Sampling designs and allocations yielding minimum cost estimators for mountain pine beetle loss assessment surveys

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SAMPLING DESIGNS AND ALLOCATIONS
YIELDING MINIMUM COST ESTIMATORS
FOR MOUNTAIN PINE BEETLE LOSS
ASSESSMENT SURVEYS

Report No. 83-3

3400

November 1982

SAMPLING DESIGNS AND ALLOCATIONS YIELDING MINIMUM COST
ESTIMATORS FOR MOUNTAIN PINE BEETLE LOSS ASSESSMENT SURVEYS

1/ 2/
Nancy X. Sharpnack— and John Wong-'

U.S.D.A. Forest Service

ABSTRACT

Tables of optimum sample size for each of three stages are presented for
estimating mountain pine beetle loss in ponderosa and lodgepole pine forests
in the western United States . These are listed for varying levels of
precision and are based on data collected during surveys conducted between
1977 and 1980.

INTRODUCTION

Information from previous mountain pine beetle loss assessment surveys has
provided the opportunity to improve future sampling designs and allocations.
The material for this study was obtained from several surveys of ponderosa and
lodgepole pine mortality that were conducted by the Forest Service in the
western Regions between 1977 and 1980. These surveys provided a sufficient
data base from which variance components were estimated, costs were assessed
and various strategies could be evaluated.

The procedures used in the previous surveys and their results have been
documented in various reports (Hostetler and Young 1979, Bennett and Bousfield
1978, Bennett et al . 1980, and Lister and Young 1981). The steps involved:

1. Aerial sketchmapping.

2. Stratification based on intensity of mortality per acre as
determined from the sketchmapping.

3. Random sample of aerial photos within each stratum (Stage 1).

4. Subsample of aerial photos chosen with probabilities proportional
to photo interpreted dead tree counts (Stage 2).

5. Sample of ground plots selected with probabilities proportional
to dead tree counts (Stage 3).

1 Mathematical Statistician, Pacific Southwest Forest and Range
Experiment Station, Berkeley, California.

2 Formerly Mathematical Statistician, Forest Pest Management, Methods
Application Group, Davis, California. Presently Mathematical
Statistician, Information Systems Management Staff, Pacific Southwest
Region, San Francisco, California.

2

The objectives of this study were to evaluate whether or not stratification
is effective, and to find those allocations of sampling units between the
three stages which costs the least for a fixed percent standard error. The 20%
standard error of the estimate is a requirement specified in the Forest Insect
and Disease Information System Implementation Plan (FIDIS) (Ciesla and
Yasinski 1980). We also considered the optimum allocation for 10%, 15% and
25% standard errors whenever these were attainable. We were constrained to
considering 40 and 90 acre photo plots and 2.5 acre ground plots since all of
the usable data available from the previous surveys fell into these
categories. We considered the effects of taking one, two or three ground
plots per photo plot in the third sampling stage where traditionally two plots
have been taken.

METHODS

We used data from four previous mountain pine beetle surveys (Table 1).

Cost factors applicable to the analysis were obtained from data furnished by
Dayl e Bennett (personal communication) in Region 3 and confirmed by Richard
Myhre of the Rocky Mountain Forest and Range Experiment Station, Ft. Collins,
Colorado. These costs are in units of the number of person hours to do photo
interpretation and obtain ground measurements (Table 2). The cost estimates
include plot set-up time and travel time. As expected, the time required to
make a ground measurement is significantly higher than to interpret a photo,
so that it is evident that a major cost element in sampling is associated with
the time spent on the ground.

For each of the previous surveys, the variance components associated with
each of the three stages were computed. From these it was possible to
estimate the variance which would have resulted for any allocation of sampling
units of interest. The estimate of total variance was computed using the
following formula:

v(y) = (N-n/N) (l/n)v(y1) + ( 1/m) v(y2 ) + (l/ml)v(y3)

in which

y = the estimate of total mortality

v(y) = the estimate of the variance of the estimate of total
mortal i ty

v(y^ ) = the estimate of stage i variance
N = the total number of possible photo plots in the stratum
n = the number of photo plots to be sampled at Stage 1
m = the number of photo plots to be subsampled at Stage 2
1 = the number of ground plots per photo plot at Stage 3

3

TABLE 1. Summary of Mountain Pine Beetle data sets used in generating
estimates of variances for this study.

