A Weighted Surveillance Approach for Detecting Chronic ... · prion, sampling, weighted...

A Weighted Surveillance Approach for Detecting Chronic Wasting

Disease Foci

Daniel P. Walsh1,2 and Michael W. Miller1

1 Wildlife Health Program, Colorado Division of Wildlife, Wildlife Research Center, 317 West Prospect Road, Fort Collins,Colorado 80526-2097, USA2 Corresponding author (email: [email protected])

ABSTRACT: A key component of wildlife disease surveillance is determining the spread andgeographic extent of pathogens by monitoring for infected individuals in regions where cases havenot been previously detected. A practical challenge of such surveillance is developing reliable, yetcost-effective, approaches that remain sustainable when monitoring needs are prolonged orcontinuous, or when resources to support these efforts are limited. In order to improve theefficiency of chronic wasting disease (CWD) surveillance in Colorado, United States, wedeveloped a weighted surveillance system exploiting observed differences in CWD prevalenceacross demographic strata within infected mule deer (Odocoileus hemionus) populations. We usedfield data to estimate sampling weights for individuals from eight demographic strata distinguishedby differences in apparent health, sex, and age. In this system, individuals from a sample sourcewith high prevalence and low inclusion probability (e.g., clinical CWD ‘‘suspects’’) received $10.3times more weight than those from a source with low prevalence and high inclusion probability(e.g., apparently healthy, hunter-harvested individuals). We simulated use of this alternativesurveillance system for a deer management unit in Colorado and evaluated the potential effects ofusing biased weights on the probability of failing to detect CWD and on relative surveillance costs.We found that this system should be transparent, cost-effective, and reasonably robust to theinadvertent use of biased weights. By implementing this, or a similar, weighted surveillancesystem, wildlife agencies should be able to maintain or improve current surveillance standardswhile, perhaps, collecting and examining fewer samples, thereby increasing the efficiency and cost-effectiveness of ongoing CWD surveillance programs.

Key words: Chronic wasting disease, disease detection, mule deer, Odocoileus hemionus,prion, sampling, weighted surveillance.

INTRODUCTION

Surveillance is a key component ofeffective wildlife disease monitoring andcontrol. Surveillance typically focuses onestimating current levels of disease in areasof known occurrence and on monitoringboth peripheral and distant areas to deter-mine the geographic distribution and estab-lishment of new disease foci. For apparentlyemerging wildlife diseases, determiningtheir broad geographic distribution may beparticularly critical to assessing implicationsand prospects for control. It follows thatdeveloping efficient approaches for detect-ing new disease foci could be valuable towildlife managers and agencies responsiblefor surveillance within their jurisdictions.

A recently emerging prion disease ofwildlife, chronic wasting disease (CWD),was first recognized in captive mule deer(Odocoileus hemionus; Williams and

Young, 1980) and elk, and was subse-quently diagnosed in free-ranging elk(Cervus elaphus nelsoni), mule deer,white-tailed deer (Odocoileus virginia-nus), and moose (Alces alces) in scatteredfoci across North America (Spraker et al.,1997; Williams, 2005; Baeten et al., 2007).Since 2002, considerable resources havebeen spent by wildlife management andanimal health agencies and their partnersaround the world in conducting surveil-lance to better define the geographicdistribution of CWD. Current CWDsurveillance efforts in North Americafocus mainly on monitoring regions wherethe disease has as not yet been detectedand on estimating prevalence in knowninfected areas; however, approaches foraccomplishing the latter seem morestraightforward and efficient than for theformer (Samuel et al., 2003). In Colorado,United States, for example, monitoring for

Journal of Wildlife Diseases, 46(1), 2010, pp. 118–135# Wildlife Disease Association 2010

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CWD in populations where cases have,thus far, not been detected involves sam-pling cervids from a variety of sourcesincluding vehicle-killed animals, animalsculled or recovered dead as CWD ‘‘sus-pects,’’ hunter-killed (or ‘‘harvested’’) ani-mals, and from other sources such aspredator-killed or confiscated animals(Miller et al., 2000; Hibler et al., 2003;Krumm et al., 2005). Among submissionsfrom hunters, there are generally numerousindividuals from various demographic seg-ments of the state’s hunted cervid popula-tions. Past surveillance methods treatedthese various sources that contributed tothe surveillance stream—hereafter referredto as ‘‘strata’’—separately for purposes ofcalculating needed sample sizes and did notalways combine data from these strata.Consequently, this traditional approachfailed to capitalize on available informationconcerning rather large differences inapparent CWD prevalence among thesestrata (Miller et al., 2000, 2008; Miller andConner, 2005; Krumm et al., 2005).

To improve the efficiency of CWDsurveillance in Colorado, and perhapselsewhere, we developed a weighted sur-veillance system that makes use of allavailable information on stratum-specificprevalence. Similar approaches have beenespoused by Cannon (2002) and have beenspecifically applied to bovine spongiformencephalopathy monitoring in EuropeanUnion member states (Wilesmith et al.,2004). For any, user-specified probabilityof detecting disease (e.g., 95% probabilityof detecting at least one case whereprevalence is $1%), weighted surveillancenot only provides rigorous estimates ofsample sizes to demonstrate that a region isnominally ‘‘disease-free’’ using combinedsample sources, but it should also be morecost-effective than traditional approaches.The weighted surveillance system de-scribed here is intended to be used inaggregating data from a stratified samplecollected from primary sampling units,selected a priori, based on some biologi-cally-relevant spatial sampling scheme.

Although we recognize the importance ofhaving a spatial sampling scheme toaccount for the spatial variability and focalnature of CWD (Samuel et al., 2003;Conner and Miller, 2004; Joly et al., 2006;Nusser et al., 2008), describing the designof such a spatial sampling scheme is beyondthe scope of this paper. Consequently, welimit our discussion to the design andapplication of a weighted sampling systemfor detecting new CWD foci.

MATERIALS AND METHODS

Derivation of the weighted surveillance system

The first step in developing this weightedsurveillance system was to estimate theweights for each specific stratum in theCWD surveillance ‘‘stream.’’ To estimate theseweights, we made the following assumptions:The number of positive cases at the time of thesurvey in each of ith strata was independentlydistributed as Poisson (li) random variables,individuals were randomly selected within theith stratum for sampling, and relative preva-lence within each stratum was constant acrossdifferent population prevalence levels. Basedon these assumptions, the maximum likelihoodestimates for the weights (wi) were:

wwi~ppi

pp0

, ð1Þ

where pi5xi/ni was the maximum likelihoodestimate of the prevalence for the ith stratumand p0 was the corresponding estimate for thebaseline stratum. The weight (w0) for the‘‘baseline stratum’’ was defined to be 1; thisbaseline stratum was considered the referencestratum to which the relative weights for theremaining strata were scaled. Both pi and p0

can be empirically estimated from historicdata, when available. For an in-depth deriva-tion of the estimated weights and variances,interested readers are referred to Appendix A.

