CPM Fits at a Boundary

1. Overview

This vignette explains what a boundary solution in a circular process model (CPM) fit is, and what to do when one appears. Section 2, “When a fit sits at a boundary”, refits the jz2017 model that “Evaluating Circumplex Structure” fits and reads its boundary note. It then glosses the five markers that cpm_fit() records and states what the package’s validation measured about them. It ends with four steps to take when one fires. The Wrap-up lists what the page covered and names related pages.

2. When a fit sits at a boundary

This page starts from the fit of the jz2017 octants that “Evaluating Circumplex Structure” makes. The same call refits it here, with the same seed and the same two warnings.

data("jz2017")
set.seed(12345)
cpm <- cpm_fit(jz2017, scales = PANO(), angles = octants(), boots = 500)
#> Warning: CPM Hessian is ill-conditioned (condition number 1.83e+14): angles
#> may be clustered or parameters weakly determined.
#> Warning: 11 of 500 bootstrap resamples were excluded (0 with a degenerate or
#> non-positive-definite correlation matrix, 11 failing the convergence
#> acceptance criterion); the confidence intervals are based on the remaining
#> 489 replicates and are conditional on estimability.
summary(cpm)
#> 
#> Circular Process Model (Browne, 1992) 
#> Model:             quasi-circumplex 
#> Harmonics (m):     3 
#> Sample size (N):   1166 
#> Reference scale:   PA 
#> CI method:         bootstrap 
#> Confidence level:  0.95 
#> 
#> # Estimated angles and communality indices
#> 
#>  Scale Theory   Angle     lci     uci  Zeta   lci   uci
#>     PA     90  90.000  90.000  90.000 0.767 0.684 0.860
#>     BC    135 125.074 113.548 137.069 0.931 0.872 1.000
#>     DE    180 170.353 156.943 185.174 0.780 0.733 0.826
#>     FG    225 195.425 185.811 206.523 0.861 0.822 0.910
#>     HI    270 250.721 244.660 258.758 0.956 0.935 0.977
#>     JK    315 269.491 261.813 279.412 0.942 0.929 0.957
#>     LM    360 294.230 286.626 303.019 0.806 0.761 0.850
#>     NO     45  11.305   2.426  20.913 1.000 1.000 1.000
#> 
#> # Correlation-function weights
#> 
#>  k  Beta Beta_lci Beta_uci
#>  0 0.450    0.418    0.481
#>  1 0.440    0.411    0.466
#>  2 0.074    0.065    0.087
#>  3 0.036    0.015    0.057
#> 
#> # Fit indices
#> 
#>   χ²(10) = 81.169, p = <1e-04
#>   RMSEA = 0.078 [0.063, 0.094] (90% CI)
#>   SRMR  = 0.042
#>   CFI   = 0.984    TLI = 0.956
#>   AIC   = 117.169    BIC = 208.273
#> 
#> # Residuals
#> 
#>   Largest absolute residual: 0.134 (PA – HI)
#> 
#> # Diagnostics
#> 
#>   Note: a communality index reached its upper boundary (ζ > 0.995, a
#>   Heywood-type solution).
#>   Note: 11 of 500 bootstrap resamples were excluded (0 degenerate, 11
#>   non-convergent); the intervals are based on 489 replicates and are
#>   conditional on estimability.
#> 
#>   Note: boundary/weak-identification markers fired: Heywood communality;
#>   small correlation-function weight; ill-conditioned Hessian.
#>   What has been measured about these markers covers analytic intervals only,
#>   and not every marker was measured; they are not validated as predictors of
#>   the bootstrap intervals shown here (see the vignette section 'When a fit
#>   sits at a boundary').

The fit above sits at a boundary, and it says so. Its # Diagnostics block prints a note that a communality index reached its upper boundary. The results table shows which one: the NO scale’s communality index is 1.000, with a confidence interval of [1.000, 1.000]. A zero-width interval is not a precise estimate. It is an absent one. Both endpoints equal the estimate. So at least the middle 95% of the retained bootstrap resamples also sat on the boundary. That left the percentile interval no width to report. The same chunk also emits a warning that the fit’s Hessian is ill-conditioned, a second signal from the same regime. When octant scales share a strong general factor, as interpersonal problem scales do, fitted solutions frequently sit at or near a parameter boundary. This is the estimator meeting real data at finite \(n\), not a data-entry error.

cpm_fit() records five such markers, and summary() names the ones that fired:

• Heywood communality: a communality index estimated at its upper boundary (\(\hat\zeta > 0.995\)). The circle accounts for that scale essentially completely.
• boundary harmonic removed: a correlation-function weight sat at zero and was dropped from the model, with the degrees of freedom adjusted.
• small correlation-function weight: the smallest weight is below 0.10, near enough to zero to matter for inference.
• ill-conditioned Hessian: the curvature matrix that the analytic standard errors are computed from is close to singular. Angles may be clustered or parameters weakly determined.
• competing near-tied optima: the search found more than one solution of nearly equal fit, so the reported one may not be uniquely determined.

