種群的空間分布型及抽樣李典謨中科院動物研究所Email:lidm@2005年10月(一）空間間分布型型1.意義種群生態(tài)態(tài)特性：：空間是是聚集分分布還還是隨隨機分布布，解解決抽樣樣方法，，提供理理論依據據。2.分類隨機分布布：泊松松(Poisson)分布聚集分布布：負二二項分布布(negativebinomialdistribution))奈曼分布布(Neyman)泊松二項項分布Thesimplestviewofspatialpatterningcanbeobtainedbyadoptinganindividualorientation,,andaskingthequestion,Giventhelocationofoneindividual,,whatistheprobabilitythatanotherindividualisnearby?Therearethreepossibilities:1.Thisprobabilityisincreased—aggregatedpattern2.Thisprobabilityisreduced——uniformpattern3.Thisprobabilityisunaffected——randompatternRandomAggregatedUniformFigure4.3Threepossibletypesofspatialpatterningofindividualanimalsorplantinapopulation.3.頻次分布布理論公公式（1）泊松（（普阿松松）分布布例：蝗蝻蝻的田間間分布02050101200112(1)普阿松分分布（Poisson分布）例：對公公共汽車車客流進進行調查查，統(tǒng)計計某天上上午10∶30—11∶47左右每隔隔20秒鐘來到到的乘客客批數，，共得到到230個記錄。。來到批數i01234總共頻數ni100813496230頻率0.430.350.150.040.030.420.360.160.050.01普阿松分分布的意意義已經發(fā)現現許多隨隨機現象象服從普普阿松分分布（1）社會生生活，服服務行業(yè)業(yè)如：電話話交換臺臺中來到到的呼叫叫數公共汽車車站來到到的乘乘客數（2）物理學學放射性分分裂落到到某區(qū)域域的質點點數（3）昆蟲個個體的空空間分布布普阿松分分布的特特點以交換臺臺電話呼呼叫數為為例（1）平衡性性在[t0,t0+t]中來到的的呼叫數數只與時時間間隔隔長度t有關，而而與時間間起點T0無關（2）獨立增增量性（（無后效效性）在[t0,t0+t]內來到k個呼叫這這一事件件與時刻刻T0前發(fā)生的的事件獨獨立（3）普通性性在充分小小的時間間間隔中中，最多多只來到到一個呼呼叫例：蝗蝻蝻分布型型調查，，共取樣樣408個蟲數x頻率ff*x02250113013024080310304312408252計算方法法另樣的理理論數n*p0=408*0..5391=219..09有一頭蟲蟲的樣本本的理論論數n*p1=135.9觀察值與與理論值值比較蟲數x觀察值(o)理論值(c)0225219.90.111130135..90.2624042.20.093108.70.2149自由度=n-2=3，失去兩兩個自由由度（1）用來限限制實際際樣本數數N((2))用來估計計

意味不是是一個小小概率事事件（p>0..05)),沒有理由由否定假假設要求各組組內的預預計數都都不少于于5，當某組組的Y少于5時，須把把它和相相鄰的一一組或幾幾組合并并直到Y大于5，然后再再用上式式計算x2值。檢驗的理理論與方方法1公式

