Morphometric Summary Statistics Are Not Parallel Indicators of Latent Nuclear Attributes: A Cross-Sectional Structural Equation Analysis of the Wisconsin Diagnostic Breast Cancer Data

Article

Morphometric Summary Statistics Are Not Parallel Indicators of Latent Nuclear Attributes: A Cross-Sectional Structural Equation Analysis of the Wisconsin Diagnostic Breast Cancer Data

Abstract: Background/Objectives: Image-derived features are often treated as fallible indicators of latent attributes, but whether several summaries of the same segmented objects behave in this way is seldom examined. We asked whether a hypothesised three-factor model of nuclear morphology reproduces the covariance among thirty indicators. Methods: Cross-sectional secondary analysis of the public Wisconsin Diagnostic Breast Cancer dataset: 569 fine-needle aspirates, ten nuclear attributes each summarised as mean, standard error and worst value, no missing cells. Because multivariate normality was rejected, the correlation matrix was fitted by maximum likelihood with a Satorra-Bentler scaled statistic and sandwich standard errors. Nine further specifications were fitted, one an unrestricted benchmark; admissibility was defined by a stated rule. Key models were validated against an established structural equation modelling package. Results: The a priori model was rejected: uncorrected chi-square 20968.6 on 402 degrees of freedom (scaled 7288.7), p < 0.001; comparative fit index 0.482 uncorrected (0.490 scaled); root mean square error of approximation 0.300 uncorrected (90% interval 0.297 to 0.304; 0.174 scaled); standardised root mean square residual 0.171. Multitrait-multimethod alternatives fitted better (uncorrected comparative fit index up to 0.830), but none converged to an admissible solution: residual variances lay on their bound and the radius-perimeter trait correlation reached 0.995. Conclusions: In this dataset, size indicators within a summary mode behave as near-algebraic transforms of one another, and summaries of the same attribute share covariance across modes that an attribute-factor model cannot represent; the indicators do not behave as parallel fallible measurements. Redundancy among image-derived indicators should be examined before a latent-variable model is specified, and a solution's admissibility should be reported beside its fit.

Keywords: structural equation modelling; confirmatory factor analysis; improper solution; multitrait-multimethod matrix; nuclear morphometry; breast neoplasms; cross-sectional studies; reproducibility of results

1. Introduction

Quantitative features extracted from digital images are a routine source of variables in clinical research, in radiomics and in digital pathology [1,2]. Many such features are highly correlated or redundant, and their values depend on acquisition and processing [3,4], which has motivated standardised feature definitions [5]. A common analytic step is to treat several features as fallible indicators of an underlying attribute in a latent-variable model. That reflective model assumes that each indicator carries the attribute plus its own independent error [6,7]. When indicators instead define or compose the attribute, a reflective specification is misspecified, and such misspecification has been found to be common in applied work [8]. Whether summaries computed from the same segmented objects meet the reflective assumption is seldom examined directly.

The Wisconsin Diagnostic Breast Cancer dataset offers a clean setting in which to examine it. Ten morphological attributes of the cell nuclei on a digitised fine-needle aspirate slide were measured, and each was summarised over the nuclei on that slide as a mean, a standard error and the mean of the three largest values, which the original investigators called the worst value [9]. The original investigators used these features to build and validate diagnostic classifiers [10,11], selecting small subsets of features by exhaustive search [12], which itself reflects the redundancy among them. Here the features are examined as indicators, not as predictors. The result is a fully crossed layout of ten attributes by three summary modes.

There is a reason to doubt that they behave as parallel indicators, and it is geometric. Three of the ten attributes are the radius, the perimeter and the area of the same outline; within a summary mode, the perimeter and the area of a roughly convex nucleus are close to fixed functions of the radius, and compactness and concavity are defined from them. Across summary modes the dependence is different: the standard error summarises dispersion within the slide, and attributes such as texture are not algebraic functions of one another, so covariance between modes of the same attribute reflects a shared attribute rather than an identity. Neither kind of dependence is represented by a measurement model with independent residuals.

This analysis asks whether a three-factor measurement model of nuclear size, nuclear shape and nuclear texture, hypothesised over all thirty indicators, reproduces their covariance matrix. Two further objectives were: to compare that model with rival specifications of the same covariance, and to describe the association between the latent attributes and the recorded diagnosis. The hypothesis was that the a priori model would not fit, because it contains no method factor and no summary-mode residual covariances.

