RMreliabilityCurve: Conditional measurement precision across the latent scale

In easyRasch2: Psychometric Analysis with Rasch Measurement Theory

Conditional measurement precision across the latent scale

Description

Plots (or tabulates) how precisely a scale measures at each point of the latent continuum, rather than collapsing precision to the single number reported by RMreliability(). The curve is the conditional standard error of measurement SEM(\theta) = 1/\sqrt{I(\theta)}, the test information I(\theta) it comes from, or the conditional reliability derived from either.

Usage

RMreliabilityCurve(
  data,
  statistic = "sem",
  method = "WLE",
  benchmark = NULL,
  reference = "marginal",
  items = NULL,
  item_params = NULL,
  boot = FALSE,
  boot_iter = 200,
  conf_int = 0.95,
  parallel = TRUE,
  n_cores = NULL,
  seed = NULL,
  show_density = TRUE,
  theta_range = NULL,
  n_nodes = 161L,
  verbose = FALSE,
  output = "ggplot"
)

Arguments

data	A data.frame or matrix of item responses. Items must be scored starting at 0 (non-negative integers). Missing values (NA) are allowed.
statistic	Character. The quantity on the y-axis: "sem" (default, the conditional standard error in logits), "information" (test information), or "reliability" (see Details for the formula used, and why). This is the argument that chooses what the figure shows, not method.
method	Character. Person-location estimator, "WLE" (default) or "EAP". This does not change the curve, which is always built from test information. It governs only the respondent locations behind the density overlay and the benchmark percentage, and is ignored when show_density = FALSE and benchmark = NULL.
benchmark	Numeric in (0, 1) or NULL (default). When supplied, the region of the scale whose conditional reliability reaches benchmark is shaded, and the percentage of respondents located inside it is reported. There is deliberately no default value; see Details.
reference	Character. Horizontal line showing the flat summary that a single coefficient implies: "marginal" (default) or "none".
items	Optional character vector of column names, or numeric column indices, selecting a subset of items. Useful for comparing a short form against the full scale.
item_params	Optional pre-specified item parameters, used in place of the thresholds estimated from data. Either a named list of Andrich threshold vectors or a long-format data.frame from RMitemParameters(), the same two forms RMpersonParameters() accepts. Supply this to draw the curve from anchored or previously published parameters. As in RMpersonParameters(), supplied thresholds are used as given and are not re-centred, so their origin defines the origin of the theta axis.
boot	Logical. If TRUE, add a bootstrap confidence band to the curve by resampling respondents. Default FALSE.
boot_iter	Integer. Bootstrap iterations when boot = TRUE. Default 200.
conf_int	Numeric in (0, 1). HDCI width for the bootstrap band. Default 0.95.
parallel	Logical. Use mirai for the bootstrap if available. Default TRUE.
n_cores	Integer or NULL. Number of parallel workers. When NULL, getOption("mc.cores") is checked first; if neither is set, the bootstrap falls back to sequential.
seed	Integer or NULL. Random seed for the bootstrap. See easyRasch2-reproducibility.
show_density	Logical. Draw the respondent location distribution as a background band. Default TRUE.
theta_range	Numeric length-2 vector, or NULL (default) for \pm 3\sigma.
n_nodes	Integer. Number of points on the theta grid. Default 161.
verbose	Logical. Print a progress bar for the bootstrap. Default FALSE.
output	Character. "ggplot" (default), "dataframe" for the curve, or "kable" for a summary table.

Details

What the curve is. Test information I(\theta) = \sum_i \mathrm{Var}_i(\mathrm{score} \mid \theta) is summed from the CML item thresholds (psychotools), and the conditional standard error follows as 1/\sqrt{I(\theta)}. Both are properties of the item set alone. The respondent distribution is drawn behind the curve because precision only matters where people actually are, but it does not enter the curve.

