Average Variance Extracted (AVE) and HTMT

Q: What is a good AVE value?

0.50 or above is the conventional benchmark, meaning the construct explains more variance in its indicators than measurement error does. Treat it as a benchmark rather than a hard pass/fail line — it is a convention, not a derived constant.

Q: How do I calculate AVE by hand?

Square each standardized factor loading, sum the squares, and divide by the sum of the squared loadings plus the summed error variances. In a fully standardized solution this is just the mean of the squared loadings.

Q: What is the difference between AVE and composite reliability?

Both use the same loadings and error variances, but CR squares the sum of the loadings while AVE sums the squares. AVE is always the more demanding criterion, which is why a construct can show CR of 0.82 and AVE of 0.54 on identical data.

Q: Can AVE be used for formative constructs?

No. AVE assumes a reflective measurement model in which the latent construct causes its indicators. Formative constructs are evaluated on indicator weights and collinearity instead.

Q: Is the Fornell-Larcker criterion still acceptable?

It is still widely reported and many journals still expect it, but the methodological literature has moved on: Henseler et al. (2015) showed it does not reliably detect discriminant-validity problems, and Rönkkö and Cho (2022) argue it is also biased toward false positives. Report HTMT, and report Fornell-Larcker alongside it if required rather than instead of it.

Q: Should I use 0.85 or 0.90 as the HTMT threshold?

0.85 for conceptually distinct constructs, 0.90 where the constructs are conceptually similar. Both are conventions rather than derived values. Bootstrap inference — testing whether the confidence interval for HTMT includes 1 — is more defensible than either fixed cut-off, because 1 is a value with an actual interpretation.

Q: Why does my AVE differ from the one in the published paper?

The most common cause is mixing standardized and unstandardized solutions. The simplified “mean of squared loadings” formula is only valid for a fully standardized solution; otherwise the actual residual variances must be used.

Average variance extracted (AVE) is the mean proportion of variance that a latent construct explains in its own indicators, rather than measurement error explaining it. It is the statistic behind two separate judgements in a measurement model: whether a construct shows adequate convergent validity, and — through the Fornell-Larcker criterion — whether it is discriminant from the other constructs in the model.

This page gives the calculation, shows where the 0.50 cut-off actually comes from, and works through the discriminant-validity tests that consume AVE. The part most content omits: the Fornell-Larcker criterion is still presented almost everywhere as the standard discriminant-validity test, and the methodological literature has been arguing against that for a decade. If you are reporting a measurement model, the section on what is wrong with Fornell-Larcker is the one that changes what you do.

AVE is computed from the standardized factor loadings of a measurement model, so it presupposes one. If you have not fitted one yet, start with CASRAI’s guide to confirmatory factor analysis — every number on this page comes out of a CFA (or a PLS-SEM outer model), and AVE reported without a stated measurement model is not interpretable.

How AVE Is Calculated

For a reflective construct with n indicators and standardized factor loadings λ_i:

AVE = Σλ_i² / ( Σλ_i² + ΣVar(ε_i) )

In a fully standardized solution the error variance of each indicator is Var(ε_i) = 1 − λ_i², so the denominator collapses to n and the formula reduces to the mean of the squared loadings:

AVE = Σλ_i² / n

That reduction only holds when the solution is fully standardized. If you are working from an unstandardized solution, use the full formula with the actual residual variances — this is a routine source of wrong AVE values in submitted manuscripts.

Worked Example

A four-item reflective construct with standardized loadings of 0.82, 0.76, 0.71 and 0.64:

Item	λ	λ² (communality)	1 − λ² (error)
1	0.82	0.6724	0.3276
2	0.76	0.5776	0.4224
3	0.71	0.5041	0.4959
4	0.64	0.4096	0.5904
Sum	2.93	2.1637	1.8363

AVE = 2.1637 / (2.1637 + 1.8363) = 2.1637 / 4 = 0.541

The construct explains about 54% of the variance in its four items; measurement error accounts for the remaining 46%.