2. Materials and Methods

2.1. Study design and reporting standards

This is a cross-sectional secondary analysis of an openly distributed dataset. Every variable was recorded from one fine-needle-aspirate slide at one time; the unit of observation is the slide.

Reporting follows the STROBE statement for cross-sectional studies [13]; the completed checklist is supplied as a supplementary file. Reporting of the structural equation models also follows the recommendations of McDonald and Ho [14].

A covariance-structure model fitted to cross-sectional data reproduces a covariance matrix; it does not identify a direction or an effect. The arrows from the latent attributes to the diagnosis in the structural model below are a choice of parameterisation, not a finding.

The primary model (M1) is the hypothesised three-factor model; it was designated primary on substantive grounds, not on the basis of fit. The analysis plan was not registered or time-stamped.

2.2. Setting, data source and participants

The data are the Breast Cancer Wisconsin (Diagnostic) dataset, computed at the University of Wisconsin from digitised images of fine-needle aspirates of breast masses [9]. The dataset is distributed by the UCI Machine Learning Repository under a Creative Commons Attribution licence [12] and was read from the copy bundled with scikit-learn.

Every record in the distributed file was eligible: a digitised fine-needle aspirate of a breast mass with a recorded diagnosis and all thirty measurements. No record was excluded. The recruitment period, referral pathway and sampling frame of the original series are not recorded in the file and are not asserted here.

The analysed file contains 569 slides, 212 malignant and 357 benign (Table 1).

Table 1. The analysed sample, the structure of the indicator set, the conditioning of the analysed matrix and the distributional assessment that chose the estimator.

Values are computed from the analysed file. Skewness and kurtosis are Mardia's coefficients; the standardised kurtosis is the excess over its expectation divided by its standard error.

2.3. Variables and how the indicators were treated

Ten attributes of the segmented nuclei were measured on each slide: radius, perimeter, area, compactness, concavity, concave points, texture, smoothness, symmetry and fractal dimension. Each was summarised over the nuclei on the slide as the mean, the standard error and the worst value, giving 30 continuous indicators. The mean and the worst value summarise location; the standard error summarises dispersion within the slide and is not a repeated measurement of the same quantity.

All thirty indicators entered the models as continuous variables, untransformed and without parcels, each loading on one factor. The matrix analysed was the product-moment correlation matrix of the thirty indicators.

The diagnosis is binary and was not treated as continuous; its handling is described in Section 2.12. No variable was measured before another, so there is no exposure, confounder or effect modifier in the usual sense.

In the a priori model, nuclear size carries the radius, perimeter and area; nuclear shape the compactness, concavity and concave points; and nuclear texture the texture, smoothness, symmetry and fractal dimension, each in all three modes.

2.4. Data sources and measurement

Every indicator was computed by the original investigators from the same segmented boundaries by one feature-extraction procedure [9]. No value was altered; columns were renamed to state the attribute and the mode, and a malignant diagnosis was coded 1.

The consequence of a single shared measurement procedure is the substantive point of this paper and is stated at the outset: indicators computed from one segmented object by one procedure cannot have independent errors of measurement, whatever a measurement model assumes about them.

2.5. Missing data

There were no missing values: all 17,070 indicator cells (569 slides by 30 indicators) and the diagnosis were complete, so no imputation or deletion was needed.

2.6. Sources of bias

Four sources of bias were identified before the analysis; none can be removed by it.

The first is selection into the distributed file. The slides come from one institution, and the balance of 212 malignant to 357 benign records is a property of that series rather than of any population; standardised loadings depend on the factor variances in the sample.

The second is the construction of the indicators. Because the perimeter and the area are close to fixed functions of the radius within a summary mode, and compactness and concavity are defined from them, part of the covariance is geometric rather than biological. This pushes size loadings towards unity and cannot be corrected by a specification that treats those indicators as independent measurements.

The third is the direction of the arrows. Nuclear morphology is a consequence of the disease process; the diagnosis is written downstream only because that is how the question is usually posed.

