Model-assisted Estimators Research Articles

Continuous Monitoring of Forests in Wetland Ecosystems with Remote Sensing and Probability Sampling

With the drastic reduction in wetland areas, it is essential to conduct an annual monitoring of the biomass or carbon content of wetland ecosystems to support international initiatives and agreements focused on sustainable development, climate change, and carbon equity. Forests in wetland ecosystems play a crucial role in carbon sequestration; however, the monitoring of small, fragmented forest components in wetlands remains insufficient, leading to an underestimation of their ecological and carbon sequestration functions. This study utilizes a model-assisted (MA) estimator, a monitoring procedure that is asymptotically design-unbiased and incorporates remote sensing, to assess the status and trends in the above-ground biomass (AGB) of forest components in wetlands, while also proposing a method of optimizing the sample size to enable continuous monitoring. Based on the population of the forest component of Baiyangdian wetland, major findings indicate that: (1) neglecting the forest component of Baiyangdian wetland will lead to an underestimation of the total aboveground biomass by 224.34 t/ha and 243.64 t/ha in the years 2022 and 2023, respectively; (2) in either year-specific monitoring or interannual change monitoring, the MA estimator is more cost-effective than the expansion estimator, a comparable procedure that relies solely on field observations; (3) the method used to optimize sample size can effectively tackle the cost-related concerns of subsequent continuous monitoring. Overall, the neglect of forest components is inevitably bound to give rise to an underestimation of wetlands, and use of an MA estimator and optimizing the sample size could effectively address the cost issue in continuous monitoring. This holds significant importance when developing management strategies to prevent the further degradation of wetland ecological functions and carbon sink capabilities.

Evaluating Model-assisted Estimators: A Comparative Study in High-dimensional Survey Data

Model-assisted estimators have gained significant attention due to their ability to efficiently utilize auxiliary information during the estimation process. These estimators rely on a working model that links the survey variable to the auxiliary variables, which is then fitted to the sample data to generate predictions. These predictions are subsequently integrated into the estimation procedures. In this study, were explores various model-assisted estimators including Generalized Regression (GREG), Ridge regression, Lasso regression, CART (Classification and Regression Tree), Random Forest, Cubist and Principal Components Regression (PCR) estimator. The analysis involved 2,000 samples of size 50 (n/N ≈ 10%) and employed a stepwise variable selection method to determine the most significant auxiliary variables, incrementally adding them to the model. The performance of these estimators was assessed using relative bias (RB), relative root mean square error (RRMSE) and relative efficiency (RE). Our findings reveal that tree-based models like CART and Random Forest and penalized regression estimators such as Ridge and Lasso display robustness with increased number of auxiliary variables. Among all the estimators, Random Forest consistently yielded the lowest RRMSE, particularly with five auxiliary variables, demonstrating superior efficiency. Conversely, the GREG estimator exhibited poor performance as the number of auxiliary variables increased. This study underscores the importance of selecting suitable model-assisted estimation procedures tailored to the data characteristics and the relationship between survey and auxiliary variables within this high-dimensional dataset.

Model-assisted estimation of domain totals, areas, and densities in two-stage sample survey designs

Model-assisted, two-stage forest survey sampling designs provide a means to combine airborne remote sensing data, collected in a sampling mode, with field plot data to increase the precision of national forest inventory estimates, while maintaining important properties of design-based inventories, such as unbiased estimation and quantification of uncertainty. In this study, we present a comprehensive set of model-assisted estimators for domain-level attributes in a two-stage sampling design, including new estimators for densities, and compare the performance of these estimators with standard poststratified estimators. Simulation was used to assess the statistical properties (bias, variability) of these estimators, with both simple random and systematic sampling configurations, and indicated that 1) all estimators were generally unbiased. and 2) the use of lidar in a sampling mode increased the precision of the estimators at all assessed field sampling intensities, with particularly marked increases in precision at lower field sampling intensities. Variance estimators are generally unbiased for model-assisted estimators without poststratification, while model-assisted estimators with poststratification were increasingly biased as field sampling intensity decreased. In general, these results indicate that airborne remote sensing data, collected as an intermediate level of sampling, can be used to increase the efficiency of national forest inventories in remote regions.

Smoothed model-assisted small area estimation of proportions.

