United States 

Department of 

Agriculture 

Forest Service 

Southern 

Research Station 

Research Paper 

SRS-29 

A Pilot Sampling Design for 

Estimating Outdoor Recreation Site 

Visits on the National Forests 

SJ. Zarnoch, S.M. Kocis, H.K. Cordell, 

and D.B.K. English 
The Authors 

S. J1 Zarnoch, Mathematical Statistician, USDA Forest Service, Southern 

Research Station, Asheville, NC; S.M. Kocis, Recreation Planner, USDA 

Forest Service, Washington Office (detached), Star Route Box 300, 

Bridgeville, CA; H.K. Cordell, Project Leader and D.B.K. English, 

Research Social Scientist, USDA Forest Service, Southern Research 

Station, Athens, CA, respectively. 

October 2002 

Southern Research Station 

P.O. Box 2680 

Ashevitle, NC 28802 
A Pilot Sampling Design for Estimating 

Outdoor Recreation Site Visits 

on the National Forests 

S.J. Zarnoch, S.M. Kocis, H.K. Cordell, and D.B.K. English 

Abstract ranged from absolute knowledge to little more than guesses. 

Other national recreation surveys have been conducted by 

A pilot sampling design is described for estimating site visits to telephone, but none has included use information specific to 

National Forest System lands. The three-stage sampling design 

consisted of national forest ranger districts, site days within ranger the national forests. Many may be of questionable statistical 

districts, and last-exiting recreation visitors within site days. validity because they usually were taken at selected 

Stratification was used at both the primary and secondary stages. 

Ranger districts were stratified based on Bailey's ecoregions, while site 

days were stratified based on site type, season, and day type. 

Statistical methodology is presented to derive site-visit estimates at 

the site day, ranger district, and national levels. Results are presented 

to illustrate the magnitude of the site-visit estimates, their variability, 

and confidence intervals. With such_ infomation, an evaluation of the 

stratification variables is presented using the design effect and the 

relative hypothet~cal efficiency. Sample size analysis is performed to 

provide recommendations for future sample surveys to meet specified 

locations and targeted only one or, at best, a few types of 

recreation users. Additionally, there may be unresolved 

problems of nonrespondent bias and sample selection. 

Moreover, none has given much attention to the variability 

of such estimates, which is necessary to quantify precision 

and reliability. Cregoire and Buhyoff (1999) presented 

statistically based methodologies for estimating recreation 

use, and those methodologies should provide a good basis 

levels of precision. for future surveys. 

Keywords: National forests, outdoor recreation, sampling, site visits. 

Introduction 

Outdoor recreation is becoming increasingly important on 

the national forests of the United States. In many areas, 

forest managers are giving higher priority to recreation 

resources than to timber harvests. Although the U.S. 

Department of Agriculture Forest Service (USDA Forest 

Service) has conducted periodic inventories of forest timber 

resources for over 70 years, the Agency neither has 

developed nor used a statistically based, nationally applied 

analog for estimating the volume of recreation use on 

national forests. In large measure, this is due to the 

complexity of sampling issues. However, dependable, 

accurate recreation-use estimates are important for national-, 

regional- and forest-level decisionmaking and planning. 

Specifically, they are needed to determine benefits that 

accrue from recreational use, as well as the impacts of such 

use on other forest resources and on local economies. 

Moreover, use estimates are needed to identify trends in 

outdoor recreation and assess the effectiveness of Federal 

govemment programs. 

In an attempt to obtain statistically valid estimates and 

develop a sampling methodology for monitoring levels of 

outdoor recreation, the National Recreation Use Pilot Study 

was formulated by the USDA Forest Service in 1996. Its 

rnajor objectives were to develop and execute a national pilot 

sample survey designed to estimate the total annual number 

of recreation site visits to the national forests and to provide 

information for developing a more efficient sampling design. 

It included the evaluation of stratification variables, 

estimates of strata variances, and sample size 

recommendations to achieve required precision levels. In 

addition, information was collected on visitors (age, gender, 

and geographic origin) and characteristics of their visit 

(where and when). 

Our objectives were to: (1) describe the stratified three-stage 

sampling design that was used; (2) present average daily 

site-visit and total annual site-visit estimates, sample 

variances, and confidence intervals at the ranger district and 

national level; (3) evaluate the stratification variables; and 

(4) determine appropriate sample size recomendations. We 

have focused on the statistical aspects of estimating site 

visits but have not included results or discussions about 

visit and visitor characteristics. 

Traditionally, national forest recreation-use estimates were 

developed from individual ranger district estimates, which 

then were aggregated upward to produce national estimates. Methodology 

In 1965, the Agency established a reporting system called 

Recreation Inventory Measurement. It required site-level The National Forest System 

reporting of recreation visits. A study conducted by Kocis 

(1 986) showed that the ranger district estimates were derived The National Forest System lands comprise approximately 

by numerous and diverse methods, producing results that 192 million acres across the continental United States, 
Alaska, and Puerto Rico. Administratively, the USDA Forest 

Service is composed of nine regional offices that constitute 

nunnerous national forests, each of which may have several 

ranger districts. Within any USDA Forest Service region, or 

even within a national forest, there may be a great variety of 

forest types and, therefore, any number of outdoor 

recreation opportunities. This could lead to large variability 

in recreation-use estimates within a region. Although 

regions and national forests may be potentially convenient 

ad~nistrative units, they may not form homogeneous 

groups of ranger districts necessary for effective 

stratification in a statistical estimation process. An 

alternative is to group ranger districts by the environmental 

characteristics described in Bailey's classification of 

ecoregions (Bailey 1995, Bailey and others 1994), which 

divides the continental United States into homogeneous 

regions based on forest type, geology, and weather patterns. 

To the extent that the amount and type of recreation is 

determined by these factors, ranger districts within each 

ecoregion may have similar recreation-use patterns and, 

consequently, less variation in recreation-use estimates. 

Therefore, we used the ecoregion as a stratification variable 

in our sample design. 

The Site Visit 

An essential element of any sampling design is precise 

definition of what is to be measured or estimated. We define 

a site visit as one individual traveling to a recreation site in a 

national forest for a variable length of stay for the purpose 

of recreating, and then departing. For example, an individual 

camping 1 week at a campground constitutes one site visit, 

while a father and son fishing one Saturday on a river 

constitutes two site visits. Nonetheless, although some 

national forest lands have discrete individual sites, others do 

not. Therefore, a clear definition of sites is required. 

Personnel from the ranger districts selected for our study 

were asked to categorize each site or area in their jurisdiction 

into one of five mutually exclusive site types. These were 

used as stratification variables in an attempt to reduce 

variation. 

Day-Use Developed Sites (DUDS)--are intended for day use 

only and include boating areas, picnic sites, fish-viewing 

sites, fishing sites, information sites, interpretive sites, 

observation sites, playground-park sport sites, ski areas 

(alpine and Nordic), wildlife viewing areas, visitor centers, 

museums, swiming areas, and winter sport sites. Generally, 

DUDS provide visitor comfort, convenience, and educational 

opportunities, but they are available only on a day-use 

basis. 

Overnight-Use Developed Sites (0UDS)-include 

campgromds, cabins, hotels, lodges, resorts, horse camps, 

organization sites, and any other overnight facility on 

national forest lands, whether they are owned and/or 

managed by the USDA Forest Service or are a private 

concession. 

Water-Based Area Sites (WBAS)--are used exclusively for 

water activities. Other uses normally associated with DUDS 

may occur occasionally, but are of only minor consequence. 

In addition, the comfort and convenience usually associated 

with DUDS are minimal or absent. 

General Roaded Dispersed Sites (GRDS)-include forest 

areas not included in DUDS, OUDS, or WBAS that are 

accessible by roads. 

Unroaded Dispersed Sites (URDS)-include forest areas not 

included in DUDS, OUDS, WAS, or GRDS, but which are 

not accessible by road. 

