Sequentially Observed Periodic Surveys 

sf Management Compartments 

to Monitor Red-cockaded Woodpecker Populations 

Robert GI Hooper, Research Wildlife Biologist 

U.S. Department of Agriculture, Forest Sewice 

Southeastern Forest Experiment Station 

Charleston, South Carolina 2941 4 

H. David Muse, Professor of Mathematics 

University of North Alabama 

Florence, Alabama 35632 
ABSTRACT 

Management of the red-cockaded woodpecker (Picordes borealis) requtres knowledge of size and trend of rndtvtdual populations Perrodtc entry rnto management 

comparlmenls for thrnn~ng and regeneral~on of stands provrdes considerable tnformatron on ~ndrv~dual cav~ty trees and colontes The statrstical rat~onale and formulas 

for uslng th~s rnformatron to est~mate populaton size and trend are presented. W~lh 

add~t~onal field work, these data may provide better est~mates of popuiat~on srze and 

trend than perrodtc random samples of compartments In adbitton, estrmates can be 

made yearly ~nstead of at 5- to 10-year rntervais 

Keywords: Compartment prescr~ptton, endangered specles, P~caldes boreahs, populat~on trend. 

A major task in the management of the endangered red-cockaded woodpecker IS the monitoring of populations Monttoring shows whether a 

population is stable, Increasing or decreasing over t~me Without such 

knowledge recovery efforts are bl~nd and no long-term assessment of the 

effeci~veness of management can be attained Past monitoring efforts emphas~zed rangew~de trends:"7 nat~onal forests, w~ldlrfe refuges and mrlitary bases, each ~nvolv~ng thousands of hectares, were sampled Consrderrng the enormous task and associated costs, the rangewrde approach was 

historicalty justified However, w~th the rangewide information in hand, most 

experts now think knowledge of individual populations are needed Populations are disjunct and independent. The effects of geographic rsolation are 

compounded by d~verse land-use histor~es and phys~ograph~c differences 

Thus, the demographic status of a particular population may not be shared 

by other populations Sampling intensity of the rangewide survey, by 

design, is not intense enough to detect trends in indivrdual populations 

One solution IS a periodrc random sample of management compartments 

at an intensity great enough to detect a specrfrc level of population 

change This approach IS good and IS being used, but it does have shortcomings To detect a small change even over a long ttme span (say 10 

years) may requlre a sample so large that rt is not practical to obtain on 

a one-time basis. A random period~c sample requires a separate rnrtiat~ve 

by the manager over and above the normal budget. Also populat~on estlmates are obtained only at 5- to 10-year ~ntewals. These data have Irttle 

value beyond the one-time populatton estlmate and cannot be used for 

future est~mates. 

'~ennartz Michael R , Gerssler Paul H Harlow Richard F Long Randait G 

ChiWood Kenneth M Jackson Jerome A 1983 Status of the red-coekadsd 

woodpecker on Federal lands in the South In Wood Don A ed Red-cockadeo 

woodpecker symposium 2 Procsedlngs 1983 January 27-29, Panama City, FL 

Tallahassee, FL State of Florrda Game and Fresh Water Fish Gommrsston, 

1983 7-12 

'U S Fish and Wildl~fe Service 1985 Red-coekaded woodpecker recovery 

ptan U S Fish and Wildlife Service Atlanta GA 
An alternative to periodic random sampling could be to augment and use 

data that are collected rn the normal course of forest management. Timber sales and other activities are based on compartment prescriptions that 

are made approximately every 10 years for a given management cornpartment. All stands lo be thinned or regenerated are rigorously searched 

for red-cockaded woodpecker cavity trees as part of the process. These 

data are necessary to coordinate timber management with woodpecker 

management. For our alternative method to work, it is necessary to be 

able to delineate all colonies in a given compartment. Thus, potential habitat 

in ;a compartment not searched during the prescription process, would 

need searching in order to have complete? informailon for that compartment. 

This paper provides the statistical rationale for integrating the monitoring of 

red-cockaded woodpecker populations into the compartment prescription 

process. Th~s integrated approach has several advantages: (1) most of the 

data used would have been collected anyway, (2) survey data from all compartments are used to estimate the population, (3) precision should be 

considerably greater than with a periodic random sample, (4) annual estimates of the population are practical, and (5) the procedure may be tolerant of minor violations in assumptions. 

