TY - JOUR
AU - Fan, Zhaofei
AB - An estimate of volume or weight variation from preliminary plots, previous inventories, or experience is required when one is beginning a cruise for a timber sale. This variation can be expressed as the coefficient of variation (CV) or variance and is used to estimate the number of plots needed to implement the cruise to a desired level of precision at a stated level of confidence. Using a Mississippi Forestry Commission inventory sales database, two models were derived from inputs of simple stand observations to aid field personnel in determining acceptable estimates of CV without installing costly preliminary plots. The models estimate the number of 0.10-acre plots needed to sample a stand to within a ±10% allowable error at the 90% confidence level for total tonnage. The average sized timber sale was 110 acres, and the models correctly estimated 77–82% of the number of required plots to within 15 plots of actual and 99–100% within 30 plots of actual. Tables by dbh, trees per acre, and variance class were constructed to aid in determining the number of required plots. These tables eliminate costs associated with installing preliminary plots and errors resulting from inappropriate rule-of-thumb methods. coefficient of variation, sample size estimation, Visual Basic code Designing a good timber inventory for the purpose of a timber sale is a complicated task requiring not only conceptual knowledge but also skills acquired through years of experience. One of the most common initial questions about a timber sale cruise is “How many plots do I measure?” Inventory sampling methods vary, but often agency or company employees erroneously rely on a 10% cruise rule or use a company “standard,” not knowing its origin or why it is used (Hamilton 1979). Because most revenue in forestry business revolves around the timber cruise and subsequent sale of timber, inventory estimation is of the utmost importance to the forester's company or client. Poorly designed or implemented cruises for a sale or for management and planning purposes can cause large errors and result in high costs to individual landowners (Borders et al. 2008). An estimate of the volume or weight variation from preliminary plots, previous inventories, or experience is required when one is beginning a cruise for a timber sale. This variation can be expressed as the coefficient of variation (CV) or variance and is subsequently used to estimate the number of plots needed to implement the cruise to a desired level of precision (allowable error) at a stated level of confidence. A priori estimation of CV can feasibly save the forester from installing precruise plots, as well as give the cruise planner an estimate of the expected time and financial resources required to properly cruise the timber sale to be within specified confidence limits. After a forester selects an appropriate cruise method and design for the timber type, the statistically valid number of plots that should be installed ultimately depends on the inherent variation in the targeted volume or weight measurements. The method for determining the required number of sample plots for a stated level of confidence and maximum allowable error uses equations for infinite or finite populations (Cochran 1977). To predict sample size, a measure of variation must be estimated and a maximum allowable error must be specified. If the population is infinite (point sampling), the iterative form of the sample size calculation (Equation 1) (Cochran 1977) is used with CV as the measure of variation.   where ni+1 is the number of plots needed, CV is the coefficient of variation expressed as a percentage, E is the allowable error expressed as a percentage of the mean, tni−1, α/2 is the Student's t-value at ni − 1 df, and α = 1 − p/100 is the probability of a type I error for a stated percent confidence of P. The iteration of the formula is terminated when ni+1 = ni. If the population is finite (plot sampling), the iterative form of the sample size estimation, Equation 2 (Cochran 1977), is used to calculate sample size. CV and population and total stand acres affect the required number of plots.   