# Axes Reliability

    library(circumplex)

## 1. What is axis reliability?

A circumplex instrument places its scales around a circle and summarizes a person (or a profile of correlations) by their position on two orthogonal axes — here communion (the X axis, at 0°) and agency (the Y axis, at 90°). Because those axis scores drive everything downstream — a person’s projected location, the displacement and amplitude of a Structural Summary Method profile — it is worth asking how reliably the instrument measures each axis.

`axes_reliability()` answers that question with the estimator of Strack, Jacobs, and Grosse Holtforth (2013). It fits an item-level measurement model that decomposes each item’s variance into orthogonal pieces — a general factor common to all items, the two circumplex axes, a scale-specificity component, item error, and, for an instrument administered in blocks, a block-specificity component — and reads axis reliability off the axes component alone. Reliability is then the Spearman–Brown “list-length” reliability of a composite of that length built from items that share only their axes variance.

This is a different question from the other reliability-adjacent tools in the package. `ssm_sem()` disattenuates a scale-level SSM profile for measurement error; `fit_structure()` evaluates whether a correlation matrix has circumplex structure at all. `axes_reliability()` instead reports a single, interpretable number per axis: how well the instrument measures communion and agency.

## 2. A worked example

The package ships `simulated_items`, a synthetic dataset of 1–7 Likert responses from 500 respondents on 32 items — four items on each of the eight octant scales, in the order that `octants()` returns. The items were drawn from a five-component population (a general factor, two equal axes with axes variance .18, a shared scale-specificity component of .10, no block specificity — the instrument is not blockwise, so that component is zero here — and free item error) that implies an axis reliability of about .78.

`axes_reliability()` needs three things: the data, a map from items to scales, and the scales’ angles. The map is a list with one character vector of item names per scale, in the same angle order as the angles you pass:

    data("simulated_items")

    # Four items per octant scale, in octants() order (PA, BC, DE, ..., NO).
    items <- split(names(simulated_items), rep(1:8, each = 4))

    res <- axes_reliability(simulated_items, items = items, angles = octants())
    #> axes_reliability(): 500 complete case(s) used.
    res
    #>
    #> Circumplex Axes Reliability (Strack, Jacobs & Grosse Holtforth, 2013)
    #> Input:        item data
    #> Items:        32 (8 scales)
    #> Complete N:   500
    #> SEm scale:    std
    #>
    #> # Per-axis reliability
    #>
    #>  Axis item_n Reliability SEm   NB_Reliability
    #>  X    16     0.773       0.476 0.822
    #>  Y    16     0.773       0.476 0.823
    #>
    #>   Note: the two axes share one axes-variance estimate and, with equal
    #>   items per axis, carry the same reliability -- expected, not an error.
    #>
    #>   Note: the model is fit to the item correlation matrix as if it were a
    #>   covariance matrix (Cudeck, 1989), and both sides of that mismatch are
    #>   corrected -- so these numbers differ from LISREL's, and from lavaan's
    #>   own, by design.
    #>   The component standard errors are adjusted to the correlation metric
    #>   and are calibrated; they are typically smaller than the values printed
    #>   by Strack et al. (2013), whose LISREL output carries no correction.

If your items belong to one of the package’s built-in instruments, you can pass the `instrument` object instead of `items` and `angles` — it supplies both the scale angles and the item membership, exactly as `score()` does. (The example above uses the explicit map because `simulated_items` is not a registered instrument.)

The header confirms how many complete cases were used, and the per-axis table reports, for each axis, the effective test length (`item_n`), the Strack axis `Reliability`, its standard error of measurement (`SEm`), and the Nunnally–Bernstein reliability (`NB_Reliability`) for comparison. For a balanced instrument the two axes share one axes-variance estimate and carry equal `item_n`, so they report the same reliability — expected, not an error.

The recovered reliability (about .77) lands close to the .78 built into the simulated population, and the axes-variance estimate (below) recovers the population value of .18.

