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The purpose of this note is to propose a variance estimator under non-measurable designs that exploits the existence of an auxiliary variable well correlated with the survey variable of interest. Under non-measurable designs, the Sen–Yates–Grundy variance estimator generates a downward bias that can be reduced using a calibration weighting based on the auxiliary variable. Conditions of approximate unbiasedness for the resulting calibration estimator are given. The application to systematic sampling is considered. The proposal proves to be effective for estimating the variance of the forest cover estimator in remote sensing-based surveys, owing to the strong correlation between the reference data, available from a systematic sample, and the satellite map data, available for the whole population and hence exploited as an auxiliary variable. Supplementary materials accompanying this paper appear online.

Large-area forest inventories are often undertaken following a stratified random or systematic design. Yet the strata rarely correspond to the reporting areas of interest (domains) over which the country wants to report specific variables. The process is exemplified by a country aiming to use national forest inventory data to obtain average biomass estimates per forest type for GHGI international reporting, where activity data (areas of land use or land use changes) and emission factors (carbon coefficients) are typically compiled from disparate sources and estimated using different sampling schemes. This study aims to provide a decision tree for the choice of the estimator to be used in forest surveys to draw conclusions about population sub-groups created after (and independently of) the sample selection. This manuscript describes two unbiased estimators that can be used to estimate reporting-strata means, regardless of the sampling design adopted, and extends the result to the common situation in which the reporting-strata are spatially explicit, where a nested group estimator outperforms in terms of both bias and precision other more traditional estimators. From this estimator, an optimal sample allocation scheme is also derived.

Post-stratified model-assisted (MA) and hybrid (HY) estimators are used with repeated airborne laser scanning (ALS) strip sampling and national forest inventory field data for stratum-wise and overall estimation of aboveground biomass (AGB) stock and change. The study area covered the southern portion of the Hedmark County in Norway. Both MA and HY estimation substantially reduced the uncertainty in AGB change when compared with estimation using the field survey only. Relative efficiencies (relative variance) of 4.15 (MA) and 3.36 (HY) for overall estimates were found. The results suggest the MA estimator for single-time estimation and the HY as more appropriate for change estimation by cover class. With the HY estimator, a nested post-stratification scheme is demonstrated, combining cover classes with change classes, which enables detailed reporting for change according to cause within each cover class, and has the potential to improve the estimation precision. Finally, parametric bootstrapping is demonstrated as an empirical alternative to estimate the model-error component in the HY estimator. The model error estimated with parametric bootstrapping converged to the analytically determined value of the HY estimator within 1000 bootstrap samples.

Forest resource assessments utilizing remotely sensed auxiliary data are becoming increasingly important due to their ability to provide precise estimates of forest parameters at low cost. In presenting results from such surveys, it is important to provide not only estimates of the target parameters, but also their confidence intervals, which provide the range of values wherein the true value is located with a certain level of confidence. If such an interval is narrow the point estimates from the survey can be considered very reliable. In estimating the confidence interval the variance of an estimator must first be estimated. Unbiasedness, i.e. that an estimator on average coincides with the true value, is an important property also for variance estimators. Another important property is that the variance estimator itself has low variance, not least in cases when the variance estimates obtained with the estimator may not be strictly positive. One such important case is when two-stage designs are used to first allocate sample clusters in the form of strips from which auxiliary data, such as metrics derived from airborne laser scanning, are obtained; field data are then derived from sample plots beneath each sample strip in a second stage. In this article we compare two variance estimators for such surveys. The first estimator is a standard estimator suggested in reference textbooks on model-assisted sampling. The second estimator is proposed by the authors, and utilizes the auxiliary data to a larger extent. Through Monte Carlo simulation we show that both variance estimators are approximately unbiased, but that the new estimator is more stable (i.e., has lower empirical variance) and provides empirical confidence interval coverage rates that coincide more closely with the nominal coverage rates.

The need to provide transparent and reliable Greenhouse Gas (GHG) emission estimates is strongly emphasized in the context of international reporting under the United Nations Framework Convention on Climate Change (UN-FCCC) and the Paris Agreement. Yet it is difficult to find specific guidance about what information is really needed to evaluate the quality of the emission factors or activity data used for GHG emission estimates. The most commonly used indicator of the reliability of an estimation procedure (and one of the few indicators explicitly mentioned in the 2006 IPCC guidelines) is the so-called confidence interval, usually at a confidence level of 90% or 95%. This interval, however, is unlikely to be a meaningful indicator of the quality of the estimate, if not associated with additional information about the estimation and survey procedures (such as on the sampling design, measurement protocols or quality control routines, among others). We provide a review of the main sources of error that can have an impact on the precision and accuracy of the estimation of both emission factors and activity data and a list of the essential survey features that should be reported to properly evaluate the quality of a GHG emission estimate. Such list is also applicable to the reporting of national forest inventories and of area estimation of activity data, and includes the case in which confidence intervals are obtained using error propagation techniques.

