Learn / Inside the pipeline / 08 · confidence

Observed vs supported: two independent confidence checks

An observed interval says a test at year t cleared the threshold. A supported interval says the uncertainty around that test also clears it. The workflow attaches two independent support flags to every interval, and they deliberately come from different statistical worlds:

GLM support fits the robust probit within the storage series and predicts germination at year t with a 95% confidence band (computed on the link scale, then transformed — never build the band on the percentage scale). Support = the lower bound of the band is still ≥ the threshold. This uses the whole series.

Wilson support ignores the series entirely and asks about the single test at year t: is the Wilson lower confidence bound of that one binomial observation ≥ the threshold? This uses only the counts at that year, so it can't be fooled by a bad curve — and the curve can't be fooled by one lucky test.

An interval with both flags is solid. One flag, interpret with care. Neither — it's still an observation, but it's one plate of seeds; report it as such.

Build it yourself
predict_prob_ci <- function(m, t, level = 0.95) {
  pr <- predict(m, newdata = data.frame(YearsDifference = t),
                type = "link", se.fit = TRUE)
  z <- qnorm(1 - (1 - level) / 2)
  c(fit = pnorm(pr$fit),
    lo  = pnorm(pr$fit - z * pr$se.fit),   # band built on the LINK scale
    hi  = pnorm(pr$fit + z * pr$se.fit))
}

wilson_lo <- function(x, n, level = 0.95) {
  z <- qnorm(1 - (1 - level) / 2); p <- x / n; den <- 1 + z^2 / n
  ((p + z^2 / (2 * n)) - z * sqrt((p * (1 - p) + z^2 / (4 * n)) / n)) / den
}

# per interval: glm_support   = predict_prob_ci(m, t)["lo"] * 100 >= threshold
#               wilson_support = wilson_lo(germ_at_t, sown_at_t) * 100 >= threshold
The math behind this step: Support: when the uncertainty agrees with the point · A germination test is binomial evidence
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