where ni+1 is the number of plots needed, N is the population sample size (total acres/plot size in acres), E is the allowable error expressed as a percentage of the mean, tni−1, α/2 is the Student's t-value at ni − 1 df, α = 1 − p/100 is the probability of a type I error for a stated percent confidence of P, and CV is the coefficient of variation. CV is a relative measure of the SD in relation to the mean (Freese 1962). It is a unitless number that is useful in many statistical analyses for comparing two samples that may be different in size, method of collection, or other characteristic. Lower CVs indicate that the required sample size will be lower than that with a high CV. In forested stands, a CV of ≥30 for a target variable is usually considered low (a uniform stand) and would have consistent trees per acre (TPA) counts as well as similar average quadratic dbh (QMD), basal area per acre (BA), and product designations. Many young stands would fall into this category, having never been thinned or experienced a major event (such as fire, hurricane, or insect damage) that altered stand makeup. Alternately, stands with a CV of >50 for a target variable have a higher degree of variability of the target variable measured among plots. Equations 1 and 2 can be used to estimate sample size when the allowable error (E) is in absolute units, such as board feet, by replacing CV with the SD in the same volume units as the absolute allowable error. Methods The Mississippi Forestry Commission (MFC) collects cruise data for estimating total and per acre weight (tons) timber stand sales with T-Cruise Mobile (World Wide Heuristic Solutions, Inc. 2010) software. Sufficient plots are installed to achieve a precision goal of ±10% sampling error at the 90% confidence level (Skidmore 2010). Data from 88 MFC timber sale inventories (based on 0.10-acre plots) were obtained for fiscal year (FY) 2010 and 2011 (Tables 1 and 2). Timber sales occurred for different products and species, but pine and hardwood are the two major species groups used. Products for pine included pulpwood, Chip-N-Saw, and sawtimber. Hardwood products are categorized as pulpwood, pallet, or sawtimber. Although some local variations existed, field data collection generally followed product classification rules defined in Table 3. Tree level field data included dbh, product class, merchantable height, and total height. Tree volumes and weights were calculated using species group profile equations, measured tree dbh, and height. Plot variables included species group, TPA BA, QMD, cover type, years since last disturbance, acres in sale, estimated variance, and number of plots estimated by T-Cruise to achieve the desired precision level. Because most sale tract sizes exceeded 30 acres and plot sizes (0.10 acre) were reasonably small, population size could be considered infinite for all practical purposes. Allowable error (E) in all MFC sales is fixed at 10%. Sales were conducted and information processed on a total tonnage basis. CV estimates derived from LandMark Reports for T-Cruise (F4 Tech 2010) were produced for the entire cruise as opposed to estimates by species or product. Equations 1 and 2 were implemented as user-defined functions in a Microsoft Excel Visual Basic Editor macro enabled worksheet (xlsm). This worksheet also calculates confidence intervals, percent sampling errors, standard errors of measurement, and t-values and has a visual interface for optimum stratified sample size estimation and allocation.1 In the Excel xlsm, “User Defined” functions TC_RequiredPointsFromCV and TC_RequiredPlotsFromCV implement the iterative Equations 1 and 2, respectively. Number of observations for 88 MFC timber sale inventories (based on 0.10 acre plots) obtained for FY 2010 and 2011 by CV of green tons per acre, QMD, and number of TPA classes. Table 1. Number of observations for 88 MFC timber sale inventories (based on 0.10 acre plots) obtained for FY 2010 and 2011 by CV of green tons per acre, QMD, and number of TPA classes. View Large Table 1. Number of observations for 88 MFC timber sale inventories (based on 0.10 acre plots) obtained for FY 2010 and 2011 by CV of green tons per acre, QMD, and number of TPA classes. View Large Number of observations for 88 MFC timber sale inventories (based on 0.10-acre plots) obtained for FY 2010 and 2011 by SD of green tons per acre, QMD, and number of TPA classes. Table 2. Number of observations for 88 MFC timber sale inventories (based on 0.10-acre plots) obtained for FY 2010 and 2011 by SD of green tons per acre, QMD, and number of TPA classes. View Large Table 2. Number of observations for 88 MFC timber sale inventories (based on 0.10-acre plots) obtained for FY 2010 and 2011 by SD of green tons per acre, QMD, and number of TPA classes. View Large Product top and diameter limits used by the Mississippi Forestry Commission during timber sale inventories in FY 2010 and 2011. Table 3. Product top and diameter limits used by the Mississippi Forestry Commission during timber sale inventories in FY 2010 and 2011. View Large Table 3. Product top and diameter