The fourth is our own specification. The four multitrait-multimethod specifications (M6, M7, M8, M8b) were added after the a priori model had been fitted and seen to fail, in response to the crossed trait-by-method layout of the data. M1 remained primary, but the model space was explored after seeing fit, so statistics for M6, M7, M8 and M8b are exploratory and their comparison with M1 is not a confirmatory test.

2.7. Study size and the number of free parameters

The study size is that of the distributed file. M1 has 63 free parameters (9.03 cases per parameter) and the analysed matrix 465 non-redundant elements (1.22 cases per element). No power calculation was performed. With this sample size the chi-square test detects small misspecification, so conclusions rest on approximate-fit indices, residuals and admissibility rather than on the chi-square alone.

2.8. Distributional assessment and the estimator

Mardia's coefficients of multivariate skewness and kurtosis were computed on all thirty indicators [15] before the estimator was chosen.

Because multivariate normality was rejected, the models were estimated by maximum likelihood with a Satorra-Bentler scaled test statistic and sandwich standard errors [16]. Every chi-square is computed as (N - 1) times the minimised maximum-likelihood discrepancy, with N = 569. The scaled incremental indices apply the same scaling factor to the baseline model, which is an approximation, and the correction treats the sample standard deviations as fixed.

2.9. The a priori measurement model and its identification

M1 has three correlated factors over all thirty indicators. Every indicator loads on one factor, no cross-loading or residual covariance is free, and the factor variances are fixed to unity. Each factor has at least three indicators, so the model is identified; it has 63 free parameters against 465 non-redundant elements, leaving 402 degrees of freedom.

The largest projected gradient at the optimum, the reciprocal condition number of the expected information matrix and the smallest eigenvalue of the analysed matrix are reported as checks on empirical identification.

2.10. Fit indices, and the source of every cutoff

The same fit statistics are reported for every model: the chi-square with its degrees of freedom and p value, the comparative fit index, the Tucker-Lewis index, the root mean square error of approximation with its 90% noncentrality interval [17,18], computed with a noncentral chi-square distribution function [19], and the standardised root mean square residual.

Only cutoffs with a named source adjudicate anything: those proposed by Hu and Bentler [20], namely comparative fit and Tucker-Lewis indices near 0.95, a standardised root mean square residual of 0.08 or below, and a root mean square error of approximation of 0.06 or below. The ratio of the chi-square to its degrees of freedom and the change-in-CFI rule are reported as numbers only, because their proposing sources could not be verified.

2.11. Alternative models and the admissibility rule

Nine further specifications were fitted (Table 2). Three were specified with the primary model: one general factor (M2), three summary-mode factors (M3) and a second-order factor above the attribute factors (M5). A model of the ten mean-mode indicators (M4) and an unrestricted three-factor benchmark with every cross-loading free (M0) were fitted as reference points. Four multitrait-multimethod specifications were added later (Section 2.6).

Table 2. The eleven models fitted, what each specifies, and whether it was specified before the a priori model was fitted or added afterwards.

A rule for deciding which model is primary was written down before the fits of the four later models were inspected. A candidate is admitted only if the optimiser converged to the stated tolerance, the solution is proper and the model is empirically identified. A solution is proper when no residual variance is on its lower bound, no factor correlation reaches 0.99 in absolute value, and the factor correlation matrix is positive definite. Among admitted candidates the smallest root mean square error of approximation is taken, ties broken on the comparative fit index. M0 was excluded from the rule.

A solution outside the admissible parameter space indicates that the matrix is not one a model of that form can produce; it is not a better description of the data.

Nested comparisons use the strictly positive scaled difference statistic of Satorra and Bentler [21,22], which replaces the original scaled difference test [23]. A restriction that is a sign condition rather than a linear one has no chi-square reference distribution, and no p value is quoted for it.

2.12. The diagnosis and the structural model

The diagnosis enters as a latent response with a single threshold fixed at the observed margin. Its association with each indicator is a polyserial correlation estimated by maximum likelihood [24]. The augmented matrix of thirty-one variables was fitted with the M1 measurement model plus regressions of the latent response on the three factors.