In felled-tree studies, ratio and regression estimators are commonly used to convert more readily measured branch characteristics to dry crown mass estimates. In some cases, data from multiple trees are pooled to form these estimates. This research evaluates the utility of both tactics in the estimation of crown biomass following randomized branch sampling (RBS). Synthetic generalized regression (GREG) estimators are developed, and their properties examined against standard estimators. It is shown that synthetic GREG estimators with zero or low design-bias can be obtained, and that the variance of a design-unbiased class of GREG estimators can be unbiasedly estimated. Simulated sampling from 20 censused crowns of two Rocky Mountain species indicated that improvements in accuracy can be obtained through GREG estimation following RBS. Simulations also showed that synthetic GREG estimators that pool data from multiple trees can stabilize coefficients of multivariate regression models to provide improved accuracy over direct GREG estimators. However, for the univariate regression models that proved most adept for the censused crowns, direct GREG estimation provided the lowest average root mean squared error (RMSE) for RBS. Simulations also showed that model-based branch aggregation estimators have generally low RMSE but can be heavily design-biased. For the crown forms studied, use of branch auxiliary information at both the design and estimation stages through RBS and GREG estimation appears to be more efficient than using the information only at the estimation stage following simple or stratified random sampling.

Individual tree heights are needed in many situations, including estimation of tree volume, dominant height, and simulation of tree growth. However, height measurements are tedious compared to tree diameter measurements, and therefore height–diameter (H–D) models are commonly used for prediction of tree height. Previous studies have fitted H–D models using approaches that include plot-specific predictors in the models and those that do not include them. In both these approaches, aggregation of the observations to sample plots has usually been taken into account through random effects, but this has not always been done. In this paper, we discuss four alternative model formulations and report an extensive comparison of 16 nonlinear functions in this context using a total of 28 datasets. The datasets represent a wide range of tree species, regions, and ecological zones, consisting of about 126 000 measured trees from 3717 sample plots. Specific R-functions for model fitting and prediction were developed to enable such an extensive model fitting and comparison. Suggestions on model selection, model fitting procedures, and prediction are given and interpretation of the predictions from different models are discussed. No uniformly best function, model formulation, or model fitting procedure was found. However, a 2-parameter Näslund and Curtis function provided satisfactory fit in most datasets for the plot-specific H–D relationship. Model fitting and height imputation procedures developed for this study are provided in an R-package for later use.

Large scale forest inventories are often undertaken following a stratified random or systematic design. Yet the strata rarely correspond to the reporting areas of interest (domains) over which the country wants to report specific variables. The process is exemplified by a country aiming to use national forest inventory data to obtain average biomass estimates per forest type for GHGI international reporting, where activity data (areas of land use or land use changes) and emission factors (carbon coefficients) are typically compiled from disparate sources and estimated using different sampling schemes. This study aims to provide a decision tree for the use of data obtained from forest surveys to draw conclusions about population sub-groups created after (and independently of) the sample selection. While bias can arise whenever activity data and emission factors are calculated independently, it can be eliminated in case of a simple random or simple systematic design if properly weighted estimators are provided. This manuscript describes two unbiased estimators that can be used to estimate reporting-strata means, regardless of the sampling design adopted, and extends the result to the common situation in which the reporting-strata are spatially explicit, where a nested group estimator outperforms in terms of both bias and precision other more traditional estimators. From this estimator, an optimal sample allocation scheme is also derived.

Auxiliary information provided by airborne laser scanners (ALS) is expected to increase the accuracy of biomass estimation in large-scale forest surveys. Because acquisition of “wall-to-wall” ALS data over large areas is not economically feasible, a systematic sampling approach using ALS as a strip sampling tool was used to supplement a conventional field-based inventory in a large-scale biomass survey in Hedmark County (HC), Norway. However, the complexities of these surveys render prohibitive the analytical determination of the properties of the resulting estimators and of the estimators of their sampling variances. To overcome the problem, the statistical properties of the estimators were empirically investigated in this paper using simulated sampling from an artificial population. Through this approach, estimators with desirable properties can be identified and used for inference in real applications. By combining biomass estimates from Norwegian National Forest Inventory plots in HC, ALS measurements and Landsat 5 TM imagery, an artificial population at the scale of HC was created. Using this artificial population as “ground-truth”, we demonstrate how simulated sampling can be used for assessing the statistical properties of regression estimators and of their variance estimators under two-phase post-stratified systematic sampling (SYS) and simple random sampling without replacement (SRSwoR) designs, considering design- and model-dependent inferential frameworks. The results were assessed using a purely ground-based systematic design with a Horvitz–Thompson (HT) estimator as benchmark. The real overall precision of the ALS-aided systematic survey was nearly five times overestimated when using the design-based variance estimators developed for SRSwoR, while under model-dependent inference the overestimation of the real standard errors was around 40%. Compared to ground-based inventory, the estimated standard errors of the systematic ALS survey doubled while in reality the standard errors were 55% lower. Using successive differences variance estimators greatly improved the precision of the systematic ALS-aided survey and produced valid 95% confidence intervals under the design-based inference. The most satisfactory results for the ALS-aided survey in terms of analytical variances occurred under design-based inference with successive difference variance estimator, closely followed by the model-dependent estimators. Using simulations, the cost efficiency of the ground based and ALS-aided surveys was assessed by evaluating accuracy against inventory cost for various sampling intensities. The results indicated that the ALS-aided surveys can be a cost-efficient alternative to traditional field inventories.