The next step in implementing this surveil-lance system was to determine when anadequate number of samples had been col-lected (i.e., to calculate a stopping point forsurveillance). To facilitate this, we employed apoints system (Cannon, 2002; Wilesmith et al.,2004). Under this system, every sampleentering the surveillance stream received anumber of weight points (wi), based on itsrespective stratum membership, and samplingcontinued until the total number of pointssummed to the target (t). The target, orcumulative number of points needed before

WALSH AND MILLER—WEIGHTED CHRONIC WASTING DISEASE SURVEILLANCE 119

enough samples have been collected to ceasesurveillance, was a function of the desiredprobability of detecting at least 1 CWD-positive case (12a) and of the specified designprevalence for the baseline stratum. This valuewas calculated (Dohoo et al., 2003) as

t~{ln að Þpdesignf

, ð2Þ

where pdesign was the specified design preva-lence for p0 (i.e., the prevalence at which thesurveyor wishes to detect at least 1 CWD-positive case in the baseline stratum), 12a wasthe probability of detecting at least 1 CWD-positive case, and f was the sensitivity of thetest. Thus, under weighted surveillance, sam-ple collection from the various strata continuesuntil the following is true:

t~Xm

i~0

niwwi, ð3Þ

where ni5the number of samples in thesurveillance stream collected from the ithstratum and wi5the estimated weight for theith stratum.

A potential concern in using weightedsurveillance is the effects of biased weights(i.e., what happens when the estimated weightsare above or below the unknown true weights).Such bias can lead to an over- or underestima-tion of the probability of detecting the disease.The amount of increase or reduction in diseasedetection probability, and the associated num-ber of samples needed to reach the target value,depends on both the bias in the individualweights and the number of samples from eachstratum. In Appendix B, we provide equationsfor estimating the bias in the number of samplesrequired for achieving the target value as well asthe bias in disease detection probability whenweights are under- or overestimated relative tothe true weight. Using simulations, we alsoexamined, in more detail, the effects of varyinglevels of bias (shown below).

Several important factors about this weight-ed surveillance system merit further consider-ation. First, the weights, as described, areanalogous to risk ratios. These types of ratioshave been extensively studied (Cox, 1972;Arnanda-Ordaz, 1983; Bedrick et al., 1997).Second, we assumed thereafter that sensitivityof the CWD diagnostic test was one (i.e., f51).Third, the pdesign parameter used to calculatethe target value is specified by the user andcorresponds to the minimum prevalence levelat which the user wishes to detect CWD in thebaseline stratum (e.g., ‘‘a goal of detecting atleast one case where CWD prevalence is $1%in adult males’’). Fourth, the user also needs to

choose the baseline population segment towhich the specified design prevalence appliesand to which other strata are scaled (‘‘adultmales’’ in the previous example). Although anystratum can be selected arbitrarily as thebaseline, we recommend that this be apopulation stratum for which sample sizesare consistently largest, or a stratum that issensitive to changes in prevalence. Based onthe foregoing guidance, we chose harvestedadult ($2 yr old) male mule deer as thebaseline stratum in the following exampleusing Colorado field data.

Estimating weights using Colorado mule deer data

To illustrate use of the weighted surveil-lance approach, we calculated samplingweights for CWD surveillance in mule deerin Colorado, using data collected from 2003–2006, in parts of Colorado where CWD isknown to occur. Data (Table 1) were from20,400 deer from various sources that enteredthe surveillance stream and were tested forCWD as described elsewhere (Miller et al.,2000; Hibler et al., 2003); samples from live-animal testing, captive facilities, or culling aspart of research studies were not includedbecause these represented few samples andwere from sources that contributed onlysporadically to the surveillance stream. Wedivided the submitted cases into eight stratadistinguished by differences in apparenthealth, sex, and age of deer included in eachstratum: 1) clinical CWD ‘‘suspect’’ females.1 yr old; 2) clinical CWD suspect males.1 yr old; 3) harvested ‘‘adult’’ ($2 yr old)males (the baseline stratum); 4) harvestedadult females; 5) harvested ‘‘yearling’’ (.1 but,2 yr old) males; 6) harvested yearlingfemales; 7) harvested ‘‘fawns’’ (,1 yr old, ofeither sex); or 8) all ‘‘other’’ dead deer (of bothsexes and all ages except fawns). The ‘‘other’’stratum included individuals recovered asvehicle-kills, predator-kills, and poaching sei-zures. Each of these strata received a uniqueweight (Table 1) calculated using equation (1);we estimated an associated SE for each weightusing equation (9) in Appendix A.

Simulations to investigate properties of theweighted surveillance system

To examine the potential performance ofthe weighted surveillance system, we firstexamined the effects of employing biasedweights (i.e., weights that were either greateror less than the true weight) on the probabilityof detecting CWD where true prevalence50.01 in harvested adult male deer, our baselinestratum. Initially, we used the simplest case

120 JOURNAL OF WILDLIFE DISEASES, VOL. 46, NO. 1, JANUARY 2010

(i.e., where all samples in the surveillancestream came from one stratum) to illustratethese effects using the stratum weight fromTable 1 as the ‘‘true’’ simulation weight. It isimportant to note that we only assumed thatestimated weights from Table 1 were the ‘‘true’’weight for simulation purposes. We also inves-tigated the effects of these biases on relativetotal surveillance costs. We repeated thisanalysis using two different strata, suspectfemale deer and harvested yearling males,because these strata represented the highestand lowest (aside from fawns) estimatedweights. We examined biases of 0–50% in theestimated weights. We used equations (11) and(13) in Appendix B to calculate the bias in 12a,sample size, and the relative costs arising fromthe use of biased weights. To calculate relativecost differences, we assumed an average cost of$75.00 USD per submission, which was basedon the estimated average testing and processingcost for a sample submitted into the ColoradoDivision of Wildlife’s (CDOW) surveillancestream. We did not estimate or include costsof acquiring samples from various strata becauseCDOW field and laboratory personnel acquiresuch samples (e.g., vehicle kills, culled clinicalsuspects, poaching cases, etc.) as part of normalwildlife health monitoring and law enforcementactivities. Thus, in Colorado, there are relativelyfew differences in acquiring samples fromdifferent strata.