SURVEY

TREE

SPECIES

STRATUM

: number of
photos
(Stage 1)

:photo
plot
si ze
( acres)

:number
photos
subsampl ed
(Stage 2)

1978

LP

L

< 4.9 trees/acre

94

40

24

Beaverhead/

M

T.0-9.9

24

16

Gal 1 ati n

H

>_ 10.0

26

16

1979

LP

L

<10.0 trees/acre

174

40

18

Montana

M

10.0-41.0

87

19

H

>41.0

81

20

1978

PP

M

109

90

50

Black Hills

H

77

50

1979

PP

Conti guous

175

90

20

Col orado
Front Range

TABLE 2. Cost per plot for photo interpretation and ground measurements (in
person hours)

Ground Plot

Photo Plot Size Size

Host Species Stratum 40 Acres 62.5 Acres 90 Acres 2.5 Acres

LP

L

0.39

0.53

0.73

10.60

M

0.48

0.66

0.85

12.40

H

1.04

1.11

1.40

13.10

PP

L

0.33

0.39

0.49

10.60

M

0.40

0.51

0.63

12.40

H

0.65

0.81

1.01

13.10

4

The percent standard error of the estimate is computed as 100 v(y)/y.
Formulas for computing variance components are given in the appendix.

A computer program used iterative methods to solve for a fixed percent
standard error while varying the Stage 2 allocation for numerous values of n
and for 1 = 1, 2, and 3. For each solution, the cost was computed with the
formul a:

C = n CpI + ml CG

in which

C = total cost for this allocation

Cpj = cost of interpreting a photo plot
CG = cost of measuring a ground plot
the minimum cost was then selected from the possible combinations of n, m, 1.

In order to combine strata, weights were used. The weight for each stratum
was the ratio of the expected number of plots that would have fallen in the
stratum had no stratification been imposed and the actual number of plots that
were sampled in the stratum in the previous survey. The variance component
for stage i, v was computed using the formula:

in which

v. = the variance component for stage i
k = the number of strata

w. . = the weight for stage i, stratum j

■ vJ

v.. = the variance component for stage i, stratum j

* J

Likewise, since the costs were dependent upon whether the plots were from
the light, medium, or heavy stratum, weighted costs were used in the combined
strata analysis.

RESULTS

The results are summarized (Tables 3 through 6) to provide a readily usable
tool with which to better plan future surveys. For each host species, those
sampling allocations which yielded the minimum cost for the various levels of
precision are provided. In addition, the costs for allocations other than the
optimum are presented. This is for planning purposes. Often, cost is not the

5

TABLE 3. Optimal Sampling Allocations for the 1978 Beaverhead-Gallatin
Survey in Lodgepole Pine.

STRATUM

% Std.
Error

One Ground
Per Stage 2

Plot

Plot

Two Ground Plots

Per Stage 2 Plot

Three Ground
Per Stage 2

PI ots
Plot

No. of
PI

Plots

No. of
PI Sub
Plots

Cost
- ($)

No. of
PI

Plots

No. of
PI Sub
PI ots

Cost
- ($)

No. of

PI

Plots

No. of
PI Sub
Plots

Cost
- ($)

L

10%

300

21

339

450

14

471

400

13

569

M

200

19

331

250

12

416

250

10

491

H

200

18

574

250

16

679

250

13

770

TOTAL

700

58

1244

950

42

1566

900

36

1830

NO STRATIFICATION

300

28

517

350

18

626

350

15

733

L

15%

150

9

154

150

7

207

200

6

258

M

80

8

137

60

6

177

60

5

215

H

100

13

274

150

7

339

100

7

379

TOTAL

330

30

565

360

20

723

360

18

852

NO STRATIFICATION

140

12

241

140

8

282

150

7

331

L

20%

80

5

84

70

4

112

70

3

123

M

50

4

73

40

3

93

40

3

131

H

60

7

154

60

4

179

70

4

203

TOTAL

190

16

311

170

11

384

180

10

457

NO STRATIFICATION

80

7

130

90

4

158

90

4

184

L

25%

50

3

51

50

2

62

50

2

83

M

20

3

47

30

2

64

20

2

84

H

40

4

100

40

3

115

40

2

130

TOTAL

110

10

198

120

7

241

110

6

297

NO STRATIFICATION

50

5

84

50

3

102

50

2

120

6

TABLE 4. Optimal Sampling Allocations for the 1979 Montana Survey in
Lodgepole Pine.

