--- title: "SEM-Based SSM Analysis" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{SEM-Based SSM Analysis} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ``` r library(circumplex) ``` **Level:** Advanced. Read "Structure Tests and Ipsatization" first. ::: {.alert .alert-warning role="alert"} **Not yet peer reviewed.** The latent Structural Summary Method this page teaches is the package's own proposal. Its authors have not yet published it in a peer-reviewed venue. Read it as a research tool, and state that status when you report results from it. ::: ## 1. Overview The other vignettes model *observed* circumplex scores. These are the mean profile of a group, or the profile of correlations between the circumplex scales and an external measure. This vignette introduces a **latent** (not directly observed) counterpart, built on a structural equation model (SEM) of the circumplex scales. `ssm_sem()` and its helpers expose it. The method has two products: the latent profile of a measure, and the invariance-gated latent contrast between groups. This page teaches the first. It shows how the profile's confidence intervals are constructed, and the assumptions that make them interpretable. The next page, "Latent Group Contrasts", teaches the second. The latent-level Structural Summary Method appears to be new. A search of the SEM–circumplex literature (2019 onward) found related work on the latent structure of circumplex *scales* (e.g., Wendt et al., 2019). It also found work on inference for disattenuated correlations, which are correlations corrected for measurement error (Moss, 2026). But it found no prior work applying the SSM decomposition (elevation, amplitude, displacement, and fit) to a measure's *latent* profile with circular-aware intervals. `vignette("introduction-to-ssm-analysis")` defines these four parameters. Treat this layer as a research tool whose assumptions you should understand before relying on it. Section 2, "Why a latent SSM?", explains attenuation. Section 3, "The measurement model", shows the lavaan model that `ssm_sem_syntax()` writes. Section 4, "Estimating a latent profile", fits it with `ssm_sem()` and compares the latent profile with the observed one. Section 5, "Where the confidence intervals come from", and Section 6, "What the parameters mean now", explain the intervals and the disattenuated parameters. The Wrap-up lists what the page covered and names the next page, and the References list the sources cited. ## 2. Why a latent SSM? An observed correlation profile is *attenuated* by measurement error. Each circumplex scale is imperfectly reliable, so the correlations between the scales and an external measure are pulled toward zero. The scales are not equally reliable, so the correlations are pulled toward zero by *different amounts*. The observed SSM parameters inherit both effects. Amplitude is deflated by the average unreliability. Displacement is rotated by the *heterogeneity* of reliability around the circle. The latent SSM estimates the profile that the measure would show against the **latent circumplex content** of the scales. This profile is the disattenuated analog of the observed correlation profile. Zimmermann and Wright (2017) defined the observed version, and this is its latent counterpart. The latent SSM fits a measurement model in which each scale loads on latent factors placed at that scale's *fixed theoretical angle*. It then reads the measure's correlations with the common circumplex content, and it summarizes them with the usual SSM transform. Two consequences are worth stating up front, because they shape everything below: - The latent profile is **model-conditional**. Every latent parameter is conditional on the fixed-angle measurement model being an adequate description of the data. A poorly fitting model does not make the latent parameters merely imprecise. It makes them uninterpretable. Global fit is therefore reported alongside every result. - The angles are **theoretical claims, not estimates**. `ssm_sem()` never estimates an angle. If an instrument's real geometry departs from theory, that departure is absorbed into misfit, not into the angles. To *examine* circumplex geometry and let the angles be free, use `cpm_fit()`. It fits Browne's (1992) circumplex model. Examining geometry is a different question, and `cpm_fit()` is a different tool. ## 3. The measurement model `ssm_sem()` builds and fits a lavaan measurement model for you, but it is worth seeing the model it