To further examine the properties andefficacy of this technique using a more realisticscenario, we created a modeled population ofmule deer and estimated, via simulation, theprobability of detecting at least one CWD-positive individual (12a) with this weightedsurveillance system when prevalence was set at0.01 for the harvested adult male population.For this simulation, we used mule deer

population estimates, sex-age ratios, mortality,and harvest data for 2006 from a data analysisunit (DAU) located in southwestern Colorado(DAU D-19; B. Banulis, CDOW, pers.comm.). We chose this DAU because it wasapparently CWD-free (Colorado Division ofWildlife, 2009) and relatively good demo-graphic data were available. The populationsize was estimated at ,40,000 deer. Wepartitioned this population into the eight stratadescribed above using the 10-yr average of sexand age ratios and the cause-specific mortalityestimates from radio-collared deer (CDOWBig Game Harvest Survey, unpubl. data).Because some data were limited for maleswhen stratifying the population into thevarious demographic strata, cause-specificmortality probabilities for males were basedon estimates for females, except for harvestprobabilities which were estimated for bothsexes. To determine the probability of anindividual entering the surveillance streamfrom the ith stratum, we used 2006 statewideestimated harvest rates in conjunction with2006 CWD test submission rates from DAUswhere CWD had been confirmed. We usedrepresentative harvest and surveillance datafrom elsewhere because few samples enteredthe surveillance stream from DAU D-19 in2006, thereby precluding reliable estimation ofDAU-specific submission parameters. Wesimulated sampling individuals from the pop-ulation based on these sampling probabilities(Table 2), with each individual being assignedto one of the eight strata. Once an individualwas selected, it was determined to be ‘‘posi-tive’’ if a uniform random variable was lessthan or equal to the stratum-specific preva-lence; the stratum-specific prevalence wasbased on the prevalences calculated from theColorado data described above (Table 1), with

TABLE 1. The stratum-specific sample size, sample prevalence, estimated weights, and associated standarderrors, based on surveillance results for mule deer in Colorado from 2003–2006, for use in a weightedsurveillance system.

Stratum Prevalence Sample size PositivesTotal sample

sizeTotal

positives WeightsSE of

weights

Suspect—female 0.36 111 40 20,400 595 11.57 1.60Suspect—male 0.32 125 40 20,400 595 10.27 1.46Other 0.06 1,300 77 20,400 595 1.90 0.24Harvest—adult male 0.03 10,046 313 20,400 595 1.00 NAa

Harvest—adult female 0.02 5,782 104 20,400 595 0.58 0.06Harvest—yearling female 0.01 645 9 20,400 595 0.45 0.15Harvest—yearling male 0.01 1,392 11 20,400 595 0.25 0.08Harvest—fawn 0.00 999 1 20,400 595 0.03 0.03

a NA 5 Not applicable (Baseline stratum weight is defined to be 1.00).


prevalence50.01 assumed in the harvestedadult male stratum.

To evaluate the properties of the weightedsurveillance system, we first used the estimatedweights from Table 1 as the ‘‘true’’ weights forour simulations, as truth must be establishednominally to examine bias effects. Each sampledindividual was weighted according to thecalculated weight for its assigned stratum. Wesampled until the sum of the weights forsampled individuals equaled the target valueof 300 (i.e., the cumulative number of weightpoints needed to assure detection of at least onecase with 95% confidence when prevalence is0.01 among harvested adult males). We ran 500repetitions of this simulation. We then calculat-ed both the probability of detecting a CWD-positive individual (12a) in the sample and themean number of samples needed to meet thetarget. We estimated the cost differential underthe weighted surveillance system based on themean number of samples required to reach thetarget and on a cost of $75 USD for testing eachindividual submitted. We also examined thedistribution of the ‘‘waiting time,’’ which we

interpreted to be the cumulative number ofsamples required before detecting the firstpositive case, and we used a normal kerneldensity estimator to provide a smoothed fre-quency distribution. We then repeated theprocedure and used weights with biases rangingfrom 610250% the true simulation weights inorder to investigate the performance of thesystem when weights were biased.

In addition, we conducted simulations incor-porating a mixed bias procedure wherein thebias again ranged from 10250%, but wasnegative (i.e., underestimated the ‘‘true’’ value)for suspect males, suspect females, and the‘‘other’’ stratum and was positive for theremaining strata. Within the mixed bias simu-lations, we also evaluated use of a traditionalsurveillance system that assumed every stratumwas equally weighted (i.e., all the weights wereone regardless of sample source).

Finally, to study the effects of increasedsampling from the higher-prevalence strata, weexamined the effects relative to the samplingprobabilities, as described above (subsequentlyclassified as ‘‘no increase’’ in the simulation

TABLE 2. Stratum-specific prevalence and sampling probabilities based on 2006 demographic data frommule deer (Odocoileus hemionus) in Data Analysis Unit (DAU) D-19 used in the simulations evaluating theproperties of the weighted surveillance system for detecting chronic wasting disease in Colorado muledeer populations.

Manipulation of samplingprobability Stratum identification Prevalence Sampling probability

No increase Suspect—female 0.116 0.010Suspect—male 0.103 0.010Other 0.019 0.109Harvest—adult male 0.010 0.392Harvest—yearling male 0.003 0.042Harvest—adult female 0.006 0.279Harvest—yearling female 0.005 0.016Harvest—fawn 0.000 0.142

1% increase Suspect—female 0.116 0.020Suspect—male 0.103 0.020Other 0.019 0.119Harvest—adult male 0.010 0.386Harvest—yearling male 0.003 0.036Harvest—adult female 0.006 0.273Harvest—yearling female 0.005 0.010Harvest—fawn 0.000 0.136

5% increase Suspect—female 0.116 0.060Suspect—male 0.103 0.060Other 0.019 0.159Harvest—adult male 0.010 0.349Harvest—yearling male 0.003 0.032Harvest—adult female 0.006 0.236Harvest—yearling female 0.005 0.006Harvest—fawn 0.000 0.099


results), by increasing the sampling probabilityof the suspect male, suspect female, and otherstrata by 1% and 5% each, while commensu-rately decreasing sampling probabilities acrossthe remaining strata by a total of 3% and 15% toensure, for simulation purposes, that the totalsampling probability was 1 (Table 2). Subse-quently, we examined the statistics previouslydescribed. All analyses were performed in SAS9.2 (SAS Institute Inc., 2008).

