Applied Sampling for Wildlife
Applied Sampling for Wildlife FW 580
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This 42 page Class Notes was uploaded by Maximilian Lynch on Monday September 21, 2015. The Class Notes belongs to FW 580 at Colorado State University taught by Paul Doherty Jr in Fall. Since its upload, it has received 42 views. For similar materials see /class/210051/fw-580-colorado-state-university in FISH at Colorado State University.
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Date Created: 09/21/15
FVV 580 Applied Sampling for Fish and Wildlife Studies Adapted from Stuart 1984 Thompson 2002 x Eml wus Habitat fragmentation and loss ngyAuk Comm J nlAvian Biology on Man niEcnlngical Economics Habitat fragmentation and loss recent projec mmvwm Vietnam Overseas swam ngram V05 mama value m m eamev new 5 Curadvised W m VichDreilZCO nw mwuume mamaimn mass m Vielna Behavioral E cnlngV nshua Dnnley MS Enema mmsmrlzance an a in optimize wldlile area do Waterfowl ecology Breeding site memwaiea W m rm Sandersr co Div mwumue mm Zimmerman cause 2 av band 51tu and methndstn address band lass mm Dis Kendall andwhne uck behavior sure veguiaimna Community ecology ITesting macroecological patterns d o d ecles Proc Nat Ac d Extinction rates at edge and interior ofrange Ann Zool Fenn incorporate new methodological opmenis to estimate communitylevel parameters with phylogenetic analyses Seabird biology 39 a and conservation 7 I gecies 3 am my ma may magnum ma We mama quotoptima man 2cm unwade 2W Walla para a 2nd alaammw miEal munitions Papulati Ehrveyd B Collaborators uwsDoDLsesFuv aaaauummw hodologV way mu m at an modeling and smsitivity analysis esign mum mam stamina 5mm 5mm Venn Waugh tan 9w mmm ml mm Lebrelon m Seabird biology and conservion current x I Dr Sarah Converse v Working on population modeling estimatio design for aibarross Dr Bill Kendall Also manames Dr Mike Runge I International Albatross Working Group I Ongoing collaborations Dr reiber DrsWaug FWSUSGS Disease ecology current IWest Nile virus Paul Oesterle MS Dr Jeff Hall Role of swallow bugs in transmission cycle 1 f39 IAvian influenza study design advisory committee USDA NWRC Southern California projects lFIattailed horned lizrad Abundance esumam and errece of oHy Ty er Grant 7 MS IPalm Springs ground squirrel habitat ssociations JWMDr Ball and MacDonald ICurrent Caifornia gnatcatcher abundance and habitat associations ridnell About you IWhere are you from IWhat are your research interests IWhy are you taking the course IWhat topics are of most interest to you Syllabus ITexts IReadings lHomework IOther topics of interest Definitions IWhat is a sample MerriamWebstei s Collegiate Dictionary 2001 a nite part of a statistical population whose properties are studied to gain information about the whole ICensus complete enumeration of the population ie the whole Definitions IWhat is sampling The act process or technique of selecting a representative part ofa population for the purpose of determining parameters or characteristics of the whole population MerriamWebster s 2001 There are dangers in sampling IWe will discuss dangers associated with various sampling schemes IWe will recognize the potential error and attempt to minimize associated with convenience sampling Sampling error ISampling error comprises the difference between the sample and the population due solely to the sampling units having been selected ITwo sources 1 Chance protection large sample size 2 Sampling bias Tendency to favor the selection of units having particular characteristics JWhy sample I 1 Often physically prohibitive to census a population because many populations are I2 Often scally prohibitive I3 Even if a census can be taken the census is not ef cient for needs Le a sample will suff39ce eaper to view a part rather than the whole I4 Timeliness samples can be conducted quicker than a census I5 Destructive sampling in some instances sampling is destructive Sampling error and bias IA sample is expected to be representative of a population IHowever there is no guarantee and chance may dictate a disproportionate number of untypical observations IIn practice hard to know when a sample is unrepresentative Sampling error I Bias Usually the result of a poor sampling plan Nonresponse no chance of some units appearln INever enough to not detect bias must assure