The augmented matrix was not positive definite and was replaced by the nearest positive-definite correlation matrix [25], so this route is conditional on smoothing. Structural standard errors and intervals come from a nonparametric case bootstrap that repeats the whole procedure in every replicate. Two further routes were used to examine stability: a probit regression on factor scores from M1, and, as a route that avoids smoothing, diagonally weighted least squares on the unsmoothed augmented matrix with the diagnosis treated as categorical through its polyserial correlations [26], weighted by the inverse bootstrap variances of the correlations (500 replicates, percentile intervals).

No indirect effect was specified or estimated.

Composite reliability (omega) [27] and average variance extracted [28] were computed for each factor from the standardised solution.

2.13. Subgroups, sampling design and sensitivity analyses

No subgroup, interaction, multiple-group or invariance analysis was undertaken, and the file records no sampling design, weights or clusters.

Four sensitivity analyses were specified: the alternative models in Table 2; two treatments of the geometric redundancy among the size indicators (dropping the perimeter and the area, M8, and merging radius, perimeter and area into one trait, M8b); and re-estimation of the structural coefficients by the alternative routes above. The propriety profile of M5 was computed to establish whether its difference from M1 is a boundary artefact.

2.14. Software, data and code

The original analysis used Python 3.11.15 with NumPy 2.4.4, SciPy 1.17.1, pandas 3.0.2 and scikit-learn 1.8.0. For this revision, every model was re-fitted with separately written code in Python 3.13.5 (NumPy 2.5.4, SciPy 1.18.1, pandas 2.3.3, scikit-learn 1.9.1), and M1, M2, M3 and M4 were validated against the semopy package, version 2.3.11 [29].

The discrepancy function, scaled statistic, sandwich standard errors, noncentrality interval, polyserial correlations, nearest positive-definite projection and modification indices were implemented for this analysis.

The analysed file, the analysis code, the complete standardised residual correlation matrix of M1, the modification indices and the parameter and fit tables are provided as Supplementary Code S1.

2.15. Use of generative artificial intelligence

Artificial intelligence was not used in the preparation of this manuscript.

3. Results

3.1. Cases analysed and completeness of the record

All 569 slides were eligible and analysed; none was excluded and there were no missing data, so a flow diagram is not drawn.

Table 3. Fit of every model fitted, with the admissibility of each solution.

CFI, comparative fit index; TLI, Tucker-Lewis index; RMSEA, root mean square error of approximation; SRMR, standardised root mean square residual; AIC, Akaike information criterion; df, degrees of freedom. Chi-square = (N - 1) times the minimised ML discrepancy; all values are uncorrected and every p < 0.001. Chi-square/df adjudicates nothing. AIC is not comparable for models fitted to different observed variables. The MS row describes fit to the smoothed augmented matrix, so no interval is given. Proper: no residual variance on its bound, no factor correlation of 0.99 or more in absolute value, positive-definite factor correlation matrix. The M2 row was corrected in revision (Section 3.8). M6 and M7 did not converge, and the revision re-fit reached lower minima (chi-square 7038.6 and 15986.5), so their values are not stable.

3.2. Distributional assessment of the indicators

Multivariate normality was rejected. Mardia's skewness was 943.619 (test statistic 89486.5 on 4960 df, p < 0.001) and kurtosis 2378.4 against an expectation of 960 (standardised value 386.1, p < 0.001).

For M1 the scaling factor is 2.877. The scaled chi-square is 7288.7, with a scaled comparative fit index of 0.490, Tucker-Lewis index of 0.449 and root mean square error of approximation of 0.174. Correction does not rescue the model; uncorrected and scaled values are both reported and labelled.

3.3. Fit of the a priori measurement model

The a priori model does not fit. The uncorrected chi-square is 20968.6 on 402 df, p < 0.001 (chi-square/df 52.16). The comparative fit index is 0.482 and the Tucker-Lewis index 0.439, against a proposed 0.95 [20]; the root mean square error of approximation is 0.300 (90% interval 0.297 to 0.304) against 0.06; and the standardised root mean square residual is 0.171 against 0.08.

The solution is nevertheless proper. The largest projected gradient at the optimum is 8.5 parts per million, no residual variance is on its bound, the smallest eigenvalue of the factor correlation matrix is 0.027, and the reciprocal condition number of the expected information matrix is 4.22 parts per million.

The analysed correlation matrix is close to singular (smallest eigenvalue 0.000133; condition number 99,828), reflecting the near-algebraic dependence among radius, perimeter and area; standard errors of the size loadings should be read with that in mind.