Simulations to investigate samplesize requirements

For jurisdictions in which CWD has alreadybeen detected, responsible management agen-cies may want to develop their own weightedsurveillance system rather than use valuesestimated for Colorado mule deer. Therefore,we conducted additional simulations to esti-mate the number of samples that must enterthe surveillance stream in order to provideadequate information to estimate strataweights accurately. We used the strata-specificsampling probabilities, calculated as describedabove, and set the true simulation weights foreach stratum equal to our estimated weightsfor Colorado (Table 1). We set prevalence inthe harvested adult male stratum, our baselinestratum, to 0.03, which seemed to be areasonable estimate for CWD-endemic areasin other jurisdictions. We then created sur-veillance datasets of 1,000, 5,000, 11,000, and15,000 samples, calculated the estimatedweights for each stratum using equation (1),and repeated this process for 1,000 replica-tions. To examine the effects of small samplesize on error in the estimated weights, wecalculated the mean absolute error, meanpercent error, and the probability that thedirection of the bias would be positive acrossthe 1,000 repetitions for each sample size. Weused the mean absolute error, instead of meanbias, to look at the mean magnitude of errordue to small sample size because the latterwould be obscured by averaging positive andnegative bias values. We focused on theprobability of the bias being positive becausepositive bias will lead to fewer samples thannecessary being collected for surveillance,resulting in a decreased disease detectionprobability. Also, we expected strata with smallweights would tend to be negatively biased.

RESULTS

Estimated weights using Colorado mule deer data

Estimated weights and their associatedstandard errors for the eight different

strata in our weighted surveillance systemare in Table 1. The greatest weights wereattributed to female and male CWDsuspects (about 12 and 10 points, respec-tively), and the least weight was associatedwith harvested fawns (0.03), as would beexpected given the prevalence and overallsample sizes of these strata. We used theseestimated weights as the ‘‘true’’ weights insubsequent simulations.

Simulations to investigate properties of theweighted surveillance system

In the simplest simulation cases, whereall samples came from either CWD suspectfemales (Fig. 1A) or from harvested year-ling males (Fig. 1B), 12a was not biased(i.e., equaled 0.95) when unbiased (‘‘true’’)weights were used. However, using biasedweights in simulations influenced bothdisease detection probability and the costsof associated surveillance; simulated under-estimation of true simulation weights in-creased the probability of detecting at leastone case (12a) and overestimation de-creased that probability (Fig. 1A, B). Con-versely, relative surveillance cost increasedwith negative bias in the weights, decreasedwith positive bias, and was most pro-nounced in the simulation based on har-vested yearling males (Fig. 1B).

More detailed simulations, based onsampling a hypothetical deer populationwhere 1% of the adult males were infected,yielded similar results. When the ‘‘true’’simulation weights were used, 12a was un-biased, but as bias in the weights increased,the bias in the probability of detecting acase also increased (Fig. 2). However, insimulations where sampling probability wasnot increased and weights were only biasedwithin 30% of true values, the bias in 12awas relatively small, and the probability ofdetecting a case remained $0.9 (Fig. 2).Increasing sampling probabilities in thehigher prevalence strata did not affect theoverall pattern in the simulated effects ofusing biased weights (Fig. 2), and 12aremained unbiased when the true simula-tion weights were used, regardless of the


FIGURE 1. Effects of using biased weights on the probability of detecting a positive case (12a; black line)and the associated surveillance cost (‘‘relative cost’’; gray line) in a simulated chronic wasting disease (CWD)-infected mule deer population sampled under the proposed weighted surveillance system. For comparison,the influences of bias are illustrated under scenarios where all samples entering the surveillance stream camefrom either (A) CWD suspect female deer or (B) harvested yearling male deer, representing the highest andsecond-lowest weighted demographic strata in the proposed system (Table 1). In all simulations, prevalence inthe baseline stratum (adult males) was 0.01, and the nominal target detection probability was 0.95.


sampling probabilities (Fig. 2). When pos-itively biased weights were employed, theresulting bias in 12a tended to be slightlygreater in simulations with increased sam-pling of high-prevalence strata (Fig. 2). Theincreasing relative surveillance costs associ-ated with negatively biased weights weresomewhat offset in simulations wheresampling probabilities were increased forhigh-prevalence strata (Fig. 2).

Comparable patterns occurred in simu-lations where the direction of the bias wasmixed among the strata (Fig. 3). The overallbias in 12a was relatively negligible, insimulations where weights were negatively

biased for the suspect and other strata, andwere positively biased for the rest of thestrata (Fig. 3). The effects on relative costswere also diminished in simulations withmixed bias in the weights (Fig. 3).

The traditional surveillance approach(i.e., all weights being equal) represents anextreme case where the direction of thebias in weights is mixed across strata.Comparing our weighted surveillance sys-tem, using unbiased weights, to the tradi-tional system revealed that implementingthe weighted system in a large ColoradoDAU would cost $315 USD more than thecurrent system. However, as the probability

FIGURE 2. Effects of using weights biased in a constant direction across strata (x axis) on the probability ofdetecting a positive case (12a; black lines) and on the associated surveillance cost (‘‘relative cost’’; gray lines)in a simulated chronic wasting disease (CWD)-infected mule deer population sampled under the proposedweighted surveillance system. The three line styles represent different levels of emphasis placed on samplingfrom ‘‘high-prevalence’’ strata (CWD suspect males, CWD suspect females, and ‘‘other’’): No increase insampling probabilities of high-prevalence strata (solid lines), sampling probabilities of these strata increasedby 1% (dashed lines), or sampling probabilities of these strata increased by 5% (dotted lines); Table 2 lists thestratum-specific sampling probabilities used in simulations under these three scenarios. In all simulations,prevalence in the baseline stratum (adult males) was 0.01, samples entered the surveillance stream frommultiple sources, and the nominal target detection probability was 0.95.


of sampling higher-prevalence strata wasminimally increased by 1% and 5%—reflecting a shift in emphasis towardcollecting clinical CWD suspect and ‘‘oth-er’’ dead deer—weighted surveillance be-came considerably more cost-effective, withprojected savings of $3,863 and $11,682USD, respectively. Simulations also dem-onstrated that fewer samples were requiredbefore detecting the first positive case whensampling effort focused on higher-preva-lence strata: The distribution of ‘‘waitingtimes’’ shifted closer to zero, with increasedemphasis on sampling higher-prevalence

strata; mean values were 79, 69, and 49 forno increase, and a 1% and 5% increase insampling probabilities, respectively (Fig. 4).