Assumptions and Requirements 

1. Compartments are searched repeatedly over time such that each compartment is entered every k years, k is simply the modal elapsed time 

in years between compartment entries for a given compartment. On the 

'I 0-year entry schedule commonly in use, k equals 10. In practice, reentry of a given compartment usually varies from 8 to ? 2 years, and is 

sometimes longer. 

2. All active colonies in a given compartment are determined by systematically searching all habitat with the potential for cavity trees during 

each compartment entry. 

3. Compartments entered in a given year are interspersed throughout the 

forest. 

4. Average characteristics among sets of compartments entered are sim- 

~lar from year lo year. For example, if compartments are mostly young 

plantations one year and mature sawtimber the next, the estimates could 

be biased. 

5. The change in the number of active colonies per compartment is direct!~ propsrlional to the number of years of elapsed time since the 

previous compartment entry. 
Current Data 

Let C denote the set of compartments for wh~ch current dala are available. 

Let j denote the number of years elapsed between the previous and current suwey for a given cornparlmenl. Also, let xIoi and yliil denote the number of active cctlonies found dunng the prevrous and current sumey, 

respectively, for the ifi RcompaPlrnent within the set of cclrnparlmenls with an 

efapsed census time of j years. 

Next fe2 

where nj denotes the number of compartments wi"r an elapsed suwey 

lime of j years. Then, 

X, - X xSi and y, = 2 ysi 

i i 

denote the respective totals of previous and current surveys of active colonies 

for the compartmenls contained in C. Note that the case involving current data 

for compa~ments that lack previous data, is covered under interim Estlmation Procedures. 

Noncurrent Data 

Let Co denote the set of comparlments For which current dala are not 

availabte. Note Ihah 42 and 8 comprise the entire Forest, Let xIbj denote the 

number OF colonies in the ith compartment contained rn the set of 

cornparlrnents surveyed j years ago. if n, denotes the total number of 

compartments surveyed j years ago, let 

denote the total number of then-active colonies for "ris setook compaflments, 
Estimation Based on a Ratis Approach 

If =: k for all compartments in C, an appropriate estimate for the ratio 

R is given by 

Then, 

n n 

Y,(k) = R X,(k) 

denotes the ratio estimate for the i(k)" censused compartment. 

If 

- then the estimated variance of is 

- where N denotes the number of compartmentsin the forest. Since ? = rh. Rx,;nk, it follows that the estimated variance of R is 

A *AX 2 

var (R) = nk var (Y) i x,. (8) 

If j + k for some com_partments of C, then the estimation of A and the associated variance of R involves a closer examination of the previous survey data. Specifically, the relationship between j and x,~t should be 

explored. 

If Assumption 4 is valid, then j and x,~) should be essentially uncorrelated. 

Of this is the case, then farrnuia (4) applies. If j and x,~) are linearly reiated, 

then each x,, j 4= k, should be adjusted to approximate previous survey 

data ts k years elapsed time. Therefore let 
where 

and 

- 

d =: ( 2 n,(x,!n, - xs$nk)i(j - k)) / ( Z n,) 

li--k j+k 

(11) 

denotes an average annual adjustment increment per compartment for j f 

k. Note that formula (9) reduces to (4) when j - k for all compartments of 

C. 

If the relationship between j and Xi(j) is curvilinear, which we think is 

unlikely, then a problem with Assumption 4 or 5 could be indicated. If so, 

then use of the ratio estimation procedure may be inadvisable; otherwise, 

curvilinear regression techniques may be needed to adjust each x,, j f k, 

to approximate previous survey data to k years elapsed time. 

Using formula (4) or (9), whichever is applicable, let j(fixiQ, - xiO))/k denote the estimated population change for the i(jIth compartment adjusted 

to k years elapsed time. Then, the addition of this term to xi() yields 

A 

y.. = x.. iQ) 10) (j 3 + - j)/k (I 2) 

which denotes the ratio estimate for the i(i)th censused compartment adjusted for k years elapsed time. If 

- where n, = Z ni, then the estimated variance of? is 

i 

.-. - n "I .-. var (Y) = (N-n,) 2 I: (Y,~) -~,~))~/(n: - n,),N J 1=1 

"I "i 

= (N - n,) Z( E y$) + (j + k - jlZ Z x&/k2 

j i-1 i=l 
where is defined by (12). Since 

the estimated variance sf ff is 

h /-. h x "I 

var (R) = n: var (Y) i ( Z j Z ~~~)/k)~. 1 r=l 

Note that forwruras (16) and (8) agree if j = k for all compafimenls in C. 

