United States

Department

of Agriculture

Forest Service

Rocky Mountain

Research Station

General Technical

Report RMRS-GTR-66

October 2000

Annual Design-Based

Estimation for the

Annualized Inventories

of Forest Inventory and

Analysis: Sample Size

Determination

Hans T. Schreuder

Jin-Mann S. Lin

John Teply
Rocky Mountain Research Station

324 25th Street

Ogden, UT 84401

Abstract __________________________________________

Schreuder, Hans T.; Lin, Jin-Mann S.; Teply, John. 2000. Annual design-based estimation for the

annualized inventories of forest inventory and analysis: sample size determination. Gen. Tech.

Rep. RMRS-GTR-66. Ogden, UT: U.S. Department of Agriculture, Forest Service, Rocky Mountain

Research Station. 3 p.

The Forest Inventory and Analysis units in the USDA Forest Service have been mandated by

Congress to go to an annualized inventory where a certain percentage of plots, say 20 percent, will be

measured in each State each year. Although this will result in an annual sample size that will be too small

for reliable inference for many areas, it is a sufficiently large sample to draw annual inferences for large

areas in terms of what is there and how much has changed on an annual basis. This is extremely useful

in estimating the state of the forest in the country over time in terms of health, timber supply, and so forth.

The formula for estimating sample size required is: n ts p = ( ) 22 22 / θ

ˆ where s2 is the estimated

variance for parameter estimates ˆ

θ or θθ θ

ˆˆ ˆ = 2 1 − , and p is the desired level of precision expressed as

percent of ˆ

θ . We conclude that for p = 5 percent, based on sampling 1 ha plots on all National Forests

in Oregon and Washington, that a sample size of 9,400 plots with a corresponding area of 22,560,000

ha, about the size of the State of Idaho, is sufficient to draw annual inferences. Alternatively, an area

of 1,000,000 ha requires a sample size of 417 plots. Clearly p = 5 percent is a tough specification. If we

go to p = 10 percent, which may be adequate for many purposes, we require one-fourth the sample size

above, or 2,350 plots representing an area of about 5,640,000 ha, about the size of West Virginia (24,070

square miles). With this specification, a sample of 2,350 plots would give adequate precision.

Keywords: reliable probabilistic estimation, sample size determination, area required, key parameters

Authors _____________________________

Hans T. Schreuder is a Mathematical Statistician, Rocky Mountain

Research Station, USDA Forest Service, located at the Natural

Resources Research Center, 2150 Centre Avenue, Building A, Fort

Collins, CO 80526.

Jin-Mann S. Lin is a Graduate Research Assistant, Department of

Statistics, Colorado State University, Fort Collins, CO.

John Teply, formerly Forester (retired), Region 6, Portland, Oregon.
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-66. 2000 1

Introduction ____________________

The USDA Forest Service has been mandated by

Congress to go to an annualized inventory (Agricultural Education Reform Act of 1998), measuring 20

percent of the plots in each state each year. The plot

used is a 1 ha circular plot subsampled by four 1⁄24 acre

circular subplots. Forest Inventory and Analysis (FIA)

will be using a simple average of measured plots,

called the moving average, to publish statistics for

each state every 5 years. Although mandated to measure 20 percent per year it appears that this percentage will only be measured each year in the South; other

areas may only have 14 or 10 percent measured each

year. Multiple imputation and related model-based

methods have been proposed to predict values for plots

not remeasured in a given year to generate a cleanlooking database that is easy to use by user groups.

Schreuder and Reich (1998) note that public databases should be defensible even in court, and it would

be better to draw inferences only on actual data using

design-based (also called probabilistic) methods until

procedures such as multiple imputations have been

thoroughly tested.

This note recommends sample sizes and corresponding areas that are large enough so that probabilistic

inferences can be safely drawn without resorting to

models.

Methods _______________________

By design-based inference we typically mean that

inference is assumed to be essentially model-free—

essentially, because one can argue that any kind of

inference assumes some underlying model. In general

there are no design-based alternatives to the moving

average method right now given the limitations of

sample size, and we should encourage proper use of

model-based methods by various user groups that may

want to make their own assumptions to generate the

complete data sets they want to use. Because results

are likely to be contested bitterly, we need the best

defensible estimates possible.

The problem then is: We have x percent of the plots

measured each year, presumably x = 20 percent. For

the last year of data, what sample size is needed to

generate estimates of acceptable precision for all variables of interest? We assume we have t consecutive

years of data, where t ≤5. This is where the idea of a

simple average is proposed using the data from the t

years of the annualized inventory. Basically then, if

we have t years of data with t ≤5, using the simple

average, we generate estimates for the period covered

by the t years. This approach will be used at this time

for publishing 5 year state reports by FIA. But users

will use annual data. To do so from a scientifically

satisfactory point of view we ask: For what sample

sizes or areas can we use a straight probabilistic

approach?