summary() prints that list when markers fire, but on the analytic path it prints the list only within a sample-size window. On the bootstrap path, as in the fit displayed above, the list prints as a descriptive note at every sample size. Beside the note, a statement says that what has been measured about the markers covers analytic intervals only. It also says that not every marker was measured. What a fired marker does and does not tell you, below, has the particulars. So the markers are not validated as predictors of the bootstrap intervals they accompany. When the intervals are analytic, the list instead arrives inside a coverage caution. That caution prints when a marker fires and \(N\) is at least 2000 and below 50000. Below 2000, an unconditional caution prints without the list. At 50000 and above, neither prints. To see that caution-embedded form, we simulate a larger sample from the structure just estimated with cpm_simulate(). Then we refit it on the correlation-matrix path, where the intervals are analytic.

set.seed(2026)
sim <- cpm_simulate(cpm, n = 2500) # draws from the structure estimated above
demo <- cpm_fit(
  cormat = cor(sim), scales = colnames(sim), angles = octants(), n = 2500
)
summary(demo)
#> 
#> Circular Process Model (Browne, 1992) 
#> Model:             quasi-circumplex 
#> Harmonics (m):     3 
#> Sample size (N):   2500 
#> Reference scale:   PA 
#> CI method:         analytic 
#> Confidence level:  0.95 
#> 
#> # Estimated angles and communality indices
#> 
#>  Scale Theory   Angle     lci     uci  Zeta   lci   uci
#>     PA     90  90.000  90.000  90.000 0.785 0.748 0.823
#>     BC    135 125.057 120.571 129.542 0.881 0.852 0.911
#>     DE    180 167.802 161.988 173.616 0.799 0.777 0.820
#>     FG    225 193.344 187.711 198.976 0.878 0.854 0.901
#>     HI    270 250.958 245.112 256.805 0.958 0.944 0.971
#>     JK    315 269.528 263.351 275.706 0.948 0.938 0.958
#>     LM    360 293.619 286.622 300.617 0.803 0.778 0.829
#>     NO     45  12.673   7.017  18.329 0.929 0.836 1.023
#> 
#> # Correlation-function weights
#> 
#>  k  Beta Beta_lci Beta_uci
#>  0 0.442    0.422    0.462
#>  1 0.453    0.436    0.470
#>  2 0.068    0.058    0.078
#>  3 0.037    0.030    0.044
#> 
#> # Fit indices
#> 
#>   χ²(10) = 2.229, p = 0.994
#>   RMSEA = 0 [0, 0] (90% CI)
#>   SRMR  = 0.005
#>   CFI   = 1    TLI = 1.002
#>   AIC   = 38.229    BIC = 143.062
#> 
#> # Residuals
#> 
#>   Largest absolute residual: 0.012 (FG – NO)
#> 
#>   Note: this solution is near a parameter boundary or weakly identified
#>   (small correlation-function weight); analytic (Wald) confidence intervals
#>   mis-covered for such fits in validation even at N in the tens of thousands.
#>   Interpret them with caution and prefer the bootstrap on the raw-data path
#>   when available.

One marker fires here, a small correlation-function weight, and summary() names it. Notice what did not happen. The population that this sample was drawn from carries NO’s communality at the boundary, yet this draw produced no Heywood case. A boundary in the population does not guarantee a boundary in every sample from it. In the same way, a quiet fit does not guarantee that there is no boundary behind the data.

A lower discrepancy beside the fit. cpm_fit() reports the lowest discrepancy that its starts from the theoretical angles reach (the cpm_fit() help section “Which solution is reported”). It then runs a fixed set of extra starts, most of them from that solution. If one reaches a discrepancy lower by more than 0.01 on the \(\chi^2\) scale, print() and summary() add a note that begins “extra starts reached a solution whose \(\chi^2\) is lower by”. The note is not one of the five markers, and the analytic caution does not read it. It changes nothing that the fit reports: the estimates, the fit indices and the intervals stay those of the reported solution. What it tells you is that the reported \(\chi^2\) is larger than at the lower solution, so at the same degrees of freedom the \(p\)-value is smaller and RMSEA and CFI are no better than they would be there. It also says whether the lower solution keeps the scales’ order around the circle, and how many more or fewer communality indices it has above 0.995. The package’s simulations covered normal data at two truths of eight octant scales with three harmonics. There, such lower solutions estimated the true angles and weights worse on average than the reported one, though not on every fit. Neither fit above prints the note. A fit with no note can still have a lower discrepancy, because the extra starts do not find every one. The check runs only for the default "quasi-circumplex" model with unit scaling, and check_lower = FALSE turns it off.