generates. `ssm_sem_syntax()` returns that model as a string: ``` r scales <- c("PA", "BC", "DE", "FG", "HI", "JK", "LM", "NO") syntax <- ssm_sem_syntax( scales = scales, angles = octants(), measures = "NARPD" ) cat(syntax) #> # circumplex SSM measurement model (generated by ssm_sem_syntax()) #> # scales: PA, BC, DE, FG, HI, JK, LM, NO #> # angles (degrees): 90, 135, 180, 225, 270, 315, 360, 45 #> # model tier: scaled #> #> # general factor: free per-scale saturations #> g =~ NA*PA + a1*PA + a2*BC + a3*DE + a4*FG + a5*HI + a6*JK + a7*LM + a8*NO #> # circumplex plane: loadings free but with each scale's angle fixed #> cx =~ NA*PA + lx1*PA + lx2*BC + lx3*DE + lx4*FG + lx5*HI + lx6*JK + lx7*LM + lx8*NO #> cy =~ NA*PA + ly1*PA + ly2*BC + ly3*DE + ly4*FG + ly5*HI + ly6*JK + ly7*LM + ly8*NO #> # fixed-angle direction constraints: sin(a)*lx - cos(a)*ly == 0 #> 0 == 1*lx1 - 0*ly1 #> 0 == 0.70710678118654757*lx2 - -0.70710678118654746*ly2 #> 0 == 0*lx3 - -1*ly3 #> 0 == -0.70710678118654746*lx4 - -0.70710678118654768*ly4 #> 0 == -1*lx5 - 0*ly5 #> 0 == -0.70710678118654768*lx6 - 0.70710678118654735*ly6 #> 0 == 0*lx7 - 1*ly7 #> 0 == 0.70710678118654746*lx8 - 0.70710678118654757*ly8 #> # isotropic orthonormal plane metric (plane scale absorbed by loadings) #> g ~~ 1*g #> cx ~~ 1*cx #> cy ~~ 1*cy #> cx ~~ 0*cy #> # general-plane covariances fixed to zero: with free per-scale #> # saturations, freeing these is locally unidentified exactly at #> # phi_g = 0 (the trade a_i +/- d*c_i*cos/sin(angle_i) <-> phi_g is #> # first-order flat there), so they cannot be estimated. To model a #> # general factor leaning into the plane, use the strict tier, whose #> # fixed loadings leave the full factor covariance matrix free. #> g ~~ 0*cx #> g ~~ 0*cy #> #> # external measure(s): related to circumplex factors #> NARPD ~~ mg1*g #> NARPD ~~ mcx1*cx #> NARPD ~~ mcy1*cy #> #> # NOTE: amplitude (a) and displacement (d) are deliberately NOT defined #> # here. They are nonlinear (sqrt / atan2) and their intervals must be #> # built in-package through circular quantiles, never via lavaan := or #> # delta-method CIs (which ignore the angular branch cut). ``` Three latent factors carry the structure: a general factor `g` and the two plane axes `cx` and `cy`. Each scale loads on (has a weight on) all three. But its plane loadings are tied to its **fixed angle** by a direction constraint (`sin(θ)·lx − cos(θ)·ly == 0`). Here `lx` and `ly` are the scale's loadings on `cx` and `cy`. So the scale's location in the plane is theoretical, while its *saturation* (how strongly it expresses the circumplex) is free. This is the default **`"scaled"`** tier, one of two versions of the model. A stricter **`"strict"`** tier instead fixes every loading to the unit cosine pattern and frees the full factor covariance matrix. It is the fully theoretical benchmark. It is also what remains identified (the data pin down its parameters) when the number of scales is small. The header and comments in the generated syntax record two design decisions that matter statistically: - Under the scaled tier the general factor is fixed **orthogonal** to the plane (`g ~~ 0*cx`, `g ~~ 0*cy`). Freeing those covariances alongside free per-scale saturations is locally unidentified exactly at the null they would test. So they cannot be estimated in this tier. A general factor that leans into the plane is expressible only under the strict tier (or it surfaces as misfit under the scaled tier). This matters for real interpersonal data. Wendt et al. (2019) found a general–agency correlation of roughly −.3, replicated across four samples. So on instruments in the family of the Inventory of Interpersonal Problems (IIP), the orthogonality that the scaled tier assumes is known to be violated. - The final `NOTE` says that amplitude and displacement are **deliberately not defined** in the lavaan syntax. That is the subject of Section 5. You rarely call `ssm_sem_syntax()` directly, because `ssm_sem()` does. But it is the escape hatch for respecifications that you then fit yourself and hand back through `ssm_sem_parameters()`. One example is partial invariance, where only some parameters are held equal across groups. ## 4. Estimating a latent profile The everyday entry point is `ssm_sem()`. Its arguments mirror `ssm_analyze()`: the data, the `scales` and their `angles`, and one or more `measures`. The `measures` argument is required in the single-group case, because the single-group latent SSM *is* the correlation