Simulations to investigate samplesize requirements

Our simulations revealed that, with asample size of at least 5,000 CWD testresults, it appears that the mean percenterror for strata with true simulation weights.1 will be #20% (Table 3). Strata withlarger mean percent error values wereharvested fawns, harvested yearling males,and harvested yearling females; although

FIGURE 3. Effects of using weights biased in mixed direction across strata (x axis) on the probability ofdetecting a positive case (12a; black lines) and on the associated surveillance cost (‘‘relative cost’’; gray lines)in a simulated chronic wasting disease (CWD)-infected mule deer population sampled under the proposedweighted surveillance system. Weights were biased negatively for the three ‘‘high-prevalence’’ strata (CWDsuspect males, CWD suspect females, and ‘‘other’’) and were biased positively for the remaining fivedemographic strata in the proposed weighted surveillance system. The three line styles represent differentlevels of emphasis placed on sampling from high-prevalence strata: No increase in sampling probabilities ofhigh-prevalence strata (solid lines), sampling probabilities of these strata increased by 1% (dashed lines), orsampling probabilities of these strata increased by 5% (dotted lines); Table 2 lists the stratum-specificsampling probabilities used in simulations under these three scenarios. In all simulations, prevalence in thebaseline stratum (adult males) was 0.01, samples entered the surveillance stream from multiple sources, andthe nominal target detection probability was 0.95.


these strata had much larger mean percenterror values ($0.68), their mean absoluteerrors were small (i.e., #0.33) at a samplesize of 5,000 CWD test results, most likelybecause prevalence in these groups waslow. Thus, for strata with small weights, amuch-larger number of samples must beacquired before error drops significantly.Simulation results also demonstrated that,across most sample sizes and strata weights,on average results tend to be negativelybiased, although the probability of a posi-tively biased weight for most strata andsample sizes is nearly 50% (Table 3). Thiswas expected because equation (1), whichwas used to estimate the weights, is anasymptotically unbiased estimator.

DISCUSSION

The weighted CWD surveillance systemand simulation analyses described here

provide statistical justification for theintuitive weighting of individuals fromvarious demographic strata, based on theestimated apparent prevalence and sam-pling probabilities of those strata as theyenter the surveillance stream and aretested. In our system, demographic strataof deer that have a higher prevalence andlower probability of being sampled, rela-tive to the baseline stratum, receive moreweight than deer from strata with lowerprevalence and from which samples aremore common (Table 1). The probabilityof detecting at least one case of CWDamong mule deer from strata whereinfection is relatively rare is lower thanthe probability of detecting an infectedindividual from strata where infection ismore likely. Therefore, larger numbers ofdeer must be sampled from ‘‘low-preva-lence’’ strata in order to achieve the highprobability of detection typically ascribed

FIGURE 4. The distribution of ‘‘waiting time’’ (expressed as cumulative number of samples) until the firstpositive deer was detected in a simulated chronic wasting disease (CWD)-infected mule deer populationsampled under the proposed weighted surveillance system. The three line styles represent different levels ofemphasis placed on sampling from ‘‘high-prevalence’’ strata (CWD suspect males, CWD suspect females, and‘‘other’’): No increase in sampling probabilities of high-prevalence strata (solid lines), sampling probabilities ofthese strata increased by 1% (dashed lines), or sampling probabilities of these strata increased by 5% (dot-dashed lines); Table 2 lists the stratum-specific sampling probabilities used in simulations under these threescenarios. In all simulations, unbiased weights (Table 1) were used, prevalence in the baseline stratum (adultmales) was 0.01, samples entered the surveillance stream from multiple sources, and the nominal targetdetection probability was 0.95.


to such surveillance efforts (Fig. 5). Ourweighted approach provides a frameworkfor combining data from various samplesources in order to more transparentlyevaluate and compare the overall efficacyof surveillance activities.

Our simulations demonstrated the utilityof maximizing the collection and submissionof mule deer from demographic strata witha higher weight to increase the probabilityof detecting disease and to minimize theoverall economic commitment towardCWD surveillance efforts: On average,surveys based on collecting and examiningclinical CWD suspects would only beexpected to encounter a case by examiningabout one tenth the number of submissionsneeded if only harvested animals were

collected and examined. By design, theweighted surveillance system promotessampling from higher-prevalence (and thushigher-risk) strata because more points areassigned to such strata, thereby motivatingusers to reach the target value with fewersamples and, thus, reduce overall surveil-lance effort. The shift in the distribution ofcumulative cases examined before detectingthe first CWD case (Fig. 4) further empha-sizes the benefits of sampling more heavilyfrom higher-prevalence strata. The costsavings associated with this system couldbe substantial because the number ofsamples required for testing decreasessignificantly as sampling is focused less onharvested (and typically healthy) animalsand more on collection of ‘‘suspect’’ or

TABLE 3. Stratum-specific mean absolute error, mean percent error, probability of estimating positivelybiased weights, and true weights based on 2006 demographic data from mule deer (Odocoileus hemionus) inData Analysis Unit (DAU) D-19 used in the simulations evaluating the effects of sample size on estimating theweights for a weighted surveillance system.

Stratum identificationNumber of

samplesTrue

weightMean absolute

errorMean percent

errorProbability ofpositive bias

Suspect—female 1,000 11.57 5.60 0.48 0.51Suspect—male 10.27 5.32 0.52 0.48Other 1.90 0.79 0.42 0.48Harvest—adult female 0.58 0.28 0.48 0.47Harvest—yearling female 0.45 0.80 1.78 0.21Harvest—yearling male 0.25 0.38 1.52 0.28Harvest—fawn 0.03 0.06 1.94 0.13Suspect—female 5,000 11.57 2.29 0.20 0.50Suspect—male 10.27 2.10 0.20 0.52Other 1.90 0.31 0.17 0.50Harvest—adult female 0.58 0.11 0.19 0.48Harvest—yearling female 0.45 0.33 0.73 0.43Harvest—yearling male 0.25 0.17 0.68 0.46Harvest—fawn 0.03 0.03 1.09 0.49Suspect—female 11,000 11.57 1.51 0.13 0.49Suspect—male 10.27 1.43 0.14 0.50Other 1.90 0.22 0.12 0.48Harvest—adult female 0.58 0.07 0.13 0.48Harvest—yearling female 0.45 0.23 0.52 0.45Harvest—yearling male 0.25 0.11 0.43 0.46Harvest—fawn 0.03 0.02 0.65 0.40Suspect—female 15,000 11.57 1.26 0.11 0.53Suspect—male 10.27 1.19 0.12 0.50Other 1.90 0.19 0.10 0.52Harvest—adult female 0.58 0.06 0.11 0.50Harvest—yearling female 0.45 0.20 0.44 0.44Harvest—yearling male 0.25 0.09 0.36 0.47Harvest—fawn 0.03 0.02 0.55 0.50