Conditional reliability, and which formula. Two definitions circulate:

\rho(\theta) = \frac{\sigma^2}{\sigma^2 + SEM(\theta)^2} \qquad \mathrm{and} \qquad \rho(\theta) = 1 - \frac{SEM(\theta)^2}{\sigma^2}

with \sigma the latent SD estimated by marginal maximum likelihood. This function uses the first, the ratio form, for three reasons. It is bounded in (0, 1), whereas the subtractive form returns negative values whenever SEM(\theta) > \sigma, which is common at the floor of a skewed scale and at both tails of a short one. It treats \sigma^2 as true-score variance, which is what the MML estimate of the latent SD is. And it tracks the reliability of the observed scores far more closely. Milanzi et al. (2015) report the subtractive coefficient falling to -0.120 against a true value of 0.480 for a 1PL with \sigma^2 = 0.25. Replicating their design (dev/milanzi_check.R, 1000 replications per cell) gives a mean absolute error against the exact expected-sum-score reliability of 0.22 for the subtractive form and 0.018 for the ratio form, with the subtractive form negative in 99.7% of replications of their lowest-variance cell. For polytomous Rasch data at 5 to 20 items the corresponding errors are 0.18 and 0.015.

The advantage is concentrated where it matters rather than uniform. Where information is high relative to the trait variance the two forms are close and either can be nearer the truth. The gap comes from the low-information cases, where the subtractive form does not merely lose accuracy but leaves the (0, 1) interval altogether.

marginal_ratio is the latent-density-weighted mean of the reliability curve rather than the ratio formed from the averaged error variance. The two differ by Jensen's inequality, and the curve mean is the more accurate of the pair (mean absolute error 0.015 against 0.018 for binary data, 0.010 against 0.015 for polytomous).

RMreliability() reports the same ratio-form coefficient in its "Marginal (curve mean)" row, so the scalar and the curve agree. The superseded subtractive value is still returned as marginal_green for comparison with easyRasch2 1.2.0 and earlier, and with mirt::marginal_rxx() and similar software.

Limits. The information-based standard error is asymptotic, and is only an approximation for short scales; Milanzi et al. (2015) find the worst behaviour with six items. The curve describes the precision of the person location estimate on the logit scale, which is not the same quantity as the reliability of a raw sum score, and reliability computed on a latent scale is consistently higher than its manifest counterpart. Use RMscoreSE() for the raw-score view.

No default benchmark. benchmark is NULL unless asked for, in keeping with the package's treatment of fixed rules of thumb. McNeish and Dumas (2025), whose respondent-weighted summary this borrows, are explicit that their own interpretive percentages are heuristic and should not be used as cutoffs.

Value

• If output = "ggplot": a ggplot object.

• If output = "dataframe": a data.frame with one row per grid point and columns theta, information, sem, reliability, plus ⁠<statistic>_lower⁠ / ⁠<statistic>_upper⁠ band columns when boot = TRUE. Attributes carry the summary quantities: sigma, marginal_ratio (the latent-density-weighted mean of the reliability curve, matching RMreliability()), marginal_green (the superseded subtractive coefficient), sem_average (root mean error variance), benchmark, benchmark_range, benchmark_percent and n_not_estimable.

• If output = "kable": a knitr_kable summary table of those quantities.

References

Green, B. F., Bock, R. D., Humphreys, L. G., Linn, R. L., & Reckase, M. D. (1984). Technical Guidelines for Assessing Computerized Adaptive Tests. Journal of Educational Measurement, 21(4), 347-360. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.1745-3984.1984.tb01039.x")}

McNeish, D., & Dumas, D. (2025). Reliability representativeness: How well does coefficient alpha summarize reliability across the score distribution? Behavior Research Methods, 57(3), 93. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3758/s13428-025-02611-8")}

Milanzi, E., Molenberghs, G., Alonso, A., Verbeke, G., & De Boeck, P. (2015). Reliability measures in item response theory: Manifest versus latent correlation functions. British Journal of Mathematical and Statistical Psychology, 68(1), 43-64. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/bmsp.12033")}

See Also

RMreliability(), RMscoreSE(), RMtargeting()

Examples

if (requireNamespace("ggplot2", quietly = TRUE)) {
  # Conditional SEM across the scale
  RMreliabilityCurve(phq9[, 1:9])

  # Conditional reliability, with the region reaching .8 shaded
  RMreliabilityCurve(phq9[, 1:9], statistic = "reliability",
                     benchmark = 0.8)
}

# Summary quantities
RMreliabilityCurve(phq9[, 1:9], benchmark = 0.8, output = "kable")

easyRasch2 documentation built on Sept. 13, 2026, 1:07 a.m.