against bias Nonsampling error measurement error I produced by participants or an innocent byproduct of sampling plans IInaccurate measurements due to malfunctioning instruments or poor procedures The interviewer effect questionnaire design Respondent effect lying 4 Integrity of the sample ISample integrity is often overlooked Little can substitute for sample integrity lEconomy and reliability are competing en 5 lSampling error can be controlled by careful sample selection methods this class Types of samples IThree types of samples Differ in how the sampling units are chosen IConvenient sample results when most convenient units are chosen from a population IJudgment sample obtained at the discretion of someone who is familiar with the relevant characteristics of a population Random sample lMost important type of sample IAllows a known probability of each unit being chosen from a popu ation sometimes referred to as a probability samp e IMany forms Simple random systematic stratified cluster etc Central Paradox of sampling IImpossible to know from examination of a sample whether it is good or not od39 representative and free from bias IThe sample itself cannot tell us whether it is free from bias only the process by which it was selecte Cbtherwise 39dogged by the shadow of selection laS Implies dgiven the exact same sample we regard it ifferently depending on how it was c osen Evaluate the credentials of a sample lMust understand basic notions of sampling theory IFree from bias IReliability Credentials I good credentials give creedence ICourt example A saint and a sinner are before a court both present the exact same evidence Whom are you more likely to believe lIf a sample has good credentials more believable lRandom sample has great credentials A Sampling Case Study EIK in NW Colorado Paul M LUKaCS Case Study IE 2 EIK Case Study IE 2 EIK importer 1t ir n NW EIK are very ecor nomically Color ado Ir n 2008 CDOW had ah Objective Of 12000 1 5000 elk ir n IE 2 Modeled populatior n Size approachihg Objective Therefore licer nse S to be reduced Case Study IE 2 EIK residehts argued there were more Local thar n 391 5000 elk Didr n t war 1t to lose hUhtir ng revehue Didr n t war 1t game damage to livestock feed was per Formihg poorly ir n abur ndahce Populatior n model suggestihg little COhfider nce esti mate populatior n model demor nstr atior I M r I 39 39 7 n quotquot39 39 rr rirrgww 7 M a Case Study IE 2 EIK 80K questior n I IOW mar 1y elk are there Framing the problem What do we I r now about E Z It s big 2000 sq miles of Wir ntsr rahgs It s diverse Mour ntair shrub sagebrush ripariar n Gor nifsrous forest agriculture SIQIZgt 77 1 Asa f tgm EIK Winter Range HOW would design a survey of E Z What would you like to Khow HOW would design a survey of E Z lil s to l r ncw best available Khowlsdgs 12 20K What would you Approximate of sIK Elk distributior n Desired precisior n ZOOC CV Budget 60K available all CDQW psrsoririsl arid squipmsr t Outcomes of past survey methods quads work well for deer limited successful use for sIK distahcs samplir ng has hot beer effective Cluster size distributior n Habitat sslsctior use all types lt lRlgt arid alfalfa Stratificatior quot3 77 39 i 7 39 quot 7 gagge rj sgbw wr iym 7 7 n V l l 5OC elkgroup coricsritrats occasiorially or39n HOW would design a survey of E Z What would you do first HOW would design a survey of E Z What would you do first DEFINE THE QBJECTIVE DEFINE SAMPLING FRAME Ider ntify strata Select samplir ng method Ider ntify samplir ng uhit selectior n method r r f 7 I vH 77 39 Vuiam HOW would design a survey of E Z HOW big Should the sample Size be Samplir ng Frame What we did for better or worse Split frame all wir nter rar nge ir n IE 2 ir ntO 8 strata based or n cover type arid relative elk der nsity Oper n cover 2 mile quads Der nser cover quotl r r lile quads a t a r t S 9 m Sampl Sampling Design Helicopter survey 2 Jet Rahger S Fixed wir ng air plahe scoutihg Used GR I S to select samplihg Uhits 391 Soloy Iquot kHii quot4 39 7 f 4rtam Generalized Random I essellation Stratified Designs Sample locatior IS Spatial balar nce VS rahdom selectior n GRIS Spatially balahced samplihg Local heighbor hood var iahce estimator Free sample selectior n software R package for ahalysis 3 fr 7 77 n r quotquotquot399 39 a 7 if A i Sample Sizes Optimal desiqr n Semi 0Dtimal