In countries where population census data are limited, generating accurate subnational estimates of health and demographic indicators is challenging. Existing model-based geostatistical methods leverage covariate information and spatial smoothing to reduce the variability of estimates but often ignore the survey design, while traditional small area estimation approaches may not incorporate both unit-level covariate information and spatial smoothing in a design consistent way. We propose a smoothed model-assisted estimator that accounts for survey design and leverages both unit-level covariates and spatial smoothing. Under certain regularity assumptions, this estimator is both design consistent and model consistent. We compare it with existing design-based and model-based estimators using real and simulated data.

Model-Assisted Complier Average Treatment Effect Estimates in Randomized Experiments with Noncompliance

ABSTRACT Noncompliance is a common problem in randomized experiments in various fields. Under certain assumptions, the complier average treatment effect is identifiable and equal to the ratio of the intention-to-treat effects of the potential outcomes to that of the treatment received. To improve the estimation efficiency, we propose three model-assisted estimators for the complier average treatment effect in randomized experiments with a binary outcome. We study their asymptotic properties, compare their efficiencies with that of the Wald estimator, and propose the Neyman-type conservative variance estimators to facilitate valid inferences. Moreover, we extend our methods and theory to estimate the multiplicative complier average treatment effect. Our analysis is randomization-based, allowing the working models to be misspecified. Finally, we conduct simulation studies to illustrate the advantages of the model-assisted methods and apply these analysis methods in a randomized experiment to evaluate the effect of academic services or incentives on academic performance.

Assessing and mitigating systematic errors in forest attribute maps utilizing harvester and airborne laser scanning data

Cut-to-length harvesters collect useful information for modeling relationships between forest attributes and airborne laser scanning (ALS) data. However, harvesters operate in mature forests, which may introduce selection biases that can result in systematic errors in harvester data-based forest attribute maps. We fitted regression models (harvester models) for volume (V), height (HL), stem frequency (N), above-ground biomass, basal area, and quadratic mean diameter (QMD) using harvester and ALS data. Performances of the harvester models were evaluated using national forest inventory plots in an 8.7 Mha study area. We estimated biases of large-area synthetic estimators and compared efficiencies of model-assisted (MA) estimators with field data-based direct estimators. The harvester models performed better in productive than unproductive forests, but systematic errors occurred in both. The use of MA estimators resulted in efficiency gains that were largest for HL (relative efficiency, RE = 6.0) and smallest for QMD (RE = 1.5). The bias of the synthetic estimator was largest for N (39%) and smallest for V (1%). The latter was due to an overestimation of deciduous and an underestimation of spruce forests that by chance balanced. We conclude that a probability sample of reference observations may be required to ensure the unbiasedness of estimators utilizing harvester data.

Precision of subnational forest AGB estimates within the Peruvian Amazonia using a global biomass map

• Reproducible approach for AGB estimation from global biomass maps and NFI data. • Hybrid-inference accounted for uncertainties in map-plot data integration. • Within-block variability was the main source of uncertainty. • The use of a global biomass map slightly increased the precision of AGB. • The modest gain in precision was due to small map-plot correlations. National forest inventories (NFI) provide essential forest-related biomass and carbon information for country greenhouse gas (GHG) accounting systems. Several tropical countries struggle to execute their NFIs while the extent to which space-based global information on aboveground biomass (AGB) can support national GHG accounting is under investigation. We assess whether the use of a global AGB map as auxiliary information produces a gain in precision of subnational AGB estimates for the Peruvian Amazonia. We used model-assisted estimators with data from the country’s NFI and explored hybrid inferential techniques to account for the sources of uncertainty associated with the integration of remote sensing-based products and NFI plot data. Our results show that the selected global biomass map tends to overestimate AGB values across the Peruvian Amazonia. For most strata, directly using the map in its published form did not reduce the precision of AGB estimates. However, after calibrating the map using the NFI data, the precision of our map-assisted AGB estimates increased by up to 50% at stratum level and 20% at Amazonia level. We further demonstrate how different sources of uncertainties can be incorporated in the map-NFI integrated estimates. With the hybrid inferential analysis, we found that the small spatial support of the NFI plots compared to the remote sensing-based sample units of aggregated pixels (within block variability) contributed the most to the total uncertainty associated with the AGB estimates from our map-NFI integration. Uncertainties caused by measurement variability and allometric model prediction uncertainty were the second largest contributors. When these uncertainties were incorporated, the increase in precision of our calibrated map-assisted AGB estimates was negligible, probably hindered by the great contribution of the within block variability to our map-plot assessment. We developed a reproducible method that countries can build upon and further improve while the global biomass products continue to evolve and better characterize the AGB distribution under large biomass conditions. We encourage further cross-country case studies that reflect a wider range of AGB distributions, especially within humid tropical forests, to further assess the contribution of global biomass maps to (sub)national AGB estimates and finally GHG monitoring and reporting.