STRATUM

% Std .
Error

One Ground
Per Stage 2

PI ot
Plot

Two

Per

Ground Plots
Stage 2 Plot

Three Ground
Per Stage 2 1

PI ots
Plot

No. of
PI

PI ots

No. of
PI Sub
Plots

Cost
- ($)

No. of
PI

Plots

No. of
PI Sub
Plots

Cost
- ($)

No. of

PI

Plots

No. of
PI Sub
PI ots

Cost
- ($)

L

10%

500

no

1360

500

70

1678

500

56

1975

M

600

177

2480

700

104

2913

700

81

3339

H

400

40

944

400

34

1318

500

29

1676

TOTAL

1500

327

4784

1600

208

5909

1700

166

6990

NO STRATIFICATION

900

114

1904

1100

77

2519

1200

66

3113

L

15%

250

48

606

300

30

752

250

25

892

M

250

81

1121

300

47

1315

300

37

1508

H

150

20

421

200

15

594

200

14

754

TOTAL

650

149

2148

800

92

2661

750

76

3154

NO STRATIFICATION

400

51

852

500

34

1127

500

30

1392

L

20%

110

28

339

80

18

413

100

14

484

M

50

45

634

150

27

744

200

20

855

H

60

11

238

100

9

335

120

8

426

TOTAL

220

84

1211

330

54

1492

420

42

1765

NO STRATIFICATION

250

28

480

250

20

636

300

17

785

L

25%

70

18

218

70

11

260

90

9

321

M

100

29

407

120

17

479

150

13

555

H

60

7

152

70

5

215

80

5

275

TOTAL

230

54

777

260

33

954

320

27

1151

NO STRATIFICATION

150

18

307

200

12

408

200

11

503

7

TABLE 5.

Optimal Sampling
Ponderosa Pine*

Allocations for

“ the

1978 Black

Hi 1 1 s

Survey in

STRATUM

% Std.

One

Ground

Plot

Two

Ground Plots

Three

Ground

Plots

Error

Per Stage 2

Plot

Per Stage 2 Plot

Per Stage 2 Plot

No. of

No. of

Cost

No. of No. of

Cost

No. of

No. of

Cost

PI

PI Sub

- ($)

PI

PI Sub-

($)

PI

PI Sub-

($)

PI ots

PI ots

PI ots

PI ots

PI ots

PI ots

M

10%

1400

62

1646

1400

37

1797

1500

27

1944

H

400

52

1084

400

32

1246

500

23

1407

TOTAL

1800

114

2730

1800

69

3043

2000

50

3351

M

15%

700

35

870

800

18

953

800

14

1027

H

200

25

530

200

15

607

200

12

687

TOTAL

900

60

1400

1000

33

1560

1050

26

1714

M

20%

400

22

528

500

11

577

500

8

621

H

110

15

309

120

9

355

130

7

397

TOTAL

510

37

837

620

20

932

630

15

1018

M

25%

300

13

347

300

8

378

300

6

411

H

70

10

201

80

6

230

80

5

260

TOTAL

370

23

548

380

14

608

380

11

671

^Strata were not combined since no data were available for the light condition.

8

TABLE 6. Optimal Sampling Allocations for the 1979 Colorado Survey in
Ponderosa Pine.

STRATUM % Std.

Error

One Ground
Per Stage 2

Plot

Plot

Two Ground Plots

Per Stage 2 Plot

Three Ground Plots
Per Stage 2 Plot

No. of
PI

PI ots

No. of
PI Sub
PI ots

Cost
- ($)

No. of
PI

PI ots

No. of
PI Sub-
Pi ots

Cost

($)

No. of
PI

PI ots

No. of
PI Sub-
Pi ots

Cost

($)

10%

1400

352

5240

1600

228

6654

1800

186

8044

15%

600

165

2421

800

104

3079

900

85

3725

20%

400

91

1378

400

61

1763

500

49

2135

25%

200

62

894

200

35

894

300

38

1130

only consideration in allocation of resources. Scheduling, amount of training
required, travel restrictions, etc., often play a part. Therefore, it is
important to be able to know how much will be sacrificed in having a design
which is suboptimum in some respect.

Some of the results were consistent over all of the previous survey data
sets. In no case was stratification beneficial. This is probably because in
variable probability sampling the measurement upon which the variance depends
is the ration of the next stage measurement to the previous stage measurement.
This would not be necessarily more homogeneous within strata defined by
intensities of mortality. In every case, sampling with only one ground plot
per photoplot in the final stage was best.

