Academic Reporting ·
Good Model Fit, Poor Discriminant Validity: When Two Constructs Won't Separate
Ten fit indices accepted, and the Fornell-Larcker table still failed: the attitude-intention latent correlation came in above the square root of AVE for both. What caused it, which item to revise, and what the run said it had not tested. Demo data, 287 complete cases.
Good model fit but poor discriminant validity is a normal result, not a contradiction: in this run all ten fit indices were marked acceptable, yet the Fornell-Larcker table failed because the attitude-intention correlation (0.854) sat above the square root of AVE for both constructs (0.811 and 0.821).
Fit indices answer one question — does the model reproduce the observed covariances. Whether the constructs inside the model are actually different things is a separate question, checked on a separate table. This page is about the case where the first table is all green and the second one isn't.
What This Page Covers
This is not a walkthrough of how to specify and estimate a confirmatory factor analysis. Model specification, estimator choice, what the output contains and what its stated limits are — those are covered in the broader questionnaire reliability and validity workflow, which is where to start if the analysis hasn't been run yet. This page covers one situation only: the fit table passes on every row, the discriminant-validity table does not, and you need to know how to read that, what caused it, and what you can and cannot write in the paper.
Two terms, in plain English. A latent variable is something the questionnaire can't measure directly, so several items are asked and the construct is inferred from how they move together. Discriminant validity asks whether two latent variables are separate things in the data: when respondents answer these two blocks of items, are they making two judgments or one?
Demo data throughout — a simulated survey, not real thesis data. The model was estimated by maximum likelihood on 287 complete cases after listwise deletion of 13 responses with missing item values, using the 20 observed items and five correlated latent factors: PU (perceived usefulness), PEOU (perceived ease of use), SI (social influence), ATT (attitude), and BI (behavioral intention).
First, Confirm the Fit Indices Really Did All Pass
They did. Ten indices, every one of them marked "Accepted" in the tool's own acceptability column.
| Index category | Index | Criterion | Result |
|---|---|---|---|
| Absolute fit | GFI | > 0.8 | 0.947 |
| Absolute fit | AGFI | > 0.8 | 0.930 |
| Absolute fit | RMSEA | < 0.08 | 0.012 |
| Absolute fit | SRMR | < 0.10 | 0.034 |
| Incremental fit | NFI | > 0.8 | 0.948 |
| Incremental fit | IFI | > 0.8 | 0.998 |
| Incremental fit | CFI | > 0.8 | 0.998 |
| Incremental fit | TLI | > 0.8 | 0.998 |
| Parsimonious fit | chi-square/df | < 3 | 1.040 |
| Parsimonious fit | PGFI | > 0.5 | 0.721 |
Source: ChatSRS Model Fit Indices table, simulated survey data, 287 complete cases, maximum likelihood estimation. The criteria column is the tool's own stated threshold set, reproduced as printed.
The tool's summary: "All reported fit indices met the stated acceptance criteria. Thus, the proposed five-factor covariance structure reproduces the observed data very well at the global model-fit level." Note the qualifier at the end of that sentence — at the global model-fit level. That is the whole point of this page.
The Table That Didn't Pass
The Fornell-Larcker table puts the square root of each construct's AVE on the diagonal and the correlations between constructs off it. Every off-diagonal number is supposed to be smaller than the two diagonal numbers it sits between.
| PU | PEOU | SI | ATT | BI | |
|---|---|---|---|---|---|
| PU | 0.798 | ||||
| PEOU | 0.454 | 0.798 | |||
| SI | -0.031 | 0.064 | 0.651 | ||
| ATT | 0.501 | 0.128 | 0.073 | 0.811 | |
| BI | 0.532 | 0.159 | 0.000 | 0.854 | 0.821 |
Source: ChatSRS Discriminant Validity (Fornell-Larcker Criterion) table, simulated survey data, 287 complete cases. Diagonal values are the square root of each construct's AVE; off-diagonal values are latent correlations.
One cell breaks the rule. ATT-BI is 0.854, and it sits between a diagonal of 0.811 (ATT) and 0.821 (BI). The tool's reading: "Here, 0.854 is higher than both 0.811 and 0.821. Therefore, the ATT-BI pair fails the Fornell-Larcker criterion, even though both constructs individually have good CR, AVE, and item loadings." It also reported the shared variance directly: r² = 0.729. Every other pair passed — so this isn't a broken model, it's one specific pair of constructs the data won't pull apart.
Why the Two Tables Can Disagree
AVE (average variance extracted) is the share of its items' variance a construct accounts for; CR (composite reliability) is how consistently those items hang together. Both were healthy for the pair that failed:
| Construct | AVE | CR |
|---|---|---|
| PU | 0.637 | 0.875 |
| PEOU | 0.637 | 0.875 |
| SI | 0.424 | 0.718 |
| ATT | 0.658 | 0.885 |
| BI | 0.674 | 0.892 |
Source: ChatSRS Reliability and Validity table, simulated survey data, 287 complete cases. The tool's stated criteria are CR > 0.70 and AVE > 0.50.
ATT and BI clear both criteria comfortably, and their items load strongly — ATT from 0.749 to 0.879, BI from 0.791 to 0.853, all significant at p < 0.001. Nothing in the reliability evidence is wrong, and that is exactly why the fit indices look fine: a model where two latent factors correlate at 0.854 can still reproduce the observed covariance matrix beautifully, because the correlation is a parameter the model estimated, not a misfit it failed to absorb. The tool put this in one sentence, and it's the sentence worth carrying into a discussion section: "Excellent fit from the current five-factor model does not prove that ATT and BI are distinct, because the model can still fit well while two latent factors are excessively correlated."