Table 3 gives the fit of every model, and Figure 1 shows M1 with its standardised estimates.

Figure 1. The a priori measurement model (M1). Ellipses are latent factors, rectangles observed indicators and small ellipses residuals; single-headed arrows are standardised loadings and dashed double-headed arrows factor correlations, labelled with the estimate and its sandwich standard error in parentheses. N = 569 slides.

3.4. Parameter estimates and measurement quality

Standardised loadings with sandwich standard errors are given in Table 4 and drawn with 95% intervals in Figure 2. The largest are on mean-mode size indicators: perimeter (mean) 0.999, radius (mean) 0.998 and area (mean) 0.988. A loading of 0.999 indicates that the indicators are close to deterministic functions of one another, not that the construct is well measured.

Table 4. Standardised loadings of the a priori model, with sandwich standard errors, and the reliability of each factor.

The last three rows give composite reliability and average variance extracted. Standard errors are sandwich standard errors. A loading close to one is a redundancy diagnostic, not evidence of good measurement.

Figure 2. Standardised loadings of M1 with 95% intervals (estimate plus or minus 1.96 sandwich standard errors), grouped by factor. N = 569 slides. No cross-loading was freed.

The factor correlations are 0.704 (size-shape), -0.026 (size-texture) and 0.652 (shape-texture); none approaches the propriety boundary.

Composite reliability is 0.974 for size and 0.954 for shape (9 indicators each), with average variance extracted of 0.809 and 0.700. For texture, composite reliability is 0.807 over 12 indicators but average variance extracted is 0.312: the factor accounts for less of its indicators' variance than their residuals do.

The texture indicators themselves load weakly: texture (mean) 0.074, texture (standard error) 0.130 and texture (worst) 0.133. The factor mainly carries smoothness and fractal dimension.

3.5. Alternative models and the model comparison

Table 3 should be read as a whole; Figure 3 draws it as one row per model, with a cutoff only where a source proposed one.

Figure 3. Uncorrected fit indices of every measurement model (Table 3). Dashed lines mark the cutoffs proposed by Hu and Bentler [20]; filled markers meet the cutoff and open markers do not. No cutoff is drawn for chi-square/df. M4 and M8 are fitted to fewer indicators. M2 shows the corrected values.

The misfit is not an artefact of assigning attributes to the wrong factors. M0, the unrestricted three-factor benchmark, is not a candidate and enters no test; it is a descriptive reference for how well any three-factor structure could fit. Its likelihood surface was multimodal across sixty seeded starts and the optimiser did not meet the convergence tolerance, so its values describe the best solution located rather than a certified optimum; the revision re-fit located the same value. Even M0 gives chi-square 15008.3 on 348 df, a comparative fit index of 0.631 and a root mean square error of approximation of 0.272, so three dimensions are not enough for this matrix.

Among the original specifications, the attribute structure fits better than the single general factor: the uncorrected chi-square difference between M2 and M1 is 5781.4 on 3 df, and the strictly positive scaled difference [21,22] is 439.8 on 3 df, p < 0.001. Against the summary-mode factors (M3), with the same degrees of freedom, the attribute structure fits better by 1467.2 in chi-square. Attributes organise this covariance better than summary modes do, and neither organises it adequately.

The second-order model (M5) was expected to be fit-equivalent to M1, because with three first-order factors a second-order factor generically imposes no constraint. That prediction was wrong: a second-order structure requires the product of the three factor correlations to be positive, and here it is -0.01200, so M5 fits worse by 526.1 in chi-square at the same degrees of freedom, with its shape loading on the propriety boundary. Because this is a sign condition, no p value is quoted.

The four multitrait-multimethod specifications fit far better than any of the original specifications, and every one is inadmissible. The correlated-trait, correlated-method model (M6) reached chi-square 7085.4 on 327 df, with a comparative fit index of 0.830 and a root mean square error of approximation of 0.191, and the ten-trait model (M7) 16091.0 on 360 df (0.604 and 0.277). Neither optimiser met the convergence tolerance, and the revision re-fit located lower minima for both (7038.6 and 15986.5). M1 is nested in M7 under 42 constraints; the chi-square difference is at least 4877.6 on 42 df (p < 0.001) on either solution, so the ten attributes do not collapse into three constructs.