Composite reliability (CR) is computed from the same quantities but squares the sum of the loadings rather than summing the squares:

CR = (Σλ_i)² / [ (Σλ_i)² + ΣVar(ε_i) ] = 2.93² / (2.93² + 1.8363) = 8.5849 / 10.4212 = 0.824

Note what this shows: the same four items yield CR = 0.82 and AVE = 0.54. AVE is always the more demanding of the two, which matters for the disputed citation discussed below.

Where the 0.50 Threshold Comes From

The 0.50 benchmark traces to Fornell and Larcker (1981), Evaluating Structural Equation Models with Unobservable Variables and Measurement Error, Journal of Marketing Research 18(1), 39–50, where it was introduced as a convergent-validity criterion.

Its logic is a single readable sentence rather than a derivation: at AVE = 0.50 the construct accounts for exactly as much variance in its indicators as measurement error does. Below 0.50, error explains more of the items than the construct they are supposed to measure — which makes the indicators hard to defend as measures of that construct.

This is a convention, not a derived constant. Nothing in measurement theory makes 0.50 a discontinuity; it is a memorable break-even point that became a reporting norm. An AVE of 0.49 and an AVE of 0.51 do not describe materially different instruments, and treating the cut-off as a pass/fail gate — rather than as one piece of evidence in an accumulating construct validity argument — is a misuse of it. The same caution applies to every threshold on this page.

The 0.50 convention is also why PLS-SEM guidance (Hair and colleagues) puts the indicator-loading rule of thumb at 0.708 rather than a round 0.70: 0.708² = 0.501, so an item loading at 0.708 contributes exactly half its variance to the construct. An instrument whose loadings all sit at or above 0.708 will clear AVE = 0.50 automatically. The two rules of thumb are the same rule expressed on different scales.

The AVE < 0.50 / CR > 0.60 Citation — Check It Before You Repeat It

A very widely propagated claim in applied papers runs: “Fornell and Larcker (1981) state that if AVE is below 0.50 but composite reliability exceeds 0.60, convergent validity is still adequate.” It is normally cited to page 46 of the original article, and it is used to justify retaining constructs that fail the AVE threshold.

Treat that citation with suspicion. The passage it points to is making a warning, not granting a licence: it observes that composite reliability is the less conservative of the two statistics, and that a researcher looking at CR alone may conclude convergent validity is adequate even when more than half the variance is error. Read in context, that is an argument for reporting AVE precisely because CR can flatter a weak construct — which is the opposite of how the sentence is usually deployed.

We were not able to re-verify the exact wording of page 46 against the original JMR article, which is paywalled, so this page does not quote it. If you intend to rely on this claim in a manuscript, read the primary source yourself rather than citing it second-hand from another applied paper — this is one of the most-copied and least-checked citations in the measurement-model literature, and reviewers increasingly ask about it. The safer and more defensible move is to report the low AVE openly and justify the construct on other grounds, not to cite a threshold exemption.

AVE, Composite Reliability and Cronbach’s Alpha: What to Report

These three statistics are computed from overlapping inputs and answer different questions. Reporting all three is normal and correct; substituting one for another is not.

Statistic	Question it answers	Common convention
Cronbach’s alpha	Internal consistency, assuming all items are equally good indicators (essential tau-equivalence)	≥ 0.70
Composite reliability (CR)	Internal consistency without assuming equal loadings	0.70–0.90; above 0.95 suggests redundant items
Average variance extracted (AVE)	How much indicator variance the construct explains vs. error	≥ 0.50

Alpha assumes something a CFA does not require, and it is the weakest of the three when loadings differ appreciably — as they do in the worked example above. CR and AVE both come free once you have fitted the measurement model. Note the upper bound on CR: a value above roughly 0.95 is a warning that items are near-duplicates of each other, not a better result. See CASRAI’s guide to reliability in research for how these relate to test-retest and inter-rater forms, and reliability vs. validity for why a highly reliable instrument can still be measuring the wrong thing.

