Confirmatory Factor Analysis (CFA)

Author

Riccardo Volpato

options(readr.show_col_types = FALSE)

source("read_qualtrics_data.R")
d <- load_data("data/pilot/TPSai_17+March+2026_16.59.csv")
cols_tpsai <- d$cols_tpsai

# switch to d$items_data after data collection
cfa_data <- read_csv("data/pilot/simulated_cfa_data.csv") %>%
  select(all_of(cols_tpsai))

Confirmatory factor analysis (CFA) tests whether a hypothesised factor structure — derived from theory and/or EFA results — replicates in the data. Unlike EFA, which lets the data reveal structure freely, CFA is a hypothesis-testing procedure: we specify in advance which items load onto which factors and which cross-loadings are constrained to zero, then evaluate how well that model fits the observed covariance matrix.

We use the WLSMV (weighted least squares, mean- and variance-adjusted) estimator throughout. WLSMV is the recommended estimator for ordinal Likert data because it treats items as ordered-categorical rather than continuous, avoiding the underestimation of factor loadings and the distorted fit indices that arise when standard maximum-likelihood estimation is applied to non-normal ordinal responses.

Build Models

trust_1f_model <- cfa(paste("TR =~", paste(d$cols_trust, collapse = " + ")),
                      cfa_data, estimator = "WLSMV", ordered = TRUE)

distrust_1f_model <- cfa(paste("DT =~", paste(d$cols_distrust, collapse = " + "), " + ",
                               paste(d$cols_company, collapse = " + ")),
                      cfa_data, estimator = "WLSMV", ordered = TRUE)

trust_efa_eq <- '
  MR1 =~ TR_02 + TR_04 + TR_06 + TR_09 + TR_10 + TR_12 + TR_13 + TR_14 + TR_15 + TR_16 + TR_18 + TR_20 + TR_22
  MR2 =~ TR_01 + TR_03 + TR_05 + TR_07 + TR_08 + TR_11 + TR_17 + TR_19 + TR_21
'
trust_efa_model <- cfa(trust_efa_eq,
                      cfa_data, estimator = "WLSMV", ordered = TRUE)

distrust_efa_eq <- '
MR1 =~ DT_01 + DT_02 + DT_04 + DT_06 + DT_10 + DT_11 + DT_13 + DT_14 + CD_04 + CD_05
MR2 =~ DT_03 + DT_05 + DT_07 + DT_08 + DT_09 + DT_12 + CD_01 + CD_02 + CD_03 + CD_06 + CD_07
'
distrust_efa_model <- cfa(distrust_efa_eq,
                        cfa_data, estimator = "WLSMV", ordered = TRUE)

Model Fit

We evaluate fit using the indices recommended by Boateng et al. (2018) and (briones2026?). No single index is sufficient; good fit requires convergent evidence across indices.

Index	Good fit	Acceptable	Poor fit
CFI	≥ .95	.90–.94	< .90
TLI	≥ .95	.90–.94	< .90
RMSEA	≤ .06	.06–.08	> .10
SRMR	≤ .06	.06–.08	> .08

extract_fit <- function(fit, subscale, model_label) {
  fi <- fitMeasures(fit, c("chisq", "df", "pvalue",
                            "cfi", "tli",
                            "rmsea", "rmsea.ci.lower", "rmsea.ci.upper",
                            "srmr"))
  tibble(
    Subscale = subscale,
    Model    = model_label,
    `χ²`    = round(fi["chisq"], 2),
    df       = fi["df"],
    # p        = round(fi["pvalue"], 3),
    CFI      = round(fi["cfi"],   3),
    TLI      = round(fi["tli"],   3),
    RMSEA    = round(fi["rmsea"], 3),
    `90% CI` = paste0("[", round(fi["rmsea.ci.lower"], 3),
                      ", ", round(fi["rmsea.ci.upper"], 3), "]"),
    SRMR     = round(fi["srmr"],  3)
  )
}

bind_rows(
  extract_fit(trust_1f_model, "Trust", "1F"),
  extract_fit(trust_efa_model, "Trust", "EFA"),
  extract_fit(distrust_1f_model, "Disrust", "1F"),
  extract_fit(distrust_efa_model, "Disrust", "EFA"),
) %>%
  kable()

Subscale	Model	χ²	df	CFI	TLI	RMSEA	90% CI	SRMR
Trust	1F	5696.62	209	0.656	0.619	0.296	[0.29, 0.303]	0.239
Trust	EFA	5135.44	208	0.691	0.657	0.281	[0.275, 0.288]	0.230
Disrust	1F	6075.73	189	0.573	0.525	0.323	[0.316, 0.33]	0.260
Disrust	EFA	5061.13	188	0.646	0.605	0.294	[0.287, 0.301]	0.240

We can also compare models with a scaled chi-square difference test.

We prefer more complex models when both criteria are met: the difference test is significant (p < .05) and ΔCFI > .01. Using both criteria guards against retaining a complex model purely on the basis of a statistically significant but practically trivial improvement in fit.

lavTestLRT(trust_1f_model, trust_efa_model, model.names = c("1F", "EFA"))


Scaled Chi-Squared Difference Test (method = "satorra.2000")

lavaan->lavTestLRT():  
   lavaan NOTE: The "Chisq" column contains standard test statistics, not the 
   robust test that should be reported per model. A robust difference test is 
   a function of two standard (not robust) statistics.

     Df AIC BIC  Chisq Chisq diff  RMSEA Df diff Pr(>Chisq)    
EFA 208         5135.4                                         
1F  209         5696.6     127.28 1.3665       1  < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

lavTestLRT(distrust_1f_model, distrust_efa_model, model.names = c("1F", "EFA"))


Scaled Chi-Squared Difference Test (method = "satorra.2000")

lavaan->lavTestLRT():  
   lavaan NOTE: The "Chisq" column contains standard test statistics, not the 
   robust test that should be reported per model. A robust difference test is 
   a function of two standard (not robust) statistics.

     Df AIC BIC  Chisq Chisq diff  RMSEA Df diff Pr(>Chisq)    
EFA 188         5061.1                                         
1F  189         6075.7     145.13 1.8381       1  < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

References

Boateng, Godfred O., Torsten B. Neilands, Edward A. Frongillo, Hugo R. Melgar-Quiñonez, and Sera L. Young. 2018. “Best Practices for Developing and Validating Scales for Health, Social, and Behavioral Research: A Primer.” Frontiers in Public Health 6 (June). https://doi.org/10.3389/fpubh.2018.00149.