The GRDS and URDS are associated with hiking, hunting, 

and dispersed camping. They were created by mapping 

watersheds or land areas accessed by major roads or trails. 

Originally, we wanted to create GRDS and URDS of 2,000 to 

5,000 acres; but, due to differences among ranger districts, 

size actually varied from 1,000 to 100,000 acres. Thus, each 

ranger district potentially had many GRDS and URDS, and 

each was treated as an individual site in the same manner as 

were DUDS, OUDS, and WAS. 

Site Days 

Generally, the most basic component of a sample survey is 

the population of sampling units from which a sample is 

drawn according to the criteria of a particular sampling 

design. In order to estimate the number of annual site visits, 

let sampling variable y be the number of last-exiting 

recreationists on a given site day. Last-exiting recreationists 

are those who are leaving the site for the last time and will 

not return during the site visit. This is in contrast to 

recreationists who merely are leaving the site for a particular 

reason, e.g., shopping, sightseeing, but will return later 

during their site visit. For our purposes, a site day is defined 

as the 24-hour period in which a site is open for recreation. 

Thus if a site is open throughout the entire year, it 

represents 365 site days. If it is open only on weekends, it 

has 104 site days. The total collection ofy units over all sites 

days in all ranger districts is the population of sampling 

units, i.e., the sum of the y's is the true total number of site 

visits. Hence, the sampling problem was to design an 

efficient method of sampling the population of site days over 

the entire National Forest System. 
It should be noted that instead ofy being the number of lastexiting recreationists, it just as easily could have been firstentering recreationists. If an estimate of site visits is all that 

is required of a survey, it makes no difference whether 

sampling is at the time of first entry or last exit. However, if 

any ancillary sampling variables associated with the visit are 

desired, as was true in our study, exit interviewing is 

preferred for several reasons. If intewiewed at the start of 

the visit, individuals can only give predictions about what 

they expect from the visit. Exit interviews capture more 

precise information especially about such thngs as length of 

stay, facilities used, and recreation activities. Moreover, 

entry interviews may directly affect the recreation visit. 

Experience also has shown that many visitors are eager to 

begin their recreation visit and are less willing to be 

interviewed as they begin their visit. For these reasons we 

designed our survey to count and interview visitors on their 

last exit. 

Sampling Design 

We used a three-stage sampling design to estimate the total 

number of site visits. The population of primary sampling 

units comprised all 606 ranger districts in the National Forest 

System. The secondary sampling unit was the collection of 

all site days (the 24-hour period during which a recreation 

site was open for visitation). The tertiary sampling unit was 

the last-exiting recreation visitors sampled and interviewed 

using a short, 2-minute questionnaire. 

The selection of sampling units for the three-stage sampling 

design could follow a variety of methods, e.g., random, 

stratified, and systematic. However, in most situations 

stratification is advantageous for numerous reasons 

(Cochran 1977, Kish 1965). Stratification can increase 

precision of estimates if it is possible to divide a 

heterogeneous population into strata that are internally more 

homogeneous. In addition, if the strata represent meaningful 

subdivisions of the population, estimates can be obtained 

for each stratum. Often it is administratively more convenient 

to use stratification because it will ensure the sample not 

only is spread over the whole population but also is divided 

into manageable subpopulations that may be sampled 

locally. Finally, since the most appropriate sampling 

methodology may differ across the population, stratification 

allows for different sampling designs among strata. 

In an attempt to reduce variation, we stratified the primary 

sampling units (ranger districts) into 16 ecoregion strata 

based on Bailey's ecoregion classification (Bailey 1995, 

Bailey and others 1994). Each stratum is composed of entire 

ecoregion divisions or parts thereof, delineated to form a 

contiguous piece of land. This resulted in 16 ecoregion 

strata formed in 10 of Bailey's 14 ecoregion divisions. Table 1 

defines the strata by Bailey's ecoregion division and 

indicates the specific ecoregion provinces that they contain. 

We felt that this would result in more homogeneous strata 

than if arbitrary administrative units, such as national 

forests, were used. Due to limited budgets, we could sample 

only 32 ranger districts over all ecoregion strata, resulting in 

a 5.3-percent sample. We used proportional allocation to 

determine the number of sampled ranger districts per 

ecoregion stratum, stipulating that each had at least one 

ranger district. The original selection of ranger districts was 

random; however, some selected ranger districts could not 

participate. To ensure the appropriate sample size, we 

replaced them with other randomly selected ranger districts. 

Table 1 shows the number of ranger districts in each 

ecoregion stratum1 and the number sampled based on 

approximate proportional allocation. During our study, the 

two ranger districts in ecoregion 7 dropped out, resulting in 

sampling of 30 ranger districts in 15 eeoregions. 

The unique and diverse character of recreation sites, as well as 

the variety of and large expected differences in the volume of 

activities during different days and seasons, allowed us to 

stratify the secondary sampling unit (site day). Those strata 

were formed by the five site types, along with strata defined by 

two seasons and two day types. Season and day-type 

stratification variables were included to reduce variation 

caused by seasonal and daily fluctuations in site visits. We 

classified each site day within a given site type as belonging to 

either high or low seasonal use, not necessarily coinciding 

with the calendar year four seasons. The day-type strata 

classified site days as either weekdays or weekendholidays. 

Thus, there were 5x2~2 = 20 possible strata for the site days of 

each ranger district. We anticipated that these strata had the 

potential to classify site days into homogeneous groups that 

would reduce variance of the estimates. Available fknding 

allowed us to sample approximately 70 site days per ranger 

district. The actual number of site days sampled for each 

ranger district, disregarding those that were rnissed for various 

reasons, e.g., dangerous weather conditions, personnel issues, 

is shown in table 2. Propohonal allocation was used to 

distribute the sample of site days over the 20 potential strata 

within a ranger district, after allocating 2 site days per stratum. 

The tertiary sampling unit was the recreation visitor, which 

was sampled by selecting vehicles exiting for the last time. 

We conducted a 2-minute interview while allowing other 

vehicles to exit and used a 24-hour vehicle counter to tally 

' Ecoregion stratum will be referred to as simply ecoregion in the 

remainder of this manuscript. 
Table l-Eeore@on strata, Bailey's division and Bailey's province, total number of ranger 

districts, and number of ranger districts sampled 

Bailey's ecoregion Ranger districts 

Ecoregion 

stratum Division Province Total Sampled 

Marine (240) 

Marine (240) 

Mediterranean (260) 

Temperate desert (340) 

Tropical/subtropical steppe (3 10) 

Tropical/subtropical desert (320) 

Temperate steppe (330) 

Temperate steppe (330) 

Tropical/subtropical steppe (3 10) 

Temperate steppe (330) 

Temperate desert (340) 

Tropical/subtropical steppe (3 10) 

Temperate steppe (330) 

Tropicalisubtropical steppe (3 10) 

Prairie (250) 

Temperate steppe (330) 

Warm continental (2 10) 

Hot continental (220) 

Hot continental (220) 

Subtropical (230) 

Hot continental (220) 

Subtropical (230) 

Warm continental (2 10) 

Hot continental (220) 

16 Hot continental (220) 

Total 

" These two ranger districts dropped out of the survey. 
Table 2-Individual ranger district average daily site visit and total annual site-visit estimates, standard errors, 

and coefficients of variation 

Ecoregion 

stratum 

Site Vehicles Total 

days site 

sampleda Stopped Interviewed daysc 

- - - - - - - - - - - - - - Number - - - - - - - - - - - - - 

80 47 1 177 4,878 

74 53 1 42 0 8,239 

50 360 234 7,070 

55 219 107 8,073 

65 526 394 13,331 

65 26 1 135 13,118 

74 313 129 6,606 

13 43 16 3,558 

98 566 3 92 9,573 

77 797 647 14,947 

60 130 7 1 5,450 

72 111 69 4,785 

60 109 86 5,64 1 

48 180 94 2,9 12 

47 600 507 2 1,784 

74 616 43 7 6,590 

69 652 509 12,182 

8 1 1,628 1,444 10,487 

8 5 508 334 8,164 

70 157 97 6,560 

8 5 229 126 8,396 

44 342 16 1 3,83 1 

7 5 483 268 7,726 

79 249 5 5 11,184 

66 387 242 6,593 

82 473 302 7,923 

86 1,079 447 9,189 

80 309 219 7,92 1 

5 8 45 1 334 1 1,682 

59 86 3 2 7,782 

Average 

daily 

site visits 

Total 

annual 

site visits 

SE = standard errors; CV = coefficients of variation. 