limits used by the Mississippi Forestry Commission during timber sale inventories in FY 2010 and 2011. View Large To categorize the total tonnage variability class (VARC), SDs of plot weights were divided into three classes (low, medium, and high) based on a 25%–50%–25% distribution of total plots and the closest SD whole number multiple of five for ease of field recognition. Low variability stands accounted for 27% (SD ≤25.0 tons/acre), medium variability 56% (25.1 tons/acre < SD ≤40.0 tons/acre), and high variability 17% (SD ≥40.1 tons/acre) of the observed data. Scatterplots and SAS 9.2 (SAS Institute, Inc. 2009) linear and nonlinear regression analyses were used to examine relationships between CV and single independent variables. The natural log of the stand quadratic average QMD (ln QMD), natural log of stand average TPA (ln TPA), and variance class indicator variable (VARC) emerged as the most significant predictor variables. Multiple predictor model forms that included QMD, TPA, VARC, BA, and transformations and interactions of these independent variables were examined. Interactions did not contribute to the models, and BA did not prove to be a significant variable after QMD and TPA were added to the model. Because the model forms of Equations 3–5 are related by the definitions,     and   they allow the substitution of alternative predictors that give root mean square error (RMSE) and adjusted R2 values identical to those in Equations 6 and 7.       where ln QMD is the natural log of QMD, ln TPA is the natural log of TPA, ln BA is the natural log of BA, and a, b, and c are parameters to be estimated. TPA is both a measure of BA and a surrogate of stand age in this data set and can also be more easily visually estimated than BA. Therefore, the model forms that include QMD and TPA predictors, with and without VARC, were logical choices for providing equations in which the predictors can be visually estimated without the need for expensive precruise plots. VARC takes on the values, 0, 1, and 2 for SD ≤ 25.0 tons/acre, 25.1 tons/acre < SD ≤ 40.0 tons/acre, and SD ≥ 40.1 tons/acre, respectively, as an indicator variable, I(VARC), in regression Equation 6. Results Two sets of multiple linear regressions were constructed to predict CV, one using all three predictors (Equation 6) and one leaving out VARC (Equation 7) to reduce the necessity of estimating this variable in the field. VARC requires a judgment call to be made by the cruiser. Predicted CV was then used in Equations 1 and 2 to calculate the number of plots required for infinite and finite populations, respectively. Equation 6 predicts the CV of an inventory sale conducted on a total tonnage basis in percentage form.     where ln QMD is the natural log of the stand quadratic average QMD, ln TPA is the natural log of the stand average TPA, and VARC is the plot weight variation class (low = 0, average = 1, or high = 2), where I(0) = −11.625, I(1) = 0, and I(2) = 5.883. Alternate Equation 7 was developed for cases in which VARC could not be estimated by field personnel or in which a less precise initial estimate was allowable.     Percentages of correctly classified CVs were slightly higher when VARC was included as a predictor variable (Table 4). Percentage of correctly classified total tonnage predicted CVs, within selected ranges for equations with (Equation 6) and without (Equation 7) estimates of plot tonnage variance class, on 88 0.10-acre Mississippi Forestry Commission timber sales inventories in FY 2010 and 2011. Table 4. Percentage of correctly classified total tonnage predicted CVs, within selected ranges for equations with (Equation 6) and without (Equation 7) estimates of plot tonnage variance class, on 88 0.10-acre Mississippi Forestry Commission timber sales inventories in FY 2010 and 2011. View Large Table 4. Percentage of correctly classified total tonnage predicted CVs, within selected ranges for equations with (Equation 6) and without (Equation 7) estimates of plot tonnage variance class, on 88 0.10-acre Mississippi Forestry Commission timber sales inventories in FY 2010 and 2011. View Large The infinite population sample size estimation equation performed slightly better than the finite equation in test runs (Tables 5 and 6). In the case in which VARC was included in the model, the infinite population equation estimated 100% of cruises within 30 plots compared with 99% for the finite population equation. Equation 6, which includes the VARC variable, was embedded in a Microsoft Excel spreadsheet application to create table versions of the sample size estimator (Tables 72760–9) using the infinite population sample size equation (Equation 1). Tables 72760–9 contain the estimated number of plots needed by QMD and TPA classes in stands of low, medium, and high total tonnage variance, respectively. Similarly, Table 10 was created using Equation 7 for the case in which no variance levels are estimated by the cruiser. Estimates associated with Table 10 are, in general, less precise than those associated with Tables 72760–9 because of the model's higher RMSE. Percentage of correctly predicted number of required sample plots, within selected ranges, using Equation 6 (with tonnage variance class) to estimate CV for infinite (Equation 1) and finite (Equation 2) populations. Table 5. Percentage of correctly predicted number of required sample plots, within selected ranges, using Equation 6 (with tonnage variance class) to estimate CV for infinite (Equation 1) and finite (Equation 2) populations. View Large Table 5. Percentage of correctly predicted number of required sample plots, within selected ranges, using Equation 6 (with tonnage variance class) to estimate CV for infinite (Equation 1) and finite (Equation 2) populations. View Large Percentage of correctly estimated number of required sample plots, within selected ranges, using Equation 7 (without tonnage variance class) to estimate CV for infinite (Equation 1) and finite (Equation 2) populations. Table 6. Percentage of correctly estimated number of required sample plots, within selected ranges, using Equation 7 (without tonnage variance class) to estimate CV for infinite (Equation 1) and finite (Equation 2) populations. View Large Table 6. Percentage of correctly estimated number of required sample plots, within selected ranges, using Equation 7 (without tonnage variance class) to estimate CV for infinite (Equation 1) and finite (Equation 2) populations. View Large Estimated number of 0.10-acre plots needed in a low tonnage variance stand for infinite population sample sizes. Table 7. Estimated number of 0.10-acre plots needed in a low tonnage variance stand for infinite population sample sizes. View Large Table 7. Estimated number of 0.10-acre plots needed in a low tonnage variance stand for infinite population sample sizes. View Large Estimated number of 0.10-acre plots needed in an average tonnage variance stand for infinite population sample sizes. Table 8. Estimated number of 0.10-acre plots needed in an average tonnage variance stand for infinite population sample sizes. View Large Table 8. Estimated number of 0.10-acre plots needed in an average tonnage variance stand for infinite population sample sizes. View Large Estimated number of 0.10-acre plots needed in a high tonnage variance stand for infinite population sample sizes. Table 9. Estimated number of 0.10-acre plots needed in a high tonnage variance stand for infinite population sample sizes. View Large Table 9. Estimated number of 0.10-acre plots needed in a high tonnage variance stand for infinite population sample sizes. View Large Estimated number of 0.10-acre plots needed without requiring an estimate of weight variance class for infinite population sample sizes. Table 10. Estimated number of 0.10-acre plots needed without requiring an estimate of weight variance class for infinite population sample sizes. View Large Table 10. Estimated number of 0.10-acre plots needed without requiring an estimate of weight variance class for infinite population sample sizes. View Large Discussion Finite and Infinite Population Equations MFC database timber sales averaged 110 acres (ranging from 28 to 291 acres) and for practical purposes all could be considered infinite populations given the small plot size. When the population size (N) is >300, 1/N in the finite population equation (Equation 2) becomes effectively 0, and the finite and infinite population formulas are essentially equivalent. Only small differences in the percentage of correctly classified CVs between finite or infinite population estimation equations were observed. In general for both finite and infinite equations, stands with small diameter trees and low TPA exhibited larger variance and required more samples than stands with large diameter trees having high TPA (Tables 727602760–10). Fewer plots were required to achieve a desired precision in stands of larger diameter trees and high TPA. Accessing Stand Weight Variability Class If the timber cruiser can visually determine the plot weight variance class (low, medium, or high), a greater percentage of correctly classified plot sample sizes can be obtained using Equation 6 or Tables 72760–9. In the field, timber cruisers must be trained to identify low (SD ≤ 25.0 tons/acre), medium (25.1 tons/acre < SD ≤ 40.0 tons/acre), and high (SD ≥ 40.1 tons/acre) variance classes. Consistency of QMD and TPA among plots within a stand can be used as a rule of thumb for classifying variability. A low variability stand will have a consistent number of trees per plot, and diameters of tally trees should be tightly grouped. Product classification, species, and merchantable heights should also be consistent among plots. High variability stands exhibit highly variable diameters and numbers of trees. These stands may have scattered groups of merchantable trees or tree species and highly variable merchantable heights. Average variability stands will fall in between the low and high variability extremes and will most likely represent the largest proportion of stands. Tracts composed of multiple stands should be stratified by stand and an optimum allocation of plots to each stratum made to take advantage of the reduced sample size afforded by optimum allocations (Freese 1962, Cochran 1977, Schultz et al. 2006). Alternative Plot Sizes Because CV decreases with increasing plot size, care should be taken in using the models developed here on inventories with plot sizes other than 0.10 acre. A method for adjusting CV from one plot size to another has been suggested by Freese (1984) and is given in Equation 8.   where s1 and CV1 are the SD and coefficient of variation of plot size 1, s2 and CV2 are the SD and coefficient of variation of plot size 2, P1 is plot size 1, and P2 is plot size 2. Conclusion Current forest inventory databases may be used to customize/update CV regression Equations 6 and 7 for estimating required inventory sample sizes. Equations correctly estimate the number of 0.10-acre plots (±10% allowable error at the 90% confidence level for total tonnage) to within 15 plots of actual, 77–82% of the time, and to within 30 plots of actual, 99–100% of the time. Ideally, inventory databases should be used to monitor stand variance patterns and update CV prediction equations on a continuous basis. The CV prediction equations are based on easily observed stand attribute variables, and their use can eliminate costs associated with installing preliminary plots and errors resulting from inappropriate percent sampling or rule-of-thumb methods not using preliminary plots. Rules of thumb usually result in excessive oversampling of stands exceeding 30 acres. Percent sampling rules typically (not always) result in extreme oversampling: the larger the acreages, the more extreme the oversampling. The equations may also be helpful during initial cruise design for estimating numbers of required plots, cruise time, and cost. Endnote 1. Download the worksheet from www.timbercruise.com/Downloads/Utility/TCruiseStatisticsOptimumAllocation.xlsm. Literature Cited Borders B.E. Harrison W.M. Clutter M.L. Shiver B.D. Souter R.A.. 2008. The value of timber inventory information for management planning. Can. J. For. Res . 38: 2287– 2294. Google Scholar CrossRef Search ADS   Cochran W. 1977. Sampling techniques , 3rd ed. John Wiley & Sons, New York. 428 p. Freese F. 1962. Elementary forest sampling . USDA For. Serv., Agri. Handbk. No. 232, Washington, DC. 91 p. Freese F. 1984. Statistics for land managers . Paeony Press, Jedburgh, Scotland. 176 p. Hamilton D.A. 1979. Setting precision for resource inventories: The manager and the mensurationist. J. For . 77( 10): 667– 670. F4 Tech. 2010. T-Cruise reports . F4 Tech, Tallahassee, FL. SAS Institute, Inc. 2009. SAS version 9.0 . SAS Institute, Inc., Cary, NC. Schultz E. Matney T. Evans D. Fujisaki I.. 2006. A Landsat stand basal area classification suitable for automating stratification of forest into statistically efficient strata. In Proc. of the 1st international conference on object-based image analysis, 2006 July 4–5, Salzburg University, Austria, Lang S. Blaschke T. Schopfer E. (eds.). Int. Soc. Photogramm. Remote Sens . XXXVI-4/C42. 6 p. Skidmore J. 2010. Forest management manual of the Mississippi Forestry Commission: 2010 edition . Mississippi Forestry Commission, Internal Document, Jackson, MS. 139 p. World Wide Heuristic Solutions, Inc. 2010. T-Cruise and T-Cruise Mobile, version 5.0 . World Wide Heuristic Solutions, Inc., Starkville, MS. Copyright © 2015 Society of American Foresters
TI - Estimation of Forest Inventory Required Sample Sizes from Easily Observed Stand Attributes
JF - Forest Science
DO - 10.5849/forsci.12-630
DA - 2015-02-01
UR - https://www.deepdyve.com/lp/springer-journals/estimation-of-forest-inventory-required-sample-sizes-from-easily-OpI0ahUmsL
SP - 123
EP - 127
VL - 61
IS - 1
DP - DeepDyve
ER -