Run the exploratory circumplex-structure criteria of Acton and Revelle (2004) on a set of scales and return one object bundling all of the tests. Four criteria are computed from the first two unrotated principal-axis factors of the scales' correlation matrix – the Fisher Test of equal axes, the Gap Test of equal spacing, the Variance Test (VT2) and Rotation Test of interstitiality – and each statistic is classified against simulation-derived, scoring- and scale-count-specific cutoffs. A fifth test, RANDALL (Hubert & Arabie, 1987; Tracey, 1997), evaluates the hypothesised circular order of the scales with a randomization test that yields a genuine p-value.
Usage
fit_structure(
data,
scales,
scoring = c("deviation", "raw"),
ridge = 0,
n_perm = NULL,
listwise = TRUE
)Arguments
- data
A data frame (or matrix) containing the circumplex scales.
- scales
A character vector of column names (or a numeric vector of column indexes) selecting the circumplex scales, in hypothesised circular order (the order is RANDALL's order hypothesis). At least four scales are required.
- scoring
Either
"deviation"(the default; row-mean-center the scales before analysis) or"raw"(analyze the scores as given). Selects the matching interpretive cutoffs.- ridge
A non-negative ridge added to the diagonal of the correlation matrix (then rescaled to a unit diagonal) to repair a non-positive-definite matrix before factoring; default
0, which matches the cutoff calibration. Raise it only if factoring fails, noting that a nonzero ridge moves the statistics off the calibrated scale.- n_perm
NULL(the default) to compute RANDALL's p-value by exact enumeration, available for up to nine scales; otherwise a single positive whole number of Monte Carlo relabelings (required for ten or more scales). The Monte Carlo path draws from the global RNG stream, so set a seed withset.seed()beforehand for reproducibility. Ten or more scales requiren_perm(exact enumeration is infeasible); supplying it is validated up front, before any criteria are computed.- listwise
A logical indicating whether missing values are handled by listwise deletion (
TRUE, the default) or pairwise deletion (FALSE), matchingssm_analyze(). Listwise deletion gives all five tests one complete-case correlation matrix, which is the metric the interpretive cutoffs were calibrated on; pairwise deletion can yield a non-positive- definite matrix and moves the statistics off that calibrated scale.
Value
An object of class circumplex_structure with print(),
summary(), and plot() methods. Its components are results (a data
frame with one row per factor-analytic criterion: statistic, cutoffs, and
interpretive category), randall (the RANDALL index, p-value, and method),
loadings (the two unrotated principal-axis factors), and details.
Details
The four factor-analytic criteria have the most power when there is no large
general factor, which deviation scoring (centering each respondent on their
own mean across the scales, exactly what ipsatize() does) approximates by
removing it (Acton & Revelle, 2004, p. 9). Deviation scoring is therefore the
default and is applied to all five tests; pass scoring = "raw" to leave the
scores untouched. The two scorings carry different cutoffs, matched
automatically.
The interpretive cutoffs are heuristic likelihood classifications read off
simulated distributions, not significance tests, and they are specific to
the number of scales. Only eight scales (the canonical octant instrument) are
calibrated; with any other count the statistics are still reported but no
interpretation is attached (see print()/summary()). The cutoffs were
re-derived under Acton and Revelle's own generating model at eight scales;
see vignette("evaluating-circumplex-structure"). RANDALL needs no cutoffs:
with up to nine scales its null distribution is enumerated exactly, so its
p-value is available at any scale count of four or more.
References
Acton, G. S., & Revelle, W. (2004). Evaluation of ten psychometric criteria for circumplex structure. Methods of Psychological Research Online, 9(1), 1-27.
Hubert, L., & Arabie, P. (1987). Evaluating order hypotheses within proximity matrices. Psychological Bulletin, 102(1), 172-178.
Tracey, T. J. G. (1997). RANDALL: A Microsoft FORTRAN program for a randomization test of hypothesized order relations. Educational and Psychological Measurement, 57(1), 164-168.
See also
cpm_fit() for a confirmatory circumplex model; ipsatize() for
deviation scoring.
Examples
data("jz2017")
scales <- c("PA", "BC", "DE", "FG", "HI", "JK", "LM", "NO")
res <- fit_structure(jz2017, scales = scales)
res
#>
#> Circumplex Structure Tests (Acton & Revelle, 2004)
#> Scales (nv): 8
#> Scoring: deviation (row-mean centered)
#>
#> # Exploratory criteria
#>
#> Test Statistic Interpretation
#> Fisher 0.102 equal axes: at least 3x as likely as the alternative
#> Gap 0.152 equal spacing: at least 3x as likely as the alternative
#> Variance 0.180 interstitiality: almost certain
#> Rotation 0.325 interstitiality: at least 3x as likely as the alternative
#>
#> # Order hypothesis (RANDALL)
#>
#> Correspondence index = 0.868, p = 0.000397 (exact, 5040 relabelings)
#>
#> Interpretations are heuristic likelihood classifications from simulation, not
#> significance tests (Acton & Revelle, 2004). RANDALL's p-value is exact.
summary(res)
#>
#> Circumplex Structure Tests (Acton & Revelle, 2004)
#> Scales (nv): 8
#> Scoring: deviation (row-mean centered)
#> Ridge: 0
#>
#> # Exploratory criteria
#>
#> Test Statistic Almost Thrice Twice Verdict
#> Fisher 0.102 0.07 0.12 0.15 3x+ likely
#> Gap 0.152 0.15 0.40 0.46 3x+ likely
#> Variance 0.180 0.19 0.59 0.64 almost certain
#> Rotation 0.325 0.32 0.64 0.67 3x+ likely
#>
#> # Estimated scale geometry
#>
#> Scale Angle Communality
#> PA 339.635 0.642
#> BC 359.858 0.571
#> DE 48.584 0.500
#> FG 81.337 0.522
#> HI 161.662 0.690
#> JK 183.339 0.713
#> LM 215.503 0.388
#> NO 287.119 0.474
#>
#> # Order hypothesis (RANDALL)
#>
#> Correspondence index = 0.868, p = 0.000397 (exact, 5040 relabelings)
#>
#> Interpretations are heuristic likelihood classifications from simulation, not
#> significance tests (Acton & Revelle, 2004). RANDALL's p-value is exact.
#>