SI is the contrast case — the one row that is a measurement problem. Its AVE of 0.424 sits below the 0.50 line, driven by a single weak indicator, SI4: standardized loading 0.198, squared multiple correlation 0.039, which the tool read as "the SI factor explains approximately 3.9% of its variance." Different failure, different fix; the item-level version is worked through in reading a factor loading matrix when one item doesn't fit.
What Caused the Overlap
Not measurement error. The tool went to item wording, and named the item responsible:
ATT4, "I am willing to keep using AI tools throughout my studies," is phrased as a future behavioral commitment. It is therefore very close to behavioral intention.
BI1 — "I intend to keep using AI tools in the future" — is, in the tool's words, "almost directly parallel to ATT4." BI4 also measures anticipated future behavior, while ATT3 ("I enjoy trying out AI tools") is flagged as genuinely different in kind: affective rather than forward-looking.
The diagnosis: "the questionnaire defines attitude and intention with overlapping future-oriented language. The very high latent correlation indicates that respondents may be treating a positive attitude toward AI use and an intention to continue using AI use as nearly the same judgment." Which is worth sitting with, because the number wasn't produced by the estimator. It was produced by the questionnaire, months earlier, when someone wrote a willingness statement into the attitude block.
What the Run Said It Had Not Tested
This is the part that decides how strongly you're allowed to write.
HTMT was not computed. The tool named it as the next check rather than a result: "HTMT would be a useful supplementary test." So the discriminant-validity evidence here is Fornell-Larcker and nothing else, and a paper should say so rather than implying two criteria agreed.
The comparison models were listed, not fitted. The recommendation was to fit three: the current five-factor model, a four-factor model combining ATT and BI, and a revised five-factor model after relocating the overlapping item. That comparison hasn't happened in this output, so "the five-factor model is better than the four-factor model" is not a claim this run supports — there is no model comparison in it to cite.
The estimator wasn't the ordinal one. The items use five-point ordinal responses, and the tool flagged that the revised CFA "should also ideally be estimated with an ordinal estimator such as WLSMV as a robustness check." The fit indices above came from maximum likelihood.
None of this makes the result unusable. It makes it a bounded result, with the boundaries stated in the tool's own words rather than bolted on afterwards.
What To Do — and What Not To Do
The one thing not to do: collapse ATT and BI into one construct because it makes the table pass. Merging changes what the study claims to have measured. The tool's framing keeps that honest — combining them "may be defensible if the research question concerns general acceptance or continued acceptance of AI tools. However, this would change the conceptual model and should be stated explicitly." And on model choice: "The preferred model should be selected using both model fit and construct interpretability." Fit alone doesn't get a vote on what your constructs mean.
What the run recommends instead, in order: revise ATT4 first, since attitude items should stay evaluative or affective and a commitment to keep using belongs in intention; make BI consistently behavioral — plans, likelihood, frequency, continuation, recommendation, not restatements of a positive attitude; fit and compare the three models above; and add a second discriminant check, HTMT alongside an ordinal-estimator robustness run.
The honest write-up for a paper in this state: report the fit indices, report the Fornell-Larcker table including the cell that failed, name the item-content overlap as the likely cause, and state which additional tests were not run. That is a defensible limitation. A silent merge is not.
When This Doesn't Apply
- Fornell-Larcker is one discriminant-validity criterion, not the only one, and journals differ on which they expect — check the target journal before deciding what "passing" means. The thresholds in the fit table are likewise the tool's stated criteria, reproduced as printed; conventions vary by field.
- Whether two constructs should be distinct is a theory question, not a statistics one. If the model requires attitude and intention to be separate, a high correlation is a measurement problem to fix; if it doesn't, it may be a finding.
- Reliability and validity are different checks, and a good result on one says nothing about the other. If the open question is internal consistency rather than construct separation, see how to run, interpret, and report Cronbach's alpha.
Frequently asked questions
All my fit indices passed — doesn't that mean the measurement model is fine?
No. Fit indices assess how well the specified structure reproduces the observed covariances; they do not test whether two latent factors are distinct from each other. In this run every one of the ten fit indices was accepted while one construct pair still failed the Fornell-Larcker criterion, because a model can fit well with two factors correlated at 0.854.
How do I read the Fornell-Larcker table?
The diagonal holds the square root of each construct's AVE; the off-diagonal cells hold the correlations between constructs. Each correlation should be smaller than both diagonal values it sits between. Here the attitude-intention correlation was 0.854 against diagonals of 0.811 and 0.821, so that one cell failed while the rest of the table passed.
My two constructs correlate higher than their AVE square roots — should I just merge them?
Only if the theory supports it, and only if you say so explicitly. Merging changes what the study claims to have measured, so it is a conceptual decision rather than a repair for a failing table. The documented first move here was to revise the attitude item whose wording had drifted into intention, then compare the revised model against a combined one.
Can I say the five-factor model is better than the four-factor model?
Not from a run like this one. The three comparison models were listed as the next step, not fitted, so there is no model comparison in the output to cite. You can report that the five-factor model fit the data well; you cannot report that it outperformed an alternative you never estimated.
Which item usually causes this kind of overlap?
The one whose wording crossed a construct boundary. Here it was an attitude item phrased as a future commitment — "I am willing to keep using AI tools throughout my studies" — which the output called almost directly parallel to an intention item. Reading the item text next to the correlation is faster than reading more statistics.
Bottom Line
Good model fit but poor discriminant validity means the model reproduced the data and two of its constructs still aren't separate things. Read the fit table and the Fornell-Larcker table as two independent verdicts, find the item whose wording crossed the boundary, and state which discriminant checks were not run rather than merging constructs until the numbers cooperate.
Read your own CFA output in ChatSRS — get the fit indices, the Fornell-Larcker diagonal, and the limits of the run stated in wording you can adapt for a results section.