Understanding variance

How pyFIA computes standard errors using FIA’s stratified, design-based variance — and how it’s validated against EVALIDator.

Every pyFIA estimate comes with a measure of uncertainty. This page explains how that standard error is calculated, why FIA’s formula looks different from a textbook one, and how closely pyFIA matches the official EVALIDator tool.

EXPNS is not a survey weight

The key to FIA variance is the expansion factor EXPNS:

EXPNS = total acres in stratum / number of plots in stratum

Unlike a standard survey weight, EXPNS already contains the inverse of the sample size (1/n_h). That single fact is why FIA’s variance formula differs from the stratified-sampling formulas in most textbooks.

The domain-total variance formula

For tree-based estimates (volume, biomass, tree count, growth, mortality, removals), pyFIA uses the stratified domain total variance:

$$V(\hat{Y}) = \sum_h w_h^2 \, s^2_{yh} \, n_h$$

where for each stratum $h$:

• $w_h$ = EXPNS (acres per plot)
• $s^2_{yh}$ = sample variance of the plot-level values (with ddof=1)
• $n_h$ = number of plots

Multiplying by $n_h$ looks backwards compared to a textbook mean-variance formula that divides by it. It’s correct here because EXPNS already carries $1/n_h$, and we’re estimating a total, not a mean. The per-acre standard error is then derived as SE_total / total_area.

Two principles that keep it unbiased

1. Include every plot, with zeros. Plots without any trees in your domain still contribute — as zeros — to the stratum variance. Dropping them biases the result. pyFIA joins all evaluation plots and fills missing values with zero before computing variance.

2. Single-plot strata contribute no variance. Variance is undefined for $n_h = 1$, so those strata are set to zero variance rather than dropped.

Reading the output

Every estimate includes standard-error columns (suffix _SE) — for example VOLCFNET_TOTAL_SE for volume or AREA_SE for area. In addition, area() and volume() emit variance columns (AREA_VARIANCE; VOLUME_ACRE_VARIANCE and VOLUME_TOTAL_VARIANCE), each equal to the corresponding standard error squared. The other estimators currently return standard errors only:

volume(db, totals=True)
# ...VOLCFNET_TOTAL, VOLCFNET_TOTAL_SE, VOLUME_TOTAL_VARIANCE
biomass(db, totals=True)
# ...BIO_TOTAL, BIO_TOTAL_SE   (standard errors only)

Avoid the variance=True parameter — it does not reliably switch the output from standard errors to variances. For most estimators it has no effect, and for tpa() it currently drops the standard-error columns entirely. Use the _SE columns (and the *_VARIANCE columns where present) directly. Tracked in #109.

There are no separate confidence-interval columns — construct one from the estimate and its standard error (for a 95% interval, ±1.96 × SE).

Validation against EVALIDator

EVALIDator is the official USFS estimation tool and pyFIA’s reference for correctness. Validated against Georgia (EVALID 132301 / 132303):

Estimator	pyFIA SE	EVALIDator SE	Difference
Forest area	138,928	136,048	2.1%
Volume (growing stock)	545,617,192	549,272,904	0.67%
Biomass (aboveground)	14,204,093	14,256,973	0.37%
TPA (live)	204,196,873	199,352,867	2.4%
Growth	35,406,501	35,445,004	0.11%
Mortality	18,549,971	18,517,419	0.18%
Removals	58,202,325	58,906,897	1.2%

Standard errors typically land within 1-3% of EVALIDator. Differences come from rounding, finite-population corrections, and edge cases like single-plot strata.

You can run this comparison yourself — see validate_pyfia_estimate().

A note on temporal methods

FIA defines several temporal estimators. pyFIA implements TI (Temporally Indifferent) — using all available data in an evaluation — which is EVALIDator’s default. Annual and moving-average methods (SMA, LMA, EMA) are not currently implemented.

References

• Bechtold, W.A. & Patterson, P.L. (2005). The Enhanced FIA Program. Gen. Tech. Rep. SRS-80, Chapter 4. DOI: 10.2737/SRS-GTR-80
• Westfall, J.A.; Patterson, P.L.; Coulston, J.W. (2011). Post-stratified estimation: within-strata and total sample size recommendations. Can. J. For. Res. 41(5): 1130-1139.
• USDA Forest Service (2018). Population Estimation User Guide.