desiqr n e p m a S d e t C e e S Data Cour Its Meah 3383 elkquad Rahge 0 546 SD 7958 Medial 5 Mode 0 orrlmv 0 rmmlNN mELOCKnWAUJ gmm mONANm mN mmm vomlNN w U QEmm mumj v w mem Hulk100 mHO PMMLM E JHmLHm GRIS VS SR8 Estimate SE CV ICI UCI GRIS 32205 5578 O l 7 2299 l 45 I IO SR8 32205 6523 020 2 I739 477 IO FW580 Sampling IAny questions ITake home messages from last time 4 Stratified Sampling IWe spent the last few weeks talking about SRS With no information we can t do better than a SRS IHowever often we have other information Like what Supplementary information IDifferent kind of individuals or plots Sometimes vague IOften can group population into a number of subgroups stratum or strata p iIf we take a SRS from each of these strata we rarely lose precision and we may gain considerably Stratified Random Sampling IWill drive us in the right direction increasing precision Strength will depend upon our skills of picking strata IExcel example same example Sample size of4 but with 2 strata Sample Size 4 pick 2 from each stratum Sampling distribution Val nsts 1m distribution 4 Stratification 2 steps IPicking strata IChoice of sample size We used uniform sampling fractions What if we used variable sampling fractions ie more intensively sampled one stratum lTake 3 of our 4 samples from Stratum 1 and only 1 from Stratum 2 3 members from stratum 1 and 1 member from stratum 2 Weighted average reciprocal of fraction I Units in stratum 1 had 3 times as large a chanc n x man Suwling m on Variable sampling fractions IYou might object to sampling all of one strata but often done nothing wrong lAs long as we know what we have done and account for it ICompare this with an arbitrary sampling scheme what is different Variabe sampling fractions lWe do not need to insist on each member of the population having the same chance of being in a sample IWe do need each member to have a positive chance though remember the papers from last time Strati cation with variable sampling fractions enables us to vary the chance of selection for different subg roups of the population Oversampling more variable strata I sample size 4 3 from strata 1 1 from strata 2 IComparison of precision 067 vs 083 IStratum 1 average 6 var 16 IStratum 2 average 10 var 4 IStratum 1 is more variable We got rid of this variance by completely sampling n39aDJm 1 All variance between 3 possible sampls is due to stratum 2 It pays to oversample the more variable strata What about undersampling more variable strata IExcel example ISampljng variance is 4x as large as when sampling fraction IS reversed l3x as large as uniform sampling fraction I25x as large as SRS IPenalty of oversampling the less variable strata can be severe IVariable sampling fraction is double ed ed r if we concentrate the sample int e wrong s rata How can this be avoided Maximum precision ITo maximize precision the sampling fraction in each stratum should be proportional to the square root of the variance in that stratum Stratification recap lStrati cation with uniform sampling fraction almost always increases precision IVariable sampling fractions can increase or ecrease precision Incrase will result ifoversample in more variable strata IIs there a mathematical rule to enable us to achieve maximum precision In our example IStratum 1var 16 IStratum 2 var 4 IThus a 41 ratio We should choose our sampling fraction as the square root of this ratio so 21 IThis is the sampling fraction not the sample size IIn our example of n 4 can not achieve this exactly but n 3 and n2 1 is close What if cost of observation differs between strata and we ave to maximize precision for a ixed total cost IWe need only replace variance in the rule by the ration variancecost of an observation 4 It follows IA uniform sampling fraction will only achieve max precision if every stratum has the same variancecost ratio INevertheless a uniform sampling fraction is often used because it is a convenient compromise Quota sampling IIn practice random sampling can be expensive IOne way used to avoid this cost is quota sampling IIndividuals are not preselected but strata are and selection of individuals within strata is left to the observers Quota sampling I Observer freedom Selection bias is present Selection procedure is illdefined no valid method of estimating SE of sample estima or I But quota sampling always achieves sample