∙ Key message The study presents novel model-based estimators for growing stock volume and its uncertainty estimation, combining a sparse sample of field plots, a sample of laser data, and wall-to-wall Landsat data. On the basis of our detailed simulation, we show that when the uncertainty of estimating mean growing stock volume on the basis of an intermediate ALS model is not accounted for, the estimated variance of the estimator can be biased by as much as a factor of three or more, depending on the sample size at the various stages of the design. ∙ Context This study concerns model-based inference for estimating growing stock volume in large-area forest inventories, combining wall-to-wall Landsat data, a sample of laser data, and a sparse subsample of field data. ∙ Aims We develop and evaluate novel estimators and variance estimators for the population mean volume, taking into account the uncertainty in two model steps. ∙ Methods Estimators and variance estimators were derived for two main methodological approaches and evaluated through Monte Carlo simulation. The first approach is known as two-stage least squares regression, where Landsat data were used to predict laser predictor variables, thus emulating the use of wall-to-wall laser data. In the second approach laser data were used to predict field-recorded volumes, which were subsequently used as response variables in modeling the relationship between Landsat and field data. Results ∙ The estimators and variance estimators are shown to be at least approximately unbiased. Under certain assumptions the two methods provide identical results with regard to estimators and similar results with regard to estimated variances. ∙ Conclusion We show that ignoring the uncertainty due to one of the models leads to substantial underestimation of the variance, when two models are involved in the estimation procedure.

Different approaches for estimation of change in biomass between two points in time by means of airborne laser scanner data were tested. Both field and laser data were collected at two occasions on 52 sample plots in a mountain forest in southeastern Norway. In the first approach, biomass change was estimated as the difference between predicted biomass for the two measurement occasions. Joint models for the biomass at both occasions were fitted using different height and density variables from laser data as explanatory variables. The second approach modelled the observed change directly using the change in different variables extracted from the laser data as explanatory variables. In the third approach we modelled the relative change in biomass. The explanatory variables were also expressed as relative change between measurement occasions. In all approaches we allowed spline terms to be entered. We also investigated the aptness of models for which the residual variance was modeled by allowing it to be proportional to the area of the plot on which biomass was assessed. All alternative models were initially assessed by AIC. All models were also evaluated by estimating biomass change on the model development data. This evaluation indicated that the two direct approaches (approach 2 and 3) were better than relying on modeling biomass at both occasions and taking change as the difference between biomass estimates. Approach 2 seemed to be slightly better than approach 3 based on assessments of bias in the evaluation.

To meet the increasing need for reliable and timely timber resources and carbon stock estimates at intermediate and local decision levels, a sampling approach using airborne laser scanning (ALS) as a strip sampling tool has been proposed as a supplement to the conventional field-based National Forest Inventory system. This idea led to a large-scale biomass survey project undertaken in Hedmark County, Norway, an area encompassing 27390 km2. The field biomass estimates were provided by the Norwegian NFI, and the ALS measurements were acquired in parallel strips using a systematic (SYS) design. The ALS-based biomass estimation was performed using regression estimators under design-based and the model-based inferential frameworks. A possible approach to assess the validity of inference when complex designs are involved is to use a sampling simulator where an artificial population represents the ‘ground truth’ and the properties of the estimators are investigated via simulated sampling. To create the artificial population, a large multivariate dataset containing NFI field observations and ALS metrics was generated using a copula function fitted to the empirical observations, and then it was generalized over the study area using satellite imagery and nearest neighbor imputations. The properties of several design-based model-assisted and model-based variance estimators were investigated using simulated sampling and the accuracy of ALS-based and ground-based estimates under simple random sampling without replacement (SRSwoR) and SYS designs were compared. The simulation results indicated that the ALS-based survey produced valid inference under design-based and model-based frameworks. The variance estimators performed well under two-phase SRSwoR, but the real standard errors were overestimated approximately 4.7 times under two-phase SYS. Compared to the pure ground-based inventories, the estimated standard errors of the ALS-based estimates were approximately 1.8 times larger, while the real accuracy (in terms of root mean squared error) improved with 59%.

In this paper two sampling and estimation strategies for regional forest inventory were investigated in detail and results were presented for various geographical scales. Airborne laser scanner (ALS) data were acquired to augment data from a systematic sample of National Forest Inventory (NFI) ground plots in Hedmark County, Norway (27,390 km 2 ). Approximately 50% of the NFI field plots were covered by the systematic ALS sample of 53 parallel flight lines spaced 6 km apart. The area was stratified into eight cover classes and independent log-transformed regression models were developed for each class to predict total above-ground dry biomass (AGB). The two laser-ground estimation strategies tested were a model-dependent (MD), two-phase approach that rests on the assumption that the predictive models are correctly specified, and a model-assisted (MA) approach with a two-stage probability sampling design which utilizes design-unbiased estimators. ALS AGB estimates were reported by land cover class and compared to the NFI ground estimates. The ALS-based MA and MD mean estimates differed from the NFI AGB estimates by about 2% and 8%, respectively, for the entire County. At the county level the smallest estimated standard error (SE) for the estimates was obtained using the field data alone. However, the SEs calculated from field and ALS data were based on unequal numbers of ground plots. When considering only the NFI plots in the ALS strips, the smallest SEs were obtained using the MD framework. However, we also illustrated the sensitivity of the estimates of applying different plausible models. All the applied estimators assumed simple random sampling while the selection of flight lines as well as ground plots followed a systematic design. Thus, the estimates of SE were most likely conservative. Simulated sampling undertaken in a parallel research effort suggests that the overestimation of the SEs was probably much larger for the ALS-based estimates compared to the NFI estimates. ALS-based estimates were also derived for sub-county political units and thereby demonstrated how limited sample sizes affect the standard error of the biomass estimates.