Comparison of Model-Assisted Endogenous Poststratification Methods for Estimation of Above-Ground Biomass Change in Oregon, USA

Quantifying above-ground biomass changes, ΔAGB, is key for understanding carbon dynamics. National Forest Inventories, NFIs, aims at providing precise estimates of ΔAGB relying on model-assisted estimators that incorporate auxiliary information to reduce uncertainty. Poststratification estimators, PS, are commonly used for this task. Recently proposed endogenous poststratification, EPS, methods have the potential to improve the precision of PS estimates of ΔAGB. Using the state of Oregon, USA, as a testing area, we developed a formal comparison between three EPS methods, traditional PS estimators used in the region, and the Horvitz-Thompson, HT, estimator. Results showed that gains in performance with respect to the HT estimator were 9.71% to 19.22% larger for EPS than for PS. Furthermore, EPS methods easily accommodated a large number of auxiliary variables, and the inclusion of independent predictions of ΔAGB as an additional auxiliary variable resulted in further gains in performance.

Bias of area counted from sub-pixel map: Origin and correction

With the increasingly widespread use of sub-pixel mapping techniques in land cover/use mapping, more accurate area information is often required for a specific land cover type in a particular study region. However, the bias of area counted from sub-pixel maps (called area bias below), and the inadequate understanding of the area bias's origin and influential factors pose a challenge to using this information accurately. Traditional model-assisted estimators combining the map and the reference sample showed unreliable performances in the case of small sample sizes collected in target regions. This work presented a theoretical analysis of the origin of area bias. It then proposed a novel bias-adjusted estimator which can effectively deal with the small sample sizes. The theoretical analysis illustrated that area bias mainly originates from two terms, i.e., the abundance-dependent error and the probability distribution of abundances. We next developed a stratified bias-adjusted area estimator named the two-term method (TTM) by incorporating the sub-pixel map and a reference sample obtained from both target and external regions. We validated the effects of different sub-pixel mapping methods, different spatial resolutions, the varying spatial structures of statistical units on area bias, and the performance of TTM in correcting the biased areas in multiple cases. The results showed that area bias varied from zero to approximately 20% with the variation of three influential factors. TTM effectively corrected the biased area values to nearly the true values, showing approximate equivalence with the traditional stratified regression estimator (STRE) when adequate reference samples are collected sorely inside target regions. However, in cases of small samples from target regions, TTM showed significant superiority over STRE in reducing the variance and MSE due to the incorporation of external reference samples. We conclude that the theoretical analysis resulted in a better understanding of area bias counted from sub-pixel maps and an improved area estimator for dealing with the cases of small sample sizes inside target regions.

Model-Assisted Estimators with Auxiliary Functional Data

Model-Assisted Estimators with Auxiliary Functional Data

Small Area Estimates for National Applications: A Database to Dashboard Strategy Using FIESTA