size as compared to RS Conceals problems of nonresponse rather than attempting to solve them Quota sampling IWith the reduction in cost comes a reduction of the credibility of the sample Selection bias Nonresponse bias No valid SE Formation of strata IHow to choose strata smartly Maximize precision absolutely How many strata later How to form strata from population Formation of strata INote Not obliged to have strata of equal sizes IExcel example strata with equal average Formation of strata ISampling variance is 133 vs 067 from before and 106 from SRS ICan not blame this on misdirected oversampling same sample size Formation of strata IThe effect of gain in precision from strati ed sampling with uniform sampling fraction depends on the variation between stratum averages The greater the variation between stratum averages the greater the gain in precision Note the loss in precision seen last slide is a result of large sampling fractions Loss of precision will rarely happen Strata with equal means equal sampling fraction 2 from each stratum 1 avg a stratum 2 avg a Strata with unequal means equal sampling fraction 2 from each Stratum 1 avg s 33 Stratum 2 avg 10 57 Formation of Strata maximize difference Formation of strata maximize difference IThe extra reward for extreme skill in forming strata as compared to moderate skill is substantial 4 Formation of strata principle ITo maximize precision of estimation we should construct strata so that their averages are as different as possible and their variances are as small as possible Choice of number of strata ICan take strati cation principle to the extreme where each strata has a single member IHowever we need to estimate variability in each stratum and should have at least 2 members ISee text for formula for estimates of total averages and associated variances and CI s Poststratification I So far sample size in each siramm is xed I Sometimes this is not possible and we can not group ampling units into strata before se ection May want in group units into strata after data collection say using a S I On the face this may appear the same a stratifcation but something crucial has changed Don t know sample sizes a pri Need to remove some part of strati cation See Ext ori me gain resul ng from proper Sample vs Population Variance IWhat is relation between sample and population variance TjPopVar SampleVar l v n N Where n sample size and N is population size Questions IWhat were take home messages from last lecture Questions IExplain the difference between sampling and population variance IWhat is the sampling variance for a judgment sample Sample Variance lAs sample size approaches population size NnN1 goes from 1 to zero IThis should make intuitive sense If the sample size equals population size then there is no sampling variance lN n n N l Sampling Variance IIf sample size n is negligible in relation to population size then N nN1 reduces to N1 and sampling variance is population variance multiplied by 1n lN n n N l Influence of Finite Population on Sampling Variance ISay you have 2 populations with the same variance One has 1000 individuals the other 1000000000 individuals Take a sample of 10 from each to estimate population average using sample average Po ulalion 1 2 N 1000 1000000000 n i N7 n nN 001 000000001 n N71 1ln 01 01 NnlN1 0990990991 0999999991 Sample size and variance IThe fact the second population is a million times larger than the first increases the sampling variance by about 1 lIt is the sample size NOT the fraction of the population sampled called the sampling fraction which almost entirely determines the precision of estimation once the variability of the population is given IHelps explain why a survey of 5000 people can predict the votes of a population of 280 million Effect of Sample Size increase sample size from 2 to 4 in our example Sample slze n 2 4 I With a sample gm of 2 Sample average Freguenc Freguene 4 l variance ofsampling 45 distribution 426 555 l I With a sample size of4 e 2 as 2 variance ofsampling 7 392 2 distribution 107 75 3 a 2 2 as 2 a a 2 as l in 2 l lEIVS ll 2 Tmal l5 l5 So far we have known truth ie the population mean and variance lWe have unbiasedness of the point estimate ie average IWe know sampling distribution variance is related to population variance by the previously discussed factor however we usually do not