Variance estimation for mean growth from successive double sampling for stratification

Double sampling for stratification (2SS) is a sampling design that is widely used for forest inventories. We present the mathematical derivation of two appropriate variance estimators for mean growth from repeated 2SS with updated stratification on each measurement occasion. Both estimators account for substratification based on the transition of sampling units among the strata due to the updated allocation. For the first estimator, sizes of the substrata were estimated from the second-phase sample (sample plots), whereas the respective sizes in the second variance estimator relied on the larger first-phase sample. The estimators were empirically compared with a modified version of Cochran’s well-known 2SS variance estimator that ignores substratification. This was done by performing bootstrap resampling on data from two German forest districts. The major findings were as follows: (i) accounting for substratification, as implemented in both new estimators, has substantial impact in terms of significantly smaller variance estimates and bias compared with the estimator without substratification, and (ii) the second estimator with substrata sizes being estimated from the first-phase sample shows a smaller bias than the first estimator.

Download Full-text

Combining double sampling for stratification and cluster sampling to a three-level sampling design for continuous forest inventories

European Journal of Forest Research ◽

10.1007/s10342-013-0743-9 ◽

2013 ◽

Vol 133 (1) ◽

pp. 89-100 ◽

Cited By ~ 3

Author(s):

Nikolas von Lüpke ◽

Joachim Saborowski

Keyword(s):

Sampling Design ◽

Double Sampling ◽

Cluster Sampling ◽

Forest Inventories ◽

Continuous Forest ◽

Double Sampling For Stratification

Download Full-text

Evaluation of mapped-plot variance estimators across a range of partial nonresponse in a post-stratified national forest inventory

Canadian Journal of Forest Research ◽

10.1139/cjfr-2021-0159 ◽

2021 ◽

Author(s):

James A. Westfall ◽

Andrew J. Lister ◽

Charles T. Scott

Keyword(s):

Forest Inventory ◽

Variance Estimation ◽

Phase 1 ◽

National Forest Inventory ◽

Forest Area ◽

Variance Estimator ◽

Adjustment Factor ◽

Primary Methods ◽

Wide Range ◽

Variance Estimates

When conducting a forest inventory, sometimes portions of plots cannot be measured due to inaccessibility. Two primary methods have been presented to account for partial nonresponse in the estimation phase: 1) use a ratio-to-size estimator, or 2) apply an adjustment factor to all plot observations in proportion to the missing area. Both approaches provide identical estimates of the population mean, but the estimates of variance differ when partial nonresponse is present. Variance estimator performance was examined for a range of population forest area and partial nonresponse proportions in the sample. The ratio-to-size variance estimator performed unbiasedly with respect to simulation results, but the adjustment factor variance estimates were biased with the magnitude and direction dependent upon the forest area proportion and amount of partial nonresponse. The bias is relatively small when the partial nonresponse is small, which is often the case; however, the ratio-to-size method is preferred to ensure accurate variance estimation for a wide range of circumstances.

Download Full-text

An approximate point-based alternative for the estimation of variance under big BAF sampling

Forest Ecosystems ◽

10.1186/s40663-021-00304-0 ◽

2021 ◽

Vol 8 (1) ◽

Author(s):

Thomas B. Lynch ◽

Jeffrey H. Gove ◽

Timothy G. Gregoire ◽

Mark J. Ducey

Keyword(s):