## 3. Reading the components

`summary()` adds the estimated variance components and the model’s global fit:

    summary(res)
    #>
    #> Circumplex Axes Reliability (Strack, Jacobs & Grosse Holtforth, 2013)
    #> Input:        item data
    #> Items:        32 (8 scales)
    #> Complete N:   500
    #> SEm scale:    std
    #>
    #> # Per-axis reliability
    #>
    #>  Axis item_n Reliability SEm   NB_Reliability
    #>  X    16     0.773       0.476 0.822
    #>  Y    16     0.773       0.476 0.823
    #>
    #>   Note: the two axes share one axes-variance estimate and, with equal
    #>   items per axis, carry the same reliability -- expected, not an error.
    #>
    #>   Note: the model is fit to the item correlation matrix as if it were a
    #>   covariance matrix (Cudeck, 1989), and both sides of that mismatch are
    #>   corrected -- so these numbers differ from LISREL's, and from lavaan's
    #>   own, by design.
    #>   The component standard errors are adjusted to the correlation metric
    #>   and are calibrated; they are typically smaller than the values printed
    #>   by Strack et al. (2013), whose LISREL output carries no correction.
    #>
    #> # Variance components
    #>
    #>  Component         Estimate SE
    #>  general           0.051    0.005
    #>  axes              0.175    0.009
    #>  scale_specificity 0.093    0.008
    #>  item              0.680    --
    #>
    #> # Global fit
    #>
    #>   chi-square(493) = 488.27,  RMSEA = 0.000,  CFI = 1.000
    #>
    #>   The global fit statistics chisq, pvalue, rmsea and cfi are scaled to
    #>   that metric (Satorra & Bentler, 1994), which removes a distortion that
    #>   flatters fit; df and srmr are unchanged. The scaled test can modestly
    #>   over-reject at typical sample sizes -- it over-flags misfit rather than
    #>   flattering it; see ?axes_reliability for the measured rates.
    #>   They follow lavaan's *.scaled definitions, not its *.robust ones, and
    #>   differ from what fitMeasures() reports for an equivalent ML fit.

The variance components show the decomposition the reliability rests on: the `axes` component is the only one that feeds reliability, while `general`, `scale_specificity`, `block_specificity` (when blocks were supplied), and item error are isolated from it. This is precisely why the Nunnally–Bernstein figure printed alongside runs higher than the Strack reliability: N–B charges scale-specificity variance to the axis rather than isolating it, so it overestimates axis reliability whenever scale specificity is non-trivial (Strack et al., 2013, Figure 3). The gap between the two numbers is a direct read-out of how much scale-specific variance the simpler formula would have miscredited to the axes.

## 4. Starting from a published correlation matrix

You do not always have the raw data. A paper may print an item correlation matrix and nothing else, and that matrix is enough: pass it as `cormat` together with the sample size it was computed from, in place of `data`.

    R <- cor(simulated_items)
    axes_reliability(
      cormat = R, items = items, angles = octants(), n = nrow(simulated_items)
    )
    #>
    #> Circumplex Axes Reliability (Strack, Jacobs & Grosse Holtforth, 2013)
    #> Input:        correlation matrix
    #> Items:        32 (8 scales)
    #> Sample N:     500
    #> SEm scale:    std
    #>
    #> # Per-axis reliability
    #>
    #>  Axis item_n Reliability SEm   NB_Reliability
    #>  X    16     0.773       0.476 --
    #>  Y    16     0.773       0.476 --
    #>
    #>   Note: the two axes share one axes-variance estimate and, with equal
    #>   items per axis, carry the same reliability -- expected, not an error.
    #>
    #>   Note: the Nunnally-Bernstein comparison needs the raw item scores
    #>   (scale alphas and the axis-composite variance), so it is NA on the
    #>   correlation-matrix path.
    #>
    #>   Note: the model is fit to the item correlation matrix as if it were a
    #>   covariance matrix (Cudeck, 1989), and both sides of that mismatch are
    #>   corrected -- so these numbers differ from LISREL's, and from lavaan's
    #>   own, by design.
    #>   The component standard errors are adjusted to the correlation metric
    #>   and are calibrated; they are typically smaller than the values printed
    #>   by Strack et al. (2013), whose LISREL output carries no correction.