For each age set of comparSments in 0, the current total of active celonies may be estimated using Assumption 5 together with the applicable 

estimate of R given above. Consequently, th_e estimated change for the set 

of cornpaflmerats suweyed j years ago is j(Rxj - xj)/k. Therelare, the sum 

of Xj and the eslimated change yields the associated current el3limated 

tot&[ 

Therefore, the estimated current total of active cotanies for O is 

A 

Yo, = 2 qj, = G2jxj,'k + Z(k - j)xj/k 

I i 1 

A-. 

= &r x, -i- Z(k - j)x{k 

I 

(1 8) 

where x,, - Z jqk. 

j 

Using formulas (2) and (181, the total forest estimate for the number of 

currently active colonies is 

The ass~~iatedstimated variance of ? is either 

A A A A 

var (Y) I= (x,, + x,12 var (Fa) (20) 

depending on whethe~ormttla (4) or Carmuia (91, respectively, were used 

in the comg=.ralalion of R. 
Estimation Based on Average Change 

Using the notation introduced in the Definitions Section Car data set G logether with formula (1 1 ), let 

denote the change in number of active colonies for the i6lth "compartment 

adjusted to an elapsed time of k years when j + k. If j -- k far all cornpadments in C, then note that formula (22) reduces to 

Next, determine 

and 

where n, denotes the number of eornparlrnents in C and x,,is given by 

(1 0). 

Far each age set sf csmga~menls in the noncurrent data set 8, ie% 

Then, the estimated total Car O is 

where no, --- X jnjik;and x, - 2i xi, 

j I 

Using formulas (24")and (27). the associated tstai fores"reslimate of active red-csekaded woodpecker colonies is 

h A 

Y=Yoa+y,=x,+~,,+(n,+n,,)~, (28) 

The corresponding estimated variance of ? is 

A A 

var (Y) = (N - n,) (n, + ~2nd~. (29) 
Interim Estimation Prscedutes 

Formulas (I ) through (29) assume that lor every compartment in the current data set, C, a comparable data set exists from an earlier period of 

time. This ideal situation will not always exist initially. If not, then it would 

take up to 13 years to be able to make complete use of the above procedures, Information is needed in that interim period and the procedures and 

formulas that follow are for that purpose. 

Let n, and n denote the number of compartments in C for which previous survey data are available and are not available, respectively. Also, let 

no and N - n - n, - no denote the number of comparlments in 0 for which 

previous survey data are available and are not available, respectively. If 

n>O, then let j =. 0, and let y,(,) denote the current survey for the associated i(o)" compartment. 

Then, 

If no = 0, it follows from Assumptions 3 and 4 that the estimate of total redcockaded woodpecker colonies for the entire forest is 

h 

Y = N (y, + y,)/(n, i. n) = N y. 

The associated variance of ? is 

A h "i 

var (Y) = N (N - n, - n) ( 2: L: y& - (n, + n)y2)/(n, + n)!(n, + n - 1). 

j 1=1 

If n, + 0 and n, is small, then total forest estimation based on formulas 

(31) and (32) is advisable. Whether or not n, is considered small is best 

decided by the user. An approximate definition of small might be <20 comparr;ments or <30 percent of N. If n, f O and n, is not too small, a total 

forest estimate may be obtained by using 

where? is given either by formula (19) or (28). Then, u%ng the associated estimated variance of Y, the estimated variance of Y, is given by 

n A A A 

var (Y,) = N~ ( (Z Y&,) - n g) (N - n)iNi(n - 1) + var (Y) )/ 

(n, + n i- (349 

where ji, = y,,/n. Since the variance of Tt is sensitive to the size of n, it 

may be wotlhwhile to compare it to the current-data-only estimate [formulas (31) and (32)], and utilize the one having the smafler estimated 

variance. 
Detection of Population Change 

A major goal of population monitoring is to detect significant changes in 

the population. To compare current survey data to an appropriate previous survey, let y,! denote the survey for the ith compartment in the set of 

compartments surveyed j years ago and nj denote the number of compartments in that data set. Note that j ranges from zero to k or more. Then 

the estimate of active red-cockaded woodpecker colonies for the total forest based on the set of compartments surveyed j years ago is given by 

The associated variance of 7 is estimated by 

A c^s. 

var (Y,) = N(N - nj) sfin, 

where 

Therefore, for any two age sets of compaPlm2ntsf_and j* where j"=j*, the 

corresponding estimated change is given by Y, - Yj*. The associated estimated variance is given by 

A A A A A h 

var (Y, - q*) = var (Y,) + var (Yi*). 