‘‘other’’ deer that inherently tend to beunhealthy. The mean cost savings shown inour simulation results are for one DAU inColorado; however, there are 33 of 55DAUs across the state where CWD has notbeen detected (Colorado Division of Wild-life, 2009) and where this weighted surveysystem could be applied; fiscal savings couldbe even more dramatic in other jurisdic-tions with greater numbers of cervidpopulations of unknown status that requiresampling. However, it is important toconsider that these cost savings estimatesare based on the surveillance systemestablished in Colorado and do not incor-porate differences across strata for samplecollection. This is reasonable, in our system,because field personnel are responsible forcollecting samples from these strata as partof their normal duties (i.e., culling suspectdeer, collecting vehicle-kills, etc.) and,therefore, there is little added cost forcollecting samples from these sources as

compared to hunter-submitted samples.Other jurisdictions with limited field per-sonnel, or other restrictions, may havemarked differences in sample collectioncosts and, therefore, cost savings as report-ed here may vary. On balance, however, itseems likely that using a weighted surveil-lance approach would provide some costsavings and would also provide a method formanagers to exploit all available informationon CWD epidemiology when conductingsurveillance.

Our simulations emphasized severalimportant considerations related to CWD(and other wildlife disease) surveillance.First, there is a clear relationship in CWDsurveillance between the probability ofdetecting at least one positive case and thecost associated with surveillance: Regard-less of sampling scheme, increasing CWDdetection probability comes at an in-creased cost and vice versa; however,exploiting differences in sampling andprevalence rates should lessen the relativecosts associated with increasing the likeli-hood of detecting new foci. If biasedweights are inadvertently used in theweighted surveillance system, then costswill increase or decrease depending on thedirection of the bias, as will the diseasedetection probability. Fortunately, it ap-pears that the weighted surveillance sys-tem is relatively robust to modest bias inthe weights: In our simulations, theprobability of detecting at least one casewas not less than 0.9 until bias in theweights was 40% or greater.

As with any surveillance system, werecognize that violating sampling designassumptions will likely diminish the abilityto detect CWD at the specified 12a. Thisconsequence was evident in simulationswhere bias was introduced into theweights. The assumption that CWD caseswere distributed as a Poisson randomvariable within each stratum seemedreasonable, given the low probability ofan individual being infected, but we didnot examine the effects of violating thisassumption. The assumption that individ-

FIGURE 5. The potential contributions of sam-ples from different demographic strata in detectingchronic wasting disease (CWD) are reflected in therelationships between sample size and the probabilityof detecting at least one CWD case across the eight(apparent health3sex3age) strata developed formule deer sampled in Colorado, United States. Theeight strata are arrayed from upper left to lower rightin order of descending estimated weights (Table 1),calculated as described in the text. The solid blackline is for harvested adult ($2 yr old) male muledeer, the baseline stratum to which other strataweights were referenced; assumed prevalence amongharvested adult male mule deer was 0.01.


uals were randomly selected within astratum for sampling is undoubtedlyviolated in practice (Otis et al., 1978),but this problem of individual heteroge-neity plagues all practiced CWD surveil-lance approaches of which we are aware.The assumption that relative prevalencewithin each stratum is constant acrossdifferent population prevalence levels mayalso likely be violated due to factors suchas transmission probabilities, spatial het-erogeneity, density-dependent mecha-nisms and, perhaps, other factors (Milleret al., 2000; Miller and Conner, 2005),although the difference in prevalencebetween sexes appears remarkably robustacross a wide range of prevalence (Milleret al., 2008). Our system performedreasonably well in simulations examiningthe use of biased weights that equated to asituation where prevalence for each stra-tum differed from the true simulationprevalence, suggesting it is robust tominor violations of this assumption. De-spite these potential limitations, we be-lieve that the assumptions associated withour, or similar, weighted surveillanceapproaches are more defensible than theassumption that every individual sampleentering the surveillance stream comesfrom a uniform population and is of equaldetection value, an assumption that iscentral to harvest-based CWD surveil-lance approaches (Samuel et al., 2003),yet clearly violated (Fig. 5).

One suggested advantage of the har-vest-based CWD surveillance approach isthat sampling apparently healthy harvest-ed animals is ‘‘conservative’’ compared tousing weighted surveillance. In otherwords, because in the weighted surveil-lance system 300 individual samples maynot be tested (i.e., the target value may bereached before the total number ofsamples tested equals 300), systems cur-rently in use will have a higher probabilityof detecting CWD. Examined in the sameframework as our weighted surveillancesystem, however, the contemporary CWDsurveillance approach in wide use repre-

sents a case of extreme bias in stratum-specific weights most comparable to sim-ulations incorporating mixed bias in theweights—the bias in harvest-based surveystends to be negative for high-prevalencestrata (e.g., unthrifty individuals) andpositive for low-prevalence strata. It fol-lows that such an approach may resulteither in increased probability of diseasedetection or in decreased probability ofdisease detection—the direction of thebias in this probability depends on samplecomposition. For example, if most samplesactually come from high-prevalence stra-tum, then a will be negatively biased andthe probability of detection will approach1.0; alternatively, if the majority of thesamples in the surveillance stream comefrom low-prevalence strata like yearlingmales (which are abundant in manyheavily harvested North American deerpopulations), then a will be positivelybiased and the true probability of detect-ing at least 1 positive case will fall belowthat believed to be assured, based on theoriginal survey design (Fig. 5). Undereither scenario, the contemporary har-vest-based approach will be inefficientrelative to the weighted surveillance sys-tem, because either costs will be higherthan necessary or disease detection prob-ability will be below the nominal ‘‘adver-tised’’ level. Moreover, if the majority ofsamples come from low-prevalence strata,such as yearling males, then the weightedsurveillance system would actually requiresampling more individuals than prescribedby the contemporary approach becausethe weights for this stratum are less thanone. The notion that traditional approach-es for CWD surveillance are somehowsuperior to a weighted approach seemslargely based on the premise that a greaternumber of samples will always be ‘‘better’’when, in fact, our results illustrate thatboth the number of samples and thesource of those samples can influencethe probability of disease detection(Fig. 5). By design, weighted surveillanceencourages sampling from demographic


strata with the highest probability ofinfection in order to maximize speed ofdisease detection and minimize cost—wesee no clear advantage to using approach-es that ignore information available ondifferences in prevalence and probabilityof CWD infection across demographicstrata within a deer population.