In forest inventories, regression models are often applied to predict quantities such as biomass at the level of sampling units. In this paper, we propose a model-based inference framework for combining sampling and model errors in the variance estimation. It was applied to airborne laser (LiDAR) data sets from Hedmark County, Norway, where the model error proportion of the total variance was found to be large for both scanning (airborne laser scanning) and profiling LiDAR when biomass was estimated. With profiling LiDAR, the model error variance component for the entire county was as large as 71% whereas for airborne laser scanning, it was 43% of the total variance. Partly, this reflects the better accuracy of the pixel-based regression models estimated from scanner data as compared with the models estimated from profiler data. The framework proposed in our study can be applied in all types of sample surveys where model-based predictions are made at the level of individual sampling units. Especially, it should be useful in cases where model-assisted inference cannot be applied due to the lack of a probability sample from the target population or due to problems of correctly matching observations of auxiliary and target variables.

Recent developments in remote sensing (RS) technology have made several sources of auxiliary data available to support forest inventories. Thus, a pertinent question is how different sources of RS data should be combined with field data to make inventories cost-efficient. Hierarchical model-based estimation has been proposed as a promising way of combining: (i) wall-to-wall optical data that are only weakly correlated with forest structure; (ii) a discontinuous sample of active RS data that are more strongly correlated with structure; and (iii) a sparse sample of field data. Model predictions based on the strongly correlated RS data source are used for estimating a model linking the target quantity with weakly correlated wall-to-wall RS data. Basing the inference on the latter model, uncertainties due to both modeling steps must be accounted for to obtain reliable variance estimates of estimated population parameters, such as totals or means. Here, we generalize previously existing estimators for hierarchical model-based estimation to cases with non-homogeneous error variance and cases with correlated errors, for example due to clustered sample data. This is an important generalization to take into account data from practical surveys. We apply the new estimation framework to case studies that mimic the data that will be available from the Global Ecosystem Dynamics Investigation (GEDI) mission and compare the proposed estimation framework with alternative methods. Aboveground biomass was the variable of interest, Landsat data were available wall-to-wall, and sample RS data were obtained from an airborne LiDAR campaign that produced simulated GEDI waveforms. The results show that generalized hierarchical model-based estimation has potential to yield more precise estimates than approaches utilizing only one source of RS data, such as conventional model-based and hybrid inferential approaches.

Formerly, tree height has been more difficult to measure accurately in the field than tree diameter at breast height. As a consequence, models to predict height from diameter measurements have been widely developed in the forestry literature. Through the use of airborne laser scanning technology (e.g., LiDAR), tree variables such as height and crown diameter can be measured accurately, a development which has spawned the need for models to predict diameter from airborne laser-derived measurements. Although some work has been done for fitting such models, none have incorporated spatial information to improve the accuracy of the predicted diameters. Using a simple linear model for predicting tree diameter from laser-derived tree height and crown diameter measurements, we compared the performance of ordinary least squares (OLS), generalized least squares with a non-null correlation structure (GLS), linear mixed-effects model (LME), and geographically weighted regression (GWR). Our data were obtained from 36 sample plots established in Norway. This is the first study to examine the use of spatial statistical models for tree-level LiDAR data. Root mean square prediction errors in tree diameter with LME are 3.5%, with GWR are 10%, and with OLS and GLS are 17%. LME also exhibited low variability in predicting performance across all the validation classes (based on laser-derived height). Giving the difficulties of using parametric statistical inference (such as maximum likelihood-based indices) for GWR, we used permutation tests as a way for detecting statistical differences. LME was significantly better than the other models, as well as GWR was to OLS and GLS. Our results indicate that the LME model produced the best predictions of tree diameter from LiDAR-based variables to a degree that has previously not been possible.

Forest inventory relies heavily on sampling strategies. Ratio estimators use information of an auxiliary variable (x) to improve the estimation of a parameter of a target variable (y). We evaluated the effect of measurement error (ME) in the auxiliary variate on the statistical performance of three ratio estimators of the target parameter total τ y . The analyzed estimators are: the ratio-of-means, mean-of-ratios, and an unbiased ratio estimator. Monte Carlo simulations were conducted over a population of more than 14,000 loblolly pine (Pinus taeda L.) trees, using tree volume (v) and diameter at breast height (d) as the target and auxiliary variables, respectively. In each simulation three different sample sizes were randomly selected. Based on the simulations, the effect of different types (systematic and random) and levels (low to high) of MEs in x on the bias, variance, and mean square error of three ratio estimators was assessed. We also assessed the estimators of the variance of the ratio estimators. The ratio-of-means estimator had the smallest root mean square error. The mean-of-ratios estimator was found quite biased (20%). When the MEs are random, neither the accuracy (i.e. bias) of any of the ratio estimators is greatly affected by type and level of ME nor its precision (i.e. variance). Positive systematic MEs decrease the bias but increase the variance of all the ratio estimators. Only the variance estimator of the ratio-of-means estimator is biased, being especially large for the smallest sample size, and larger for negative MEs, mainly if they are systematic.