' Site days sampled per ranger district. 

Represents vehicles the occupants of which had been recreating, were willing to be interviewed, and were exiting the site for the 

last time. 

Total site day is the total annual number of site days on the ranger district. 
all exiting vehicles. The inteniiews served to calibrate the 

24-hour vehicle count to produce an estimate of the 

sampling variable y as described in the next section. The 

total number of vehicles stopped for a potential complete 

interview and the total number of last-exiting recreating 

vehicles whose occupants were willing to complete the 

interview are shown in table 2. 

There were several advantages to using the three-stage 

sampling approach over simple random sampling. 

Transportation costs were reduced because sampling was 

clustered within several primary sampling units and not 

spread out over the entire population of ranger districts. 

Logistics and administrative details were simplified because 

sampling was made on far fewer ranger districts. By 

narrowing our sample of ranger districts, we were able to 

interact with and train personnel conducting the field 

sampling. 

To obtain an estimate of LEY, we used both sampling 

devises. Because LEV was based on last-exiting recreating 

vehicles, that number could not be detemined by a vehicle 

counter that could not distinguish nomecreating vehicles or 

recreating vehicles that were not exiting for the last time. 

Therefore, we used infomation from the questionnaire to 

estimate the proportion, P, of all exiting vehicles whose 

occupants had been recreating and were exiting for the last 

time. We assumed that P estimated from interviews would 

provide a reasonable estimate of the proportion of lastexiting recreating vehicles in a 24-hour period. 

One-way counters are preferred over two-way counters. If a 

two-way counter is used, as it was at some of the ranger 

districts, a one-way exit vehicle count can be approximated 

by dividing the count by two. Such an adjustment assumes 

that vehicles enter and exit at the same rate. If this 

assumption is faulty, a biased estimate of exiting traffic 

results. 

Site-Visit Estimates for a Site Day 

An estimate of LE V was then defined as: 

We used a double sampling technique (James 1967, James 

and Henley 1968, James and Ripley 1963) to estimate the 

number of recreation site visits per site day. Using a one-way 

exit-vehicle counter, we recorded the number of vehicles 

exiting a recreation site during a 24-hour period. To calibrate 

the 24-hour vehicle count, we interviewed a random sample 

of last-exiting visitors on the same day. An estimate of site 

visits per day was defined as: 

y"' = LEV 3 , 

where 

y(') = an estimate of the number of site visits on a given site 

day, 

LEV= the number of recreating vehicles exiting the site for 

the last time during the 24-hour period, and 

2 = the average number of people in a recreating vehicle 

exiting the site for the last time. 

We administered a questionnaire to a random sample of all 

vehicles exiting the site during the interviewing period, 

which usually lasted about 6 hours. The initial questions 

screened out all but the last-exiting recreating vehicles, to 

which a more intensive set of questions was given. This 

produced information about duration of site visit, recreation 

activities, citizenship, gender and age class, and mode of 

transportation. However, our study focused on site visits, 

not the characteristics of visits or visitors. From the 

interview data, we calculated the average number of people 

in a vehicle, X , based only on those vehicles that were 

recreating and exiting the site for the last time. 

where 

LEV= the number of last-exiting recreating vehicles, 

C = the number of exits recorded by the vehicle counter 

during the 24-hour period, and 

P = the proportion of exiting vehicles that were last-exiting 

recreating vehicles. 

To accommodate for variations in vehicle counters among 

the ranger districts, C was calculated as: 

c= END - BEGIN 7 

WA Y * INTER VAL * AXLES 

where 

BEGIN = the beginning vehicle count, 

EhrD = the ending vehicle count, 

FAY== the number of trafic directions (one way or two way) 

in which the vehicle counter operated, 

INTERVAL = the interval length (proportion of a 24-hour 

day) in which the vehicle counter was operable, and 

AXLES =; the average number of axles per vehicle estimated 

from interviews (this is set to 1 when vehicle counters do not 

count axles but count vehicles instead). 

Although the daily site-visit estimator y(u based on 

interviews and vehicle counters was preferred, difficulties 

prohibiting its use sometimes occurred. For example, it could 
not be used when no interviews had been conducted, or 

when the vehicle counter did not operate. Altemative sitevisit estimators were defined as: 

y(2j= the total number of people counted by the interviewer 

(this should be biased low), 

the total number of people counted by the interviewer 

expanded to a 16-hour day, which represents the typical time 

period when recreationists might be exiting the site, and 

yo = an alternative, reliable source such as ticket receipt 

counts (a nonestimated, observable count). 

If, on occasion, infomation based on an alternative, reliable 

source was available, such as ski rental ticket receipts or 

park entrance fee receipts, then y'4) was used instead of y('). 

If no vehicle count was available, then y(3) was used. If both 

interviews and vehicle count were available, then 9') was 

used, unless yC1) was less than y(", in which case y(2) was the 

site-visit estimate. In situations where no site-visit estimator 

was available, e.g., no one was interviewed, no vehicles were 

counted, and no alternative, reliable sources were available, 

then it was simply assigned zero. 

Site-Visit Estimates for all Site Days on a 

Ranger District 

Site-visit estimates for individual ranger districts were 

obtained using a stratified random sampling design where 20 

strata consisted of the 5 site types, 2 seasons, and 2-day 

types (see Sampling Design). Let: 

N = the total annual number of site days on the ranger 

district, 

L = the total number of strata on the ranger district, 

Nh = the total annual number of site days on the ranger 

district for stratum h (h = l,2,3, . . . ,20), 

Wh = Nh IN= stratum h weight, 

n = the total number of site days sampled in all strata, 

nh = the number of site days sampled in stratum h, and 

y,, = the site-visit estimate in stratum h for day i. 

Note that yhl may be based on the yi', $3, yo), or )i4) sitevisit estimators, depending on the data available for the 

specific site day as discussed in the previous subsection. 

Then the average daily site-visit estimate in stratum h is 

defined as: 

and the estimated variance in stratum h is: 

The overall average daily site-visit estimate for the entire 

ranger district is: 

with estimated variance: 

Approximate confidence intervals may be calculated in the 

typical manner as: 

where 

z = the z-value from the standard nomal distribution at the 

appropriate percentage point (Montgomery 19'76), for 

example, z = 1.96 at the 95-percent confidence level. 

An estimate of the total annual number of site visits for a 

ranger district is easily obtained by expanding the average 

daily site-visit estimate [equation (6)3 by the total annual 

number of site days on the ranger district. Mathematically, 

this is defined as: 

with estimated variance: 

An approximate confidence interval would be: 

Two types of sampling allocation are commonly used with 

stratified random sampling-proportional or optimum. Our 

study used proportional allocation, which is defined as: 

It assigns the number of sampling units to a stratum in 

proportion to its size. Optimum allocation, used later in the 

evaluation of the stratification variables, is defined as: 

This allocation method assigns a larger sample to a stratum if 

the stratum is larger or is more variable internally. 
Site-Visit Estimates for the Entire S, Ic the true stratum h standard deviation, 

National Forest System 

The site-visit estimate for the entire National Forest System 

was obtained by estimating site visits at the ecoregion level 

and expanding them to the national level. Let 

4 = the number of ranger districts in ecoregion i , 

then the average total annual site-visit estimate for a ranger 

district in ecoregion i is: 

- j =1 Y; = --------- , 

where n, 

= the total annual site-visit estimate for ranger district j in 

ecoregion i, and 

nl = the number of ranger districts sampled in ecoregion i. 