Another concern among prospectiveusers is whether adequate data are avail-able to create a jurisdiction-specificweighted system, as well as how toestimate the required weights. Based onour sample size simulations, it appears thatestimating weights from at least 5,000samples will yield errors #20% for stratawith weights .1, which our bias simula-tions suggest should not markedly affectdisease detection. These sample sizesimulations also demonstrate that, ifweights are biased, the biases tend to benegative and thereby provide conservativeestimates of the number of samplesneeded (i.e., 12a will be higher than thelevel prescribed by the user). Moreover,the strata most affected by small samplesize are those with low weights and, thus,these biases will have minimal effects onoverall disease detection probability.

The information provided here shouldbe sufficient to allow other jurisdictions todevelop weighted surveillance systems asthe need arises. However, as a first step indoing so, we recommend constructing ademographic model and conducting sam-ple size simulations, as described herein,based on strata-specific sampling probabil-ities and prevalence estimates from otherregions of interest. If sufficient data areavailable, but jurisdictions also want toincorporate the data from Colorado includ-ed here, we suggest using a Bayesianapproach to estimate weights. Such anapproach should be relatively straightfor-ward, given the likelihoods described inAppendix A and using weights fromTable 1 as prior values; as more local dataare acquired over time to estimate theweights, these prior values will be over-whelmed by local data. We encourage

other jurisdictions with adequate data toconsider estimating weights independently,using the maximum likelihood frameworkprovided herein for comparison to theweights we have reported to help elucidatepotential regional or species-specific dif-ferences that could be important in furtherrefining CWD surveillance approaches.

For jurisdictions lacking appropriate oradequate data (e.g., a state, province, ornation where CWD has not been detect-ed), an alternative to the traditional‘‘random sampling’’ approach is to simplyuse the weights directly from Table 1.This assumes that probabilities of samplesfrom the various strata entering thesurveillance stream from other jurisdic-tions are similar to those we reported andthat the effects and epidemiologic patternsof CWD are relatively constant betweenregions and across species, which seemreasonable, based on published observa-tions (Miller et al., 2000; Miller and Wild,2004; Miller and Conner, 2005; Williams,2005; Joly et al., 2006; Grear et al., 2006)and on similarities between the weightvalues independently estimated for muledeer (Table 1) and for elk (D. Walsh,unpubl. data) using data from Colorado.Given the relatively robust nature of theseestimated weights, as demonstrated by oursimulations, it seems unlikely that a CWDsurveillance approach based on our esti-mated weights for mule deer in northernColorado would be any less reliable thanalternatives that incorrectly assume thatthe probability of CWD infection acrossall demographic strata is equal.

The need for a weighted surveillancesystem that incorporates all availableinformation regarding stratum-specificprevalence and sampling probabilities hasbecome apparent in Colorado becausepublic interest and economic support forCWD surveillance has begun to wane. Asinterest and funding have declined, it hasbecome necessary to streamline our sur-veillance to refocus on sampling individu-als with the greatest probability of beinginfected in order to continue detecting


changes in the geographic distribution ofCWD in a timely manner. Additionally,the inability to collect the ‘‘standard 300samples’’ from high-risk managementunits due to lack of harvest, hunterparticipation, or other constraints (Colo-rado Division of Wildlife, 2009) hascompelled a shift in emphasis from harvestsubmissions to those that can be collectedby agency personnel. Using this weightedsurveillance system can reduce the totalnumber of samples that agency personnelwill need to acquire from a population inorder to achieve a nominal ‘‘disease-free’’determination at the same probability ofdisease detection as afforded under har-vest-based surveillance approaches, byexploiting variability in stratum-specificprevalence and sampling probabilitiesand by allowing all surveillance submis-sions to count toward reaching localsurveillance goals. Moreover, the clearlydefined scoring system and target shouldmake this an intuitive system for agencypersonnel to use and track. Although notdescribed further here, our weightedsurveillance system can also potentiallyuse data from multiple cervid species insituations where agencies are attemptingto detect CWD in a region rather than in aparticular species. Under such an ap-proach, samples from other susceptiblehost species (elk, white-tailed deer, andmoose) could be included in the weightingscheme and thereby contribute to theoverall confidence and probability ofdetecting CWD in a particular geographicarea (D. Walsh, unpubl. data).

In the face of shrinking budgets anddwindling public participation, we believethat our weighted surveillance system willbe a useful tool for CDOW and, perhaps,for other wildlife management agenciescharged with monitoring cervid populationsto detect new CWD foci. Weighted surveil-lance is intended to encourage local wildlifemanagers to remain actively involved inCWD surveillance by encouraging samplesubmissions from demographic sourceswhere CWD is more prevalent; the greatest

strengths of this approach are the intuitiveassignment of different values to samplesfrom different sources and the ability tocombine contributions from multiple sam-ple sources toward reaching a quota ofsurvey points. As interest shifts from CWDto other wildlife diseases in Colorado andelsewhere, surveillance systems will need tocontinue to evolve and become moreefficient in order to be sustainable. Webelieve the weighted surveillance systemdescribed here represents a step forward inthis surveillance process.

ACKNOWLEDGMENTS

The data analyzed here are the cumulativeproduct of numerous laboratory and field effortsto study CWD in Colorado, efforts that nowspan over a decade. We extend special thanks toK. Larsen, L. Baeten, I. Levan, K. Griffin, andmany, many others in the CDOW for submit-ting and sampling cervids over the years; and toE. Williams, P. Jaeger, and others at theUniversity of Wyoming and to B. Powers, C.Hibler, T. Spraker, and others at the ColoradoState University Diagnostic Laboratory fordiagnostic support. We also thank L. Creekmoreand A. Scott for their assistance. Numeroushunters facilitated this work by submittingharvested mule deer for CWD testing. Wethank B. Banulis for providing the DAU D-19deer population data used in the simulations.We thank P. Lukacs, D. Heisey, M. Samuel, andtwo anonymous reviewers for helpful commentson earlier drafts of our manuscript. Our studywas funded by the Colorado Division of Wildlifeand the Federal Aid in Wildlife RestorationProject W-153-R, with partial support providedfrom a cooperative agreement with the USDepartment of Agriculture, Animal and PlantHealth Inspection Services, Veterinary Services.

