# Pre-registered plan: does the finding that global warming accelerated survive a null simulation of the whole procedure?

## Question

Foster and Rahmstorf (2026, Geophysical Research Letters 53, e2025GL118804; "F&R") adjust five global mean surface temperature (GMST) records for El Niño (NINO3.4), volcanic aerosol and sunspot number by a lagged regression fitted to the same records, and then test the adjusted series for acceleration since 1970 in two ways:

- **PLF changepoint.** The changepoint test of Beaulieu et al. (2024): a continuous piecewise-linear trend with one changepoint, on annual means for 1970–2024, with Gaussian AR(1) errors and Monte Carlo critical values. F&R report more than 99% confidence in all five records, with the change in 2013 or 2014.
- **Quadratic.** A quadratic trend with the ARMA(1,1) correction of Foster and Rahmstorf (2011). F&R report better than 98% confidence in all five.

Both tests' critical values treat the adjusted series as if it were the data. The adjustment, however, is estimated on those same records: a 20-year lowess, four rounds of backfitting, and lags chosen by AIC among 4,225 combinations. The changepoint test's AR(1) noise model is estimated on 55 annual values. RealClimate (2026-03) and Bennedsen and Vera-Valdés (arXiv 2608.18298) note that the adjustment's uncertainty is not propagated. No published analysis we found simulates the whole procedure under a no-acceleration null.

**Question:** when the entire procedure (adjustment and test) is applied to series generated under F&R's own null hypothesis (a linear trend since 1970 plus stationary noise), how often does it declare acceleration, and how extreme are the observed statistics against that full-pipeline null distribution?

## Data

These are F&R's own inputs and outputs, fetched from Zenodo and checked by SHA-256 (`external.json`), all under CC BY 4.0:
- `AdjustData.zip` (doi:10.5281/zenodo.15608929), whose `ObservedData.csv` holds monthly NASA, NOAA, HadCRU, Berkeley and ERA5 GMST, the volcanic AOD, NINO3.4 and the sunspot number, 1880–2024;
- `GMST_adjusted.csv` (doi:10.5281/zenodo.15591644), their adjusted series.

No other data are used.

## Methods (all locked in `code/` and `parameters.json`)

1. **The adjustment.** `code/fr.py` is a numpy port of F&R's `Xogen.r`:
   - lowess with f = 20 years / record span, as R's `lowess` computes it;
   - simple lags of 0–24 months (volcanic), 0–12 (NINO3.4) and 0–12 (sunspots), the combination with the lowest AIC chosen;
   - the fit window 1950–2024, four backfitting rounds, and rounding to 4 decimals each pass, as in `Xogen.r`.

   Before this plan was registered, F&R's R code and the port were both run on their data. The port reproduces the lags R chooses and the published adjusted series to within 1e-4 °C (the rounding).
2. **The changepoint test.** `code/tmax.py` implements the test on annual means 1970–2024 (N = 55).
   - For each changepoint k with 6 ≤ k ≤ 49 (10% trimming), it fits a continuous piecewise-linear trend with Gaussian AR(1) errors by exact maximum likelihood, profiled over φ.
   - T_k is the slope change divided by its GLS standard error, and T_max = max |T_k|.
   - Its T_k agree with R's `arima(..., method="ML")` to within 0.02 on the series checked, and it reproduces the published T_43 for HadCRUT 1970–2023 (Beaulieu et al. 2024: 1.528; here 1.575 from F&R's HadCRU file).
   - Critical values follow Beaulieu et al.: AR(1) Monte Carlo at φ̂ from a linear trend plus AR(1) fitted by ML. For the observed series that means 100,000 simulations at φ̂. For simulated series it means a table of 10,000 simulations at each φ from −0.40 to 0.80 in steps of 0.02, linearly interpolated.
3. **The quadratic test.** OLS of a quadratic in time, with the standard error of the quadratic coefficient multiplied by √ν, where ν = 1 + 2 r₁ / (1 − r₂ / r₁) is computed from the residuals' lag-1 and lag-2 autocorrelations (Foster and Rahmstorf 2011). It is applied to monthly values 1970–2024 and, separately, to annual means. Two-sided normal p-values are used.
4. **The null world, for each record.** Simulated monthly temperatures = g₀(t) + E(t) + ε(t), where:
   - **g₀(t)** is a straight line fitted to the adjusted series for 1970–2024. Before 1970 it is the adjusted series' 20-year lowess, shifted to meet the line, so the pre-1970 history and the adjustment's lowess see realistic shape.
   - **E(t)** is the exogenous terms as F&R's procedure fits them to the real record (coefficients and lags, applied to the real predictor series).
   - **ε(t)** is stationary Gaussian noise from one of three models, each fitted to the real adjusted series:
     - **primary, `null_trend`:** ARMA(1,1) by ML, fitted to the adjusted series minus g₀ over 1970–2024, which is the noise as estimated under the null;
     - **sensitivity, `lowess`:** ARMA(1,1) fitted to the adjusted series minus its 20-year lowess over 1950–2024. This has less low-frequency variance, so it is more favourable to F&R;
     - **sensitivity, `long_memory`:** ARFIMA(1, d, 0) fitted to the same residuals as the primary model, with d from a Geweke–Porter–Hudak regression on the lowest ⌊√n⌋ frequencies, clipped to [0, 0.45], and an AR(1) on the fractionally differenced series. This allows decadal-scale variability.