Theoretically, the sample variance of consists of two 

components-the variances between and within ranger 

districts. However, because the number of ranger districts 

within each ecoregion was low, i.e., nl / IV, is negligible, the 

variance within ranger districts could be ignored as 

explained by Cochran (1 977, p. 279), yielding the sample 

variance as: 

The grand total annual site-visit estimate over all ecoregions 

in the National Forest System is obtained by expansion as: 

16 

SV=CN,Y,, 1=I 

(16) 

with estimated variance: 

16 

s2 (SV) = Cq2 s' (E;) i =I 

Evaluation of Stratification 

One way to evaluate the effect of stratification is to compare 

the variance of a simple random sample, V,, , and the 

variance of a stratified random sample under optimum 

allocation, yoPi . Cochran (1 977) shows that: 

where 

= the true stratum h mean, 

f= the sampling fraction, n/N, 

y = the true population mean, 

= KS, = the true population standard deviation, and h=l 

the other terrns are as previously defined. 

The tern on the far right reflects the gain due to strata that 

have different means. The middle term to the right of the 

equal sign is the difference in variance between optimum and 

proportional allocation and is associated with differences in 

the stratum standard deviations. Therefore, the criteria used 

for evaluating the effectiveness of stratified random 

sampling under optimum allocation are the stratum means 

and standard deviations. Stratum differences in either of 

these indicate that stratification is effective. The former 

could be tested with a fixed-effects analysis of variance on 

the stratum means (Montgomery 1976, SAS Institute Inc. 

1989). In simple situations, such as where only one 

stratification variable is being studied, the standard 

deviations could be tested with Bartlett's test for 

homogeneity of variances (Milliken and Johnson 1984, 

Montgomery 1976). However, we used the fixed-ef5ects 

analysis of variance to test for variance differences because 

multiple stratification variables were being considered in a 

factorial fashion. Because variances usually exhibit a 

positively skewed distribution, they were converted to rank 

statistics to better satisfy the normality assumption required 

for the analysis of variance, resulting in a nonparametric 

statistical test (Gonover 1980, SAS Institute Inc. 1 990). 

The design effect (DEFF) is another method used to compare 

the efficiency of a stratified random sampling design to a 

simple random sample (Cochran 1977). The DEFF is the ratio 

of the stratified random sampling variance of the estimate to 

the variance ofthe estimate from a simple random sample of 

the same sample size and, thus, quantifies the gain due to 

stratification. When DEFF = 1, the sampling methods are 

equivalent. However, if DEFF is < 1, the stratified random 

sampling design has reduced the estimate's variance. 

Stratified random sampling is somewhat more complex and 

costly than simple random sampling; thus, DEFF must be 

substantially < 1 if stratification is to be of practical value. In 

the unlikely event it is > I, simple random sampling is more 

precise. Rao (1962) defines an unbiased estimator of the 

variance of the mean of a simple random sample in terms of a 

stratified random sample as: 

where 

u<t = the mean estimate based on the stratified random 

sample, 
s2 (ys, ) = the variance based on the stratified random 

sample, 

and the other terms are as previously defined. 

The DEFF is then: 

s' (Ys, ) DEFF = - 

~'(7) ' 

If the stratified random sampling design was performed 

under suboptimum allocation, where the n, 's differed 

substantially from what they would be under optimal 

allocation, the DEFF may be above 1 even if stratification 

has potential. Thus, the evaluation of the stratification 

variables can be hypothetically evaluated under optimal 

allocation even though our study deviated from optimal 

allocation. The criterion is the relative hypothetical 

efficiency (RHE) (Ruark and Zarnoch 1992, Zarnoch and 

others 1993) defined as: 

where ~'(7'~ ) is the estimated hypothetical variance based 

on the variance estimates sh2 when the stratum sizes, n, , 

are determined by optimum allocation [equation (1 3)]. The 

DEFF will decrease to < 1 as the strata means become 

increasingly different. However, this is not necessarily true 

for differences in the stratum variances, because the DEFF 

will be approximately 1 under proportional allocation and will 

only diminish as optimal allocation is approached. Thus, 

RHE is useful in detecting effective stratification due to 

differences in variances when optimum allocation is used. 

Results and Discussion 

Site-Visit Estimates for Individual Ranger Districts 

Total number of site days per year per ranger district varied 

from 2,9 12 to 2 1,784 and averaged 8,539. In general, this 

should be highly correlated with not only the number of 

recreational sites and number of days the recreational sites 

are open but also with how the sites are defined in the field. 

This is particularly true of GRDS and URDS sites that were 

quite arbitrarily defined. The number of site days selected for 

sampling per ranger district varied from 62 to 98; however, 

due to various factors such as bad weather, personnel 

issues, and emergencies, the number actually sampled 

ranged from 13 to 98 (average 68). 

Estimates for average daily site visits [equation (611 and total 

annual site visits [equation (9)] for each ranger district are 

shown in table 2. The average daily site-visit estimate was 

56.9 people and varied across ranger districts from 6.1 to 

208.5. Total annual site visits per ranger district averaged 

536,378, ranging fi-011121,527 to 2,186,175. 

The precision of these estimates varied considerably and is 

reflected in the standard errors. However, it is often more 

informative to evaluate precision with the coefficient of 

variation, which puts the standard error on a relative basis to 

the parameter that is being estimated. The coefficient of 

variation ranged from 10.4, which is considered quite good in 

this type of survey, to a high of 66.9 (average 29.3). It is also 

useful to construct confidence intervals [equations (8) and 

(1 I)] from which the precision of the estimate can be 

appreciated even more easily. For example, a 90-percent 

confidence interval on the total annual site-visit estimate for 

the most precise ranger district (ecoregion 12 in table 2) is 

430,006 to 606,560, which implies a considerably tight bound 

on the estimate. However, when a 90-percent confidence 

interval is constructed for the most variable ranger district 

(second ranger district in ecoregion 6 in table 2), the bounds 

are 0 to 458,454, indicating that this estimate is practically 

useless. 

Site-Visit Estimate for Entire National 

Forest System 

The total annual site-visit estimate for National Forest 

System lands was 330,772,894 [equation (1 6)] with a standard 

error of 66,280,755 [equation (l7)], yielding a coefficient of 

variation of 20 percent. The 80-percent confidence interval 

was calculated to be 245,800,966 to 4 15,744,822, which 

indicates that the estimate was within 25.7 percent of the true 

number of site visits with a probability of 0.80. This is a 

reasonable level of precision for a study of this type. The 

goal for future recreation surveys is to be within 15 percent 

with a probability of 0.80. 

The true cost of sampling is hard to determine because we 

used a variety of full-time paid employees and volunteers for 

sampling. We asked ranger district officials to estimate the 

survey's true cost. Estimated startup costs ranged from $300 

to $2,387 and a sampled site-day cost from $58 to $240. 

Therefore, ranger district funding was about $200 per 

sampled site day, including $150 to set up a vehicle counter 

and conduct on-site interviews, and $50 for retrieving the 

counter the next day. Overhead and vehicle expenses were 

included in these costs but equipment purchases were not. 

Calculating the national site-visit estimate included a slight 

modification, because both ranger districts in ecoregion 7 

dropped out of the survey. Basically, a preliminary national 
site-visit estimate and variance were calculated from 15 

ecoregions comprising 578 ranger districts. The final national 

site-visit estimate was obtained by multiplying the 

prelianinary estimate by the expansion factor 6061578 to 

reflect the unsampled ecoregion 7, which had 28 ranger 

districts. This assumes that the recreation-use level per 

ranger district in ecoregion 7 was not different from the 

overall national average. The variance was also expanded by 

multiplying by the square of the expansion factor. 