In M6 seven and in M7 five residual variances are on their lower bound, and the radius-perimeter trait correlation reaches 0.995 in both. The two specifications that handle the geometric redundancy explicitly (M8 and M8b) are also inadmissible, with the mean-mode residual variances of compactness and concavity on their bound in both.

Applying the admissibility rule (Section 2.11) to the ten measurement models admits M1, M2, M5 and rejects M3, M4, M7, M6, M8, M8b. Among the admitted, the smallest root mean square error of approximation belongs to M1, which is the hypothesised model. The primary model is therefore the a priori model, and it remains primary not because it fits but because every model that fits better returns a solution outside the admissible parameter space.

3.6. Where the misfit lies

The residual correlations locate the misfit precisely. Of the 435 off-diagonal residual correlations of the primary model, 253 exceed 0.10 in absolute value, the mean absolute residual is 0.140 and the largest is 0.902, between texture (mean) and texture (worst). Figure 4 prints the residual correlations for the radius, the perimeter, the area and the concavity in all three summary modes; the complete matrix over all thirty indicators is provided in Supplementary Code S1.

Figure 4. Standardised residual correlations (observed minus model-implied) of M1 for the radius, perimeter, area and concavity indicators in all three summary modes. Values are printed in every cell; a heavy border and asterisk mark an absolute residual of 0.10 or more. N = 569 slides.

Modification indices were computed for all 495 candidate parameters of M1 (60 cross-loadings and 435 residual covariances) and none was acted on; every fit statistic for M1 comes from its specification as written. The full table is in Supplementary Code S1.

Of the indices, 325 exceed 10 and 376 exceed 3.84; the largest is 851.3, for the residual covariance between radius (mean) and perimeter (mean). The top of the list is dominated by residual covariances among radius, perimeter and area within one summary mode and between the same attribute across modes, the two hypothesised sources of misfit.

3.7. Association with the diagnosis

The threshold on the latent response is 0.3250. The augmented matrix was not positive definite (smallest eigenvalue -0.1383) and was replaced by the nearest positive-definite correlation matrix (largest change 0.0678; 0.0117 within the thirty indicators). The maximum-likelihood structural model fitted to that smoothed matrix gave chi-square 35041.6 on 429 df (chi-square/df 81.68), a comparative fit index of 0.371 and a root mean square error of approximation of 0.377.

Table 5 gives the unadjusted and adjusted associations side by side. The largest unadjusted polyserial correlation of a size indicator is that of perimeter (worst), 0.914, and of a shape indicator that of concave points (worst), 0.904. Adjusted for the other two factors, the maximum-likelihood standardised coefficients on the smoothed matrix are 1.140 for size (standard error 0.182, p < 0.001), -0.438 for shape (0.288, p = 0.127) and 0.693 for texture (0.220, p = 0.002). The factor correlations in this model are 0.704 (size-shape), 0.031 (size-texture) and 0.693 (shape-texture). The explained variance of the latent response, R² = b′Φb, is 0.898: the squared coefficients contribute 1.975 and the covariance terms -1.078. With the M1 factor correlations (-0.026 and 0.652 in place of 0.031 and 0.693) the same coefficients would give 0.831. A standardised coefficient above 1 is possible on correlated factors and signals collinearity rather than an error.

Table 5. Association of the three factors with the latent response underlying the diagnosis, unadjusted and adjusted, by three estimation routes.

Adjusted means adjusted for the other two factors. ML, maximum likelihood on the smoothed augmented matrix, with standard errors and intervals from a case bootstrap that repeats the whole procedure. Factor-score probit, probit regression on M1 regression factor scores (not on the standardised latent scale). DWLS, diagonally weighted least squares on the unsmoothed augmented matrix (500 case-bootstrap replicates, percentile intervals). ML quantities are conditional on the smoothing described in the Methods.