We want to determine that sample size n that will

give us the following desirable characteristics in estimating a parameter θ :

1. An estimate of θ , 

ˆ

θ , such that the standard error

(se) of 

ˆ

θ is less than or equal to p1 percent of 

ˆ

θ .

2. Estimates of 

∆ ∆ θθθ θ , ˆˆ ˆ = 2 1 − such that the se of

 

∆ ˆ

θ is less than or equal to p2% of 

∆ ˆ

θ .

3. For area and area change estimation we may be

able to use regression estimators 

ˆ

θregr where the

sample size will depend on the strength of the

correlation between ground (y) and remote sensing information (x) on the 1-ha plots.

The standard formula for sample size determination

is (Schreuder and others 1993): 

n ts p = ( ) 22 2 2 / ˆ

θ

where t is the tabulated t-value for the desired p level,

s

2 is the estimated variance for parameter estimates 

ˆ

θ

or 

ˆˆ ˆ θθ θ = 2 1 − , and p is either p1 or p2. Similarly for

the regression estimator in 3 above:

 

n ts p = ( )( ) − ( ) regr

22 2 2 2 1 ˆ / ˆ θ ρθ where r is the

correlation between y and x.

Annual Design-Based Estimation

for the Annualized Inventories of

Forest Inventory and Analysis:

Sample Size Determination

Hans T. Schreuder

Jin-Mann S. Lin

John Teply
2 USDA Forest Service Gen. Tech. Rep. RMRS-GTR-66. 2000

Table 1—Summary listing of smallest and largest coefficients of variation with required acreages for p = 0.05 for key parameters.

CV Hectares CV Hectares

Variable smallest (acres) Forest largest (acres) Forest

G 55 1,171,200 Mt. Hood 94 3,369,600 Olympic

(2,928,000) (8,424,000)

N 68 1,785,600 Umpqua 118 5,258,400 Wallowa-

(4,464,000) (13,146,000) Whitman

M 110 4,617,600 Siuslaw 176 11,923,200 Malheur

(11,544,000) (29,808,000)

Ns 55 1,152,000 Rogue River 112 4,800,000 Wallowa-

(2,880,000) (12,000,000) Whitman

Ms 107 4,406,400 Umpqua 159 9,753,600 Wallowa-

(11,016,000) (24,384,000) Whitman

∆G 58 1,286,400 Malheur 122 5,683,200 Gifford

(3,216,000) (14,208,000) Pinchot

∆N 62 1,468,800 Willamette 95 3,475,200 Wallowa-

(3,672,000) (8,688,000) Whitman

∆M 103 4,099,200 Mt. Hood 242 22,425,600 Malheur

(10,248 ,000) (56,064,000)

∆Ns 48 902,400 Siskiyou 106 4,329,600 Wallowa-

(2,256,000) (10,824,000) Whitman

∆Ms 104 4,128,000 Gifford Pinchot 200 15,379,200 Malheur

(10,320,00) (38,448,000)

For determination of 1 above, we will concentrate on

current basal area (G), number of trees excluding

seedlings (N), number of trees including seedlings

(Ns), mortality (M), and mortality including seedlings

(Ms). These variables represent key variables of interest as well as some (M and Ms) that are likely to be as

variable as any in the list of parameters of interest. All

plots will be considered as measured, regardless of

the fact that some plots may be partially or entirely

nonforest.

For 2 above we need measured plots at two times

close together to be able to estimate se (∆ 

ˆ

θ ) for a period

mimicking the annualized inventory time period of

1 year as much as possible, concentrating on ∆G, ∆N,

∆M, ∆Ns, and ∆Ms. These variables represent key

variables of interest as well as some (∆Mand ∆Ms) that

are likely to be as variable as any in the list of

parameters of interest. We assume all plots to be

measured 2 years apart even though some were

remeasured over shorter and longer (from 1 to 3 years)

periods. Paired differences are used for change

calculations.

There are no data available yet for 3 above.

