DMAR (version 1.0.0)

average_variance_extracted: Average Variance Extracted (AVE)

Description

The average variance extracted (AVE) is the mean proportion of indicator variance a factor accounts for, a standard convergent validity summary for a reflective measurement block in confirmatory factor analysis and structural equation modeling. With standardized loadings \(\ell_j\), $$\mathrm{AVE} = \frac{1}{J} \sum_j \ell_j^2.$$ The quantity itself is elementary. In a model with cross-loadings, an item contributes its loading to the AVE of every factor it loads on; the same shared variance then counts toward each factor's summary, so compare AVE values across factors of such a model with that overlap in mind. Fornell and Larcker (1981) are credited for establishing it as a validity criterion: a construct shows convergent validity when its AVE reaches the conventional 0.50 (the construct explains at least half its indicators' variance), and the Fornell-Larcker discriminant criterion compares each construct's AVE with its squared correlations with the other constructs. AVE is closely related to composite reliability (omega); the modern complement on the discriminant side is htmt.

Usage

average_variance_extracted(
  fit = NULL,
  loadings = NULL,
  conf_level = 0.95,
  ci_method = c("none", "percentile"),
  B = 1000L,
  seed = NULL
)

Value

A data.frame (class dmar_tbl) with one row per factor: factor (label), ave, and

ci_lower / ci_upper (the percentile bootstrap limits; NA when ci_method = "none").

Arguments

fit

Optional lavaan fit (for example from cfa_1 or lavaan::cfa); the standardized loadings of every factor are extracted and one AVE is reported per factor.

loadings

Optional numeric vector of standardized loadings for a single block, as an alternative to fit. Supply exactly one of fit and loadings. No interval can be constructed from loadings alone (their sampling variability is not carried by the numbers), so ci_method = "percentile" requires fit.

conf_level

Confidence level for the bootstrap interval (default 0.95); used when ci_method = "percentile".

ci_method

Interval method: "none" (the default) or "percentile". No closed-form interval for the AVE is in common use; the percentile bootstrap is the standard route in the validity literature. The cases in the fitted data are resampled with replacement B times, the model is refit to each resample, and each factor's interval is the pair of empirical quantiles of its B AVE values (Efron & Tibshirani, 1993). Replications whose refit fails or does not converge are dropped, and the interval is computed from those that return a value; a single warning reports how many were dropped.

B

Number of bootstrap replications when ci_method = "percentile" (default 1000). The default is smaller than the package's usual 10000 because every replication refits the model; raise it for a reported analysis when time allows.

seed

Optional integer seed for the bootstrap. The default NULL uses the current state of the random number generator; a supplied seed is set internally and the prior state restored on exit.

Author

Ken Kelley [email protected]

Details

The percentile bootstrap interval resamples the cases behind fit and refits the model once per replication, so its cost is B model fits. That refitting is why the examples below stop at the point estimates: even the smallest permitted B = 100 runs for several seconds on the two-factor model there. To obtain the interval, pass ci_method = "percentile" together with a seed, as in average_variance_extracted(fit, ci_method = "percentile", seed = 113), and keep the default B = 1000 or more for a reported analysis.

References

Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. New York, NY: Chapman & Hall/CRC.

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

See Also

htmt for discriminant validity; reliability_omega for the composite reliability of the same block (coefficient omega is what the composite-reliability literature computes); cfa_1 to obtain the fit.

Other multivariate and latent variable methods: bifactor_indices(), cfa_1(), cfa_2(), cfa_k(), 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

# Directly from the standardized loadings a paper reports:
average_variance_extracted(loadings = c(.8, .7, .6))

# From a fitted model, one AVE per factor (requires lavaan).
data(holzinger_swineford)
fit <- lavaan::cfa(
  "verbal    =~ t6_paragraph_comprehension + t7_sentence +
                t9_word_meaning
   deduction =~ t20_deduction + t22_problem_reasoning +
                t23_series_completion",
  data = holzinger_swineford)
ave_tbl <- average_variance_extracted(fit)
ave_tbl

# The Fornell and Larcker (1981) discriminant criterion compares each
# factor's AVE with the squared correlation between the factors: a
# factor should account for more of its own indicators' variance than
# it shares with the other factor. Here the comparison favors verbal
# and goes against deduction, whose AVE falls below the shared
# variance. Fitting with cfa_k(..., output = "measurement") puts the
# AVE values and the latent correlations in one table.
lavaan::lavInspect(fit, "cor.lv")["verbal", "deduction"]^2

# The broom verbs: one row per factor.
generics::tidy(ave_tbl)
generics::glance(ave_tbl)