The shape coefficient is not stable. By the factor-score route the coefficients are 2.549 for size, 0.350 for shape and 0.679 for texture. By DWLS on the unsmoothed matrix they are 0.739 (95% interval 0.627 to 0.848) for size, 0.117 (-0.043 to 0.279) for shape and 0.324 (0.199 to 0.445) for texture, with factor correlations of 0.743, 0.203 and 0.680 and R² 0.942; the shape coefficient was negative in 8% of bootstrap replicates, and one standardised communality slightly exceeded 1 (1.002). Size and texture are positively associated with the latent response by every route, whereas the sign and size of the shape coefficient depend on the route, which is what collinearity between the size and shape factors produces. Nothing here indicates that concavity is associated with a lower chance of malignancy. This objective is therefore best read as an illustration of collinearity, not as an estimate of the separate contributions of the factors.

3.8. Validation of the analysis

For this revision every measurement model was re-fitted with separately written code and the key models were checked against semopy 2.3.11 [29]. The re-fit reproduced the printed chi-square of M1, M3, M4, M5, M8, M8b and M0 to within 0.1. For M2 it gave chi-square 26750.0 on 405 df (CFI 0.336, TLI 0.287, RMSEA 0.338, SRMR 0.254), which replaces the previously printed values and changes the M1-M2 difference to 5781.4; semopy gave the same value (26750.0 in the (N - 1) convention), and agreed with the re-fit for M1 (20968.7), M3 (22435.8) and M4 (2443.5) to within 0.1. The scaling factor of M1 (2.877), the modification-index counts, the polyserial correlations and the structural solution were also reproduced. M6 and M7 were not reproduced, consistent with their non-convergence.

4. Discussion

4.1. Key results

The a priori three-factor model of nuclear size, shape and texture does not reproduce the covariance among the thirty indicators: the uncorrected comparative fit index is 0.482 and Tucker-Lewis index 0.439 against a proposed 0.95, the root mean square error of approximation 0.300 against 0.06, and the standardised root mean square residual 0.171 against 0.08.

Specifications that treat the data as a crossed layout of ten attributes by three summary modes fit far better; the best reaches an uncorrected comparative fit index of 0.830, but its optimiser did not converge. Every one returns a solution outside the admissible parameter space, with residual variances on their bound and a radius-perimeter trait correlation of 0.995. Under the admissibility rule, the a priori model remains primary.

The association with the diagnosis is strong jointly, but its parts cannot be separated: the shape coefficient changes sign across estimation routes, so this objective is best read as an illustration of collinearity among the factors.

4.2. Limitations

Six limitations bound what this report supports, and each is stated with the direction in which it acts.

First, the model space was widened after the a priori model had been fitted and seen to fail (Section 2.6); statistics for M6, M7, M8 and M8b are exploratory.

Second, no claim about the shape factor net of the size factor is supported. The shape coefficient is negative by the primary route and positive by the factor-score route, whereas the bivariate associations of the shape indicators with the diagnosis are strongly positive. Conditioning on a nearly collinear factor inflates the variance of both coefficients and can reverse the sign of either.

Third, the polyserial-augmented matrix was not positive definite (smallest eigenvalue -0.1383) and was smoothed before the maximum-likelihood structural model was fitted (largest change 0.0678), which acts in the direction of better apparent fit. Estimation by DWLS without smoothing gave positive size and texture coefficients and a shape coefficient whose interval included zero.

Fourth, the texture factor has an average variance extracted of 0.312 and should not be interpreted as a measure of nuclear regularity; any coefficient attached to it describes what the smoothness and fractal-dimension indicators share.

Fifth, 495 modification indices were computed and none was acted on, so this report describes how badly a specified model fits, not how well a repaired one could be made to.

Sixth, M6 and M7 did not converge, and the original and revision searches located different solutions, so their fit statistics are not stable; they are reported as properties of those specifications.

Equivalent models were not enumerated, and the conventional cutoffs used here are proposals from one simulation study [20], not laws.

4.3. Interpretation

The failure lies less in the assignment of attributes to factors than in the assumption that the indicators are independent measurements. The unrestricted benchmark still leaves a comparative fit index of 0.631; the residual correlations concentrate where hypothesised, within the size trio in one summary mode and across summary modes of the same attribute; and the models that fit best leave the admissible parameter space at parameters that describe the geometry, a trait correlation of 0.995 between radius and perimeter and zero residual variances.

