← all lessons

Scaffold S1 — "No trend" is a distribution, across five datasets

Five rounds. You are shown a time series and its fitted slope β̂. Your job: predict the 95% half-width of the null distribution of |β̂| — i.e. how big a slope would you get, by chance, if the y-values were shuffled? The page then shuffles 1000 times and shows the null distribution with your guess and the observed slope overlaid. Repeat with a different dataset.

Locked — answer the pretest above first.

Running tally — null envelope vs. observed slope (signed z-score)

What you just did has a name

Five times, you predicted the width of the null distribution of slopes under the permutation hypothesis: "the y-values are exchangeable, the x-values are not." Your guesses should have gotten better as you went — the envelope width depends on three things you can see directly from the scatter: the residual scatter σ, the range of x, and the number of observations. The formal relationship is SE(β̂) ≈ σ / (SD(x) · √(n−1)); the 95% half-width is roughly 2 · SE(β̂).

What you just did is a permutation test. It is the most defensible null model you can build: it makes no distributional assumptions, and it directly answers "would the pattern be surprising under random relabeling?" If the observed slope is far outside the permutation envelope, something non-random is going on. If it is inside, whatever trend is there might just be sampling noise.

Notice that rounds 3 (LTEE early) and 4 (LTEE plateau) use the same simulator and the same population, but a different window in time. The early window has a slope far outside the null envelope; the plateau window has a slope just barely outside — that is the famous "diminishing-returns" debate as a null-model result.