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The default separation for the distinguishing and consensus probabilities: the reliability-adjusted critical difference used to flag distinguishing statements in classical Q analysis (Brown 1980; Zabala & Pascual 2016), here generalized by computing it from the posterior dominant-factor counts. With p_f the number of participants whose posterior dominant factor is f, r_f = p_f r0 / (1 + (p_f - 1) r0) and SE_f = sqrt(1 - r_f), delta = z * mean_{k < l} sqrt(SE_k^2 + SE_l^2). The reliability is the stable population reliability of the design, not the posterior estimation spread.

Usage

critical_delta(Lambda_draws, level = 0.05, r0 = 0.8)

Arguments

Lambda_draws

Array of shape [T, N, K] of aligned loading draws.

level

Two-sided level for the critical value, z = qnorm(1 - level/2). 0.05 (default) gives z = 1.96; 0.01 gives z = 2.58. Report sensitivity over level.

r0

Conventional single-sort reliability (default 0.80; Brown 1980).

Value

A single numeric value, the critical-difference delta.

Examples

critical_delta(array(rnorm(200 * 8 * 3), c(200, 8, 3)))
#> [1] 0.822852