Evaluation of Stratification 

Overview-Our study used stratification to reduce variance 

of the estimates. Stratification was used at both stages of the 

sampling design: the primary units (ranger districts) were 

stratified based on 16 ecoregions, while the secondary units 

(site days) were stratified based on 5 site types, 2 seasons, 

and 2 day types. There were very few sampled ranger 

districts in each ecoregion (table 1). Because sample 

variances for each stratum require at least two observations, 

almost half of all ecoregions had no sample variance 

estimate, while others were based on an extremely low 

sample size. Only poor estimates of the stratum sample 

variances were obtainable; thus, no evaluation of the 

ecoregion strata was attempted. 

Stratification at the secondary stage (site days) produced a 

better estimate of the sample variance in each stratum. 

However, even here small sample sizes hindered the 

estimates. Evaluation of stratification at this stage focused 

on determining the usefulness of the site-type, season, and 

day-type strata for reducing variation in site-visit estimates. 

The analysis of variance was used to test for differences 

between stratum means and variances for the site-day strata 

on an ecoregion basis. In addition, the DEFF and RHE 

criteria were also computed and compared in a similar 

manner. 

Analysis of stratum means-Stratification effectiveness was 

assessed based on the relationship between simple random 

sampling and stratified random sampling under optimum 

allocation [equation (1 8)]. Differences between stratum 

means were tested using analysis of variance procedures for 

a randomized block design. The ranger districts were 

considered blocks, while the three stratification variables 

were site type, season, and day type. We conducted the 

analysis by ecoregion, and then pooled all ecoregions. 

The analysis of variance initially was performed on the full 

mode1 by including all two- and three-way interactions of the 

stratification variables. On an ecoregion basis, very few 

interactions were significant. Site type*season was only 

significant for ecoregions 2,3, and 13; and site type*day 

type and site type*season*day type were significant only 

for ecoregion 13. When pooled over ecoregions, site 

type*season was significant (p = 0.0223); however, site 

type*day type (p =. 0.3754), season*day type (p == 0,58401, 

and site type*season*day type (p = 0.4 19 1) were not 

significant. Only main effects were considered further 

because interactions were mostly nonsignificant and those 

that were significant would be difficult to interpret. 

Using a main effects model on an ecoregion basis, analysis 

of variance still showed very few statistical significances at 

the 0.05 level, even though the strata means appeared quite 

different (table 3). However, when all ranger districts were 

pooled across the ecoregions, all stratification variables 

showed statistical significance. Overall, trends within each 

of the stratification variables seemed reasonable, but they 

were not consistent among the ecoregions. For instance, 

within the site-type strata, DUDS had the highest overall 

mean site-visit estimate (106.8) while URDS had the lowest 

(37.2). However, this relationship only held up on a few of 

the ecoregions. In particular, in ecoregions 3,10,12, and 14, 

DUDS had the lowest mean site-visit estimate. Although this 

may be due to high variability, differences may also have 

been the result of different recreational patterns among the 

ecoregions. The seasonal strata means were different in a 

consistent manner across all ecoregions. High season was 

greater than low season in all cases except ecoregion 15. The 

day-type strata were not quite as consistent but generally 

exhibited the expected pattern, i.e., weekendiholiday days 

were greater than weekdays, although the weekendiholiday 

mean occasionally was less than the weekday mean. Thus, 

although statistical significances were for the most part 

lacking, site type and season tended to exhibit different 

strata means and, hence, appear as potentially feasible 

stratification candidates. 

Analysis of stratum variances-Stratification effectiveness 

was further tested by using analysis of variance procedures 

similar to those used for the stratum means, except that the 

variable tested was the rank statistics of the stratum 

variances. Results from the full model, including interactions 

on an ecoregion basis, revealed that season*day type was 

significant only in ecoregions 2,3, and 11; and that site 

type*season and site type*season*day type were 

significant only in ecoregions 1 1 and 13. However, when 

pooled over ecoregions, site type*season (p = 0.4995), site 

type*day type (p = 0.3706), season*day type (p = 0.1000), 

and site type*season*day type (p = 0.3968) were all not 

significant, Again, only main effects were considered further. 

For the main effects model on an ecoregion basis, the 

analysis of variance showed five significances for site type, 

five for season, and only one for day type (table 4). When 
Table 3---Analysis of variance on the stratum mean site-visit estimates for three stratification variables and 

their associated least squares means 

Analysis of variance Site type Season a Day type 

Ecoregion Source df P-value DUDS GRDS OUDS URDS WBAS H L H L 

------------- - Least squares means ofsite-visit estimates - - - - - - - - - - - - - - 

1 Site type 4 0.25 25.9 12.3 11.0 9.6 3.4 

Season 1 .07 19.1 5.8 

Day type 1 .53 14.5 10.5 

2 Site type 4 .08 216.6 97.4 84.3 17.3 66.7 

Season I .06 140.3 52.6 

Day type 1 .76 103.4 89.5 

3 Site type 4 .12 34.9 63.4 43.8 35.9 139.5 

Season 1 .2 1 78.1 48.9 

Day type 1 .84 65.8 61.2 

4 Site type 4 .36 72.5 33.9 148.5 15.4 37.4 

Season 1 .06 106.7 16.4 

Day type 1 .88 58.1 64.9 

5 Site type 4 .35 186.3 55.4 73.8 152.7 236.8 

Season 1 .42 165.9 116.1 

Day type 1 .12 188.1 93.9 

6 Site type 4 .46 18.7 29.1 14.0 32.8 48.1 

Season 1 .29 34.3 22.8 

Day type 1 .18 35.8 21.3 

8 Site type 4 .05 168.7 29.9 19.7 39.8 70.9 

Season I '04 103.8 27.8 

Day type 1 .84 62.1 69.5 

9 Site type 4 .01 524.6 124.6 29.6 80.1 242.7 

Season 1 .14 274.0 126.7 

Day type 1 .75 215.5 185.1 

10 Site type 3 .15 6.6 19.1 43.2 - 8.6 

Season 1 .20 27.1 11.6 

Day type 1 .27 26.0 12.8 

11 Site type 4 .12 53.5 94.2 44.2 39.4 19.0 

Season 1 .48 56.9 43.2 

Day type 1 .2 1 62.1 38.0 

12 Site type 4 .22 10.5 53.2 59.0 154.1 51.8 

Season 1 .26 87.6 43.9 

Day type 1 .13 94.3 37.2 

13 Site type 4 .15 12.2 13.0 34.8 9.6 19.6 

Season 1 .08 24.1 11.7 

Day type 1 .03 25.6 10.2 

14 Site type 4 .07 -3.6" 16.0 57.0 29.1 205.8 

Season 1 .13 98.8 22.9 

Day type 1 .17 92.9 28.8 

15 Site type 4 .60 15.2 5.4 12.0 5.6 33.1 

Season 1 .72 12.0 16.5 

Daytype I .2 2 6.4 22.1 

16 Site type 4 .26 5.1 169.2 196.4 28.1 -43.9" 

Season 1 '33 109.0 33.0 

Day type 1 .2 3 117.4 24.6 

All Site type 4 .01 106.8 59.8 56.9 37.2 79.8 

Season I 0 92.3 43.9 

Day type 1 .05 80.0 56.2 

df = degrees of freedom; DUDS = day-use developed sites stratum; GRDS = general roaded dispersed sites stratum; OUDS = 

overnight-use developed sites stratum; URDS = unroaded dispersed sites stratum; WBAS = water-based area sites stratum. 

" H = high season stratum; L = low season stratum. 

