SamplingandSummaryStatisticsBiology683

Lecture2

HeathBlackmon

Lastweek

• