United States

Department of 

Agriculture

Forest Service

Pacific Northwest

Research Station

Research Note

PNW-RN-532

June 2002

A Comparison of Two

Estimates of Standard Error

for a Ratio-of-Means Estimator

for a Mapped-Plot Sample

Design in Southeast Alaska

Willem W.S. van Hees

Comparisons of estimated standard error for a ratio-of-means (ROM) estimator

are presented for forest resource inventories conducted in southeast Alaska

between 1995 and 2000. Estimated standard errors for the ROM were generated

by using a traditional variance estimator and also approximated by bootstrap

methods. Estimates of standard error generated by both traditional and bootstrap methods were similar. Percentage differences between the traditional and

bootstrap estimates of standard error for productive forest acres and for gross

cubic-foot growth were generally greater than respective differences for nonproductive forest acres, net cubic volume, or nonforest acres. 

Keywords: Sampling, inventory (forest), error estimation.

Between 1995 and 2000, the Pacific Northwest Research Station (PNW) Forest

Inventory and Analysis (FIA) Program in Anchorage, Alaska, conducted a land

cover resource inventory of southeast Alaska (fig. 1). Land cover attribute estimates derived from this sample describe such features as area, timber volume,

and understory vegetation. Each of these estimates is subject to measurement

and sampling error. Measurement error is minimized through training and a

program of quality control. Sampling error must be estimated mathematically.

This paper presents a comparison of bootstrap estimation of standard error for 

a ratio-of-means estimator with a traditional estimation method. Bootstrap estimation techniques can be used when a complex sampling strategy hinders

development of unbiased or reliable variance estimators (Schreuder and others

1993.) Bootstrap techniques are resampling methods that treat the sample as 

if it were the whole population. Repeated samples are taken, with replacement,

from the original sample and the statistic of interest is calculated for each sample.

The average of the statistic over all the samples is the bootstrap estimator. 

This study was conducted to examine briefly whether or not traditional estimates

of standard error differed in some meaningful way from the bootstrap approximation. It was felt that the two estimates would produce substantially similar

results.

Willem W.S. van Hees is a research forester, Forestry Sciences 

Laboratory, 3301 C. St., Suite 200, Anchorage, AK 99503.

Abstract

Introduction
2

Figure 1—Southeast Alaska inventory region, 1995-2000.
3

In lieu of extensive simulations, multiple bootstrap estimates of standard error, along

with traditional estimates, were made for each of five population parameters in three

sampling units. These parameters, for unreserved USDA Forest Service lands, were

productive forest-land acres, nonproductive forest-land acres, nonforest acres, net

cubic-foot volume, and gross cubic-foot growth in the Chatham, Stikine, and

Ketchikan inventory units (van Hees 2001a, 2001b, 2001c). 

Land cover of the southeastern panhandle of Alaska (fig. 1) was sampled with a 

single-phase, unstratified, systematic grid sample (grid spacing was 4.8 kilometers).

Ground plots (1 hectare) were established at each grid intersection and were subsampled by a cluster of four, 7.3-meter, fixed-radius subplots. Three subplots were

sited, one each at 63.4 meters north, southeast, and southwest from the first, centrally located subplot. Each subplot was mapped for land cover (Scott and Bechtold

1995.)

Estimation of resource attributes such as area and timber volume used ratio-ofmeans estimators. The ratio-of-means estimator and associated traditional variance

estimate are defined as (Cochran 1977):

(1)

where

yi = the variable of interest on plot i,

xi = an auxiliary variable (such as area) on plot i that is correlated with yi, and

n = number of plots selected from the population, 

with variance

(2)

Zarnoch and Bechtold (2000) demonstrate use of this ratio estimator to generate

various population estimates from data collected by using a sampling design similar

to that described above. Essentially, the ratio estimate defined by equation (1) is

multiplied by some total, such as area, to estimate the amount of the total in the

condition of interest (yi ).

Methods

Sample Design

Ratio-of-Means 

Estimator 

n

Σyi

i=1 , ˆ

R = n

Σxi

i=1

1

V(Rˆ)= (Σyi

2 + ˆ

R2

Σxi

2 – 2 ˆ

RΣyi xi) . 

n(n–1) Σx

i

2

n
4

Table 1—Ratio-of-means (ROM) population estimates, traditional estimate of standard error, and bootstrap estimates of standard error bynumber of iterations, sampling unit, and population attribute

Estimated standard error 

Bootstrap 

ROM No.

Sampling unit/ population of Traditionala 500 1,000 3,000

population attribute estimate plots estimate iterations iterations iterations 

Ketchikan: 550

Productive forest, acres 1,405,362 58,036 61,359 57,410 56,880 59,002 58,449Nonproductive forest, acres 851,262 53,754 56,895 55,148 53,482 54,649 54,689Nonforest, acres 449,053 41,962 43,615 43,416 41,269 45,148 41,961Net cubic-foot volume 7,293,763,469 483,113,144 504,230,788 462,260,072 479,479,823 463,481,657 468,461,285 470,675,742 463,202,816Gross cubic-foot growth 63,725,479 3,250,717 3,516,024 3,378,918 3,461,131 3,328,462 3,414,326 3,446,209 3,400,647

Stikine: 520 

Productive forest, acres 1,085,711 60,713 62,704 58,648 63,772 57,690 62,517Nonproductive forest, acres 708,385 53,341 53,910 51,318 54,061 53,152 55,388Nonforest, acres 1,077,131 60,469 62,760 59,758 62,718 58,038 61,135 Net cubic-foot volume 5,437,872,712 456,690,704 462,024,168 457,544,019 450,882,246 429,174,042 453,095,411 453,530,368Gross cubic-foot growth 40,659,537 2,652,600 2,660,205 2,779,894 2,580,381 2,568,629 2,730,045 2,697,633

Chatham: 979

Productive forest, acres 1,302,152 72,813 74,370 69,716 74,637 70,059 71,547 71,726 Nonproductive forest, acres 709,450 56,555 56,060 60,671 56,542 56,008 58,272 59,047 Nonforest, acres 3,368,739 82,340 79,570 83,406 84,722 79,248 79,016 82,717 Net cubic-foot volume 7,561,291,112 540,235,099 534,119,374 519,603,815 541,402,469 558,017,539 515,248,933 535,130,404 536,971,963 551,209,556 568,524,342

Gross cubic-foot growth 59,953,509 3,803,595 3,856,939 3,654,326 3,786,265 4,034,644 3,728,864 3,856,871 3,863,259 3,929,279 4,036,473 

a Traditional estimate of standard error is from Cochran (1977).
Bootstrap estimation of the variance was accomplished by iterative random sampling, with replacement, of the plot list to generate a new plot list. Within each plot,

subplots were randomly selected, with replacement, to generate a new subplot list

for each plot. The ROM estimates of population totals were then recomputed. The

variance of the estimated totals provides an estimate of the sample variance. This

process was repeated 500 to 3,000 times. At least four bootstrap estimates, for each

of the five population parameters, in all three sampling units, were made by using

500 iterations. Greater numbers of iterations (1,000 and 3,000) were used fewer

times and for increasingly fewer population parameters. 

Although bootstrap samples are not independent, variance estimates resulting from

them are considered conservative approximations (Schreuder and others 1993).

Bootstrap and traditional estimates of standard errors are presented in table 1. In 

no case did the bootstrap estimate differ from the traditional estimate by more than

7.55 percent (table 2). Eleven of 92 observations were more then 5 percent different; the remaining 81 bootstrap estimates were less than 4.99 percent different from

the ROM estimate. The larger percentage differences (greater than + 5.5 percent)

are not specific to a particular inventory area or estimated population parameter

(figs. 2 through 6). 

5

Bootstrap Estimate

Results

Table 2—Differences between bootstrap and traditional estimates of standard errors as a percentage of the bootstrap estimate, by number of bootstrap iterations, sampling unit, and population attribute

Number of bootstrap iterations

Sampling unit/ 

population attribute 500 1,000 3,000

Percent

Ketchikan:

Productive forest, acres 5.42 -1.09 -2.03 1.64 0.71

Nonproductive forest, acres 5.52 2.53 -0.51 1.64 1.71

Nonforest, acres 3.79 3.35 -1.68 7.06 0.00

Net cubic-foot volume 4.19 -4.51 -0.76 -4.24 -3.13 -2.64 -4.30

Gross cubic-foot growth 7.55 3.79 6.08 2.34 4.79 5.67 4.41

Stikine:

Productive forest, acres 3.18 -3.52 4.80 -5.24 2.89

Nonproductive forest, acres 1.06 -3.94 1.33 -0.35 3.70

Nonforest, acres 3.65 -1.19 3.59 -4.19 1.09

Net cubic-foot volume 1.15 0.19 -1.29 -6.41 -0.79 -0.70

Gross cubic-foot growth 0.29 4.58 -2.80 -3.27 2.84 1.67

Chatham:

Productive forest, acres 2.09 -4.44 2.44 -3.93 -1.77 -1.52

Nonproductive forest, acres -0.88 6.78 -0.02 -0.98 2.95 4.22

Nonforest, acres -3.48 1.28 2.81 -3.90 -4.21 0.46

Net cubic-foot volume -1.15 -3.97 0.22 3.19 -4.85 -0.95 -0.61 1.99 4.98

Gross cubic-foot growth 1.38 -4.08 -0.46 5.73 -2.00 1.38 1.54 3.20 5.77
6

Ketchikan

Stikine

Chatham

Number of bootstrap iterations

Percentage difference

 6.00

 4.00

 2.00

 0.00

-2.00

-4.00

-6.00

500 500 500 1,000 3,000

Number of bootstrap iterations

Percentage difference

 6.00

 4.00

 2.00

 0.00

-2.00

-4.00

-6.00

-8.00

500 500 500 500 50 1,000 1,000 30 

Ketchikan

Stikine

Chatham

Figure 3—Differences between bootstrap and traditional estimates of standard errors as a percentage of the bootstrap estimate, by number of bootstrap iterations, for estimates of net cubic-foot 

volume on productive forest-land area within unreserved national forest lands, southeast Alaska.

Figure 2—Differences between bootstrap and traditional estimates of standard errors as a 

percentage of the bootstrap estimate, by number of bootstrap iterations, for estimates of 

productive forest-land area within unreserved national forest lands, southeast Alaska.
7

Number of bootstrap iterations

Percentage difference 6.00

 4.00

 2.00

 0.00

-2.00

-4.00

-6.00

Ketchikan

Stikine

Chatham

500 500 500 500 1,000 

8.00

Number of bootstrap iterations

Percentage difference

500 500 500 1,000 

 6.00

 4.00

 2.00

 0.00

-2.00

-4.00

-6.00

8.00

Ketchikan

Stikine

Chatham

Figure 4—Differences between bootstrap and traditional estimates of standard errors as a 

percentage of the bootstrap estimate, by number of bootstrap iterations, for estimates of nonproductive forest-land area within unreserved national forest lands, southeast Alaska.

Figure 5—Differences between bootstrap and traditional estimates of standard errors as a

percentage of the bootstrap estimate, by number of bootstrap iterations, for estimates of nonforest-land area within unreserved national forest lands, southeast Alaska.
8

For the response variables listed, there is little difference between the two estimates.

The largest percentage differences between error estimates are in the Ketchikan

unit. The forest resource of the Ketchikan unit is more variable than in the other

two units; it is ecologically transitional (approaching the southern extent of the Sitka

spruce/western hemlock zone), and logging has been more intensive and extensive

resulting in greater variety of age and size classes. Greater variability between

bootstrap and traditional estimates may indicate the number of iterations was low

for those estimates.

Relative rapidity of data processing is a significant client service consideration.

Rapid responses are important when providing clients with inventory results.

Although bootstrap estimates can be generated for any of the population estimates

FIA produces, the computational resources needed to generate those estimates for

each cell in all the output tables typical for FIA reporting exceed those needed for

ROM estimates. Use of the ROM estimator reduces production time dramatically. 

Kevin Dobelbower, Team Leader for Information Management, Anchorage, Alaska,

created and ran necessary computer programs for this study.

0.3048 meter = 1 foot 

0.0283 cubic meter = 1 cubic foot

0.4047 hectare = 1 acre 

1.609 kilometers = 1 mile

Discussion

Acknowledgments

English Equivalents

Figure 6—Differences between bootstrap and traditional estimates of standard errors as a 

percentage of the bootstrap estimate, by number of bootstrap iterations, for estimates of gross

cubic-foot growth on productive forest-land area within unreserved national forest lands, southeast Alaska.

Number of bootstrap iterations

Percentage difference 10.00

 8.00

 6.00

 4.00

 2.00

 0.00

 -2.00

 -4.00

 -6.00

Ketchikan

Stikine

Chatham

500 500 500 500 1,000 1,000 1,000 3,000
9

Literature Cited Cochran, W.G. 1977. Sampling techniques. 3rd ed. New York: John Wiley and 

Sons, Inc. 428 p.

Schreuder, H.T.; Gregoire, T.G.; Wood, G.B. 1993. Sampling methods for 

multiresource forest inventory. New York: John Wiley and Sons, Inc. 446 p.

Scott, C.T.; Bechtold, W.A. 1995. Techniques and computations for mapping plot 

clusters that straddle stand boundaries. Forest Science Monograph. 31: 46-61.

van Hees, W.W.S. 2001a. Summary estimates of forest resources on unreserved

lands of the Chatham inventory unit, Tongass National Forest, southeast

Alaska, 1998. Resour. Bull. PNW-RB-234. Portland, OR: U.S. Department of

Agriculture, Forest Service, Pacific Northwest Research Station. 12 p.

van Hees, W.W.S. 2001b. Summary estimates of forest resources on unreserved

lands of the Ketchikan inventory unit, Tongass National Forest, southeast Alaska,

1998. Resour. Bull. PNW-RB-233. Portland, OR: U.S. Department of Agriculture,

Forest Service, Pacific Northwest Research Station. 8 p.

van Hees, W.W.S. 2001c. Summary estimates of forest resources on unreserved 

lands of the Stikine inventory unit, Tongass National Forest, southeast Alaska,

1998. Resour. Bull. PNW-RB-232. Portland, OR: U.S. Department of Agriculture,

Forest Service, Pacific Northwest Research Station. 8 p

Zarnoch, S.J.; Bechtold, W.A. 2000. Estimating mapped-plot attributes with ratios 

of means. Canadian Journal of Forest Research. 30: 688-697.

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