Statistics for Terrified Biologists VanEmden: “C000” — 2008/1/31 — 16:18 — PAGE i — #1 VanEmden: “C000” — 2008/1/31 — 16:18 — PAGE ii — #2 Statistics for Terrified Biologists HelmutF.vanEmden EmeritusProfessorof Horticulture Schoolof BiologicalSciences Universityof Reading,UK VanEmden: “C000” — 2008/1/31 — 16:18 — PAGE iii — #3 ©HelmutF.vanEmden2008 BLACKWELLPUBLISHING 350MainStreet,Malden,MA02148-5020,USA 9600GarsingtonRoad,OxfordOX42DQ,UK 550SwanstonStreet,Carlton,Victoria3053,Australia TherightofHelmutF.vanEmdentobeidentifiedastheauthoroftheeditorialmaterialinthis workhasbeenassertedinaccordancewiththeUKCopyright,Designs,andPatentsAct1988. Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,or transmitted,inanyformorbyanymeans,electronic,mechanical,photocopying,recordingor otherwise,exceptaspermittedbytheUKCopyright,Designs,andPatentsAct1988,withoutthe priorpermissionofthepublisher. 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ForfurtherinformationonBlackwellPublishing,visitourwebsiteat www.blackwellpublishing.com VanEmden: “C000” — 2008/1/31 — 16:18 — PAGE iv — #4 Contents Preface xiii 1 Howtousethisbook 1 Introduction 1 Thetextof thechapters 1 Whatshouldyoudoif yourunintotrouble? 2 Elephants 3 Thenumericalexamplesinthetext 3 Boxes 3 Spare-timeactivities 4 Executivesummaries 4 Whygotoallthatbother? 4 Thebibliography 6 2 Introduction 8 Whatarestatistics? 8 Notation 8 Notationforcalculatingthemean 10 3 Summarizingvariation 12 Introduction 12 Differentsummariesof variation 13 Range 13 Totaldeviation 13 Meandeviation 13 Variance 14 Whyn−1? 16 Whythesquareddeviations? 17 Thestandarddeviation 18 Thenextchapter 20 Spare-timeactivities 20 VanEmden: “C000” — 2008/1/31 — 16:18 — PAGE v — #5 vi Contents 4 Whenaresumsof squaresNOTsumsof squares? 21 Introduction 21 Calculatingmachinesofferaquickermethodof calculating sumsof squares 21 Addedsquares 21 Thecorrectionfactor 22 Avoidbeingconfusedbytheterm“sumof squares” 22 Summaryof thecalculatormethodof calculatingdownto standarddeviation 23 Spare-timeactivities 24 5 Thenormaldistribution 25 Introduction 25 Frequencydistributions 25 Thenormaldistribution 26 Whatpercentisastandarddeviationworth? 27 Arethepercentagesalwaysthesameasthese? 29 Othersimilarscalesineverydaylife 30 Thestandarddeviationasanestimateof thefrequency of anumberoccurringinasample 31 Frompercenttoprobability 31 Executivesummary1–Thestandarddeviation 33 6 Therelevanceof thenormaldistributionto biologicaldata 35 Torecap 35 Isourobserveddistributionnormal? 36 Checkingfornormality 37 Whatcanwedoaboutadistributionthatclearlyisnotnormal? 38 Transformation 38 Groupingsamples 40 Doingnothing! 40 Howmanysamplesareneeded? 40 Factorsaffectinghowmanysamplesweshouldtake 41 Calculatinghowmanysamplesareneeded 41 7 Furthercalculationsfromthenormaldistribution 42 Introduction 42 Is“A”biggerthan“B”? 