Occasion-to-occasion SD from a test-retest study

Estimates the per-occasion SD, in logits, of a respondent's occasion-specific deviation: the variation between two administrations that is neither measurement error nor change in the construct. It is the retest_sd argument of RMpersonChange() when null = "retest".

Usage

RMretestSD(
  data_t1,
  data_t2,
  anchor = "stack",
  item_params = NULL,
  method = "WLE",
  estimator = "CML",
  sim_iter = 500,
  parallel = TRUE,
  n_cores = NULL,
  verbose = FALSE,
  boot = TRUE,
  boot_iter = 500,
  conf_int = 0.95,
  seed = NULL,
  theta_range = c(-10, 10),
  output = "kable"
)

Arguments

data_t1, data_t2: Data.frames or matrices of item responses from a test-retest study, same items in the same order, row i being the same respondent at both administrations.
anchor, item_params, method, estimator, theta_range: As in RMpersonChange(). Use the same settings there as here, since the estimate is defined relative to the measurement-error variance the model implies.
sim_iter: Integer. Iterations used to simulate the measurement-error component. Default 500.
parallel: Logical. Use mirai for the simulation if available. Default TRUE.
n_cores: Integer or NULL. Parallel workers. When NULL, getOption("mc.cores") is checked first.
verbose: Logical. Progress bar for the simulation. Default FALSE.
boot: Logical. Bootstrap a confidence interval by resampling respondents. Default TRUE.
boot_iter: Integer. Bootstrap iterations. Default 500.
conf_int: Numeric in (0, 1). HDCI width. Default 0.95.
seed: Integer or NULL. Random seed.
output: Character. "kable" (default) or "dataframe".

Value

A one-row data.frame (or its knitr_kable) with variance, sd, lower, upper, var_change (the observed variance of the change) and var_error (the simulated measurement-error variance of the change), plus n.

Details

Writing $\hat\theta_{it} = \theta_i + u_{it} + e_{it}$ with $u_{it} \sim N(0, \sigma^2_{retest})$ the occasion deviation and $e_{it}$ the measurement error, the observed change has variance $2\sigma^2_{retest} + SE_{i1}^2 + SE_{i2}^2$. So

$$\hat\sigma^2_{retest} = \frac{\mathrm{Var}(d) - \mathrm{Var}_{err}(d)}{2}$$

How the error term is obtained. The obvious choice for $\mathrm{Var}_{err}(d)$ is $\overline{SE_1^2 + SE_2^2}$ from the information-based standard errors, and that is what Caronni et al. (2026) and the classical formulations use. It is badly biased on short scales, because the information standard error is asymptotic and overstates the real sampling variance of $\hat\theta$ when items are few, so the subtraction removes too much. Simulation over a range of scale lengths, at a true occasion variance of 0.09 and n = 2000:

items	asymptotic	simulated
6	-0.023	0.072
12	0.053	0.093
20	0.086	0.099

Across eight replications at n = 1000 the simulated version is close to unbiased on a six-item scale (mean 0.002 against a truth of 0, and 0.081 against a truth of 0.09), with a replication SD near 0.015.

This function therefore obtains $\mathrm{Var}_{err}(d)$ by simulation: respondents are held at their observed locations, two response vectors are drawn with no occasion variance, and the variance of the resulting change is averaged over sim_iter iterations. The observed missingness pattern is carried into the simulated responses. var_error in the output is that simulated quantity, not the mean of the squared standard errors.

The retest interval has to be long enough for recall to fade and short enough that no real change occurs. That is an assumption about the design and the data cannot check it.

The estimate can be negative. It is a variance obtained by subtraction, so when the observed change is tighter than the measurement model says it should be, the difference goes below zero. A negative value is reported as such rather than floored, because flooring hides the signal. It points at standard errors overstated by misfit or local dependence, at regression toward a common value over the interval, or at sampling noise. sd is NA whenever variance is negative.

It is a variance estimated from n differences, so its own sampling error is roughly $\sigma^2\sqrt{2/(n-1)}$. A few dozen respondents will not pin it down, and the bootstrap interval is the honest way to see that. The bootstrap resamples respondents and recomputes $\mathrm{Var}(d)$ only, holding the simulated error term at its full-sample value, since that term is model-implied and its Monte Carlo error is small beside the sampling error of the change variance.

Converting a published test-retest coefficient. A published test-retest ICC or Pearson correlation can be converted, but it is not itself the quantity wanted here, and neither is a published test-retest SEM ($SD\sqrt{1-ICC}$), which already contains measurement error. Since $\hat\theta = \theta + u + e$ gives $r = \mathrm{Var}(\theta) / \mathrm{Var}(\hat\theta)$,

$$\hat\sigma^2_{retest} = (1 - r)\,\mathrm{Var}(\hat\theta) - \mathrm{Var}_{err}(d)/2$$

with $\mathrm{Var}(\hat\theta)$ the observed variance of the person estimates and $\mathrm{Var}_{err}(d)$ the simulated error term this function reports as var_error. Checked against known truth at n = 3000, true occasion variance 0.09:

items	r (logit)	r (sum score)	from logit r	from sum-score r
6	.808	.829	0.094	0.046
12	.873	.890	0.084	0.048
20	.903	.913	0.088	0.067

Three conditions have to hold, and the first is usually violated:

The coefficient must be on the logit metric. The score-to-logit map is nonlinear, so a coefficient computed on sum scores runs consistently higher than the same coefficient on logits. After subtraction the residual is small, so that modest gap does real damage: on a six-item scale, substituting the sum-score value roughly halves the recovered occasion variance. The error has a known direction, understating occasion noise and leaving RMpersonChange() too permissive. Nearly all published coefficients are on sum scores.
The published sample must resemble yours. A retest coefficient rises with the heterogeneity of the people measured, so it is a property of that sample rather than of the instrument. Combining someone else's r with your $\mathrm{Var}(\hat\theta)$ assumes the structure transfers.
Pearson and an agreement ICC are not interchangeable. Pearson ignores a systematic mean shift between occasions and so reports stability that practice or drift has removed. ICC(A,1) penalises it and is the closer choice here.

The conversion also needs the simulated error term, which this function produces only when given two occasions of raw data. Converting from a published coefficient and a single administration is not implemented. When that is the situation you are in, retest_sd_tip from RMpersonChange() answers the practical question from the other end: it reports how much occasion noise each flagged result would tolerate, which you can weigh against a published coefficient without converting anything.

Examples