Convergent and Discriminant Validity: AVE, CR, HTMT

Convergent validity means that items intended to measure the same construct are strongly related to each other. Discriminant validity means that a construct is empirically distinct from other constructs in the model. In survey research using confirmatory factor analysis (CFA) or PLS-SEM, convergent validity is usually checked with average variance extracted (AVE ≥ .50) and composite reliability (CR ≥ .70), and discriminant validity with the Fornell–Larcker criterion and the heterotrait-monotrait ratio (HTMT).

This guide explains each criterion, computes AVE and CR by hand from standardized loadings, and gives reporting sentences you can adapt for your measurement model section.

Where these checks fit

Both are forms of construct validity evidence based on respondent data. A typical validation sequence for a multi-item questionnaire is:

1. Content validity from an expert panel, before data collection.
2. Internal consistency (Cronbach's alpha and/or composite reliability).
3. A measurement model (CFA or the outer model in PLS-SEM) with standardized factor loadings.
4. Convergent validity: loadings, AVE, CR.
5. Discriminant validity: Fornell–Larcker, HTMT, and cross-loadings.
6. Only then, the structural model or hypothesis tests.

If the measurement model fails, results from the structural model are hard to defend, which is why reviewers look at these tables first.

Convergent validity: the three criteria

1. Standardized factor loadings

Each item should load substantially on its construct. A common guideline is a standardized loading of at least .70, because a loading of .70 means the construct explains about half of the item's variance (.70² = .49). Items between about .40 and .70 are often retained if AVE and CR remain adequate and the item matters for content coverage; items well below that are candidates for removal.

2. Average variance extracted (AVE)

AVE is the average amount of variance in the items explained by the construct:

AVE = (sum of squared standardized loadings) / (number of items)

Criterion: AVE ≥ .50 (Fornell & Larcker, 1981). An AVE of .50 or more means the construct explains more variance in its items than measurement error does.

3. Composite reliability (CR)

CR estimates internal consistency while allowing items to have different loadings (alpha assumes equal loadings):

CR = (sum of loadings)² / [(sum of loadings)² + sum of error variances]

For standardized loadings, each item's error variance is 1 − loading².

Criterion: CR ≥ .70. Values between .60 and .70 are sometimes accepted in exploratory research. Very high values (above about .95) can suggest redundant, near-duplicate items.

Worked example: AVE and CR from standardized loadings

Suppose a CFA with two constructs, Trust (4 items) and Satisfaction (3 items), produced these standardized loadings:

Trust

• Sum of λ² = .6724 + .6084 + .5625 + .4900 = 2.3333
• AVE = 2.3333 / 4 = .583
• Sum of loadings = .82 + .78 + .75 + .70 = 3.05; squared = 9.3025
• Sum of error variances = .3276 + .3916 + .4375 + .5100 = 1.6667
• CR = 9.3025 / (9.3025 + 1.6667) = 9.3025 / 10.9692 = .848

Satisfaction

• Sum of λ² = .7225 + .6400 + .5929 = 1.9554
• AVE = 1.9554 / 3 = .652
• Sum of loadings = .85 + .80 + .77 = 2.42; squared = 5.8564
• Sum of error variances = .2775 + .3600 + .4071 = 1.0446
• CR = 5.8564 / (5.8564 + 1.0446) = 5.8564 / 6.9010 = .849

Both constructs meet AVE ≥ .50 and CR ≥ .70, and all loadings are at least .70. Convergent validity is supported.

What failure looks like

Imagine a third construct with loadings .72, .65, .58, and .55. Its AVE = (.5184 + .4225 + .3364 + .3025) / 4 = .395, below .50, even though CR = 6.25 / (6.25 + 2.4202) = .721 passes. This is a common situation: CR can look fine while AVE fails, because CR rises with the number of items. Some authors accept an AVE slightly below .50 when CR is adequate, but you need to justify it; the cleaner option is to look at the weakest items (here .58 and .55), check their wording, and consider removing one if content coverage allows.

Discriminant validity: Fornell–Larcker criterion

Fornell and Larcker (1981) proposed that a construct should share more variance with its own items than with any other construct. In practice:

The square root of each construct's AVE should be greater than its correlations with every other construct.

Continuing the example, suppose the latent correlation between Trust and Satisfaction is .58.

• √AVE for Trust = √.583 = .764
• √AVE for Satisfaction = √.652 = .807

The usual table puts √AVE on the diagonal and correlations below it:

Both diagonal values exceed .58, so the Fornell–Larcker criterion is met.

