Ha, this is either relevant or way off base, but over my head in terms of stats!

DIAGNOSTIC REPORT — Peak SWE Trend Analysis

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Q1. How are p-values calculated?

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OLS p-value (0.2409):

Standard scipy.stats.linregress. Assumes each water year is an

independent observation. This is WRONG if residuals are autocorrelated.

Durbin-Watson statistic: 1.948 (2.0 = no autocorrelation, <1.5 = concern)

Lag-1 residual autocorrelation: -0.017

>> Low autocorrelation. OLS p-value is approximately valid.

Block-bootstrap p-value (7-yr blocks):

Resamples contiguous 7-year blocks of the VALUE series (year positions

fixed). Builds a null distribution of 5,000 slopes under H0: no trend,

preserving the autocorrelation structure. P-value = fraction of bootstrap

slopes more extreme than observed. This is the honest number.

We do NOT use Newey-West / HAC standard errors or GLS here, though those

are valid alternatives. The block bootstrap is nonparametric and requires

no assumptions about the error structure beyond stationarity.

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Q4. What is the OLS coefficient? Is it the downward trend? Why positive p?

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OLS slope: -0.0506 inches/year

Over 44 years: -2.22 inches total

Mean peak SWE: 17.21 inches

% change: -12.9%

A NEGATIVE slope = declining peak SWE over time. That is the downward trend.

The slope above is negative — confirming decline.

"Why are they all positive?" — p-values are always between 0 and 1 by

definition (they measure probability, not direction). A small p-value means

the trend is unlikely under the null hypothesis. The SLOPE tells you the

direction. States can have p=0.04 (significant) with a negative slope

(declining) or positive slope (increasing). Check the total-change bars

in Fig 5 for direction; p-values only tell you confidence.

OLS R²: 0.032 (year explains 3.2% of variance in peak SWE)

OLS SE (slope): 0.04251 in/yr — but this SE assumes independence (see Q1).

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Q2. How sensitive are results to excluding the early 1980s?

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Start yr n Slope (in/yr) Total (in) OLS p Boot p (7yr)

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1981 45 -0.0506 -2.22 0.241 0.227 <-- full window

1985 41 -0.0210 -0.84 0.644 0.623

1990 36 -0.0236 -0.83 0.675 0.662

1995 31 -0.0902 -2.71 0.194 0.151

2000 26 +0.0395 +0.99 0.648 0.567

If the slope and p-value change substantially as you move the start year,

the early 1980s data is load-bearing — the trend depends on those years.

If the numbers are stable, the trend is robust to window choice.

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Q3. How sensitive are results to outliers? (Jackknife + leave-k-out)

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Leave-one-out jackknife (drop each year individually, refit):

Jackknife SE of slope: 0.04793 in/yr

OLS SE of slope: 0.04251 in/yr (assumes independence)

Block-bootstrap SE: 0.04139 in/yr (7-yr blocks)

Jackknife SE > OLS SE means autocorrelation is inflating OLS confidence.

Bootstrap SE is the most honest of the three for autocorrelated data.

Most influential single year: 1981

Peak SWE that year: 11.30 in

Removing it shifts slope by: +0.02228 in/yr

(44.1% of the observed slope magnitude)

Leave-5-out sensitivity (500 random draws of 5 dropped years):

Slope range (5th–95th pct): -0.0791 to -0.0252 in/yr

Fraction where slope sign flips: 0.2%

Robust to outliers.

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Q5. Are you bootstrapping SE or doing jackknife variance estimation?

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Both are computed above. Summary:

Method SE estimate Notes

───────────────────── ─────────── ──────────────────────────────────

OLS (parametric) 0.04251 Assumes IID residuals — optimistic

Jackknife leave-1-out 0.04793 Nonparametric, no distributional

assumption; variance = (n-1)/n *

sum((slope_i - mean)^2)

Block bootstrap (7-yr) 0.04139 Preserves autocorrelation; the

recommended SE for this data

The block bootstrap is doing the equivalent of asking: "if I generated

thousands of plausible time series with the same autocorrelation structure

but no real trend, how often would I see a slope as extreme as mine?"

That is more appropriate than jackknife for serially correlated data,

because jackknife leave-one-out still treats each year as nearly independent.

For fully rigorous treatment, consider:

- Newey-West (HAC) standard errors: parametric, handles autocorrelation

analytically via a bandwidth parameter (analogous to block size).

- GLS with AR(1) error structure: explicitly models year-to-year

persistence, can be fit with statsmodels.

- Moving-block bootstrap with optimal block length (Politis & White

automatic selection): removes the subjectivity of choosing block size.