know the population variance nor do we have all possible samples lWe need to estimate the population variance from a single sample Yikes What estimator for variance should we use Correspondence 4prineiple suggests what VFZ XT Sampl Erma amen ram v em Me her Membevvauev ue sample We refquot quot23 R33 Numb r 2 Avevage l m 33 l e 4 4 a a U27 m7 2 c 5 g a 2 an no r la 2 m m 3 D B 16 e l um l A E E 3 E eclellan 5 F 7 25 L r so a c 7 l 7 e e 4 e e e 4 9 r 9 9 r in D 9 1 cups ll e 9 l 2 r re 4 13 E in D I Population variance 1067 14 r ll l 5 r l What estimator for variance should we use Correspondence principle suggests what Sam Ie average varies by as much as 50 of population average 8 range 41 Sample variance rangs from 0 to 25 more than 100 Most sample variance estimates are below population variance 0 1057 I Average sample variance 54 mo low Badly biased correspondence principle doem t work here Unbiased estimation of population variance lUnfortunately correspondence does not work for variance as it did for average IHowever with a small change we can get it to work We will divide sum of square deviations by one fewer than their number IWe redefine the de nition of population and sample variance and sample variance becomes an unbiased estimator of population variance 2 E xii Var n71 4 Further Amendments ISample Variance is now an unbiased estimator of Population Variance with our amended de nitions quot Variance old def of the Sampling Average is given by Population Variance times 71ef where fis nN n 1 Nan 1 n becomes 1 71 N71 71 N How do we use this estimate of of sampling variance to make statements about population average IVery little use when sample size is small but as sample becomes larger something remarkable happens IWe intuit that sampling variance decreases steadily as sample size increases 0 Distribution becomes less variable around average IHow does this happen Central Limit and Normal Dist I Astounding fact that with large sample size the sampling distribution ALWAYS Elks the same form a norma is r39 u 39on E Central Limit Theorem completely general es not depend on underlying dislributlon ofvalues Form of sampling dislribuu39on of me average will be closely approximated by a normal given a large enough sample I e I This new knowledge along with the average and variance allows us to make general statements Large enough IMore irregular the population distribution the larger sample size needed Central Limit Theorem relies on random sampling Without random sampling no need to go further Random sample allow us unbiasedness allow us to estimate precision also allow us to make more elaborate in r nces lWithout random sampling apparatus fall apart Central Limit Theorem IApplies to most estimators we will work with not just sample average lAllows us to construct confidence intervals Confidence intervals lFrom a normal distribution we know 68100 that the deviation of the estimate won t exceed 1 SE square root of estimated sampling variance 90100 won t exceed 164 SE 95100 won t exceed 2 SE ITurn around and say 90 times out of 100 our sample estimate is within 164 SE of the true value IExcel example sample size 4 Confidence Intervals 95Cl Sun name 39 a M557 55555 venanue vename variance Law Uppev cmeee 1 39 m 257 522 m7 e55 msz u 2 39 5 53 575 555 mm 5 3 39 575 75 m5 1 a 39 57 555 575 75 m5 1 5 39 357 Dal 555 7 e55 1 5 5 39 57 5o 25 5ee Mm 1 7 57 55 25 555 mu 1 5 57 55 25 555 my 1 e 39 75 39 33 119 ms 555 e5 1 m 39 75 e57 155 125 we mm 1 M 39 75 391957 5 25 we 5m 1 12 39 7 W57 22 w m e17 1 13 39 7 3917 m 25 55 935 1 w 57 ue7 nee 557 m 1 5 39 55 391157 ue7 nee 557 as 1 5 5 99435119 Smthlgl39 smmwgp l 2053 1315 87 H Nut 555157 smll m mama sanle SIZE Sampling with replacement IUp until now we have sampled without replacement IIn our sample size of 2 how many samples would there be with replacement I36 15 original 15 reverse order 6 same individuals Sampling with replacement ILose some precision with replacement IBut in practice sometimes easier to sample with replacement IMain reason math gets easier investigate more complex sampling systems Sampling with replacement I Sample average remains unbiased estimator I Sample variance incrmses more extremes than without replacemen I Without replacement maximum sample size is opulation size With replacement maximum sample size is in In39 e I Sampling W r replacement395 