This paper demonstrates a process for translating a database of forest measurements to interactive dashboards through which users can access statistically defensible estimates and analyses anywhere in the conterminous US. It taps the extensive Forest Inventory and Analysis (FIA) plot network along with national remotely sensed data layers to produce estimates using widely accepted model-assisted and small area estimation methodologies. It leverages a decade’s worth of statistical and computational research on FIA’s flexible estimation engine, FIESTA, and provides a vehicle through which scientists and analysts can share their own tools and analytical processes. This project illustrates one pathway to moving statistical research into operational inventory processes, and makes many model-assisted and small area estimators accessible to the FIA community. To demonstrate the process, continental United States (CONUS)-wide model-assisted and small area estimates are produced for ecosubsections, counties, and level 5 watersheds (HUC 10) and made publicly available through R Shiny dashboards. Target parameters include biomass, basal area, board foot volume, proportion of forest land, cubic foot volume, and live trees per acre. Estimators demonstrated here include: the simplest direct estimator (Horvitz–Thompson), model-assisted estimators (post-stratified, generalized regression estimator, and modified generalized regression estimators), and small area estimators (empirical best linear unbiased predictors and hierarchical Bayes both at the area- and unit-level). Auxiliary data considered in the model-assisted and small area estimators included maps of tree canopy, tree classification, and climatic variables. Estimates for small domain sets were generated nationally within a few hours. Exploring results across estimators and target variables revealed the progressive gains in precision using (in order of least gain to highest gain) Horvitz–Thompson, post-stratification, modified generalized regression estimators, generalized regression estimators, area-level small area models, and unit-level small area models. Substantive gains are realized by expanding model-assisted estimators beyond post-stratification, allowing FIA to continue to take advantage of design-based inference in many cases. Caution is warranted in the use of unit-level small area models due to model mis-specification. The dataset of estimates available through the dashboards provides the opportunity for others to compare estimators and explore precision expectations over specific domains and geographic regions. The dashboards also provide a forum for future development and analyses.

Multiply Robust Estimation of Causal Effects under Principal Ignorability

AbstractCausal inference concerns not only the average effect of the treatment on the outcome but also the underlying mechanism through an intermediate variable of interest. Principal stratification characterizes such a mechanism by targeting subgroup causal effects within principal strata, which are defined by the joint potential values of an intermediate variable. Due to the fundamental problem of causal inference, principal strata are inherently latent, rendering it challenging to identify and estimate subgroup effects within them. A line of research leverages the principal ignorability assumption that the latent principal strata are mean independent of the potential outcomes conditioning on the observed covariates. Under principal ignorability, we derive various nonparametric identification formulas for causal effects within principal strata in observational studies, which motivate estimators relying on the correct specifications of different parts of the observed-data distribution. Appropriately combining these estimators yields triply robust estimators for the causal effects within principal strata. These triply robust estimators are consistent if two of the treatment, intermediate variable and outcome models are correctly specified, and moreover, they are locally efficient if all three models are correctly specified. We show that these estimators arise naturally from either the efficient influence functions in the semiparametric theory or the model-assisted estimators in the survey sampling theory. We evaluate different estimators based on their finite-sample performance through simulation and apply them to two observational studies.

Model-assisted estimation in high-dimensional settings for survey data

Model-assisted estimators have attracted a lot of attention in the last three decades. These estimators attempt to make an efficient use of auxiliary information available at the estimation stage. A working model linking the survey variable to the auxiliary variables is specified and fitted on the sample data to obtain a set of predictions, which are then incorporated in the estimation procedures. A nice feature of model-assisted procedures is that they maintain important design properties such as consistency and asymptotic unbiasedness irrespective of whether or not the working model is correctly specified. In this article, we examine several model-assisted estimators from a design-based point of view and in a high-dimensional setting, including linear regression and penalized estimators. We conduct an extensive simulation study using data from the Irish Commission for Energy Regulation Smart Metering Project, to assess the performance of several model-assisted estimators in terms of bias and efficiency in this high-dimensional data set.

Calibration estimation for non-probability samples under two distance functions: a comparative study

Calibration estimation is a popular method for making inference from probability samples and dealing with nonresponse problems. Calibration estimation, specially model-assisted calibration estimation, has been extensively explored for inference from non-probability samples in recent years. When there are rich covariates, the variable selection method such as Adaptive LASSO can be used to select important covariates and estimate model parameters for establishing appropriate models in model-assisted calibration estimation. In addition to the established model, distance functions also have an impact on calibration estimation. Different distance functions may produce different calibration estimators. In this paper, traditional calibration estimators, estimated-control model-assisted calibration estimators and estimated-control model-assisted calibration estimators with Adaptive LASSO under the chi-square distance and the modified backward Kullback-Leibler distance are proposed when the population information of covariates is unknown. The calibrated weights based on traditional calibration, model-assisted calibration and model-assisted calibration with Adaptive LASSO under the two distance functions are also derived. Results from a simulation study are presented to compare different calibration estimators. A real data set is also adopted to confirm the performance of the proposed estimators.

Model-assisted analyses of longitudinal, ordinal outcomes with absorbing states.