O為實際觀觀測值，，E為理論推推算值。。其基本原原理是應應用理論論推算值值與實際際觀測值值之間的的偏離程程度來決決定其值值的的大小。。是理論分分布總體體的頻數數是觀察分分布總體體的頻數數兩個樣本本來自不不同的總總體分布的特特點df=1df=3df=5（1）分分布布于區(qū)間間[1，）），，偏斜度度隨自由由度降低低而增大大，當自自由度df=1時，曲線線以縱軸軸為漸近近線。（2）隨自由由度df增大，分分布趨左左右對稱稱，當df>30時，分分布接近近正態(tài)。。檢驗的基基本步驟驟（1）建立檢檢驗假設設，確定定檢驗水水平。（2）計算檢檢驗統(tǒng)計計量（3）確定概概率P值，計算算自由度度df＝k-1由和和自由度度查統(tǒng)計計表的的臨界界值（4）判斷結結果臨界值檢檢驗假設設的關系系值P假設判判斷<>0.05不拒絕差差異無顯顯著性0.05拒絕差差異有顯顯著性例：假定定某地嬰嬰兒出生生的男女女比例為為1：1。研究者抽抽取了一一個含10，000名嬰兒的的樣品，，男孩5100，女孩4900，問他是是否證實實了假設設或否定定了假設設。某地嬰兒兒出生性性比為1：1>拒絕嬰嬰兒兒性比不不為1：1注：在自自由度df＝1時，需進進行連續(xù)續(xù)性矯正正，其矯矯正的為為：適合性檢檢驗比較觀測測數與理理論數是是否符合合的假設設檢驗叫叫適合性性檢驗。。例如在在遺傳學學上，常常用檢檢驗來測測定所得得的結果果是否符符合孟德德爾分離離規(guī)律，，自由組組合定律律等。例有有一鯉鯉魚遺傳傳試驗，，以荷包包紅鯉（（紅色））與湘江江野鯉（（青灰色色）雜交交，其代代獲得得如表5-2所列得體體色分離離尾數，，問這一一資料的的實際觀觀察值是是否符合合孟德爾爾的青：：紅=3：1一對等為為基因的的遺傳規(guī)規(guī)律？表鯉魚遺傳試驗驗F2觀察結果果體色色青青灰色色紅紅色色總總數F2觀測尾數數1503991602（1）鯉鯉魚魚體色分分離符符合3：1比率。（2）取顯著著水平（3）計算青灰色理理論數紅色理論論數（4）差值值表。。df=1時，故否定，，接接受即鯉魚體體色分分離不不符合3：1比率。（2）負二項項分布正二項分分布是(p++q)n的展開式式的各項項,其中n為個體總總數，p,q為分成對對比兩類類期望的的比例。。[Student((1907)..]展開上述述式子，，于是一一個樣本本單位有有r個個體的的概率為為可以估算算出p,k。矩法由此可以以推出（二）分分布型指指數上述蝗蝻蝻例子中中說明上述述蝗蝻屬屬Poisson分布。2.David&Moore((1954)方法IndexofDispersionTest..WedefineanindexofdispersionItobeForthetheoreticalPoissondistribution,thevarianceequalsthemean,sotheexpectedvalueofIisalways1.0inaPoissonworld.Thesimplestteststatisticfortheindexofdispersionisachi-squaredone:whereI=Indexofdispersion((asdefinedinequation4.3)n=Numberofquadratscounted=valueofchi-squaredwith((n-1)degreesoffreedom.04825231蟲數頻頻率25例：取了了25個樣，調調查蚯蚓蚓的田間間分布。。由于observedchi--squared所以，我我們接受受原假設設：蚯蚓蚓田間分分布符合合Poisson分布。3.Waters((1959)提出負負二項分分布中的的Kk’的特性：：當種群群密度因因為隨機機死亡而而減小時時，k’保持不變變，表示示種群空空間分布布的內在在特點，，而與密密度無關關4.Tayloz((1961,,1965,1978)方法密度越高高，種群群分布越越均勻，，（聚集集度越低低）5.平均擁擠擠度指標標Lloyd,M.(1967)例：a1b0c2d3X1=1;x2=0X3=2;x4=3n=4A：一頭頭“獨居居”1*(1-1))B:沒有鄰居居C:有兩頭，，各以對對方為鄰鄰居；2*(2-1))=2D:每個有兩兩個鄰居居，3*（3-1）=6，總共““鄰居””數為：：0+0++2+6=8平均每個個個體有有1.33個鄰居Lloyd定義聚集度指指標：Iwao發(fā)現Theidealizedindexshouldhavethreeproperties..