42 Theyardstickfordeciding 43 Derivationof thestandarderrorof adifferencebetween twomeans 45 Step1–fromvarianceof singledatatovarianceof means 45 VanEmden: “C000” — 2008/1/31 — 16:18 — PAGE vi — #6 Contents vii Step2–fromvarianceof singledatato“varianceof differences” 48 Step3–thecombinationof Steps1and2;thestandard errorof differencebetweenmeans(s.e.d.m.) 49 Recapof thecalculationof s.e.d.m.fromthevariance calculatedfromtheindividualvalues 51 Theimportanceof thestandarderrorof differencesbetween means 52 Summaryof thischapter 52 Executivesummary2–Standarderrorof adifference betweentwomeans 56 Spare-timeactivities 57 8 Thet-test 58 Introduction 58 Theprincipleof thet-test 58 Thet-testinstatisticalterms 59 Whyt? 60 Tablesof thet-distribution 61 Thestandardt-test 64 Theprocedure 64 Theactualt-test 69 t-testformeansassociatedwithunequalvariances 69 Thes.e.d.m.whenvariancesareunequal 70 Aworkedexampleof thet-testformeansassociated withunequalvariances 73 Thepairedt-test 75 Pairwhenpossible 78 Executivesummary3–Thet-test 80 Spare-timeactivities 82 9 Onetailortwo? 83 Introduction 83 Whyistheanalysisof varianceF-testone-tailed? 83 Thetwo-tailedF-test 84 Howmanytailshasthet-test? 86 Thefinalconclusiononnumberof tails 87 10 Analysisof variance–Whatisit?Howdoesitwork? 88 Introduction 88 Sumsof squaresintheanalysisof variance 89 Some“made-up”variationtoanalyzebyAnova 89 Thesumof squarestable 91 VanEmden: “C000” — 2008/1/31 — 16:18 — PAGE vii — #7 viii Contents UsingAnovatosortoutthevariationinTableC 91 Phase1 91 Phase2 92 SqADS–animportantacronym 93 Backtothesumof squarestable 96 Howwelldoestheanalysisreflecttheinput? 96 EndPhase 97 Degreesof freedominAnova 97 Thecompletionof theEndPhase 99 Thevarianceratio 100 Therelationshipbetween“t”and“F” 101 Constraintsontheanalysisof variance 103 Adequatesizeof experiment 103 Equalityof variancebetweentreatments 103 Testingthehomogeneityof variance 104 Theelementof chance:randomization 104 Comparisonbetweentreatmentmeansintheanalysisof variance 107 Theleastsignificantdifference 108 AcaveataboutusingtheLSD 110 Executivesummary4–Theprincipleof theanalysisof variance 111 11 Experimentaldesignsforanalysisof variance 115 Introduction 115 Fullyrandomized 116 Dataforanalysisof afullyrandomizedexperiment 117 Prelims 117 Phase1 118 Phase2 118 EndPhase 120 Randomizedblocks 121 Dataforanalysisof arandomizedblockexperiment 123 Prelims 123 Phase1 125 Phase2 126 EndPhase 127 Incompleteblocks 127 Latinsquare 130 Datafortheanalysisof aLatinsquare 131 Prelims 132 Phase1 134 Phase2 134 VanEmden: “C000” — 2008/1/31 — 16:18 — PAGE viii — #8 Contents ix EndPhase 135 FurthercommentsontheLatinsquaredesign 136 Splitplot 137 Executivesummary5–Analysisof arandomizedblock experiment 139 Spare-timeactivities 141 12 Introductiontofactorialexperiments 143 Whatisafactorialexperiment? 143 Interaction 145 If thereisnointeraction 145 Whatif thereisinteraction? 147 Howaboutabiologicalexample? 148 Measuringanyinteractionbetweenfactorsisoftenthe main/onlypurposeof anexperiment 148 Howdoesafactorialexperimentchangetheformof the analysisof variance? 150 Degreesof freedomforinteractions 150 Thesimilaritybetweenthe“residual”inPhase2and the“interaction”inPhase3 151 Sumsof squaresforinteractions 152 13 2-Factorfactorialexperiments 154 Introduction 154 Anexampleof a2-factorexperiment 154 Analysisof the2-factorexperiment 155 Prelims 155 Phase1 156 Phase2 156 EndPhase(of Phase2) 157 Phase3 158 EndPhase(of Phase3) 162 Twoimportantthingstorememberaboutfactorialsbefore tacklingthenextchapter 163 Analysisof factorialexperimentswithunequalreplication 163 Executivesummary6–Analysisof a2-factorrandomized blockexperiment 166 Spare-timeactivity 169 14 Factorialexperimentswithmorethantwofactors 170 Introduction 170 Different“orders”of interaction 171 VanEmden: “C000” — 2008/1/31 — 16:18 — PAGE ix — #9