path. (A single-group latent *mean* profile is not identified, so it is not offered.) Because the confidence intervals are simulated, set a seed immediately before the call for reproducibility. ``` r data("jz2017") set.seed(12345) latent <- ssm_sem( jz2017, scales = scales, angles = octants(), measures = "NARPD", boots = 500 ) latent #> #> # Latent (SEM-based) SSM #> #> Measurement model: scaled fixed-angle circumplex #> Global fit (N = 1166, robust): chisq(17) = 300.546, p < 0.001 #> CFI = 0.93, RMSEA = 0.13, SRMR = 0.072 #> #> # Profile [NARPD]: #> #> Estimate Lower CI Upper CI #> Elevation 0.249 0.210 0.295 #> X-Value -0.009 -0.054 0.033 #> Y-Value 0.231 0.189 0.273 #> Amplitude 0.232 0.192 0.274 #> Displacement 92.132 82.515 104.461 #> Model Fit 0.975 ``` Compare this with the *observed* correlation profile of the same measure: ``` r set.seed(12345) observed <- ssm_analyze( jz2017, scales = scales, angles = octants(), measures = "NARPD" ) observed #> #> # Profile [NARPD]: #> #> Estimate Lower CI Upper CI #> Elevation 0.202 0.169 0.238 #> X-Value -0.062 -0.094 -0.029 #> Y-Value 0.179 0.145 0.213 #> Amplitude 0.189 0.154 0.227 #> Displacement 108.967 98.633 118.537 #> Model Fit 0.957 ``` The two profiles tell the same broad story: narcissistic PD (personality disorder) relates to the upper (dominant) region of the circumplex. But the disattenuated profile has a **larger amplitude** and a **higher fit**, and its displacement sits at a somewhat different angle. The amplitude increase is the removal of attenuation. The latent correlations are not pulled toward zero by scale unreliability. The displacement shift and the fit increase are the removal of *reliability heterogeneity* around the circle. (Section 6 explains why each moves.) `ssm_sem()` returns a `circumplex_ssm_sem` object, a subclass of the ordinary `circumplex_ssm` object, so the familiar table and plot functions work on it: ``` r ssm_plot_circle(latent) ``` plot of chunk latent-plot ```r ssm_table(latent, drop_xy = TRUE) ``` Table: Latent SSM profile of NARPD |Profile |Elevation |Amplitude |Displacement |Fit | |:-------|:-----------------|:-----------------|:------------------|:-----| |NARPD |0.25 (0.21, 0.29) |0.23 (0.19, 0.27) |92.1 (82.5, 104.5) |0.975 | By default `ssm_sem()` fits with a robust estimator (`estimator = "MLR"`, maximum likelihood with robust corrections) and reports robust global fit indices. This is because circumplex scale scores are typically skewed, and the naive chi-square over-rejects. The robust (sandwich) standard errors also feed the confidence intervals. Section 5 explains why that matters. ## 5. Where the confidence intervals come from Amplitude and displacement are **nonlinear** functions of the model parameters: amplitude is a square root and displacement is an `atan2`. lavaan can attach a delta-method (linear approximation) or bootstrap-percentile interval to any derived quantity. But for displacement those intervals are wrong in a way that is easy to miss. `atan2` has a branch cut, a place on the circle where its output jumps by a full turn. A displacement near 0°/360° can produce an interval that has been unwrapped across the cut, or whose endpoints have sign-flipped. So a naive percentile interval straddling the boundary points the wrong way around the circle. `circumplex` therefore **never delegates amplitude or displacement intervals to lavaan**, which is why the generated syntax refuses to define them. Instead it reuses the same machinery that the observed bootstrap uses: 1. lavaan supplies only the point estimates and their covariance (or bootstrap replicates) of the model's *free* parameters. 2. `ssm_sem()` draws from that covariance and maps each draw through the profile and the SSM transform. This gives a replicate of every SSM parameter. 3. Those replicates go through the package's existing interval assembly. The elevation, X value, Y value and amplitude get percentile intervals. Displacement gets **circular quantiles** (centered on the circular mean, unwrapped, quantiled, and re-wrapped), with contrast intervals aligned to the estimate's branch. As a result, a latent displacement interval behaves correctly at the 0°/360° pole. It can straddle the boundary contiguously, and the point estimate always sits inside its own interval. This is the same architecture that the Monte Carlo engine uses for observed profiles, with lavaan supplying the mean and covariance rather than resampling. Two engines feed step 2, selected by `ci_method`: - **`"mvn"` (default):** draw from a multivariate normal centered at the estimates, with the model's (robust) covariance. It is fast: one model fit plus vectorized draws. - **`"boot"`:** refit the model on each bootstrap resample. This is far more expensive, but it does not lean on asymptotic normality. The package's design notes report a coverage study across constructed populations. It found `"mvn"` well-calibrated at realistic sample sizes *when the covariance is the robust sandwich*. That is why robust SEs are the default. Under a misspecified fixed-angle model, plain (non-robust) SEs undercovered displacement: their intervals held the true displacement less often than nominal. The sandwich restored nominal coverage. If you supply your own lavaan fit through `ssm_sem_parameters()` and intend to use `"mvn"`, fit it with `se = "robust.huber.white"` so that the propagated covariance stays valid. ## 6. What the parameters mean now Disattenuation changes what two of the parameters *mean*. The vignette would be misleading if it did not say so. **Fit.** Under the scaled tier the latent profile is not forced to be a perfect cosine, because the scales have different saturations. So a latent fit below 1 is *informative*. It measures how far the measure's latent profile departs from a pure cosine wave. Differential saturation across scales drives that departure. Under the strict tier, anisotropy in the factor covariance (unequal spread in different directions) drives it too. What the latent fit removes, relative to the observed fit, is the contamination from **reliability heterogeneity**, not sampling error. As population quantities, the observed and the latent fit contain no sampling error at all. But their *estimates* at a finite sample size both remain noisy. So a latent fit of, say, .85 in a modest sample is not automatically substantive structure. **Displacement.** Latent displacement is the first-harmonic direction (the peak direction of the best-fitting cosine) of the *saturation-modulated* disattenuated profile. It is **not** simply "the measure's angle in the latent space." Heterogeneous saturations around the circle, or a general factor leaning into the plane, rotate it, exactly as they rotate the observed displacement. Fit can stay high while this happens. The latent layer's contribution is the removal of the *reliability* modulation that additionally rotates the observed displacement. It does not remove the saturation modulation. The removal of the reliability modulation is why the observed and latent displacements in Section 4 differ. This account, with the reliability modulation removed and the saturation modulation kept, is the honest description of what `d` estimates here. The interpretation aids you already know carry over unchanged. Amplitude is the gate for interpreting displacement. When the amplitude confidence interval's lower bound sits too close to zero relative to its width, the profile has no well-defined direction. Then the displacement is not interpretable. The low-fit dashing on plots and the displacement caution in `print()` behave exactly as they do for observed profiles. ## Wrap-up The latent SSM corrects a measure's circumplex profile for scale unreliability, both its average level and its differences across scales. `ssm_sem()` fits the measurement model, propagates its full covariance into circular-aware intervals, and gates the latent group contrast on measurement invariance. Every latent quantity is conditional on the fixed-angle model, so read the global fit first. The next page to read is "Latent Group Contrasts". It covers group differences, the invariance gate, the limitations of the method and its relation to the literature. ## References * Browne, M. W. (1992). Circumplex models for correlation matrices. _Psychometrika, 57_(4), 469–497. * Moss, J. (2026). Inference for disattenuated correlations. _Applied Psychological Measurement_. Advance online publication. https://doi.org/10.1177/01466216261440511 * Wendt, L. P., Wright, A. G. C., Pilkonis, P. A., Nolte, T., Fonagy, P., Montague, P. R., Benecke, C., Krieger, T., & Zimmermann, J. (2019). The latent structure of interpersonal problems: Validity of dimensional, categorical, and hybrid models. _Journal of Abnormal Psychology, 128_(8), 823–839. * Zimmermann, J., & Wright, A. G. C. (2017). Beyond description in interpersonal construct validation: Methodological advances in the circumplex Structural Summary Approach. _Assessment, 24_(1), 3–23.