If the correlation had been .82, it would exceed √AVE for Trust (.764), and discriminant validity would be questionable: the two constructs may be measuring overlapping content.

Limitation. Simulation research by Henseler, Ringle, and Sarstedt (2015) showed that the Fornell–Larcker criterion often fails to detect discriminant validity problems, especially when loadings are fairly similar across items. That is why HTMT is now expected in many journals, particularly for PLS-SEM.

Discriminant validity: the HTMT ratio

The heterotrait-monotrait ratio compares correlations between constructs with correlations within constructs, using the item correlation matrix:

HTMT(A, B) = mean of heterotrait correlations / √(mean monotrait correlation of A × mean monotrait correlation of B)

• Heterotrait correlations: correlations between each item of A and each item of B.
• Monotrait correlations: correlations among items of the same construct (excluding the diagonal).

Example. Suppose the average correlation between Trust items and Satisfaction items is .42, the average correlation among Trust items is .55, and among Satisfaction items is .62.

HTMT = .42 / √(.55 × .62) = .42 / √.341 = .42 / .584 = .72

Thresholds (Henseler et al., 2015):

• HTMT below .85: a conservative cut-off.
• HTMT below .90: acceptable for constructs that are conceptually close (for example, two facets of attitude).
• A stricter alternative is to check, via bootstrapping, that the confidence interval for HTMT does not include 1 (HTMT inference).

With HTMT = .72, Trust and Satisfaction are distinct under either cut-off.

Cross-loadings

A further, weaker check: each item should load more highly on its own construct than on any other construct. Reviewers in PLS-SEM sometimes request a cross-loadings table, but it should supplement HTMT, not replace it.

What to do when validity fails

• Low loading or low AVE: inspect wording, reverse-coded items, and translation issues. Remove an item only if the remaining items still cover the construct, and report the removal.
• High HTMT or Fornell–Larcker failure: check whether two constructs are conceptually distinct. Look for items that cross-load and consider removing them. If the constructs are genuinely the same thing, merging them may be more honest than forcing them apart.
• Re-estimate and re-report all indices after any change. Do not report the original model's indices with the modified model's results.

How to report it (APA 7)

A measurement model paragraph usually includes loadings, CR, AVE, and the discriminant validity tests, with a table for each.

Suggested tables:

1. Measurement model table: construct, item, standardized loading, Cronbach's alpha, CR, AVE.
2. Fornell–Larcker table: √AVE on the diagonal, correlations below.
3. HTMT matrix: HTMT values below the diagonal.

If you are preparing a full survey paper, the questionnaire-to-research-paper guide shows how these tables fit with descriptive statistics and hypothesis testing.

Common mistakes

• Using unstandardized loadings. The AVE and CR formulas above assume standardized loadings.
• Computing CR with Cronbach's alpha values. They are different statistics; report both if you like, but compute CR from loadings.
• Reporting only Fornell–Larcker. Many reviewers now expect HTMT as well.
• Comparing AVE (not √AVE) to correlations. Fornell–Larcker compares √AVE with correlations, or equivalently AVE with squared correlations. Mixing the two scales is a frequent error.
• Ignoring content. A statistically acceptable model with an item that does not belong conceptually is still a weak instrument; content validity should come first.

FAQ

What is the difference between convergent and discriminant validity?

Convergent validity shows that items of one construct agree with each other (high loadings, AVE ≥ .50, CR ≥ .70). Discriminant validity shows that different constructs are not too similar (Fornell–Larcker criterion, HTMT below .85 or .90).

Can AVE be below .50 if CR is above .70?

Some researchers accept it with justification, but it signals that more than half of the item variance is error. Check weak items first and explain your decision in the paper.

Should I use HTMT .85 or .90?

Use .85 as the default, conservative cut-off. The .90 threshold is reasonable when constructs are conceptually similar, and you should say why. Reporting bootstrapped HTMT confidence intervals is stronger still.

Do I need both Cronbach's alpha and composite reliability?

Many journals expect both. Alpha is familiar to most readers; CR accounts for unequal loadings and is the standard in CFA and PLS-SEM measurement models.

Can I assess discriminant validity with simple correlations between scale means?

That is a weak test. Use latent correlations from the measurement model for Fornell–Larcker, and the item correlation matrix for HTMT.

From survey data to a draft

If you have the full survey dataset, Data2Paper's Generate Paper can produce a paper draft with reliability, validity, and hypothesis-testing sections, plus tables and references. The first half is free to preview; unlocking the full draft is a one-time $19.90. Treat it as a draft: verify every statistic and citation before submission.