Studies of critically ill, hospitalized patients often follow participants and characterize daily health status using an ordinal outcome variable. Statistically, longitudinal proportional odds models are a natural choice in these settings since such models can parsimoniously summarize differences across patient groups and over time. However, when one or more of the outcome states is absorbing, the proportional odds assumption for the follow-up time parameter will likely be violated, and more flexible longitudinal models are needed. Motivated by the VIOLET Study (Ginde et al), a parallel-arm, randomized clinical trial of Vitamin in critically ill patients, we discuss and contrast several treatment effect estimands based on time-dependent odds ratio parameters, and we detail contemporary modeling approaches. In VIOLET, the outcome is a four-level ordinal variable where the lowest "not alive" state is absorbing and the highest "at-home" state is nearly absorbing. We discuss flexible extensions of the proportional odds model for longitudinal data that can be used for either model-based inference, where the odds ratio estimator is taken directly from the model fit, or for model-assisted inferences, where heterogeneity across cumulative log odds dichotomizations is modeled and results are summarized to obtain an overall odds ratio estimator. We focus on direct estimation of cumulative probability model (CPM) parameters using likelihood-based analysis procedures that naturally handle absorbing states. We illustrate the modeling procedures, the relative precision of model-based and model-assisted estimators, and the possible differences in the values for which the estimators are consistent through simulations and analysis of the VIOLET Study data.

Examining k-Nearest Neighbor Small Area Estimation Across Scales Using National Forest Inventory Data

National forest inventories (NFI), such as the one conducted by the United States Forest Service Forest Inventory and Analysis (FIA) program, provide valuable information regarding the status of forests at regional to national scales. However, forest managers often need information at stand to landscape scales. Given various small area estimation (SAE) approaches, including design-based and model-based estimation, it may not be clear which is most appropriate for the user’s application. In this study, our objective was to assess the uncertainty in tree aboveground live carbon (ALC) estimates for differing modes of SAE across multiple scales to provide guidance for appropriate scales of application. We calculated means and variances for ALC with design-based (Horvitz-Thompson), model-assisted (generalized regression), and model-based (k-nearest neighbor synthetic) estimators for estimation units over a range of sizes for 30 subregions in California, United States. For larger areas (10,000–64,800 ha), relative efficiencies greater than one indicated that the generalized regression estimator (GREG) generated estimates with less error than the Horvitz-Thompson estimator (HT), while the bias-adjusted synthetic estimator relative efficiency compared to either the Horvitz-Thompson or model-assisted estimators exceeded one for areas 25,000 ha and smaller. Variance estimates from the unadjusted synthetic estimator underestimated the total error, because the estimator ignores bias and thus only addresses model variance. Across scales (250–64,800 ha, 0–27 plots per area of interest), 93% of the variation in the synthetic estimator’s relative standard error was explained by forest area, forest dominance, and regional variation in forest landscapes. Our results support model-assisted estimation use except for small areas where few plots (<10 in the current study) are available for generating estimates in spite of biases in estimates. However, users should exercise caution when interpreting model-based estimates of error as they may not account for model mis-specification, and thus induced bias. This research explored multiple scales of application for SAE procedures applied to NFI data regarding carbon pools, potentially supporting a multi-scale approach to forest monitoring. Our results guides users in developing defensible estimates of carbon pools, particularly as it relates to the limits of inference at a variety of spatial scales.

Toward Better Practice of Covariate Adjustment in Analyzing Randomized Clinical Trials

In randomized clinical trials, adjustments for baseline covariates at both design and analysis stages are highly encouraged by regulatory agencies. A recent trend is to use a model-assisted approach for covariate adjustment to gain credibility and efficiency while producing asymptotically valid inference even when the model is incorrect. In this article we present three considerations for better practice when model-assisted inference is applied to adjust for covariates under simple or covariate-adaptive randomized trials: (a) guaranteed efficiency gain: a model-assisted method should often gain but never hurt efficiency; (b) wide applicability: a valid procedure should be applicable, and preferably universally applicable, to all commonly used randomization schemes; (c) robust standard error: variance estimation should be robust to model misspecification and heteroscedasticity. To achieve these, we recommend a model-assisted estimator under an analysis of heterogeneous covariance working model that includes all covariates used in randomization. Our conclusions are based on an asymptotic theory that provides a clear picture of how covariate-adaptive randomization and regression adjustment alter statistical efficiency. Our theory is more general than the existing ones in terms of studying arbitrary functions of response means (including linear contrasts, ratios, and odds ratios), multiple arms, guaranteed efficiency gain, optimality, and universal applicability. Supplementary materials for this article are available online.