9 = high (weekendsiholidays) day-type stratum; L = low (weekdays) day-type stratum. 

" Since least squares means are computed from a general linear model, negative values are possible. 
Table &Analysis of variance on the stratum rank variances of the site-visit estimates for three 

sh-atification variables along vvith the stratum average variances (1,000s) 

Analysis of variance Site type Season a Day type " 

Ecoregion Source df P-value DUDS GRDS OUDS URDS WBAS H L H L 

----------------- Estimated ai)erage ofsite-visit days - - - - - - - - - - - - - - - - 

1 Site type 4 0.33 0.6 1.4 0.5 0.4 0 

Season 1 .09 0.7 0.4 

Day type 1 .48 0.5 0.6 

2 Site type 

Season 

Day type 

3 Site type 

Season 

Day type 

4 Site type 

Season 

Day tY Pe 

5 

6 

8 

9 

10 

11 

12 

13 

14 

15 

16 

All 

Site type 

Season 

Day type 

Site type 

Season 

Day type 

Site type 

Season 

Day type 

Site type 

Season 

Day type 

Site type 

Season 

Day type 

Site type 

Season 

Day type 

Site type 

Season 

Day type 

Site type 

Season 

Day type 

Site type 

Season 

Day type 

Site type 

Season 

Day ty Pe 

Site type 

Season 

Day type 

Site type 

Season 

Day type 

df = degrees of freedom; DUDS = day-use developed sites stratum; GRDS = general roaded dispersed sites stratum; OUDS = 

overnight-use developed sites stratum; URDS = unroaded dispersed sites stratum; WBAS = water-based area sites stratum. 

a H = high season stratum; L = low season stratum. 

H = high (weekendsiholidays) day-type stratum; L = low (weekdays) day-type stratum. 
pooled over all ecoregions, site type and season were visitation. Consequently, stratum variances for each ranger 

significant; day type was not. There were no consistent district were converted into relative coefficients of variation 

trends for site type and day type among the ecoregions; (RCV) defined as: 

however, season usually exhibited greater variability in the 

high stratum. Generally, similar to the stratum means 

s RCV = 1004, v analysis, these results indicate that site type and season are i 

potentially feasible stratification candidates. where y and s, are on a ranger district basis. 

The DEFF and RHE analysis-An analysis of the DEFF 

[equation (20)] also was performed to assess strata 

usefulness (table 5). When all 20 strata were considered 

jointly, the mean DEFF was 1.0 1. On some ranger districts, 

DEFF values were close to 0.50, indicating that the strata 

were beneficial. However, caution must be used in 

interpreting DEFFs, because the stratified random sampling 

was performed under suboptimal allocation, i.e., not 

according to equation (1 3). In this situation the DEFF may be 

above 1 even if stratification has potential. To analyze the 

effect of stratification under optimum allocation, we used the 

RHE [equation (2 I)]. Our analysis showed that the average 

RHE was 0.40 and ranged from 0.12 to 0.75. Therefore, for all 

Computing the mean RCV for stratum h over all ranger 

districts in an ecoregion provides a relative measure of 

variation conditioned on the ranger district site-visitation 

level. The mean RCV for each stratum was squared and 

converted back to yield the s2,'s for a given ranger district 

by multiplying by the ranger district's specific (L i 100)~ 

factor. Thus, a set of stratum variances was obtained for a 

ranger district based on data pooled from all the ranger 

districts in one ecoregion. The mean RCV for the 20 strata for 

each of the 4 ecoregions reveal no obvious pattern within or 

between ecoregions (table 6). This is probably due to large 

sampling variability and relatively small sample sizes, even 

though ranger districts were pooled. 

30 ranger districts, the strata used have shown a significant Sample size analysis was based on computing the variance reduction in variance if optimal allocation is used to assign of the average daily site-visit estimate for each of the ranger sample sizes to the strata. districts under a stratified random sampling design using 

The DEFF and RHE were also analyzed on each of the three 

individual stratification variables separately. Each analysis 

on a stratification variable was performed by simply 

disregarding the other two strata variables (table 5). Site type 

clearly showed the most potential use for stratification 

because numerous DEFFs were between 0.50 and 0.75, while 

season and day type had none in that range. Also, the mean 

RHE for site type (0.64) was substantially lower than that for 

either season (0.54) or day type (0.89). 

optimum allocation. An estimate of the sample variance 

under optimum allocation is derived by substituting the n, 

formula for optimum allocation [equation (1 3)] into the 

general sample variance formula [equation (7)], yielding: 

where,the yg 's and N are specific to the ranger district, and Sample sizeranger district level-Determining a the sf,- 's are computed from the mean RCVs for the recommended sample size for future surveys at the ranger ecoregion (table 6). district level requires obtaining good stratum variance 

estimates [equation (5)] from the pilot study, computing the To facilitate comparisons across ranger districts, we appropriate variance formula [equations (7) and (1 O)], and converted s' (&, ) to a coefficient of variation defined as: detemining the appropriate sample size to meet a desired 

level of precision. I-foweuer, because most strata had very 

few sample days, the reliability of stratum variance estimates 

for most individual ranger districts was poor. To increase the 

sample size upon which the stratum variances were based. 

we pooled ranger districts by ecoregions. Only ecoregions 2, 

6. 13, and 14 had a complete set of stratum variances after 

pooling, so only these were used for the sample size 

analysis. In order to pool variances from different ranger 

districts within an ecoregion, we felt it was inappropriate 

simply to use an average stratum variance estimate [equation 

(5)] weighted by sample size for any given stratum because 

Sample size curves for each ranger district were developed 

by calculating cv (*F*~, ) over a range of sample sires (n = 25 

to n = 400) and planing the results (fig. 1). The curves for 

each ranger district within an ecoregion were similar. This is 

expected, because within an ecoregion the CV (yopt 

only in the stratum weights (y? 's) and mean estimates 

the ranger districts had greatly different levels of site 
Table SIndividual ranger district LIEFFs and RHEs based on all 20 strata 

(5 site types, 2 seasons, and 2 day types) jointly and then individually by site 

type, season, and day type 

Strata 

1111 20 5 Site types 2 Seasons 2 Day types 

Ecoregion DEW E DEW DEW E DEW E 

Mean 1.01 .40 .97 .64 1.01 '84 .98 .89 

DEFF = the design effect; RHE = the relative hypothetical efficiency. 
Table &Mean relative coefficients of variation for the strata in ecoregions 

2,6,13, and 14 

Ecoregion 

Site type Seasonu Day typeb 2 6 13 14 

OUDS H 

H 

L 

L 

URDS H 

H 

L 

L 

WBAS H 

H 

L 

L 

DUDS = day-use developed sites stratum; GRDS = general roaded dispersed sites stratum; 

OUDS = overnight-use developed sites stratum; URDS = unroaded dispersed sites 

stratum; WBAS = water-based area sites stratum. 

a H = high season stratum; L = low season stratum. 

H = high (weekendsiholidays) day-type stratum; L = low (weekdays) day-type stratum. 

(7) . However, the curves also are very similar across 

ecoregions; specifically, CV (K, ) was approximately 30 

percent when n =. 25, and between 5 and 10 percent when 

n = 400. Despite large variability in the mean RCV, these 

sarnple size curves are quite stable. Generally, large gains in 

precision [decreasing CV (L, ) ] are obtained as n increases 

to 100; however, beyond that it becomes increasingly 

difficult to reduce the coefficient of variation. 

These curves assume optimum allocation and require fewer 

observations than proportional allocation for the same 

coefficient of variation. Comparing our study's average 

coefficient of variation of 29.3 percent under proportional 

allocation (n = 68) with the 18 percent (fig. 1) obtained under 

optimum allocation (n = 68) reveals a reduction of about 1 1 

percent in variability. 