Results for ponderosa pine are somewhat limited in that for the Black
Hills, data from only two strata were available and for the Colorado Front
Range survey, only one stratum was usable. The results are summarized in
Tables 5 and 6. Since data were incomplete, we were not able to do the
analysis for no stratification.

9

DISCUSSION

The variances in any two surveys will not be the same. Much depends upon
the geographic location of the survey, the quality of the photos, the skill of
the interpreters, and the inherent variability in the population being
surveyed. Consequently, the results presented here should be used
conservatively as a guideline, not as an absolute rule.

It is felt that the results derived from these analyses are highly
dependent on the cost information used. If more precise answers are to be
obtained, more effort should be directed in the future to obtain and maintain
cost data. For each survey, a good estimate of costs could be obtained if the
total person hours spent doing photo interpretation and the total number of
photos interpreted were tallied, as well as the total person hours spent doing
ground work plus the number of plots measured on the gorund were recorded.

It is not necessary to take more than one plot in the final stage of
sampling in order to estimate the standard error of the estimate of total for
any single survey; however, it is impossible to evaluate the variance
component in the final stage for the optimization of future surveys if only
one plot is sampled. For this reason, it is often desirable to consider only
those designs with at least two ground plots per photo plot in the final stage
even though this may not be the most cost effective. Also, in choosing a
viable alternative for a particular survey, some provision should be made to
allow for missing or unusable data. This is another reason why it may be
better to use more than one ground plot per cell in the final sampling stage.

REFERENCES

Bennett, D.D., and W.E. Bousfield. 1979. A pilot survey to measure annual
mortality caused by the mountain pine beetle in lodgepole pine on the
Beaverhead and Gallatin National Forests. USDA For. Serv., Northern
Region, Rep. No. 79-20. 13 pp.

Bennett, D.D., and W.E. Bousfield, M.D. McGregor, and K.E. Gibson. 1980.
Evaluation of multistage sampling techniques to measure lodgepole pine
mortality caused by mountain pine beetle in Montana, 1979. USDA For.

Serv., Northern Region, Rep. No. 80-13. 11 pp.

Ciesla, W.M., and F.M. Yasinski. 1980. Forest insect and disease information
systems (FIDIS) implementation plan. USDA For. Serv., FIDM/MAG Rep. No.
80-3. 10 pp.

Hosteller, B.B., and R.W. Young. 1979. Estimation procedures for determining
annual tree mortality caused by the mountain pine beetle. USDA For. Serv.,
Rocky Mtn. Region, Tech. Rep. R2-20. 25 pp.

Klein, W.H., D.D. Bennett, R.W. Young. 1979. A pilot survey to measure
annual mortality of lodgepole pine caused by the mountain pine beetle.

USDA For. Serv., FIDM/MAG Rep. No. 78-4. 15 pp.

Lister, C.K., and R.W. Young. 1981. 1979 Colorado forest range mountain pine

beetle survey. USDA For. Serv., Rocky Mtn. Region, Tech. Rep. R2-22. 19 pp.

10

APPENDIX

The variance components of the unbiased estimator for the number of dead
trees are defined in this section. Let V(Y^) be the variance component
associated with the first stage sampling, then V(Y^) is as follows:

00

_ N1 (h* ~ n)

N 2- » “ Z.

" ^ - n v

J

A

A - 1

Let V ( Y^ ) be the variance component associated with the second stage
sampling, the V ( Y^ ) is as follows:

MV

a.

I

Let V (Y^) be the variance component associated with the third stage

sampling, the V ( Y3 ) is as follows:

V

k

k - L'
lJ - \

The definitions for the variables applicable to the above formulas as follows
N = total possible number of PI plots

N' = total number of PI plots taken

x. = total PI for plot i.

x = mean of Pi values for all photos

Vjk = ground measurement for photo j and ground plot k

Pjk = probability of ground plot k within photo plot j

l

L'

U

Pi*-

11

number of ground plots samled per photo plot

total possible number of ground plots/photo plot

probability for selecting photo plot j

number of photo plots sub-sampled during second stage in
original survey

number of photo plots to sub-sample during second stage for next
time (parameter to be optimized)

number of ground plots/photo plot to sample next time (parameter
to be optimized)

number of photos in first stage next time (parameter to be
optimized)