The estimates are identical to the raw-data run above — the raw-data path builds exactly this matrix internally and fits it the same way. `items` selects and orders the matrix’s rows by name, so its own column ordering does not matter, and it must be symmetric, positive definite, and have a unit diagonal (the model assumes unit-variance items).

Two things are unavailable here, and both for the same reason: they are properties of the respondents, not of their correlations. The Nunnally–Bernstein comparison is reported as `NA` — it needs each scale’s alpha and the axis composite’s variance, neither of which a correlation matrix carries. And `sd = "raw"` is refused, because there are no scale scores to take an observed SD from; supply the axis SDs numerically if you want SEm on a raw scale. Both are reported rather than silently omitted, so a matrix-based result cannot be mistaken for a raw-data one.

## 5. Caveats to keep in mind

Four properties of the method shape how its output should be read.

Both the standard errors and the global fit statistics are corrected for the correlation metric. Following the paper’s own practice, the model is fit to the item correlation matrix as though it were a covariance matrix. That metric mismatch leaves the component point estimates and the reliabilities correct, but it misprices anything computed from the input’s sampling variability (Cudeck, 1989).

The component standard errors are corrected for it. Normal-theory maximum likelihood prices its standard errors for a sample covariance input, while this estimator consumes a sample correlation matrix, whose diagonal cannot vary at all. Left uncorrected that mismatch overstates sampling variability by about 40% for an instrument whose axes carry a lot of variance, and understates it slightly for weak-axes, strong-general instruments — it changes sign across the range of instruments the function accepts, which is why it could not be left to a caveat. The reported SEs are adjusted to the correlation metric and are calibrated uncertainty. They are typically smaller than the standard errors printed in Strack et al. (2013), whose LISREL values carry the uncorrected approximation. What the estimator reported before the correction is kept in `details$se_uncorrected`.

The global fit statistics are corrected too, for the same mismatch running the other way. Sample correlations vary less than the covariances the reference chi-square distribution is derived for, so an uncorrected test statistic comes out too small and fit is flattered. `chisq`, `pvalue`, `rmsea` and `cfi` are therefore reported as Satorra–Bentler-type scaled values — the chi-square divided by a factor computed at the fitted matrix, with `cfi` also scaling its own baseline model (Satorra & Bentler, 1994). The factor is recomputed for every fit rather than being a constant, which is the whole point: how much fit was flattered depends on the instrument. `df` and `srmr` are unchanged, being a count of restrictions and a residual summary rather than test statistics with a reference distribution. What the estimator reported before the scaling is kept in `details$fit_uncorrected`, and the factors themselves in `details$scaling_factor`.

If you go looking for these numbers in lavaan, match the variant. They are built with the definitions lavaan calls `chisq.scaled`, `pvalue.scaled`, `rmsea.scaled` and `cfi.scaled` — the mean-adjusted Satorra–Bentler forms — and not the `*.robust` ones (`cfi.robust`, `rmsea.robust`), which apply a different (Brosseau-Liard/Savalei) adjustment and give different numbers. Because the model is estimated with plain ML and the scaling is applied by this package rather than by lavaan, `fitMeasures()` on an equivalent fit hands back the uncorrected values under the bare names `chisq`, `pvalue`, `rmsea` and `cfi` — and reports no `*.scaled` or `*.robust` measure at all, since lavaan supplies those only for a genuinely scaled estimator such as `"MLM"` or `"MLR"`, and quietly returns a shorter vector when asked for one it does not have. So a cross-check against lavaan’s bare `cfi` will disagree with `$fit$cfi`, a request for `cfi.robust` will come back empty rather than disagreeing, and neither outcome is a defect.

Read the scaled statistic for what it is. The correction makes the test statistic match its reference distribution in expectation; it is a calibration, not an exactness guarantee, and it will not rescue a model that is badly wrong. The usual cutoffs (Hu & Bentler, 1999) are no more or less appropriate than before — what has changed is that the number being compared to them is no longer systematically kind.