To determine sample size for 1 above, we used the

entire data set of 1 ha plots available for Region 6 of the

Forest Service, a total of 10,720 plots, and also used

subsets of these data for each National Forest. We also

used the one available panel of remeasured data sets

available for most forests for recent change in the

parameters of interest. This data set was used because

it is the most intensive data available for an area,

recent 2 year change data are available for a subset of

the plots, and the same basic 1 ha primary sampling

unit (psu) is used currently by FIA (FIA had used other

plot and sample designs in the past; Birdsey and

Schreuder 1992). Plots in Region 6 were established on

a 2.74 km (1.7 mile) grid, with a 1 ha psu subsampled

at five locations with the following subplots: the whole

1 ha plot for trees larger than 121.9 cm (47.9 inches) in

the western part of the State and larger than 81.0 cm

(31.9 inches) in the eastern part of the State, 0.076 ha

(

1⁄5 acre) for trees between 32.9 and 121.7 cm (13.0-47.9

inches) in the west and between 32.9 and 81.0 cm

(13.0-31.9 inches) in the east, .020 ha (1⁄24 acre) for

trees between 7.6 and 32.8 cm (3.0-12.9 inches), and

0.004 ha (1⁄100 acre) for seedlings (trees less than 7.4 cm

or 3 inches in d.b.h.) (Max and others 1996).

The results are likely to be somewhat conservative

because a grid sample as used is often more efficient

than the simple random sampling assumed for variance calculations. We also ignored possible stratification into National Forests and the fact that in wilderness areas the grid intensity was on a 3.4 mile grid

rather than the 1.7 mile grid used elsewhere. Both the

stratification and the not weighting of the 3.4 mile grid

plot locations can be safely ignored for our purposes.

Results are shown in table 1. For G, N, M, Ns, and Ms,

these results are intuitively reasonable. The within

the 1 ha plot sampling in Region 6 is geared to larger

trees to some degree (three subplot sizes with larger

subplots for the larger trees) so that one would expect

generally to have better efficiency in estimating

G relative to N. This is true for all forests. Number

of trees including seedlings, Ns, Mortality, M, and
USDA Forest Service Gen. Tech. Rep. RMRS-GTR-66. 2000 3

mortality including seedlings, Ms, should be estimated

less efficiently because they are less frequent or more

highly peaked in occurrence. For a 5 percent precision

criterion, the required sample size of G estimates

varies from 480 for Rogue River to 1,404 for the

Olympic Forest. For N they vary from 744 for the

Umpqua to 2,236 for the Wallowa-Whitman. For Ns

they vary from 480 for Rogue River to 2,000 for WallowaWhitman, for M from 1,924 for the Siuslaw to 4,968 for

the Malheur, and for Ms from 1,836 for the Umpqua to

4,064 for the Wallowa-Whitman. Based on these results it is then clear that a sample of size 5,000

(rounded up from 4,968 for convenience of recollection)

is quite adequate for sampling for these parameters of

interest given the 5 percent criterion for precision.

This translates to an area of 6,000 x 5,000 = 30,000,000

acres or about 12,000,000 ha.

The results for ∆G, ∆N, ∆M, ∆Ns, and ∆Ms are less

consistent in that ∆G is usually estimated less efficiently than ∆N. This is plausible but not as readily

obvious as for G verses N because of the possible

dynamics involved over time. Mortality can fluctuate

considerably over time, and this shows up clearly in

both M and Ms. For ∆G sample sizes required vary

from 536 for the Malheur to 2,368 for the Gifford

Pinchot, from 612 for the Willamette to 1,448 for the

Wallowa-Whitman for ∆N, from 1,708 for the Mt. Hood

to 9,344 for the Malheur with ∆M, from 376 for the

Siskiyou to 1,804 for the Wallowa-Whitman with ∆Ns,

and from 1,720 for the Gifford-Pinchot to 6,408 for the

Malheur forest with ∆Ms. Based on these results then

it is clear that a sample of size 9,344 (say 9,400 for ease

of recollection) is quite adequate for estimating these

parameters, corresponding to an area of size 6,000 x

9,400 = 56,400,000 acres or about 22,560,000 ha, about

the size of the State of Idaho. Clearly less stringent

criteria than p = 5 percent may be used frequently

resulting in a much smaller sample size requirement.

With p = 10 percent for example, a sample of size 2,350

would be adequate, about the size of West Virginia.

References _____________________

Birdsey, R.A.; Schreuder, H.T. 1992. An overview of forest inventory

and analysis estimation procedures in the eastern U.S.—with an

emphasis on components of change. USDA Forest Service, Rocky

Mountain Forest and Range Experiment Station. Gen.Tech. Rep.

RM-214. 11 p.

Max, T.A; Schreuder, H.T.; Hazard, J.W.; Teply, J; Alegria, J. 1996.

The Region 6 Vegetation Inventory and Monitoring System.

USDA Forest Service, Pacific Northwest Station. Res. Pap. PNWRP-493. 22 p.

Schreuder, H.T.; Reich, R. 1998. Data estimation and prediction for

natural resources public data. USDA Forest Service, Rocky

Mountain Research Station. Res. Note RMRS-RN-2. 5 p.

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