Summary With auxiliary information that is well correlated with the primary variable of interest, ratio estimation of the finite population total may be much more efficient than alternative estimators that do not make use of the auxiliary variate. The well-known properties of ratio estimators are perturbed when the auxiliary variate is measured with error. In this contribution we examine the effect of measurement error in the auxiliary variate on the design-based statistical properties of three common ratio estimators. We examine the case of systematic measurement error as well as measurement error that varies according to a fixed distribution. Aside from presenting expressions for the bias and variance of these estimators when they are contaminated with measurement error we provide numerical results based on a specific population. Under systematic measurement error, the biasing effect is asymmetric around zero, and precision may be improved or degraded depending on the magnitude of the error. Under variable measurement error, bias of the conventional ratio-of-means estimator increased slightly with increasing error dispersion, but far less than the increased bias of the conventional mean-of-ratios estimator. In similar fashion, the variance of the mean-of-ratios estimator incurs a greater loss of precision with increasing error dispersion compared with the other estimators we examine. Overall, the ratio-of-means estimator appears to be remarkably resistant to the effects of measurement error in the auxiliary variate.

In many situations, including forest management planning, the underlying diameter distribution of a forest stand needs to be predicted. One alternative for predicting diameter distribution involves modeling diameter percentiles. We introduce a quantile regression (QR) approach for predicting diameter percentiles, and compare it with customary linear fixed-effect and linear mixed-effects models. The customary methods involve first estimating plot-specific sample percentiles and then regressing them on stand characteristics, whereas QR directly models percentiles. We compared two prediction situations: a conditional one where a previously measured diameter sample is available from the stand of interest, and a marginal one where only some stand-specific characteristics are known. To compare the predictions to the true underlying percentiles, we conducted a simulation study. The QR approach led to slight improvements in the marginal prediction situation. In the conditional situation, the mixed-effect model led to clearly better predictions and should be preferred until QR methods have been developed for hierarchical data. It is extremely important to filter out the highly correlated sampling error from conditional predictions in order that the models predict underlying population percentiles rather than the realized sample percentiles. Copyright © 2007 John Wiley & Sons, Ltd.

A multiple linear regression model which includes 0/1 variables to indicatemembership in a group is a convenient way to model parallel regression surfaces.Building upon this, an extended model which includes predictor variables thatare the product of the indicator and other variables willcoincide with separate regressions for each group using only the other predictors.The algebraic basis for this concurrence is demonstrated.Least squares estimation is presumed.Examples with two groups and with four groups are presented.

When field plots are prevented from overlapping the boundary of the tract being sampled, the selection probabilities of population elements near the edge differ from what they are when plots are permitted to overlap the boundary. In both situations, selection probabilities of edge elements differ from those of elements located far to the interior of the tract. An analytical expression and graphical depiction of the altered selection probabilities are presented, accompanied by an empirical study of the consequences on estimation for the situation where plots are pulled back from the edge until just tangent with the boundary.

The probability of selecting a population element under line intersect sampling depends on the width of the particle in the direction perpendicular to the transect, as is well known. The consequence of this when using ell-shaped transects rather than straight-line transects are explicated, and modifications that preserve design-unbiasedness of Kaiser's (1983) conditional and unconditional estimators are presented. A case against treating multiple intersections as multiple probabilistic events is argued on the basis, also, of preserving design-unbiased estimation.

Almost any type of sample has some utility when estimating population quantities. The focus in this paper is to indicate what type or combination of types of sampling can be used in various situations ranging from a sample designed to establish cause-effect or legal challenge to one involving a simple subjective judgment. Several of these methods have little or no utility in the scientific area but even in the best of circumstances, particularly complex ones, both probabilistic and non-probabilistic procedures have to be used because of lack of knowledge and cost. We illustrate this with a marbled murrelet example.

Summary. Models to depict the tapering of a tree bole abound in the literature of forest science, and such models are widely used in forestry practice. One important use is the integration of a taper equation to predict the volume of the tree bole. The statistical properties of volume prediction from an integrated taper equation have been obscure. Based on the statistical characteristics of a taper model for the bole's cross-sectional area, we derive the first two moments of the volume predictor and the prediction error. Bias from the integration is nil. The importance of a reasonable model of the error structure is demonstrated.

Four populations were repeatedly sampled with the aim of examining interval estimation of the cumulative amount, T, of some characteristic that has a positively skewed distribution, a feature of many biological populations. With samples of various sizes, the empirical sampling distribution of the simple expansion estimator was remarkably symmetric and approximately normal, as was that of the customary ratio-of-means estimator. While the nominal coverage rate of (1 − α)100% intervals based on the Student’s t distribution was nearly achieved in all cases, there was a substantially greater tendency for such intervals to fail from below rather than above. This behavior is attributed to the strong positive correlation between the estimator of the population total and the corresponding estimator of its variance when sampling from a finite, and perforce nonnormal, population.

