The normal-approximation companion of ape_alpha() for pricing
minimum detectable effects: the error rate that minimizes the
weighted combined Type I and Type II error rate (Mudge et al., 2012;
Maier & Lakens, 2022) for the effect that is just detectable at the
target power. Under the normal approximation, the minimize-mode
first-order condition for an effect-to-standard-error ratio r is
m * cost * phi(z) = prior * phi(z - r), with z the claim's
critical value and m = 2 for the two-sided detection claim (m = 1
for the one-sided minimum-effect claim). At the detectability
frontier the ratio r equals z + qnorm(power), so the condition
collapses to
phi(z) = prior * phi(qnorm(power)) / (m * cost) – one equation
involving neither the sample size nor the standard error. The
frontier level is therefore the same at every n, which makes it
the natural companion of an MDE grid: solve the level once, pass it
as alpha to ape_mde() (or ape_power(), ape_n()), and the
whole grid is priced under one justified error rate.
Usage
alpha_frontier(
cost = 1,
power = 0.8,
prior = 1,
claim = c("detect", "minimum")
)Arguments
- cost
Relative cost of a Type I error against a Type II error (default 1).
- power
Target power defining the frontier effect (default 0.80).
- prior
Prior odds of the just-detectable effect against the claim boundary (default 1).
- claim
"detect"(default; two-sided) or"minimum"(one-sided). For equivalence runape_alpha()on stored draws.
Value
The frontier error rate (a numeric scalar), suitable as the
alpha argument of the power functions when it is below 0.5.
Details
Two anchors worth knowing. At Cohen's 4:1 weighting and 80% power the
two-sided frontier level is 0.0274. At equal costs the one-sided
frontier level is exactly 1 - power (the frontier balances
alpha = beta there).
Relation to ape_alpha(). ape_alpha() is exact for a specific
design: it optimizes over the stored draws of one simulation, applies
the conventional cap = 0.05, and can verify the realized size.
alpha_frontier() is its closed-form, scale-free limit at the
just-detectable effect; it applies no cap (levels above .05 need a
justification of their own; Maier & Lakens, 2022) and warns when the
optimum does not lie below 0.5, where it could not be used as an
alpha anyway.
References
Maier, M., & Lakens, D. (2022). Justify your alpha: A primer on two practical approaches. Advances in Methods and Practices in Psychological Science, 5(2). Mudge, J. F., Baker, L. F., Edge, C. B., & Houlahan, J. E. (2012). Setting an optimal alpha that minimizes errors in null hypothesis significance tests. PLOS ONE, 7(2).