To obtain sample size recommendations for a ranger district, 

it is best to perform a specific pilot study at that ranger 

district with sample sizes large enough to ensure good 

variance estimates. We suggest that at least 5 site days be 

taken in each stratum. However, with 20 strata, a sarnple size 

of 100 site days may be too costly. An alternative is to use 

the mean RCV from table 6 from a comparable ecoregion. If 

the strata sizes (N, 's) are known and a rough estimate of 

is assumed, then cv (G, ) could be computed. If these 

quantities are not known, as is usually the case, the 

relationships shown in figure 1 will determine the appropriate 
- Pooled 

0 100 200 300 400 

Sample size (n) 

Ecoregion 13 

....... Ranger districts 

- Pooled 

Ecoregion 6 

....... Ranger districts 

- Pooled 

Sample size (n) 

Ecoregion 14 

Sample size (n) Sample size (n) 

Figure I-Sample size curves on an individual ranger district basis and a pooled ecoregion basis assuming stratified random sampling under 

optimum allocation. 
sample size for a desired coefficient of variation. For example, 

if a coeficient of variation of 10 percent is desired, 

approximately 200 to 250 site days are required under 

optimum allocation. 

Sample size-ecoregion level-Development of sample size 

recomendations at the ecoregion level is somewhat more 

complex than at the ranger district level. This is because 

ecoregion level estimates were based on a three-stage 

sampling design. For this sampling design under simple 

random sampling, the sample variance of the estimated mean 

at the ecoregion level is: 

where 

N, M, and K = the total number of primary, secondary, and 

tertiary stage units in the population; 

n, m, and k = the number of these sampled; and 2 7 

s, , s2- , and s,' = the sample variances, respectively. 

To develop sample size recommendations from the pilot 

study, we made several simplifications. 

The first simplification assumed that the variance s,' 

associated with the site-visit estimate on a given site day 

(third term on the right) was zero. Generally, the site-visit 

estimate was );('I, which is a function of a supposedly 

known census quantity, (G), and two estimated quantities, P 

and - 2. Estimation of sT2 is complex because C, P, and 

X contribute in a nonadditive manner to the variance. 

Moreover, sometimes the other site-visit estimators );('I, 

, and y(4) were used instead, adding more complexity. In 

addition, the finite population correction (I-kiK) is unknown 

because K, the total number of last-exiting recreation 

vehicles from a ranger district, is never observed. Further, 

the finite population correction for the ranger districts is 

n i fv' - 301606 = 0.050, and for site days it is rn i ,id = 68i 

8539 = 0.008, yielding (n / N)(m I M) = 0.0004, which is 

small enough that the third component of s' (7) should 

also be small. Because of that and the difficulty in estimating 

s,' , this term was assumed to be zero in the variance 

[equation (25)f. 

The second-stage component [second tern to the right of 

the equal sign in equation (25)f was simplified by assuming 

the finite population correction (1 - mi M) was 1.0 because 

m / M - 0.008. Assuming this approximation allowed us to 

avoid the complexities of an unequal number of site days for 

various ranger districts. This tends to make the sample size 

analysis a little more conservative than that obtained by 

using the exact value because it results in a slightly larger 

estimate of s2 (7) and, consequently, a higher 

recornended sample size. Moreover, simple random 

sampling was assurned instead of stratification because (1) 

there was considerable concern that strata variances were 

poorly estimated, which would consequently lead to poor 

estimates of s2 (7) and suboptimal allocation, and (2) this 

assumption keeps the recomendations on the conservative 

side. 

Variance at the primary stage, s,' , was problematic because 

only nine ecoregions had two or more ranger districts to 

permit an estimate of the variance. Hence, variance estimates 

were analyzed at the ecoregion level only for the nine 

ecoregions where they could be calculated. Although 

sample size curves could have been computed for the nine 

ecoregions, we used only the same four ecoregions (2,6, 13, 

and 14) used in the previous section to illustrate the 

concepts. 

To develop sample size curves for an ecoregion, we 

performed a random-effects analysis of variance (Milliken 

and Johnson 1984, Montgomery 1976) using Proc 

VARCOMP (SAS Institute Inc. 1989). Results included the 

typical variance components between ranger districts, a,' , 

and within ranger districts, 0,' , which were used to estimate 

3,' = 0,' + 0,' / rn and s,' = 0,' (Marcuse 1949, Zarnoch 

and others 1993) shown in table 7. Note that ii? is an 

unequal sarnple size mean number of site days per ranger 

district for the given ecoregion computed according to 

Montgomery (1 976, p. 53) and Milliken and Johnson (1984, p. 

219). Using our simplifications, the known Nand Mvalues 

(number of primary and secondary stage units) and 

estimates of s,' and s2' , s2 (7) [equation (25)] was 

computed on an ecoregion basis for a range of n (2,4,8, and 

16 sampled ranger districts) and m (I 0 to 100 site days) and 

converted into coefficients of variation. Figure 2 illustrates 

general trends in these coefficients of variation for four of 

the ecoregions. The general result is that the coefficient of 

variation is improved little when m is greater than 40. 

However, increasing n diminishes the coefficient of variation 

substantially. For instance, the initial gain in precision by 

doubling n from 2 to 4 is almost as much as what is gained 

by quadrupling from 4 to 16. 

Although the coefficient of variation will decrease as n and 

m increase, a coefficient of variation of < 15 percent is 

probably impractical to obtain unless more than eight ranger 

districts are sampled in each ecoregion. Generally, it is better 

to increase n and decrease m for a given total sample size 

(nm). The dotted tine in figure 2 shocvs the nm = 200 line, 

which is the coefficient of variation when a total of 200 site 
Table 7-Variances s; and S; used to compute the variance 

of the mean site-visit estimate at the ecoregion level 

assuming two-stage sampling with a simple random sample 

at both stages 

Ecoregion N 7 

n s; $22 Mean 

N = number of ranger districts in the ecoregion; n = number of ranger 

districts sampled in the ecoregion. 

days are sampled with n ranger districts and m site days per 

ranger district. Drastic reductions to the coefficient of 

variation are possible for a fixed nm by increasing n and 

decreasing m. Such gains in precision at a fixed nm = 200 are 

obtainable if there is little additional cost or a fixed overhead 

cost for sampling more primary units (ranger districts). 

However, if these costs are a function of the number of 

primary units, for example, due to increased travel costs, 

then cost must be considered, and the nm line should be 

viewed with caution. In any event, it should be reassuring to 

the user of figure 2 that the relationships are similar among 

the four ecoregions. 

Sample sizenational level-Results from an ecoregion 

analysis easily can be extended to the national level by 

specifying a comon coefficient of variation for all 

ecoregions. The sample size curves (fig. 2) can be used to 

obtain an approximate n and m for all ecoregions based on 

the comon coefficient of variation. If this is done, then the 

level of precision achieved when the national estimate is 

obtained by combining the ecoregions [equation (16)] is at 

least at this level. This is easily shown as follows. Let there 

be two ecoregions where it is specified that the coefficient of 

variation of both is P percent. Thus, 

for all i 

where 

7,' = the total estimate for ecoregion i, and 

st2 = its variance. 

Solving for st' yields 

Because the total estimate is simply T= 7; + q, the 

coefficient of variation relationship becomes 

because 

for 7; and T2 2 0 . The generalization to more than two 

ecoregions is obvious. 