Model-based ideas in finite-population sampling have received renewed discussion in recent years.Their relationship to the classical ideas in sampling theorydo not appear to be universally well understood by samplers in applied disciplines such as forestry, and ecology more broadly.The two inferential paradigms are constrasted, andexplanations are supplemented with examples of discrete aswell as continuously distributed populations. The treatment of spatial structureis examined, also.

Sampling strategies to provide unbiased estimates of individual tree characteristics have received increased attention in the past decade. Sampling-based estimation is appealing for many tropical species owing to the dearth of appropriately fitted models from which one can obtain an accurate prediction of these characteristics. Two methods, namely randomized branch sampling and importance sampling, are described in this paper. Their application to estimate surface (bark) area and average stem length is elucidated. Extensions to a stratified design are briefly discussed.

For purposes of forest inventory and eventual management of the forest resource, it is essential to be able to predict the cumulative bole volume to any stipulated point on the standing tree bole, while requiring measurements of tree size that can be made easily, quickly and accurately. Equations for this purpose are typically non-linear and are fitted to data garnered from a sample of felled trees. Because the cumulative bole volume of each tree is measured to numerous upper-bole locations, correlations between measurements within a tree are likely. A mixed-effects model is fitted to account for this within-subject (tree) correlation structure, while also portraying the sigmoidal shape of the cumulative bole volume profile.

The total foliar area or mass of a tree is difficult to measure, as is its bark or cambial area, and various other components of aboveground biomass. A variety of sampling methods is proposed and estimators of these characteristics are presented. Based on probability precepts, all estimators are unbiased. An unbiased estimator of variance for each estimator also is presented. The basis in probability rather than a fitted regression equation provides some important safeguards, and is a useful alternative when fitted re- gression functions are unavailable for a particular species and physiographic condition.

By recognizing the mutually exclusive subsets of trees constituting a forest population over some interval of time, the relationship of an estimator of growth to its targeted value is made clear. Estimators are investigated in this light, and expressions of their variances are derived. The connection of traditional estimators to Horvitz-Thompson estimators is shown, and a newly proposed ratio estimator to ensure compatible growth estimators is discussed. Recent developments of potential interest include the use of control variates to achieve more precise estimation of survivor growth. The utility of generalized regression estimators for growth component estimation is suggested.

When a line transect overlaps the boundary of the sampled region, it can be reflected back on top of itself into the region, thereby making it possible to include elements near the edge twice from the same transect. A practical advantage of doing so is the reduction of field time and effort compared to the customary procedure of folding the transect back into another part of the region. An estimator is presented which accounts for this procedure in a way that preserves design-unbiased estimation.

A probabilistic sampling approach for design-unbiased estimation of area-related quantitative characteristics of spatially dispersed population units is proposed. The developed field protocol includes a fixed number of 3 units per sampling location and is based on partial triangulations over their natural neighbors to derive the individual inclusion probabilities. The performance of the proposed design is tested in comparison to fixed area sample plots in a simulation with two forest stands. Evaluation is based on a general approach for areal sampling in which all characteristics of the resulting population of possible samples is derived analytically by means of a complete tessellation of the areal sampling frame. The example simulation shows promising results. Expected errors under this design are comparable to sample plots including a much greater number of trees per plot. peerReviewed

The method of weighted least squares can be used to achieve homogeneity of variance with linear regression that has a heterogeneous error structure. A weight function commonly used when constructing regression equations to predict tree volume is [Formula: see text], where k1 ≈ 1.0–2.1. This paper examines the weight function [Formula: see text] for modelling the error structure in two loblolly pine (Pinustaeda L.) data sets and one white oak (Quercusalba L.) data set. The weight function [Formula: see text] is recommended for all three data sets, for which the k1 values ranged from 1.80 to 2.07.

Estimators of forest growth and meaningful components of growth are presented in a unified framework. The approach is based on the identification of mutually exclusive domains of the target population of trees, each of which contributes to the overall net change in the value of the resource to be estimated. Various estimators that have been proposed in the literature are reviewed and examined within this framework. An estimator of the component of growth corresponding to each domain of study is shown to relate back to the trees sampled from that domain, a portion of it, or in some cases, a union of two domains. Special attention is given to the treatment and examination of the issues of additivity of growth components and the recalibration effect which hampers additivity, both of which arise in the context of horizontal point sampling designs.

The importance and centroid sampling methods enable the volume of the bole of a standing tree or any portion of it (log) to be estimated based on a single diameter measurement. The methods eliminate the need for volume tables or equations and the bias which these might incur. They can be used anywhere for any species of tree of excurrent or deliquescent habit which has a well-defined main bole and would seem to be particularly appropriate for inventory of tropical mixed-moist forests. Programs written in BASIC and examples are provided to facilitate application of the methods in the field.

Following a test of the hypothesis that the intercept of a simple linear regression model is zero, the conditional properties of the fitted model differ from those that obtain in the absence of pretesting and from the unconditional properties in the presence of pretesting. For the estimator of the mean response at a point X 0, the conditional mean squared error may be better or worse, depending on the magnitude of the pretest hypothesis error and on the outcome of the pretest. The degree to which the conditional mean squared error is better or worse depends on the location of X 0 relative to , the level of the pretest, and the correlation between the initial least squares estimators of the intercept and slope coefficients.

