Scale reliability and validity

Reliability from the instrument

reliability_report() reads the scale definitions and reports Cronbach’s alpha and, with the optional psych package, McDonald’s omega. The published study reported strong reliability for every construct. At the higher-order level the alpha and composite reliability were 0.828 and 0.883 for digital marketing effectiveness, 0.792 and 0.906 for service quality, 0.769 and 0.843 for sustainability quality, 0.897 and 0.936 for satisfaction, and 0.921 and 0.950 for behavioural intention.

demo      <- sframe_demo_data()
instr     <- demo$instrument
responses <- demo$responses

rr <- reliability_report(responses, instr, omega = TRUE)
# as.data.frame() gives one row per scale, so no reaching into the object.
rr_df <- as.data.frame(rr)
rel_df <- data.frame(
  Scale   = paste0(rr_df$label, " (", rr_df$scale_id, ")"),
  Items   = rr_df$n_items,
  N       = rr_df$n,
  Alpha   = ifelse(is.na(rr_df$alpha),   "n/a", sprintf("%.2f", rr_df$alpha)),
  Omega_h = ifelse(is.na(rr_df$omega_h), "n/a", sprintf("%.2f", rr_df$omega_h)),
  Omega_t = ifelse(is.na(rr_df$omega_t), "n/a", sprintf("%.2f", rr_df$omega_t)),
  stringsAsFactors = FALSE)
kable(rel_df, row.names = FALSE,
      col.names = c("Scale", "Items", "N", "Alpha", "Omega h", "Omega total"),
      align = c("l", "c", "c", "r", "r", "r"),
      caption = "Scale reliability statistics")

Item diagnostics

Item diagnostics identify sparse items, weak item-total relationships, and floor or ceiling effects. These are the item-level facts behind a retention decision.

demo      <- sframe_demo_data()
instr     <- demo$instrument
responses <- demo$responses

items <- item_report(responses, instr)
first  <- items[[1]]
kable(first$diagnostics, digits = 2,
      caption = paste0("Item diagnostics: ", first$label, " (", first$scale_id, ")"))

EFA readiness

efa_report() reports the Kaiser-Meyer-Olkin measure and Bartlett’s test as a screening step before a confirmatory model.

efa_report(responses, instr)
#> EFA Readiness Diagnostics
#> 
#>   Items:          12
#>   Complete cases: 120
#>   KMO overall:    0.761
#>   Bartlett chi-sq: 665.84  df: 66  p: 0.0000
#>   Suggested factors (parallel analysis): 4
#>   Planned rotation: oblimin
#> 
#> Note: estimate the EFA solution with a dedicated modelling package.

Convergent validity from the published loadings

This step runs on the loadings alone. validity_report() computes composite reliability and average variance extracted from standardised loadings. Feeding the outer loadings reported in the paper reproduces its convergent validity, which is a direct check of the measurement model.

published_loadings <- list(
  DMRE = c(dmre_1 = .815, dmre_2 = .899, dmre_3 = .668, dmre_4 = .838),
  DMAU = c(dmau_1 = .843, dmau_2 = .915, dmau_3 = .920),
  DMEU = c(dmeu_1 = .897, dmeu_2 = .929, dmeu_3 = .932),
  DMPV = c(dmpv_1 = .818, dmpv_2 = .916, dmpv_3 = .900, dmpv_4 = .863),
  DSQA = c(dsqa_1 = .904, dsqa_2 = .920, dsqa_3 = .869),
  DSQT = c(dsqt_1 = .778, dsqt_2 = .883, dsqt_3 = .879, dsqt_4 = .811, dsqt_5 = .713),
  DSUQ = c(dsuq_1 = .780, dsuq_2 = .855, dsuq_3 = .845, dsuq_4 = .551, dsuq_5 = .529),
  TS   = c(ts_1 = .912, ts_2 = .950, ts_3 = .869),
  BI   = c(bi_1 = .937, bi_2 = .911, bi_3 = .940)
)

validity <- validity_report(published_loadings)
kable(as.data.frame(validity), digits = 2,
      caption = "Composite reliability and average variance extracted")

The average variance extracted exceeds 0.5 for every construct, and the composite reliability exceeds 0.7, so each construct explains more than half the variance in its items. The sustainability construct sits lowest, in line with the two weaker items the paper flagged (DSUQ4 and DSUQ5).

Discriminant validity

The study assessed discriminant validity in two ways. The Fornell-Larcker criterion compares the square root of a construct’s AVE with its correlations with the others. The square roots, 0.810 for digital marketing effectiveness, 0.910 for service quality, 0.726 for sustainability quality, 0.911 for satisfaction, and 0.930 for behavioural intention, each exceeded the off-diagonal correlations. The Heterotrait-Monotrait ratios were all below the 0.90 threshold, the highest being 0.766 between ease of use and accessibility. Both results support discriminant validity.

validity_report() computes both, and which one you get depends on what you give it. Construct scores give the Fornell-Larcker matrix, and the htmt element then holds the absolute inter-construct correlations, which is a different quantity from the Henseler ratio. Item-level responses per construct give the real heterotrait-monotrait ratio. The report records which it computed in htmt_method, so a write-up can name the quantity it reports.

demo <- sframe_demo_data()
scored <- score_scales(demo$responses, demo$instrument)
scale_ids <- vapply(sf_scales(demo$instrument), sf_id, character(1))
construct_scores <- scored[, scale_ids, drop = FALSE]

loadings <- lapply(sf_scales(demo$instrument), function(s) {
  stats::setNames(rep(0.8, length(s$items)), s$items)
})
names(loadings) <- scale_ids

from_scores <- validity_report(loadings, construct_scores = construct_scores)
from_scores$htmt_method
#> [1] "correlation_fallback"

Passing the items themselves is what asks for the Henseler ratio:

items_by_construct <- lapply(sf_scales(demo$instrument), function(s) {
  scored[, s$items, drop = FALSE]
})
names(items_by_construct) <- scale_ids

real_htmt <- validity_report(loadings, construct_scores = construct_scores,
                             items_by_construct = items_by_construct)
real_htmt$htmt_method
#> [1] "henseler"

A construct with a single item has no monotrait correlations, so its HTMT entries come back NA.

Collinearity

The study reported variance inflation factors below 5 for every indicator, which indicates multicollinearity mild enough to proceed. assumption_report() returns variance inflation factors when a regression is specified, which is shown in the “Analysing survey responses” vignette.

Cautious interpretation

Reliability and validity summaries are diagnostics, not automatic decisions. Read them with the questionnaire wording, the sampling context, the construct definitions, and the planned model in view.