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APPENDIX A: DERIVATION OF WEIGHTEDSURVEILLANCE SYSTEM

The first step in developing the weightedsurveillance system is to determine the meth-od for calculating the number of samplesneeded across the various strata to achievesome user-specified disease detection proba-bility (12a) at a given prevalence (pdesign). To

facilitate this, we will rely on the assumptionspreviously stated (see Materials and Methods).Based on these assumptions, the joint proba-bility density function can be formulated asfollows:

P(X0, . . . , Xm j li)~Pm

i~0

exp ({li)lXi

i

Xi!, ð1Þ


where Xi5the total number of positives fromthe ith stratum in the surveillance stream, andthe expected number of CWD-positive cases inthe ith stratum is li5Ni3di3pi3f5ni3pi3f,where Ni5population size of the ith stratum,di5sampling probability of the ith stratum (i.e.,probability an individual enters the surveillancestream), ni 5 the realized number of samples inthe surveillance stream from ith stratum, pi isthe prevalence for the ith stratum, p0 is theprevalence for the baseline stratum (i.e., thestratum with weight (w0) equal to 1), andf5the sensitivity of the test. Based on equation(1), the probability of failing to detect a positivecase during surveillance (a) is:

prob(Xm

i~0

Xi~0)~exp {Xm

i~0

nipif

!~a,

Xm

i~0

nipi~{ ln að Þ

f: ð2Þ

The probability of detecting $1 CWD-positivecase can be calculated as 12a. Using ourassumption that relative prevalence within eachstratum is constant across different populationprevalence levels, we let pi5wipdesign, where wi

5 weight for the ith stratum and pdesign5thespecified design prevalence for p0 (i.e., this isthe prevalence at which the practitioner wishesto detect at least 1 CWD-positive case withprobability 12a in the baseline stratum; com-monly 0.95 is used). Assuming f51, equation (2)can be rewritten as follows:

Xm

i~0

niwipdesign~{ln að Þ: ð3Þ

Thus, the number of samples needed from eachstratum, or from a combination of strata, toachieve a disease detection probability of 12a is:

Xm

i~0

niwi~{ln að Þpdesign

~t, ð4Þ

where t is the target value, as described earlier. Itis clear from this equation that current surveil-lance systems (i.e., wi;1 for all i) are specialcases of the weighted surveillance system. Thisequation provides the basis for determining howmany samples from each stratum or combinationof strata are required to reach a desired diseasedetection probability of (12a).

The next step is to estimate weights (wi) forthe various strata in equation (4). Once again,using the assumption that relative prevalencewithin each stratum is constant across differentpopulation prevalence levels, we let pi5wip0,with p0 representing the prevalence in the user-

specified base-line stratum. Thus, prevalencefor each stratum is scaled relative to the base-line stratum. Earlier we had set p05pdesign, butto estimate the weights we will use an estimateof p0 derived from our surveillance data. Then,given the data vector x, the likelihood functionfor wi is as follows:

L(wi j x0, . . . , xm, n0, . . . , nm, p0)

~Pm

i~0

exp ð{liÞlxi

i

xi!

~Pm

i~0

exp ð{wip0niÞwip0nið Þxi

xi!: ð5Þ

Based on our assumption, if the number ofpositive cases at the time of the survey in each ofith strata is independently distributed as Poisson(li) random variables and the realization ofli5ni3pi5ni3wi3po, then the joint likelihoodfunction can be expressed as:

L(wi, p0 j x0, . . . , xm, n0, . . . , nm)

~exp ð{p0n0Þp0n0ð Þx0

x0!

|Pm

i~1

exp ({wip0ni)wip0nið Þxi

xi!, ð6Þ

from which we generate maximum likelihoodestimates for wi as:

wwi~xi

nipp0

~ppi

pp0

, ð7Þ

and we also generate p05x0/n0, the maximumlikelihood estimate of prevalence for thebaseline stratum. It is clear from equation (7)that the weight for the baseline stratum is ;1.

The weighted surveillance system is thenemployed by collecting samples from thevarious strata until:

Xm

i~0

niwwi~t, ð8Þ

where t has been calculated before the onsetof surveillance from equation (4) using a user-specified a and pdesign.

Using the delta method, the estimate of thevariance for the individual weights can becalculated as follows:

vaar(wwi)~ppi 1{ppið Þni pp0ð Þ

2z

ppið Þ2pp0 1{pp0ð Þ

n0 pp0ð Þ4

: ð9Þ

APPENDIX B: BIAS EQUATIONS

If weights are biased, the target value is


reached too early, or too late, depending onthe direction of the bias. The difference in thetotal number of samples needed to reach thetarget value, compared to the actual numberneeded for each stratum, can be derived. Letqi5difference in number of samples from thetrue number of samples (ni) needed in the ithstratum to reach the target (t), when thereexists a bias (bi) in the estimated weightsrelative to the true weights (wi). Then, forstratum i, the following is true:

nizqið Þ| wizbið Þ~niwi~t, ð10Þ

Using simple algebra, the difference in the num-ber of samples collected based on biased weightsfrom the true number of samples needed is:

bias nið Þ~qi~{nibi

wizbið Þ : ð11Þ

In addition to affecting the number of samplesneeded to reach the target value, biased weights

will also increase or decrease the diseasedetection probability (12a) beyond the intendedlevel, depending on the direction of the bias ofthe weights. For simplicity, we present theformula for calculating this change in a due tobiased weights for 1 stratum here; the extensionto all strata follows logically. Using equation (1)and based on the true weights,

bias að Þ~ab{atrue~exp ð{ nizqið Þ|wipdesignÞ

{ exp ð{niwipdesignÞ, ð12Þ

where 12ab5the actual disease-detection prob-ability based on using biased weights, and12atrue5the actual disease-detection probabilitybased on using the true weights. Then it can beshown that:

bias að Þ~a exp {wi

nibi

wizbið Þ|pdesign

� �{1

� �:

ð13Þ


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