   Each simulated record is 1,200 months of burn-in plus the real record's length.
5. **Simulation.** For each record and noise model, 2,000 simulated records go through the whole procedure: the adjustment re-estimated, including lag search and backfitting, then both tests. All randomness comes from one seed (20261010) split into independent streams.

## Outcomes

For each record and noise model:
- **O1:** the full-pipeline size of the nominal 5% (and 1%) changepoint test: the fraction of null records whose nominal p is below 0.05 (0.01);
- **O2:** the full-pipeline p of the observed T_max: (1 + number of null T_max ≥ observed) / 2,001;
- **O3:** the full-pipeline 95% and 99% quantiles of T_max;
- **O4:** the same for the quadratic test, monthly and annual: the size of the nominal 5% test, and the full-pipeline p of the observed |z|.

We also report the reproduction of F&R's tests on their data: T_max, the changepoint year, the nominal p, the quadratic z, and the unadjusted records' T_max (Beaulieu et al. and F&R found these not significant).

## Decision rules (primary noise model, `null_trend`)

- **Changepoint finding:** "robust" if the full-pipeline p < 0.01 in all five records; "not robust" if p ≥ 0.05 in at least three; otherwise "mixed".
- **Quadratic findings (monthly; annual):** "robust" if p < 0.02 in all five; "not robust" if p ≥ 0.05 in at least three; otherwise "mixed".
- **Size:** the nominal 5% test is called calibrated if the full-pipeline size lies in [0.035, 0.065], anti-conservative above 0.065, and conservative below 0.035.

The sensitivity noise models are reported with the same statistics. They do not enter the decisions.

## Pilot disclosure

Before registration we:
- ran F&R's R code and the port on all five records, comparing adjusted series and lags;
- computed T_max on the adjusted annual Berkeley and HadCRU series and the unadjusted HadCRU 1970–2023 series, to check the test against R's `arima`. These reproduce F&R's and Beaulieu et al.'s published results (adjusted T_max 5.63 for Berkeley and 4.91 for HadCRU, change in 2014);
- computed F&R's quadratic test on the five adjusted records (two-sided confidence 96.8–99.98% on monthly values, 98.8–99.99% on annual);
- ran the full program end to end with a different seed and five or fewer null records per record, to measure runtime, without looking at their test outcomes. The fitted noise parameters were printed.

No full-pipeline null outcome (O1–O4) was computed before registration.

## Limitations stated in advance

The null world takes the adjustment's real coefficients and the real predictor series as fixed, and the noise models are fitted to one realization. A noise model can understate low-frequency variability, which is why the long-memory sensitivity is included. The analysis tests F&R's statistical procedure on its own data. It does not test whether the ENSO, volcanic and solar adjustments are physically complete, or the causes of any acceleration.

report_by: 2026-10-31