The comparison of interest might involve the current estimate (j == 0) versus a previous year estimate, say j* -. 4. Then, srilitistical inference %bout 

population change could be based on the approximate normality of Yj - A 

Yr and the associated t-statistic based on nj + n,* - 2 degrees of 

freedom. If the approximate normality assumption with regard to the number of active colonies per compartment is doubtful, then a large absolute 

value of the associated t-statistic may be used as an indicator of population change without recourse to the t-table. 

Mote that a finding of significance may indicate either a change in population size or a possible breakdown in Assumption 4. Therefore, a significant finding should be confirmed, Confirmation could include (a) analyses 

of the type indicated below [formula (39)j; (b) appropriate check samples 

if the findings under (a) were also significant; and (c) examining the prior 

trend and waiting at least one more year to see if the trend continues. An 

appropriate check sample for a sudden drop in the population wuld he a 

random sample of colonies to determine if a widespread loss of active cslonies was occurring. An appropriate confirmation response given a significant finding of population increase would be to wait at least one more 

year to see if the trend continues. 
Other comparisons might be sf interest. For example, the statistic, 

contrasts the most recent "ihree years sf data with the next oldest three 

years sf dala. "The associated variance is given by 

5 

with ( Z n, - 6) degrees of freedarn. This type of c~mparison might be 

j=O 

preferred if Assumption 4 is tenuous. 

Plan for Implementation 

If previous sumey dala are incomplete or nonexistent far some compa~- 

ments, then the interim estimation procedures [formulas (30) and (34) as 

appropr~atej could be used until previous survey da"r bbecome available 

For all csmparlmenls. 

If: previous serwey dala are cornptete for ail csmpav"ements, then both the 

ratio estimation procedure and the average change estimation procedure 

are appropriate, A total forest estimate based on either of these procedures should prove to be more accurate and precise than the corresponding estimate based on the current-data-only procedure (31). During the 

early stages of' implementation, it sl-rouid be useful $0 obtain estimates 

based on each of these procedures "igetker with assoctated variance 

estimates* If Assumptions I through 5 are valid and if previous survey 

data are correlated with current suwey data, then the ratio estimator should 

prove to be substantially more precise than either of the other Two es"6- 

mabsrs. ff however, the correlation between previous and current survey 

data 1s near zero, then "ie ratio and average change estimators should 

be competitive, If Assumption 4 is tenuous, then the: average change estimator may do well, particularly if j f k for some compartments in 61;. 

Consequently, comparison of the variances generated by these estimators 

is needed. 
In view of the longevily of red-cockaded woodpecker colonies, the current 

data set might safely include current year data together with the survey data 

lor the previous two years, Also, habitat changes resulting from a given 

prescription do not occur until several years aner the prescription and woodpecker survey are made. Consequently, the current data set could represent approximately 30 percent of the compartments in the forest on a 

continuing basis. 

Comparison to Periodic Stratified Randsrn Samples 

Periodic stratified random samples are currently used to estimate redcockaded woodpecker populations. This procedure typically employs a 15 

to 30 percent random sample of compartments at 5- to 10-year intervals. 

Considerable effort went into developing and implementing that procedure, 

and we consider it to be both sound and practical. The main reasons for 

considering an alternative are the integration of population monitoring with 

ongoing compartment prescriptions, anticipated gain in accuracy and 

precision, and possibly some cost reduction. 

Cost reduction csuid possibly occur because a high percentage of forest 

stands with potential for red-cockaded woodpecker colonies would be 

searched during the normal prescription process for a given compartment. 

By also searching remaining potential stands not searched by the regular prescription process, the data would be complete for that compartment. 

Whether or not this returns a savings depends upon the average remaining proportion to be searched among compartments entered during the prescription process, and the size and frequency of the random sample, such 

that 

where: Efliciency is proportion of compartments searched using the 

periodic random sample compared to monitoring populations by 

the compartment prescription process, 

GR = proporfion of comparlments in random sample 

Cp = mean proportion of all compartments entered during 

prescription process each year 

S = average proportion of a cornparlment searched during 

prescription process (limited to potential habitat) 

Y = number of years between random samples. 
For exampie, ~f 66 percent of a compartment rs normally searched durIng the prescription process, if the random sample rs a relatively small 15 

percent, and rf the random sample is taken every 10 years, it wrfl cost 19 

percent more to monrtor red-cockaded woodpecker populations with the 

prescription process than with a random sample of compartments. However, tf 85 percent of a compartment IS searched during the prescription 

process, ~f the random sample IS 30 percent, and if the random sampie 

is taken every 5 years, it would be 1.45 t~mes cheaper to use the enhanced 

prescription process for monitoring. Finally, rf only 66 percent of a compartment IS searched In the prescript~on process, and ~f the random sample 1s 30 percent and is taken every 10 years. the costs of the two methods are stmtlar (0.96:l.O). Cost In these examples rs expressed as 

"compartments searched." 