How well calibrated is the test, and at what sample size? The scaling fixes the metric error, and the χ² test built on it is asymptotically exact: its rejection rate approaches the nominal α as the number of distinct moments, p* = p(p+1)/2, falls relative to N. Simulated at one population (eight octant scales, three items each, axes variance .35), the rejection rate at α = .05 runs

p*/N | 0.50 | 0.25 | 0.12 | 0.06
--- | --- | --- | --- | ---
rejection rate | .092 | .079 | .062 | .054

reaching the nominal band by a p*/N of about 0.06. That is a sweep at a single population, not a general threshold.

You do not have to work out where your own fit sits on that row. The object reports both halves of the ratio: `details$n_moments` is p*, and `details$n` is the N the fit was priced at — on the raw-data paths the count of rows the estimator was actually handed, smaller than `details$n_total` wherever cases were dropped; on the correlation-matrix path the `n` you supplied. Divide the first by the second and read across.

At N = 600 the test over-rejects — measured .06 to .11 at three populations chosen to bracket the range of instruments the function accepts. The uncorrected statistic under-rejects over the same range, at .02 to .03, and moves further from nominal as N grows, because its error is asymptotic while the scaled statistic’s is a finite-sample one that shrinks away. The over-rejection at a fixed N grows with instrument size (larger `df`) and shrinks with N.

Two practical consequences. A p-value near whatever threshold you are using deserves caution at moderate N with many items — but note the direction: the scaled test over-flags misfit rather than flattering it, which is the safer error, and the opposite of what the uncorrected statistic did. And all of this evidence is complete-data: under `missing = "fiml"` the scaled statistic is calibrated in mean, but its rejection rate has not been measured, so none of these rates should be read as applying to that path.

If you check the implied matrix, you will also find that it does not reproduce the unit diagonal exactly. That too is expected: with the loadings fixed, the condition a free item error satisfies is the weighted diagonal rather than the raw one, so off-diagonal sampling misfit leaks into the implied diagonal at roughly the sampling standard error of a correlation.

Missing data: listwise by default, FIML on request. The default, `missing = "listwise"`, uses complete cases only and reports how many there were. `missing = "fiml"` instead estimates from every respondent who answered at least one item, and reports the total N with the complete-case count beside it. Pairwise-deletion correlations are never used on either setting.

The choice is a trade of assumptions, not a free upgrade. FIML requires the data to be missing at random — missingness may depend on values you observed, but not on the unobserved values themselves — and multivariate normal. Under MCAR, where missingness is unrelated to anything, listwise deletion is already consistent and merely wasteful, so FIML buys precision rather than correctness; under MAR, listwise deletion is genuinely biased and FIML is not. Under FIML the standard errors are observed-information standard errors on the standardized metric, conditional on the standardization constants. They carry the same correlation-metric correction as every other path, applied multiplicatively so the observed information’s own pricing of the missing data survives it. What the correction does not reach is the uncertainty in those standardization constants, which stays small at mild missingness but grows with the rate — at 15% cellwise missingness the reported SEs average about 7% below the estimator’s actual sampling variability, so treat heavy missingness as the regime where they are least trustworthy. Two results become unavailable, both needing items observed by every respondent: the Nunnally-Bernstein comparison is `NA` with a stated reason, and `sd = "raw"` is refused in favour of numeric axis SDs.

Note the provenance, because it differs from the rest of this vignette: Strack et al. (2013) report no missing-data analyses at all, so nothing about the FIML path rests on their results. It is certified against this package’s own synthetic oracle, where the true variance components are known by construction.

A boundary fit returns `NA`, not a clipped value. If the model estimates an axes variance outside the interval (0, 1) — at or below zero the axes carry no variance to be reliable, at or above one they carry all of it — or any negative variance component, the reliability and SEm are reported as `NA` with a warning and a boundary flag, rather than a clipped, negative, or missing number. An `NA` here is a signal that the model did not identify a usable axes-variance component in your data — not a defect to be worked around.

A blockwise instrument needs its blocks declared. Some circumplex instruments are administered in blocks — items grouped by something other than their scale — which carries a block-specificity component of its own, reported as high as 6.7% by Strack et al. (2013, Table 3). Pass `blocks` (a list of item columns, one element per block, exactly as `items` is a list per scale) and that component is estimated too: the component table gains a `zeta2` row and `details$zeta2_fitted` is `TRUE`. Nothing in the bundled instrument objects records block membership, so the map has to come from you.