Effects of lidar coverage and field plot data numerosity on forest growing stock volume estimation

ABSTRACT Forest parameter estimation is required to support the sustainable management of forest ecosystems. Currently, forest resource assessment is increasingly linked to auxiliary information obtained from remote sensing (RS) technologies. In forest parameter estimation, airborne laser scanning (ALS) data have been demonstrated to be an invaluable source of information. However, ALS data are often not available for the entire forest area, whereas images from multiple satellite systems offer new opportunities for large-scale forest surveys. This study aims to assess and estimate the contribution of field plot data and ALS data along with Landsat data to the precision of growing stock volume (GSV) estimates. We compared different approaches for model-assisted estimation of mean forest GSV per unit area using different proportions of field sample data, ALS cover data, and wall-to-wall Landsat data. Model-assisted estimators were used with NFI sample data in an Italian study area using 10 RS predictors, specifically the seven Landsat 7 ETM+ bands and three fine-resolution metrics based on ALS-derived canopy height. We found that relative to the standard simple expansion estimator, the model-assisted estimators produced relative efficiency of 1.16 when using only Landsat data and relative efficiencies as great as 1.33 when using increasing levels of ALS coverage.

A Comparison of Model-Assisted Estimators, With and Without Data-Driven Transformations of Auxiliary Variables, With Application to Forest Inventory

Forest information is requested at many levels and for many purposes. Sampling-based national forest inventories (NFIs) can provide reliable estimates on national and regional levels. By combining expensive field plot data with different sources of remotely sensed information, from airplanes and/or satellite platforms, the precision in estimators of forest variables can be improved. This paper focuses on the design-based model-assisted approach to using NFI data together with remotely sensed data to estimate forest variables for small areas, where the variables studied are total growing stock volume, volume of Norway spruce (Picea abies), and volume of broad-leaved trees. Remote sensing variables may be highly correlated with one another and some may have poor predictive ability for target forest variables, and therefore model selection and/or coefficient shrinkage may be appropriate to improve the efficiency of model-assisted estimators of forest variables. For this purpose, one can use modern shrinkage estimators based on lasso, ridge, and elastic net regression methods. In a simulation study using real NFI data, Sentinel 2 remote-sensing data, and a national airborne laser scanning (ALS) campaign, we show that shrinkage estimators offer advantages over the (weighted) ordinary least-squares (OLS) estimator in a model-assisted setting. For example, for a sample size n of about 900 and with 72 auxiliary variables, the RMSE was up to 41% larger when based on OLS. We propose a data-driven method for finding suitable transformations of auxiliary variables, and show that it can improve estimators of forest variables. For example, when estimating volume of Norway spruce, using a smaller expert selection of auxiliary variables, transformations reduced the RMSE by up to 10%. The overall best results in terms of RMSE were obtained using shrinkage estimators and a larger set of 72 auxiliary variables. However, for this larger set of variables, the use of transformations yielded at most small improvements of RMSE, and at worst large increases of RMSE, except in combination with ridge and elastic net regression.

Prediction and model-assisted estimation of diameter distributions using Norwegian national forest inventory and airborne laser scanning data

Diameter at breast height (DBH) distributions offer valuable information for operational and strategic forest management decisions. We predicted DBH distributions using Norwegian national forest inventory and airborne laser scanning data and compared the predictive performances of linear mixed-effects (PPM), generalized linear-mixed (GLM), and k nearest-neighbor (NN) models. While GLM resulted in smaller prediction errors than PPM, both were clearly outperformed by NN. We therefore studied the ability of the NN model to improve the precision of stem frequency estimates by DBH classes in the 8.7 Mha study area using a model-assisted (MA) estimator suitable for systematic sampling. MA estimates yielded greater than or approximately equal efficiencies as direct estimates using field data only. The relative efficiencies (REs) associated with the MA estimates ranged between 0.95–1.47 and 0.96–1.67 for 2 and 6 cm DBH class widths, respectively, when dominant tree species were assumed to be known. The use of a predicted tree species map, instead of the observed information, decreased the REs by up to 10%.