The usual F test of regression coincidence, which is appropriate under a homoscedastic model, is examined under a multiplicatively heteroscedastic model. The departure of the test from its nominal level is slight when the sample of explanatory variables is symmetric, but may be substantially inflated when the sample has positive skew. Conversely, the nominal level may be slightly depressed when the sample has negative skew. The size of the perturbation from the nominal level depends on the degree of heteroscedasticity, however its effect is more pronounced with positively skewed samples. Similar trends are evident for the usual F test of regression parallelism. There is no apparent pattern to the discrepancy of the level of the test with regard to the data which would permit empirical researchers to adjust their results.

Publisher Summary At some time in the mid-1780s, Pierre Laplace started to press the French government to conduct an enumeration of the population in about 700 communes scattered over the kingdom, with a view to estimate the total population of France. He intended to use, for this purpose, the fact that there was already a substantially complete registration of births in all communes, of which there would then have been the order of 10,000. He reasoned that if he also knew the populations of those sample communes, he could estimate the ratio of population to annual births and apply that ratio to the known number of births in a given year to arrive at a ratio estimate of the total French population. For various reasons, however, notably the ever-expanding borders of the French empire during Napoleon's early years, events militated against his obtaining a suitable total of births for the entire French population; therefore, his estimated ratio was never used for its original purpose. He did, however, devise an ingenious way for estimating the precision with which that ratio was measured. This was less straightforward than the manner in which it would be estimated today, but, at the time, it was a very considerable contribution to the theory of survey sampling.

Qualitative response models constitute a class of regression models used to predict one of a discrete number of mutually exclusive outcomes. These models differ from continuous regression models in that the response variable takes only discrete values. In forestry applications, the use of such models has been largely confined to mortality studies where the dependent variable is dichotomous. However, it is common in forestry to deal with variables that are either naturally discrete or continuous but recorded discretely. Consequently, there is a need for models that are appropriate for polychotomous dependent variables. Two models that appear to be suitable for forestry applications are presented, namely the ordered and unordered multinomial models, with emphasis on their theoretical justification, statistical inference, and model selection criteria. Using permanent plot data from loblolly pine (Pinustaeda L.) plantations on cutover, site-prepared areas throughout the southern United States, these models were fitted to assess the merchantability of loblolly pine trees. The results demonstrate the potential of qualitative response models for meaningful implementation in a variety of forestry applications and, also, for suggested topics for future research work.

The computed estimate of basal area was the focus of this investigation into the accuracy of tree diameter measurements. Diameter at breast height was measured on 101 trees, the breast height disc was removed from the stem, and each disc outline was then digitized to determine its actual area. Actual areas were compared with the basal areas computed with the field measurements of diameter by expressing the deviations as a proportion of the actual areas. These relative errors averaged 3.1% when basal area was computed with a taped diameter at breast height, and they averaged −2.5% when the mean of two calipered, perpendicular diameter at breast height measurements were used. Noncircularity accounted for about 2.6% of the bias, on average, in basal areas from taped diameters. Root mean square error of relative basal area errors ranged from 7.3 to 7.9% among these methods. The difference between basal areas computed with a taped diameter at breast height and with calipered diameter at breast height was computed and expressed as a proportion of the actual basal area. On average, a difference of 5% was observed. Interval estimates for these differences are presented.

Cut-off importance sampling (CIS) is introduced as a means of sampling individual trees for the purpose of estimating bole volume. The novel feature of this variant of importance sampling is the establishment on the bole of a cut-off height, He, above which sampling is precluded. An estimator of bole volume between predetermined heights Hi and Hu > Hc is proposed, and its design-based bias and mean square error are derived. In an application of CIS as the second stage of a two-stage sample to estimate aggregate bole volume, the gain in precision realized from CIS more than offset its bias when compared to the precision of importance sampling when He = HJJ.

Reliable quantitative information is required to manage forest lands. This chapter gives an overview of statistical procedures used to quantify tree volume and form, site quality (height-age curves), site occupancy (stand density), and growth-yield relationships. The quantification of tree volume and form is basic to assessing the volume and value of forest stands and to evaluating response to silvicultural treatments. Regression has been used to relate tree volume to easily measured variables, such as diameter at breast height and total tree height. The usual assumptions of normality and homogeneity of error variance frequently are not satisfied. As an alternative to predicting volume directly, taper functions, which predict diameter at specified heights, can be fitted and integrated for volume estimates. While estimates of volume from integrated taper functions have proven to be quite accurate in many instances, they are likely to be biased. Because foresters employ regression for a variety of prediction purposes (volume, diameter at given heights, etc.), simultaneous estimation of tree volume and taper functions under a joint mean square error criterion has been proposed. The site index concept (average height at a specified reference age) is used to quantify site quality. Basic assumptions of the site index method often are not met and numerous statistical problems arise when estimating the parameters in height-age curves. A natural selection bias oftentimes occurs which invalidates the usual assumption of age-site quality independence when applying the guide curve method; lack of independence between observations and the absence of an appropriate sampling protocol cause difficulty when fitting height-age curves to stem analysis data (multiple observations on each individual in a sample of trees). Stand density has been quantified by a variety of measures — many of which are mathematically equivalent. Stand density is dynamic through time and must be considered as such when projecting stand dynamics, growth and yield. A wide variety of growth and yield models — ranging from models which predict whole-stand volumes only to simulators that project the growth of each individual tree — have been developed. The most appropriate modeling approach depends on the detail needed when the models are applied for specified objective(s). Regardless of the modeling approach taken, a system of interrelated equations arise. Estimation of the parameters in these systems of equations presents a variety of challenging statistical problems.