Sample sizealternative specification-The sample size 

recommendations given above are based on sampling with a 

desirable coefficient of variation. However, the sample size 

issue is often stated in terms of being within an allowable 

percent error (D) with a specified probability. To see the 

relationship between these two specifications, note that a 

coefficient of variation of 20 percent implies that one 

standard error is 20 percent of the mean. Assuming the 

normal distribution, one is 68 percent confident that the 

mean is within D = 20 percent of the true mean. This is the 

basis underlying figure 2. To design a survey with other, 

more typically used probability levels, one can simply 

redefine the coefficient of variation as the desired percent 

error D divided by the appropriate z-value from the standard 

normal distribution (Montgomery 1976). For instance, if a 

survey is to be designed where the specified allowable error 

is D = 20 percent at the 90-percent probability level, then 

redefine the coefficient of variation to be CV= 20i1.645 = 

12.16 percent where 1.645 is the z-value that corresponds to 

the 90-percent probability level. Then select an appropriate 

ecoregion sample size curve from figure 2 and draw a 

horizontal line at the coefficient of variation = 12.16 position, 

This line will represent a set of n and rn values from which 

the number of ranger districts and site days per ranger 

district, respectively, are determined to ensure being within 

D = 20 percent error at the 90-percent probability level. 
Ecoregion 2 Ecoregion 6 

Number of Number of site-days per ranger district (m) site-days per ranger district (m) 

Ecoregion 13 Ecoregion 14 

Number of site-days per ranger district (m) Number of site-days per ranger district (m) 

Figure 2-Coefficients of variation for four of the ecoregions for estimating site visits assuming a two-stage sampling design where 

n = the number of sampled ranger districts per ecoregion and m - the number of sampled site days per ranger district. 
Conclusion Cochran, W.G. 1977. Sampling techniques. 3" ed. New York: John 

Wiley. 428 p. 

This study is the first statistically based, outdoor recreation 

survey performed on the entire National Forest System, 

Among the ranger districts sampled, daily site-visit estimates 

varied considerably and averaged 6.1, and total annual site 

visits averaged 536,378 per ranger district. The variability of 

these estimates also ranged widely with a coefficient of 

variation of 10.4 to 66.9 percent, averaging 29.3 percent, 

indicating the potential for reasonably good estimates for 

some ranger districts, but also some difficulty for others. The 

total annual site-visit estimate for the entire National Forest 

System was 330,772,894 with a coefficient of variation of 20 

percent. 

Due to great variability and small sample size, it was difficult 

to evaluate stratification variables. Generally, based on the 

analysis of variance, DEFF, and RHE, site type appeared to 

be the most important; and it should be incorporated in 

future surveys, Season and day type appeared somewhat 

less important, possibly because the variance reduction that 

they possessed jointly was partitioned individually to these 

two variables. For that reason, we suggest that season and 

day type should be combined into one use-level 

stratification variable with only two or three levels. There 

were too few ranger districts in the ecoregion stratification to 

evaluate ecoregion stratification. 

The sample size curves presented should give an estimate of 

the sample size requirements for specified levels of precision 

based on the coefficient of variation. Generally, site visits at 

the ranger district level could be estimated with a coefficient 

of variation of 15 percent by sampling 100 site days using a 

stratified random sampling design with optimum allocation. 

The sarnple size curves at the ecoregion level present 

various combinations of number of ranger districts and site 

days per ranger district required to achieve a specified level 

of precision using a two-stage sampling design. These 

sample size guidelines can be used in planning national level 

outdoor recreation surveys. 

Literature Cited 

Bailey, R.G. 1995. Descript~on of the ecoregions of the Un~ted 

States. Misc. Publ. 1391. Washinton, DC: U.S. Department of 

Agriculture, Forest Service. 108 p. + map. 

Bailey, R.G.; Avers, P.E.; King, T.; McNab, WH., eds. 1994. 

Ecoregions and subregions of the United States (map). Washington, 

DC: U.S. Department of the Interior, Geological Survey. 

1 :7,500,000; colored. [Accompanied by a supplementary table of 

map unit descriptions compiled and edited by W. Henry McNab and 

Robert G. Bailey. Prepared for the U.S. Department of Agriculture, 

Forest Service.] 

Conover, W.J. 1980. Practical nonparametric statistics. 2d ed. New 

York: John WlIey. 493 p. 

Gregoire, T.G.; Buhyoff, G.J. 1999. Sampling and estimating 

recreational use. Gen. Tech. Rep. 456. Portland, OR: U.S. 

Department of Agriculture, Forest Service, Pacific Northwest 

Research Station. 39 p. 

James, G.A. 1967. Recreation use estiniation on Forest Service lands 

in the United States. Res. Note SE-79. Asheville, NC: U.S. 

Department of Agriculture, Forest Service, Southeastern Forest 

Experiment Station. 8 p. 

James, G.A.; Heniey, R.K. 1968. Sampling procedures for 

estimating mass and dispersed recreation use on large areas. Res. 

Pap. SE-3 1. Asheville, NC: U.S. Department of Agriculture, Forest 

Service, Southeastern Forest Experiment Station. 15 p. 

James, G.A.; Ripley, T.H. 1963. Instructions for using traffic 

counters to estimate recreation visits and use. Res. Pap. SE-3. 

Asheville, NC: U.S. Department of Agriculture, Forest Service, 

Southeastern Forest Experiment Station. 12 p. 

Kish, L. 1965. Survey sampling. New York: John Wiley. 643 p. 

Kocis, S.M. 1986. The adoption and diffusion of methods for 

estimating recreation use in the Pacific Northwest Region of the 

U.S. Forest Service. Moscow, ID: University of Idaho. 161 p. M.S. 

thesis. 

Marcuse, S. 1949. Optimum allocation and variance components in 

nested sampling with an application to chemical analysis. Biometrics. 5: 189-206. 

Milliken, G.A.; Johnson, D.E. 1984. Analysis of messy data. 

Volume I: Designed experiments. New York: Van Nostrand Reinhold 

Co. 473 p. 

Montgomery, D.C. 1976. Design and analysis of experiments. New 

York: John Wiley. 418 p. 

Rao, J.N.K. 1962. On the estimation of the relative efficiency of 

sampl~ng procedures. Annals of the Institute of Statistical 

Mathematics. 14: 143-1 50. 

Ruark, G.A.; Zarnoch, S.J. 1992. Soil carbon, n~trogen, and fine 

root biomass sampling in a pine stand. So11 Sclence Society of 

Arner~ca Journal. 56(6): 1945-1 950. 

SAS Institute Inc. 1989. SAS/STAT user's guide. Version 6. 4& ed. 

Gary, NC: SAS Institute Inc. 846 p. Vol. 2. 

SAS Institute Inc. 1990. SAS procedures guide. Version 6. 3'Qed. 

Gary, KC: SAS Institute Inc. 705 p. 

Zarnoch, S.J.; Marx, D.H.; Ruehle, J.L.; Baldwin, V.C. 1393 

Sampling open-top chambers and plantations for live fine-root 

biomass of loblolly pine. Res. Note SE-368. Ashev~lle, XC: U.S. 

Department of Agriculture, Forest Service, Southeastern Forest 

Exper~ment Statlon. 19 p. 
Zarnoeh, S J.; Kocis, S.M.; Cordell, H.K.; English, D.B.K. 2002. A pilot 

sampling design for estimating outdoor recreation site visits on the national 

forests. Res. Pap. SRS-29. Asheville, NC: U.S. Department of Agriculture, 

Forest Service, Southern Research Station. 20 p. 

A pilot sampling design is described for estimating site visits to National Forest 

System lands. The three-stage sampling design consisted of national forest ranger 

districts, site days within ranger districts, and last-exiting recreation visitors within 

site days. Stratification was used at both the primary and secondary stages. Ranger 

districts were stratified based on Bailey's ecoregions, while site days were stratified 

based on site type, season, and day type. Statistical methodology is presented to 

derive site-visit estimates at the site day, ranger district, and national levels. Results 

are presented to illustrate the magnitude of the site-visit estimates, their variability, 

and confidence intervals. With such information, an evaluation of the stratification 

variables is presented using the design effect and the relative hypothetical efficiency. Sample size analysis is performed to provide recommendations for future 

sample surveys to meet specified levels of precision 

Keywords: National forests, outdoor recreation, sampling, site visits. 
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