Wlthout actual examples, it IS more drSflcult to compare accuracy and preclsion of the sequentrally observed per~odlc survey to that of a random 

sample. tjowever, given the large srze of the current data set (30 percent) 

and the fact that lnformatfon from the total populatron IS used, this method 

should be as good or better than a per~od~c random sample of the same 

size A primary advantage of the sequentially observed period~c survey is 

that yearly estrmates of the population can be made. W~th perfodlc random 

samples, estimates are made every 5 to 10 years. In addit~on, all the data 

collected durtng the prescrlptton process can be used immediately by the 

manager. Most of the tnformat~on on red-cockaded woodpecker colonies 

collected in a compartment by the random sampie procedure must be collected agarn when ~t IS time for its compartment prescription 

Randomness 

A crlticlsm of the sequentially observed perlodic survey IS that the compartments to be surveyed each year are not randomly selected from across 

the forest. Therefore, an est~mator of red-cockaded woodpecker colon~es 

for the forest could be strongly biased and the accompanying estimate of 

the varlance could be mlsleadlng. Although th~s concern rs reasonable, several factors may reduce its impact: 

I. Interspersat of survey compartments. One objective of a good random 

select~on is that compartments to be sampled be interspersed throughout the forest. If randomization fails to provide satisfactory coverage 

of the forest, then many practitioners would rerandomize to obtaan better interspersion of compartments. Therefore, one major goal of a random selection is consistent with our Assumption 3. 

2. Size of current data set. The current data set represents approximately 

30 percent of the compartments rn the forest on a conttnurng basts. In 

contrast, a perrodrc stratifred random sample would yield rnformatron on 

15 to 30 percent of the compaflments every 5 to 10 years. Consequently, the advantages of a preponderance of current data should outwergh the advantages of a periodic random sample if the compartments are indeed interspersed throughout the forest. 
3. Compartment entry scheduts, Because the enty schedule for compartment prescriptions is largely based on factors other than the redcockaded woodpecker, the set of compartments entered each year is 

unrelated to the red-cockaded woodpecker. Consequently, potential 

comparlrnent selection bias related lo the red-cockaded woodpecker 

should not be a problem, 

In view of the above factors, whether randomization can be foregone is 

a matter of professional judgment, both statistical and biologicai, Clearly, 

randomization is incompatible with the prescription process; but, in our 

opinion, potential bias from using the sequentially observed periodic survey method is of little concern if Assumptions 3, 4, and 5 are valid. 

Cautions 

Population changes resulting from management practices, based upon research and agreed upon in consultation with the U.S. Fish and Wildlife Service, and from habitat changes due to aging of forest stands, will probably 

occur at a slow rate. The sequentially observed periodic survey procedure 

offers a reliable 10-year estimate of subtle forestwide population changes 

(actually k-year changes) on a yearly basis. 

t-towever, it should be reallzed that monitoring at the population level is 

not capable of detecting the immediate effects of management. Changes 

in population resulting from new management could not be detected by 

the sequentially observed periodic survey until the next prescription. At that 

time the procedure should be highly sensitive to population changes. 

Unfortunately, the entire forest will have likely been subjected to the new 

management practice by the time of re-entry of the f~rst compartments to 

have been treated. Similar concerns hold as well under the periodic random sample approach. 

The legally required consultation process between the U.S. Fish and Wildlife Service and Federal agencies managing endangered species is critical 

for identifying potentiaily negative practices. The best safeguard against 

a new management practice with potentially negative ~mpacts on redcockaded woodpeckers is to test that practice before it is applied forestwrde. Alternatively, rt would be useful to check each colony in the afiecled 

compartments on a yearly basis, but if the questionable practice is being 

applied on a forestwide schedule and the birds are slow to respond, most 

of the forest may have been treated before changes are detected. 

Finally, rn small and/or obviously stressed populations, significant declines 

could occur but not be detected for severat years because of the small percentage of compartments with colonies. it would be wise to check such 

colonir?~ on a yearly or biennial scheduiet even though the sequentially obsewed periodic survey was being used.