental test scores. Addison-Wesley.

McDonald, R. P. (1999). Test theory: A unified treatment. Erlbaum.

See Also

cfa_1 and cfa_2 for the one and two factor convenience wrappers; plot_cfa_k to display the estimates and the equality question visually; reliability_omega, reliability_H, average_variance_extracted, and htmt for the measurement properties as standalone functions; measurement_invariance for the across-group analog of these within-factor constraints; lavaan, lavTestLRT.

Other multivariate and latent variable methods: average_variance_extracted(), bifactor_indices(), cfa_1(), cfa_2(), ci_eigenvalue(), common_method_marker(), common_method_single_factor(), dmacs(), ecvi(), htmt(), irt_grm(), irt_information(), measurement_alignment(), measurement_invariance(), procrustes_phi(), simple_structure()

Examples

data(holzinger_swineford)
hs_factors <- list(
  verbal    = c("t6_paragraph_comprehension", "t7_sentence",
                "t9_word_meaning"),
  deduction = c("t20_deduction", "t22_problem_reasoning",
                "t23_series_completion"))

# Congeneric measurement model for both factors (the default:
# nothing is constrained, and the header names the structure).
cfa_k(holzinger_swineford, hs_factors)

# Equal loadings within every factor. Because only the covariance
# structure identifies this constraint, the implied structure is
# essentially tau-equivalent, and the header says so.
cfa_k(holzinger_swineford, hs_factors, equal_loading = TRUE)

# Measurement properties: omega, ave, and H per factor (each with a
# delta method standard error and confidence interval), the latent
# correlations, and the htmt ratios.
cfa_k(holzinger_swineford, hs_factors, output = "measurement")

# Descriptors can differ by factor: here the verbal loadings are
# equated and the deduction loadings are left free, and the header
# names each factor's structure separately.
cfa_k(holzinger_swineford, hs_factors,
      equal_loading = c(verbal = TRUE, deduction = FALSE))

# Equal loadings and intercepts (tau-equivalent), then also equal
# error variances (parallel). The mean structure is added because
# equal_intercept asks about it, so the nu terms join the table.
cfa_k(holzinger_swineford, hs_factors, equal_loading = TRUE,
      equal_intercept = TRUE)
cfa_k(holzinger_swineford, hs_factors, equal_loading = TRUE,
      equal_intercept = TRUE, equal_error = TRUE)

# Ordered categorical items: two correlated factors of three
# four-category items each, generated here. The model is fit by WLSMV
# to polychoric correlations, and each ordered factor's omega is
# reported on the categorical sum score metric (Green & Yang, 2009),
# which is why the interval columns of those two rows are NA.
set.seed(113)
n <- 200
eta_a <- rnorm(n)
eta_b <- 0.5 * eta_a + sqrt(1 - 0.5^2) * rnorm(n)
lambda <- c(0.6, 0.7, 0.8)
lat <- cbind(outer(eta_a, lambda), outer(eta_b, lambda)) +
  matrix(rnorm(n * 6), n, 6) %*% diag(sqrt(1 - c(lambda, lambda)^2))
likert <- as.data.frame(apply(lat, 2, function(x)
  as.integer(cut(x, breaks = c(-Inf, -1, 0, 1, Inf)))))
names(likert) <- paste0("item_", 1:6)
cfa_k(likert,
      list(scale_a = paste0("item_", 1:3),
           scale_b = paste0("item_", 4:6)),
      ordered = TRUE, output = "measurement")

# Does the equal-loadings description hold? Two fits that differ only
# in a descriptor are nested, so output = "fit" hands them straight to
# lavaan's likelihood ratio test.
fit_free  <- cfa_k(holzinger_swineford, hs_factors, output = "fit")
fit_equal <- cfa_k(holzinger_swineford, hs_factors,
                   equal_loading = TRUE, output = "fit")
lavaan::lavTestLRT(fit_free, fit_equal)