These results separate two sources of dependence. Within a summary mode, radius, perimeter and area are close to algebraic transforms of one another, so loadings near 1 reflect geometric redundancy rather than precise measurement. Across modes, the same attribute summarised as mean, standard error and worst value shares covariance that is not an algebraic identity: the largest residual, 0.902 between texture (mean) and texture (worst), reflects a shared attribute that an attribute-factor model with independent residuals cannot represent. Neither source fits the assumption of parallel fallible indicators. This agrees with reports that many image-derived features are redundant and sensitive to how they are computed [3,4], and with the distinction between reflective and formative or composite measurement [6–8]: features that define a property of an outline are closer to composite than to reflective indicators.

Eleven models, 495 modification indices and several estimation routes were used, but no inference rests on selecting the most favourable of them; the argument rests on a pattern common to all.

For clinical image-derived measurement the practical reading is short. Image-derived features increasingly feed diagnostic and prognostic models [1,2]; when several features summarise the same segmented objects, high loadings and reliability coefficients can reflect redundancy rather than measurement quality, and composite scores built on them can overstate precision. Before specifying a model, ask whether the indicators are distinct measurements or rearrangements of one another; report admissibility beside fit; and treat a loading above 0.99 as a redundancy diagnostic.

The structural coefficients are conditional associations in one cross-section; they are not effects and describe no mechanism.

4.4. Generalisability

The finding about the diagnosis generalises least: the 569 slides come from one institution, the balance of 212 malignant to 357 benign records is a property of that series, and standardised coefficients depend on the factor variances in the sample.

The methodological finding may extend to other feature sets in which several summaries are computed from the same segmented objects and some features are algebraic functions of others, as is common in radiomics and digital pathology [3,5]. It was shown here for one dataset and one feature-extraction procedure; whether it holds elsewhere has to be tested in those data, and its size will depend on how the features are defined and summarised.

5. Conclusions

A three-factor measurement model of nuclear size, shape and texture, hypothesised over thirty morphometric indicators, is rejected on every fit criterion with a named source. Specifications that treat the data as a crossed attribute-by-mode layout fit far better and are all inadmissible, and the two that fit best did not converge.

In this dataset, morphometric summary statistics do not behave as parallel fallible indicators: size indicators within a summary mode are close to geometric transforms of one another, and summaries of the same attribute share covariance across modes. Redundancy among image-derived indicators should be examined before a latent-variable model is specified, and the admissibility of a solution should be reported beside its fit.

Supplementary Materials: Supplementary Material (Section S1, reporting instruments; Section S2 and Table S1, admissibility of every fitted solution; Section S3, files provided; Section S4, validation of the analysis); STROBE checklist; Supplementary Code S1 (analysed file, analysis code, standardised residual correlation matrix of M1, modification indices, and parameter and fit tables of every model).

Funding: This research received no funding.

Institutional Review Board Statement: This study was exempt from ethics review because the data are public (secondary analysis of public, de-identified data).

Informed Consent Statement: Informed consent was not sought by the present authors and could not be. The records analysed are de-identified numerical measurements distributed openly by a public repository, with no name, no date, no identifier and no image that could be traced to a person, and the individuals they derive from cannot be contacted through the distributed file. Consent for the original collection and for the publication of the derived measurements rests with the original investigators and their institution.

Data Availability Statement: The data are openly available and no new data were created. The Breast Cancer Wisconsin (Diagnostic) dataset is distributed by the UCI Machine Learning Repository (https://doi.org/10.24432/C5DW2B) under a Creative Commons Attribution licence and is bundled with scikit-learn, from which it was read. The analysed file, the analysis code, the standardised residual correlation matrix of M1, the modification indices and the parameter and fit tables of every model are provided as Supplementary Code S1.

Acknowledgments: The authors thank the investigators who created and described the Breast Cancer Wisconsin (Diagnostic) dataset and the repository that distributes it.

Conflicts of Interest: The authors declare no competing interests.

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Abbreviations and symbols

AIC, Akaike information criterion; CFI, comparative fit index; DWLS, diagonally weighted least squares; ML, maximum likelihood; RMSEA, root mean square error of approximation; SRMR, standardised root mean square residual; STROBE, Strengthening the Reporting of Observational Studies in Epidemiology; TLI, Tucker-Lewis index; UCI, University of California, Irvine; WDBC, Wisconsin Diagnostic Breast Cancer.