Generalized Linear Models.- Linear Models, Vector Spaces, and Residual Likelihood.- An Assessment of Approximate Maximum Likelihood Estimators in Generalized Linear Models.- Scaled Link Functions for Heterogeneous Ordinal Response Data.- Longitudinal Data Analysis.- Software Design for Longitudinal Data Analysis.- Asymptotic Properties of Nonlinear Mized-Effects Models.- Structured Antedependence Models for Longitudinal Data.- Effect of Confounding and Other Misspecification in Models for Longitudinal Data.- The Linear Mixed Model. A Critical Investigation in the Context of Longitudinal Data.- Modelling the Order of Disability Events in Activities of Daily Living Using Discrete Longitudinal Data.- Estimation of Subject Means in Fixed and Mixed Models with Application to Longitudinal Data.- Modeling Toxicological Multivariate Mortality Data: a Bayesian Perspective.- Comparison of Methods for General Nonlinear Mixed-Effects Models.- Repeated Measures Analysis Using Mixed Models: Some Simulation Results.- Spatial Data Analysis.- Object Identification Using Markov Random Field Segmentation Models at Multiple Resolutions of a Rectangular Lattice.- Comparison of Some Sampling Designs for Spatially Clustered Populations.- Using Geostatistical Techniques to Map the Distribution of Tree Species from Ground Inventory Data.- Global Analysis of Ozone Data Based on Spherical Splines.- Bounded Influence Estimation in a Spatial Linear Mixed Model.- Spatial Correlation Models as Applied to Evolutionary Biology.- Rainfall Modelling Using a Latent Guassian Variable.- Estimation of Individual Exposure Following a Chemical Spill in Superior, Wisconsin.- Flexible Response Surface Methods via Spatial Regression and EBLUPS.- Robust Semivariogram Estimation in the Presence of Influential Spatial Data Values.- Modelling Spatio-Temporal Processes.- Elephant Seal Movements: Dive Types and Their Sequences.- Models for Continuous Stationary Space-time Processes.- A Comparison of Two Spatio-temporal Semivariograms with Use in Agriculture.- Structuring Correlation Within Hierarchical Spatio-temporal Models for Disease Rates.- Modelling Messy Data.- Generalized Linear Mixed Measurement Error Models:.- Calculating the Appropriate Information Matrix for Log-linear Models When Data Are Missing at Random.- Nonparametric Regression in the Presence of Correlated Errors.- Exploratory Modelling of Multiple Non-Stationary Time Series: Latent Process Structure and Decompositions.- Modelling Correlations Between Diagnostic Tests in Efficacy Studies With an Imperfect Reference Test.- Special Topics and Future Directions.- Combining Standard Block Analyses with Spatial Analyses Under a Random Effects Model.- Spatial and Longitudinal Data Analysis: Two Histories with a Common Future?.

Equations to predict the volume of an individual tree bole between stump height and the height at which its diameter has tapered to a specified minimum are common in forestry. When fitting such a regression equation, a sample of trees which span the range of sizes needed for eventual application of the equation is selected. Bole diameter is measured at ascending heights on the bole. Each tree, therefore, contributes multiple measurements to the data fitted to the equation. In contrast to past practice, we model these data in a manner which accounts for the likely spatial correlation among measurements within a tree. The resulting mixed-effects nonlinear model is fitted by restricted maximum likelihood (REML) and also by generalized estimating equations (GEE). Results from the two approaches are nearly identical, which suggests that the computationally less demanding GEE may be acceptable as a routine alternative to a fully parameterized approach.

We consider the prediction of stem volume of a tree by means of simple linear regression wherein the explanatory variable is D2H; D is the bole diameter measured outside bark at breast height and H is the total height of the tree. Given a fitted equation, a tree whose volume is to be predicted must be measured for D and H. Two sources of error account for the deviation of predicted tree volume from V' the actual tree volume, i.e. (1) error in the estimated coefficients of the volume function and (2) variation of V around its expected value conditional upon D2H. Owing to the greater difficulty and expense of measuring H in forest conditions, a standard practice is to predict its value from a regression on diameter. Consequently the prediction of V, V say, is subject to additional sources of error due to (1) the error in the estimated coefficients of the height-diameter function and (2) the variation of H around its expected value conditional on D. An empirical study of a population of more than 14 000 loblolly pine trees was undertaken with the objective of quantifying the average proportion of overall prediction error which was due to these components. The exact mean square error of the vector of predicted volume, V, is also derived.

Confidence intervals are developed for the location parameter of a continuous, symmetric, unimodal distribution in the casev where only a single observation from the distribution is available. These intervals are similar to those given by Abbott and Rosenblatt (1963), but shorter. The result is extended to include distributions which can be standardized to have unit scale. The procedure is exemplified for the normal distribution and the power of one- and two-sided significance tests are computed under normality.