Forest Inventory ◽

Variance Estimation ◽

Basal Area ◽

Volume Estimation ◽

Delta Method ◽

Sampling Point ◽

Variance Estimator ◽

Computer Simulation Studies ◽

New Formula ◽

Estimation Of Variance

Abstract Background A new variance estimator is derived and tested for big BAF (Basal Area Factor) sampling which is a forest inventory system that utilizes Bitterlich sampling (point sampling) with two BAF sizes, a small BAF for tree counts and a larger BAF on which tree measurements are made usually including DBHs and heights needed for volume estimation. Methods The new estimator is derived using the Delta method from an existing formulation of the big BAF estimator as consisting of three sample means. The new formula is compared to existing big BAF estimators including a popular estimator based on Bruce’s formula. Results Several computer simulation studies were conducted comparing the new variance estimator to all known variance estimators for big BAF currently in the forest inventory literature. In simulations the new estimator performed well and comparably to existing variance formulas. Conclusions A possible advantage of the new estimator is that it does not require the assumption of negligible correlation between basal area counts on the small BAF factor and volume-basal area ratios based on the large BAF factor selection trees, an assumption required by all previous big BAF variance estimation formulas. Although this correlation was negligible on the simulation stands used in this study, it is conceivable that the correlation could be significant in some forest types, such as those in which the DBH-height relationship can be affected substantially by density perhaps through competition. We derived a formula that can be used to estimate the covariance between estimates of mean basal area and the ratio of estimates of mean volume and mean basal area. We also mathematically derived expressions for bias in the big BAF estimator that can be used to show the bias approaches zero in large samples on the order of $\frac {1}{n}$ 1 n where n is the number of sample points.

Download Full-text

Parametric bootstrap estimators for hybrid inference in forest inventories

Forestry An International Journal of Forest Research ◽

10.1093/forestry/cpx048 ◽

2017 ◽

Vol 91 (3) ◽

pp. 354-365 ◽

Cited By ~ 7

Author(s):

Mathieu Fortin ◽

Rubén Manso ◽

Robert Schneider

Keyword(s):

Multiple Imputation ◽

Nonlinear Models ◽

Parametric Bootstrap ◽

Simulation Studies ◽

Variance Estimator ◽

Resampling Methods ◽

Forest Inventories ◽

Point Estimator ◽

Complex Models ◽

High Level

Abstract In forestry, the variable of interest is not always directly available from forest inventories. Consequently, practitioners have to rely on models to obtain predictions of this variable of interest. This context leads to hybrid inference, which is based on both the probability design and the model. Unfortunately, the current analytical hybrid estimators for the variance of the point estimator are mainly based on linear or nonlinear models and their use is limited when the model reaches a high level of complexity. An alternative consists of using a variance estimator based on resampling methods (Rubin, D. B. (1987). Multiple imputation for nonresponse surveys. John Wiley & Sons, Hoboken, New Jersey, USA). However, it turns out that a parametric bootstrap (BS) estimator of the variance can be biased in contexts of hybrid inference. In this study, we designed and tested a corrected BS estimator for the variance of the point estimator, which can easily be implemented as long as all of the stochastic components of the model can be properly simulated. Like previous estimators, this corrected variance estimator also makes it possible to distinguish the contribution of the sampling and the model to the variance of the point estimator. The results of three simulation studies of increasing complexity showed no evidence of bias for this corrected variance estimator, which clearly outperformed the BS variance estimator used in previous studies. Since the implementation of this corrected variance estimator is not much more complicated, we recommend its use in contexts of hybrid inference based on complex models.

Download Full-text

A note on design-based versus model-based variance estimation in stereology

Advances in Applied Probability ◽

10.1017/s000186780001171x ◽

2002 ◽

Vol 34 (03) ◽

pp. 484-490 ◽

Cited By ~ 1

Author(s):

Asger Hobolth ◽

Eva B. Vedel Jensen

Keyword(s):

Mathematical Model ◽

Original Problem ◽

Variance Estimation ◽

Point Of View ◽

Systematic Sampling ◽

Variance Estimator ◽

Shape Model ◽

Model Based ◽

Versus Model ◽

Special Case

Recently, systematic sampling on the circle and the sphere has been studied by Gual-Arnau and Cruz-Orive (2000) from a design-based point of view. In this note, it is shown that their mathematical model for the covariogram is, in a model-based statistical setting, a special case of the p-order shape model suggested by Hobolth, Pedersen and Jensen (2000) and Hobolth, Kent and Dryden (2002) for planar objects without landmarks. Benefits of this observation include an alternative variance estimator, applicable in the original problem of systematic sampling. In a wider perspective, the paper contributes to the discussion concerning design-based versus model-based stereology.

Download Full-text