The component is only estimable when the blocks say something the model does not already know. Blocks that coincide with the scales, a single block holding every item, or one block per item all leave `zeta2` unidentified, and it is dropped with `details$zeta2_fitted` set to `FALSE` — the same treatment scale specificity gets on a single-item instrument.

What it costs to ignore real blocks depends on their geometry. The general factor never hands block variance back, so `xi2` is inflated under most layouts and unchanged under a few — never deflated. The axes variance, the quantity reliability is actually read from, moves only when block membership carries information about the angular distance between items, beyond what sharing a scale already tells you.

One case is clean enough to rely on: when each block draws exactly one item from every scale, every within-block pair is a different-scale pair and the blocks cover every pair of scale positions equally often. Membership then says nothing about angular distance, and `xi1`, the reliability, and the SEm are untouched — worth estimating for its own sake, but costless to omit.

Elsewhere, be careful: “the blocks are spread evenly around the circle” is not the test. Blocks pairing diametrically opposite scales are as evenly spread as blocks get — their angles average to the centre of the circle — and at eight scales they still pull `xi1` about 9% below truth, because every within-block pair sits exactly half a turn apart, which is very much information about angular distance. Blocks covering contiguous arcs pull about 12% the other way. Unless each block takes one item per scale, estimate the component rather than reasoning about the picture.

Finally, a note on the `SEm`. The standard error of measurement supports a location interval for a single profile (Strack et al., 2013, use ±1.65·SEm). By default `axes_reliability()` reports the z-standardized SEm, `sqrt(1 - reliability)`; pass `sd = "raw"` (or your own axis SDs) to put the SEm on the raw axis-score scale. Such an interval describes the measurement imprecision of one profile’s axis position; it is not a significance test of that position against any particular value.

## 6. Wrap-up

`axes_reliability()` gives a compact, per-axis answer to “how reliably does this instrument measure communion and agency?”, isolating the axes variance from the general and scale-specific components that a simpler reliability formula would conflate. Use it to characterize a circumplex instrument before leaning on its axis scores, and read its output with the correlation-as-covariance, missing-data, and boundary caveats in mind.

The examples above all use the canonical eight octant scales, but nothing in the model requires them: any equally spaced set of angles works, at any rotation and at any count from four scales upward. Equal spacing is what matters, and it is required rather than merely preferred — a quasi-circumplex, whose scales sit at slightly unequal intervals, is refused rather than approximated, because Strack et al. (2013) excluded such instruments when validating the model. Three scales are refused for a different reason: at that count every pair of scales sits the same angular distance apart, so the general, axes, and scale-specificity variances can no longer be told apart.

Scales may also carry a single item each, as Strack et al.’s single-item circumplex types do. With one item at every position no two items ever share a scale, so nothing in the data distinguishes scale specificity from item specificity: the scale-specificity component is not identified, and `axes_reliability()` drops it from the model rather than estimating a quantity the data cannot support. The components table then carries three rows instead of four, and `details$zeta1_fitted` records the drop. An instrument that mixes single-item and multi-item scales still estimates the component — one multi-item scale supplies the information, and the shared-value restriction carries it to the remaining scales. One consequence follows for the comparison figure: coefficient alpha is undefined for a one-item scale, so the Nunnally–Bernstein reliability is reported as `NA` with its reason whenever any scale has fewer than two items, exactly as Strack et al. leave that column blank for such instruments.

One further boundary is worth stating plainly, because the paper crosses it and this function does not: the model implemented here is two-dimensional. Strack et al. also analyze a spherical instrument, SYMLOG, whose items span three orthogonal dimensions; the figures they publish for it come from that three-axis model, and no configuration `axes_reliability()` accepts will reproduce them.

## References

- Cudeck, R. (1989). Analysis of correlation matrices using covariance structure models. Psychological Bulletin, 105(2), 317–327.

- Satorra, A., & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In Latent variables analysis: Applications for developmental research (pp. 399–419).

- Strack, S., Jacobs, K. A., & Grosse Holtforth, M. (2013). The reliability of circumplex axes. SAGE Open, 3(2). https://doi.org/10.1177/2158244013486115