(Elliott1977))1.Itshouldchangeinasmoothmannerasmovesfrommaximumuniformitytorandomnesstomaximumaggregation.2.Itshouldnotbeaffectedbysamplesize(n),populationdensity(()),,orbyvariationinthesizeandshapeofthesamplingquadrat..3.Itshouldbestatisticallytractable,,sothataconfidencebeltcanbespecifiedandcomparixonsbetweensamplescanbetestedforsignificance..Morisita’sIndexofDispersion(1)Morisita’sindexofdispersion樣本大小小sumofthequadnatcounts=Morisita((1962)證明隨隨機機分布的的假設下下：StandardizedMorisitaIndexUniformindex=((2)Clumpedindex=((3)=Valueofchi-squaredfromtablewith((n-1)degreesoffreedomthathas97..5%oftheareatotheright.WhenWhenWhenWhen取值以-1.0到+1.0帶著95%置信區(qū)間間隨機分布布聚集分布布均勻分布布InasimulationstudyMyers((1978))foundthestandardizedMorisitaindextobeoneofthebestmeasuresofdispersionbecauseitwasindependentofpopulationdensityandsamplesize.例：E.Sinclair在26個10公頃的樣樣點調查查大象的的數量，，其中一一個樣點點有20頭，令一一樣點30頭，還有有一樣點點10頭，其他他23點為零。。（1）計算Morisita‘sindex（2）以公式(2)((3)中計算臨臨界點。。當自由度度=n--1=25,UniformindexClumpedindex（3）計算StandardizedMorisitaindex::由于（4）因為于于是是我們得得到結論論：在置信水水平95%下，在我我們取樣樣區(qū)大象象是聚集集分布的的。（三）SampleandExperimentalDesignSamplingandexperimentaldesignarestatisticaljargonforthethreemostobviousquestionsthatcanoccurtoafieldecologist::WhereshouldItakemysamples,,howshouldIcollectthedatainspaceandtime,andhowmanysamplesshouldItrytotake?抽樣理論論及在生生態(tài)學中中的應用用W.Gosset1908年以“Student”筆名將““t-檢驗”發(fā)發(fā)表于《biometrika>上,文章中說說:“任何實驗驗可以作作為是許許多可能能在相同同條件下下作出的的實驗的的總體中中的一個個個體.一系列的的實驗則則是以從從這個總總體中所所抽得的的一個樣樣品”1.總體與抽抽樣設一塊棉棉田有N株棉株,每株上某某種害蟲蟲數分別別為X1,X2…..XN,從總體N中,隨機抽取取n株(n<N)樣本,每株蟲數數分別為為X1,X2,……,,Xn.目的:通過樣本本對總體體做出推推斷抽樣誤差差估計及及t分布1908年,“Student””發(fā)表了t分布例:棉田中隨隨機調查查50株棉株,以估計該該棉田中中害蟲的的數量.SampleSizeforContinuousVariable理論抽樣樣數模型型例:洪澤湖蝗蝗區(qū)蟲數樣本數(f)fx0170153532183631030428100127如果，我我們引入入變異系系數（coefficientofvariation）這兒，=標準差=觀察平均均數那么，絕絕對誤差差可寫成相相對誤差差，，（（以百分分比形式式）（方程1）兩個平均均數的比比較例如，我我們要比比較兩個個池塘中中同一種種魚的重重量是否否有差異異，典型型的方法法是個抽抽取一定定數量的的樣本用用t檢驗來檢檢驗兩樣樣本平均均數是否否有差異異。但是是，如何何在抽樣樣前回答答應該取取多少樣樣？SnedecorandCochran(1967,113)提出了如如下的近近似公式式：一般這兒=從兩個種種群中的的每一個個抽取的的樣本大大?。?水平為的的標準正正態(tài)離差差值（））=水平為的的Ⅱ型錯誤概概率下的的標準正正態(tài)離差差值（見見下表））=測量的方方差。（（已知，，或推測測）。=你希望以以概概率率能檢測測出的兩兩平均值值的最小小差異。。TypeⅡerrorPowerTwo-tailed0.400..600.250.200..800.840.100..901.280.050..951.640.010..992.330.0010.9992.58決策