The Fornell-Larcker Criterion

Fornell and Larcker’s second contribution was a discriminant-validity test built on AVE. The criterion:

Discriminant validity holds between two constructs when the AVE of each exceeds the squared correlation between them.

Equivalently, and as it is more often tabulated: the square root of each construct’s AVE must exceed that construct’s correlation with every other construct in the model. The intuition is that a construct should share more variance with its own indicators than it shares with any other construct.

Worked Example, Continued

Take the construct above (AVE = 0.541, √AVE = 0.735) and a second three-item construct with loadings 0.88, 0.81 and 0.77:

Σλ² = 0.7744 + 0.6561 + 0.5929 = 2.0234, so AVE = 2.0234 / 3 = 0.674, and √AVE = 0.821.

If the estimated correlation between the two constructs is 0.70, then the squared correlation is 0.49:

Construct 1: AVE 0.541 > 0.49 &check;
Construct 2: AVE 0.674 > 0.49 &check;

The pair passes. But look how little headroom there is. Had the correlation been 0.73 instead of 0.70, the squared correlation would be 0.533 and construct 1 would pass by 0.008. At a correlation of 0.74 it fails. A test whose verdict flips on the second decimal place of an estimated correlation is not a robust gate, and this sensitivity is part of why the criterion has been challenged.

Two Ways People Get This Test Wrong

1. Using the wrong correlation. The criterion is defined on the factor correlation — the latent correlation estimated by the model, which is already corrected for attenuation. Rönkkö and Cho (2022) identify the most common misapplication as comparing AVE values against the square of the scale score correlation (the correlation of summed or averaged item scores) instead. Scale-score correlations are attenuated by unreliability, so they are smaller, so the test passes when it should not. If your software hands you a table of composite correlations rather than latent correlations, check which one you are squaring.

2. Using AVE ≥ 0.50 as the discriminant test. The 0.50 cut-off is a convergent-validity criterion. Comparing AVEs against 0.50 and calling the result discriminant validity conflates two distinct judgements. Discriminant validity is the AVE-versus-squared-correlation comparison; 0.50 plays no part in it.

HTMT: The Heterotrait-Monotrait Ratio

Henseler, Ringle and Sarstedt (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, proposed HTMT as a replacement.

HTMT is the ratio of (a) the average correlation between items measuring different constructs to (b) the geometric mean of the average correlations among items within each construct. If items belonging to two supposedly different constructs correlate about as strongly with each other as they do with their own stablemates, the constructs are not empirically distinct and HTMT approaches 1.

The conventional thresholds:

HTMT < 0.85 — for constructs that are conceptually distinct.
HTMT < 0.90 — the more permissive threshold, applied where the constructs are conceptually similar.
HTMT inference — rather than a fixed cut-off, bootstrap a confidence interval for HTMT. Discriminant validity fails if the interval includes 1, since that is the value at which the two constructs are indistinguishable.

The 0.85 and 0.90 figures are conventions, exactly as 0.50 is. Rönkkö and Cho (2022) note explicitly that these thresholds are not empirically derived. They are consensus reporting norms. The bootstrap-inference variant is the more defensible option precisely because it tests against a value (1) that has an actual meaning, rather than against a number chosen for being memorable.

One practical note: HTMT is computed from the item correlation matrix, not from factor loadings. You cannot back it out of a published loadings table the way you can recompute AVE and CR. If you are reassessing someone else’s measurement model, you need their correlation matrix or their data.

Why Fornell-Larcker Is No Longer the Default

This is the part that most treatments of AVE leave out, and it is the reason the calculation above matters less than it looks.