equivalentto sampling W rout replacement om a popula 39on of infinite s39ze Symbol N changes I We use old de nition of population variance with N as the divisor Sample size I Need to define study population and sampling frame IQuestion of how large a sample is a difficult one only can design the perfect study once you know the answer Available funding may predetermine the sample size Sample Size lDepends on desired precision of estimates statistical level of con dence and variance in population I I want to be with 013 units of the population average 99 percent of the time ISE 01326 005 ISampling variance 00025 PopVar1n INeed estimate of PopVar to solve for n Proportions IFormulas are simplier IExact confidence intervals are possible ISample size calculations are easier Unequal probability sampling IHorvitzThompson Estimator for population total v f L 11 7139 Can be useful if sample inclusion is b ed on size Often need computer 4 Estimating a proportion ISO far we have focused on means and totals across plots IOften interested in a proportion Proportion of females Proportion occupied Attibute can be coded as a 1 or 0 for each plot Usual methods can be used Proportions ISample proportion is ISample variance is A A 7 N7quot M17 WIpe N j nil lNotice that the relevant statistics can be computed from the proportion alone Supplementary information Ratio estimation IWe saw that stratification is a worthwhile way of using coarse supplementary information to improve estimation lWhat about more precise supplementary data ILet s distinguish between using aux data in the design and in the estimation 4 eig eig Individual PrimaryVar SecVar A 2 4 B 6 8 C 10 8 D 8 16 E 10 16 12 20 How do we use these data 17 g it all 5 Ratio estimator 3 9 13 B31 SampAvgSampSuppAv 4 9 15 7 X l 5 85 12 850 g I 6 85 ll 850 7 E 12 ECO a s 14 63936 A y 9 75 ii 915 u m M 10 75 10 900 x ll 75 12 750 12 7 9 933 13 7 ii 700 Need to know suppl Info 14 65 9 867 39 39 15 65 me for entire population age 00 1200 906 Rawqelmn 6 Supplementary information Ratio estimation ISay we have a second variable related to the primary variable for each possible unit IExamples IExcel example height and weight Sampling distributions n4 Weig DT I L I an i c Ratio estimation IMean 806 vs 800 slightly biased IRange of Samp Var 686 933 vs 65 10 IVar 056 vs 107 Increased precision outweighs slight Ias PierreSimon Laplace IWanted to have a census of France in 1802 ISampled 30 communities and calculated a total of 2037615 people with 71866 births I203761571866 2835 people per birth Reasoned he could multiply the number of births by 2875 to obtain an estimate of population size ICrossmultiplication problem like our example 4 Ratio estimation IRarely do we go through the expense of taking a sample and measuring only 1 quantity Often the sampling frame gives us extra information to improve precision IRatio and regression estimation use variables that are correlated with the variable of interest to improve precision The stronger the correlation the better they work Regression estimation IWhat is a slope IIf regression goes through zero and describes the relationship well great predictor that is a ratio estimator IIf you believed this relationship a priori how would you design a sampling plan Regression estimation IIf regression does not go through zero then need to account for yintercept Regression estimator of population mean ISay we want to estimate the number of dead trees in an area lDivide area into 100 square plots and count dead trees quickly from photographs with error mean 113 ISRS 25 plots to visit and get actual count Yint 505 Slope 061 Regression estimator Difference estimator lIf we use this estimator IUsed when slope is known to equal 1 A auditin qumear be b1 ux I g 12ym 50606131131199 I1199 times 100 plots 1199 dead trees What is meant by design unbiased Modelbased vs designbased IDesignbasedrelies upon the randomization theory for estimate and variance IModelbased relies upon a relationship IIf Model is correct we can do extremely well Suggest sampling based on a mo e Ifa linear model is assumed where would you take samples to best estimate slope IIf Model isincorrect the estimates are only modelunbiased model variance Wil underestimate true variance Design vs Model based IHow do you balance these
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