Power越大，決決策結果果越可靠靠

不拒絕H0

拒絕H0

H0是真

決策正確確（概率率＝1－α）

I型錯誤（（概率＝＝α）

H0是假

II型錯誤（（P＝β）

決策正確確（P＝1－β）＝power

例.如果上例例中我們們希望檢檢測出的的平均數數差異是是：（從以前前的研究究中知道道）如果，則則條。2.SAMPLESIZEFORDISCRETEVARIABLESCountsofthenumbersofplantsinaquadratorthenumbersofeggsinanestdifferfromcontinuousvariablesintheirstatisticalproperties.Thefrequencydistributionofcountswilloftenbedescribedbyeitherthebinomialdistribution,,thePoissondistributionorthenegativebinomialdistribution(Elliott1977)).Thesamplingpropertiesofthesedistributionsdiffer,sowerequireadifferentapproachtoestimatingsamplesizesneededforcounts.ProportionsandPercentagesProportionslikethesexratioorfractionofjuvenilesinapopulationaredescribedstatisticallybythebinomialdistribution.Alltheorganismsareclassifiedintotwoclasses,andthedistributionhasonlytwoparameters::ProportionoftypesinthepopulationProportionoftypesinthepopulationIfsamplesizeisabove20,wecanusethenormalapproximationtotheconfidenceinterval::WhereObservedproportionValueofStudent’st-distributionforn-1degreesoffreedomStandarderrorofThusthedesiredmarginoferrorisSolvingforn,thesamplesizerequirediswheren=Samplesizeneededforestimatingtheproportionpd=DesiredmarginoferrorinourestimateAsafirstapproximationforwecanuseWeneedtohaveanapproximatevalueofptouseinthisequation.Priorinformation,,oraguess,,shouldbeused;theonlyrule-of-thumbisthatwhenindoubt,pickavalueofpcloserto0.5thanyouguess.Thiswillmakeyouranswerconservative..Asanexample,supposeyouwishtoestimatethesexratioofadeerpopulation..Youexpectptobeabout0.40,andyouwouldliketoestimatepwithinanerrorlimitofwith..Fromequation(2)CountsfromaPoissonDistributionSamplesizeestimationisverysimpleforanyvariablethatcanbedescribedbythePoissondistribution,inwhichthevarianceequalsthemean.FromthisitfollowsthatorThusfromequation,,(1))assuming::whereSamplesizerequiredforaPoissonvariableDesiredrelativeerror(aspercentage)Coefficientofvariation==Forexample,,ifyouarecountingeggsinstarlingnestsandknowthatthesecountsfitaPoissondistributionandthatthemeanisabout6..0,thenifyouwishtoestimatethismeanwithprecisionof(widthofconfidenceinterval),,youhave:nestsEquation((2)canbesimplifiedforthenormalrangeofrelativeerrorsasfollows::Forprecision3.STATISTICALPOWERANALYSISDecisionStateofrealworldDonotrejectnullhypothesisRejectthenullhypothesisNullhypothesisisCorrectdecisionTypeⅠerroractuallytrue((probability=1-))((probability==))NullhypothesisisTypeⅡⅡerrorCorrectdecisionactuallyfalse((probability==))((probability==(1-))=power)Mostecologistsworryabout,,theprobabilityofaTypeⅠerror,butthereisabundantevidencenowthatweshouldworryjustasmuchormoreabout,,theprobabilityofaTypeⅡⅡerror((Peterman1990;;Fairweather1991)).Poweranalysiscancarriedoutbeforeyoubeginyourstudy(apriori,orprospectivepoweranalysis)orafteryouhavefinished((retrospectivepoweranalysis)..Herewediscussaprioripoweranalysisasitisusedfortheplanningofexperiments..Thomas((1997)discussedretrospectivepoweranalysis.Thekeypointyoushouldrememberisthattherearefourvariablesaffectinganystatisticalinference::samplesizeProbabilityofaProbabilityofaTypeⅠerrorTypeⅡⅡerrorMagnitudeoftheeffect==effectsizeThesefourvariablesareinterconnected,,andonceanythreeofthemarefixed,thefourthisautomaticallydetermined..Lookedatfromanotherperspective,,givenanythreeofthese,youcandeterminethefourth.Figure7.3Anexampleofhowpowercalculationscanbevisualized..Inthissimpleexample,at-testtobecarriedouttodetermineiftheplantnitrogenlevelhaschangedfromthebaselevelof3..0%((thenullhypothese))totheimprovedlevelof3..3%((thealternativehypothese)).Givenn=100,sSUMMARYThemostcommonquestioninecologicalresearchis,howlargeasampleshouldItake?Thischapterattemptstogiveageneralanswertothisquestionbyprovidingaseriesofequationsfromwhichsamplesizemaybecalculated.Itisalwaysnecessarytoknowsomethingaboutthepopulationyouwishtoanalyzeunlessyouuseguessworkorpriorobservations.Youmustalsomakesomeexplicitdecisionabouthowmucherroryouwillallowinyourestimates((orhowsmallaconfidenceintervalyouwishtohave)..Forcontinuousvariableslikeweightorlength,wecanassumeanormaldistributionandcalculatetherequiredsamplesizesformeansandforvariancesquiteprecisely.Forcounts,weneedtoknowtheunderlyingstatisticaldistribution—binomial,Poisson,ornegativebinomial——beforewecanspecifysamplesizesneeded..Poweranalysisexplorestherelationshipsbetweenthefourinterconnectedvariables((probabilityofTypeⅠⅠerror)),((probabilityofTypeⅡⅡerror)),effectsize,andsamplesize.Fixingthreeoftheseautomaticallyfixesthefourth,andecologistsshouldexploretheserelationshipsbeforetheybegintheirexperiments.Significanteffectsizesshouldbespecifiedonecologicalgroundsbeforeastudyisbegun..SamplingDesigns:Random,,AdaptiveandSystematicSampling(1)SimpleRandomSampling(2)StratiliedRandomSampling(3)AdaptiveSampling(4)SystematicSamplingSimplerandomsamplingistheeasiestandmostcommonsamplingdesign.Eachpossiblesampleunitmusthaveanequalchanceofbeingselectedtoobtainarandomsample.Alltheformulasofstatisticsarebasedonrandomsampling,,andprobabilitytheoryisthefoundationofstatistics.Thusyoushouldalwayssamplerandomlywhenyouhaveachoice.Insomecasesthestatisticalpopulationisfiniteinsize,andtheideaofafinitepopulationcorrectionmustbeaddedintoformulasforvariancesandstandarderrors.Theseformulasarereviewedformeasurements,ratios,andproportion.Oftenastatisticalpopulationcanbesubdividedintohomogeneoussubpopulations,andrandomsamplingcanbeappliedtoeachsubpopulationseparately.Thisisstratifiedrandomsampling,andrepresentsthesinglemostpowerfulsamplingdesignthatecologistscanadoptinthefieldwithrelativeease.Stratifiedsamplingisalmostalwaysmoreprecisethansimplerandomsampling,andeveryecologistshoulduseitwheneverpossible.Samplesizeallocationinstratifiedsamplingcanbedeterminedusingproportionaloroptimalallocation.Touseoptimalallocation,youneedroughestimatesofthevariancesineachofthestrataandthecostofsamplingeachstrata.Optimalallocationismoreprecisethanproportionalallocation,,andistobepreferred..Somesimplerulesarepresentedtoallowyoutoestimatetheoptimalnumberofstratayoushoulddefineinsettingupaprogramofstratifiedrandomsampling..Iforganismsarerareandpatchilydistributed,youshouldconsiderusingadaptiveclustersamplingtoestimateabundance..Whenarandomlyplacedquadratcontainsararespecies,adaptivesamplingaddsquadratsinthevicinityoftheoriginalquadrattosamplethepotentialcluster.Thisadditionalnonrandomsamplingrequiresspecialformulastoestimateabundancewithoutbias..Systematicsamplingiseasiertoapplyinthefieldthanrandomsampling,butmayproducebiasedestimatesofmeansandconfidencelimitsifthereareperiodicitiesinthedata..Infieldecologythisisusuallynotthecase,andsystematicsamplesseemtobetheequivalentofrandomsamplesinmanyfieldsituations.Ifagradientexistsintheecologicalcommunity,systematicsamplingwillbebetterthanrandomsamplingfordescribingit.SystematicSamplingWhatisthelikelihoodthatproblemslikeperiodicvariationwilloccurinactualfielddata?Milne((1959)attemptedtoanswerthisquestionbylookingatsystematicsamplestakenonbiologicalpopulationsthathadbeencompletelyenumerated..Heanalyzeddatafrom50populationsandfoundthat,inpractice,,therewasnoerrorintroducedbythatacentricsystematicsampleisasimplerandomsample,andusingalltheappropriateformulasfromrandomsamplingtheory..Step1..Calculatetheaverageabundanceofeachofthenetworks:(8.35)

where==Averageabundanceofthei-thnetwork=Abundanceoftheorganismineachofthekquadratsinthei-thnetwork=Numberofquadratsinthei-thnetwrokStep2..Fromthesevaluesweobtainanestimatorofthemeanabundanceasfollows:(8.36)whereUnbiasedestimateofmeanabundancefromadaptiveclustersamplingNumberofinitialsamplingunitsselectedviarandomsamplingIftheinitialsampleisselectedwithreplacement,thevarianceofthismeanisgivenby:(8.37)whereEstimated