Henseler and colleagues’ central contribution was not the HTMT formula but a Monte Carlo simulation testing whether the established criteria actually detect discriminant-validity problems that were known, by construction, to exist in the simulated data. The finding: the Fornell-Larcker criterion and the examination of cross-loadings do not reliably detect a lack of discriminant validity in common research situations, while HTMT does. Cross-loadings performed particularly badly. (Specific per-condition detection rates are reported in the original article; we have not reproduced them here because we could not verify the exact figures against the paywalled primary source, and approximate numbers are not worth quoting on a methods page.)

The awkward consequence is that a large body of published research reports Fornell-Larcker tables showing discriminant validity established, where the test used had limited power to find a problem in the first place. Passing Fornell-Larcker is weaker evidence than it appears.

The 2022 Update: HTMT Is Not the End of the Story Either

Content that stops at “use HTMT instead of Fornell-Larcker” is roughly seven years out of date. Rönkkö and Cho (2022), An Updated Guideline for Assessing Discriminant Validity, Organizational Research Methods, reviewed the whole family of techniques and reached a more uncomfortable set of conclusions:

Fornell-Larcker is not merely insensitive — it is biased. They characterise it as severely biased with a high false-positive rate, flagging discriminant-validity problems in conditions where most researchers would not consider one to exist. That is a different criticism from Henseler’s, and it cuts in the opposite direction.
HTMT is a familiar statistic under an unfamiliar name. They show it is equivalent to a disattenuated correlation of unit-weighted composites using parallel reliability, and argue the “heterotrait-monotrait” label is misleading because it implies a connection to multitrait-multimethod analysis that is not really there. They propose renaming it accordingly.
Their own recommendation is neither test. They recommend CICFA(sys), based on confidence intervals in confirmatory factor analysis, and χ²(sys), based on nested model comparisons in CFA — on the grounds that these have appropriate statistical power, manageable false-positive rates, and can be adapted to different cut-off thresholds rather than being locked to one.

Both CFA-based recommendations reinforce the point at the top of this page: these are measurement-model questions, and the model is the unit of analysis, not the summary statistic.

What to Actually Do

A defensible reporting procedure for a reflective measurement model:

Fit and report the measurement model first. Standardized loadings with standard errors, and fit indices. AVE without the model behind it cannot be checked by a reader.
Report AVE and CR per construct, alongside alpha if your field expects it. State whether the solution is standardized — this determines which AVE formula applies.
Treat AVE ≥ 0.50 as a benchmark, not a gate. If a construct falls below it, say so plainly and address it, rather than citing a threshold exemption.
Use HTMT with bootstrap inference for discriminant validity, not Fornell-Larcker alone. If a journal or reviewer requires a Fornell-Larcker table, report both — and confirm you are squaring the latent correlations.
Say which criterion you applied and which threshold, including whether you used 0.85 or 0.90 and why. An unstated threshold is unreproducible.
If the constructs are central to your argument, consider a CFA-based test (nested model comparison, or confidence intervals on the factor correlation) rather than relying on a single ratio.

When AVE Falls Below 0.50

The honest options, in rough order of preference:

Inspect the loadings. A low AVE is usually one or two weak items dragging down an otherwise sound construct, not uniform weakness. Removing a genuinely poor indicator is legitimate; removing items until the number clears the threshold is not, and leaves an audit trail in the item count.
Check whether the construct is actually unidimensional. Low AVE frequently means two dimensions have been forced into one factor. That is a specification problem, and no amount of item deletion fixes it.
Reconsider whether the indicators are reflective at all. AVE, CR and Fornell-Larcker all assume a reflective measurement model, where the construct causes the indicators. They do not apply to formative constructs, which are assessed on collinearity and indicator weights instead. Applying AVE to a formative block is a category error — see multicollinearity and VIF for the diagnostics that do apply.
Report it and defend it. A construct with AVE of 0.46, disclosed, with the weak items identified and the limitation stated, is more credible than one that reached 0.51 through undisclosed item deletion.

Frequently Asked Questions

What is a good AVE value?