What a fired marker does and does not tell you. The package’s validation simulations measured what these markers predict, and the measurement is narrower than the markers are. It covered analytic (Wald) intervals fitted from a correlation matrix. That marker study measured interval coverage (how often an interval contained the truth), not bias in the point estimates.

• Communality indices take the worst of it when an ill-conditioning, a Heywood, or a near-tied-optima marker fires. All three covered \(\zeta\) well below nominal, and ill-conditioning was the worst of them. Two failure modes show up in the output rather than hiding in it. First, the analytic standard errors can come back missing. This happens for the communality indices, the angles and the weights at the same time. The reason is that one singular curvature matrix takes all three families down together. Second, an interval can collapse to zero width, because a negative asymptotic variance was clamped at zero. NO’s zero-width interval above arrives by a third route: it is a percentile interval whose resamples sat on the boundary. But it reaches the reader the same way, as an interval with no usable width.
• Angles. Taking all marker-firing fits together, angle intervals actually degraded somewhat more than communality intervals did. So the ranking above (communality indices worst under those three markers) belongs to those markers, not to the parameter families in general. The strongest single angle signal was the ill-conditioning marker. The small-weight marker was next, and the Heywood marker was the weakest of the three.
• Correlation-function weights moved least. Marker-firing fits covered \(\beta\) only slightly less often than quiet ones, so a fired marker tells you least here. The real \(\beta\) problem is a different one, and it does not depend on any marker. In the package’s bootstrap validation, the percentile intervals missed the largest weight, \(\beta_0\), at least as often as any other weight. Almost every miss fell below the true value. The misses of two smaller weights, \(\beta_2\) and \(\beta_3\), fell mostly above it. These shortfalls did not shrink as the sample size grew, up to the largest sample size at which the package’s own bootstrap was studied. They appeared at both weight patterns studied. Those simulations drew normal data for eight octant scales with equal communalities. There, the cause is a bias in the weight estimates themselves, not a weight that sits near zero. Refitting the same data with the angles fixed at their theoretical values, which in the simulations were the true angles, removed that bias. The bootstrap reproduces only part of the bias, so its intervals stay off center.

The record does not support four things, and no reading of these markers should assume them. First, the marker study measured nothing about bias in the point estimates. Second, the marker study fitted analytic intervals only. So the markers are not validated as predictors on the bootstrap path that this vignette uses by default. Third, the boundary harmonic removed marker showed no evidence of predicting mis-coverage at all. Fits carrying it covered as well as fits without it. It is kept because it names a real feature of the solution, not because it forecasts trouble. Fourth, the numbers behind Heywood communality, ill-conditioned Hessian and competing near-tied optima come almost entirely from one simulated configuration, chosen deliberately to provoke them. In that configuration, those markers fire often enough to measure. Even there, near-tied optima fired only rarely. Read them as what happens where those markers fire, not as how often they fire.

What to do when one fires.

1. Locate it. Read the results table beside the note, and find the scale or weight sitting at the boundary, as NO is above. A marker with no identified owner is not yet a diagnosis.
2. Re-fit to find out what the boundary is about. The default "quasi-circumplex" model is the least constrained of the variants that “Evaluating Circumplex Structure” compares. So a boundary that appears only under a constrained variant was put there by the constraint. A boundary already present in the default fit belongs to the data and the model family, not to an assumption you added. If you fitted from a correlation matrix, refitting from raw data also buys you bootstrap intervals in place of analytic ones.
3. Report it. Quote the marker alongside the estimate it belongs to: which scales sat at the boundary, and that their intervals are degenerate or missing. An angle or communality reported without its diagnostic note is a number stripped of its own caveat.
4. Keep using what still holds. Three things remain usable. The first is the point estimates, as the solution the model actually fitted. Their bias was measured only at the validation’s own weight patterns, where the weight estimates were biased (see above). So read them as estimates, not to the digit. The second is the fit indices and the residual summary. A boundary changes how the intervals around them behave. It does not change how the fit indices and the residual summary are computed. Read them with the caution that “Evaluating Circumplex Structure” gives under Reading the fit indices, which is about field sample sizes, not about markers. The third is parameter intervals that came back with ordinary width. Read them knowing that marked fits covered less well than unmarked ones across all three families. For the weights, also keep in mind the shortfall described under Correlation-function weights above, which appeared whether or not a marker fired. One thing does not remain usable: a parameter interval that is missing or zero-width. (The demonstration fit above prints RMSEA = 0 [0, 0], which is not that failure. It is a fit statistic at its own floor, because the refit recovers almost exactly the structure it was simulated from.)

Wrap-up

A boundary solution is the estimator meeting real data, and a fired marker is a finding to locate, refit, report and keep in view. No page follows this one. “Evaluating Circumplex Structure” compares the model variants that a refit chooses among. “Confidence Interval Accuracy” asks whether the intervals of an SSM analysis can be trusted at your sample size.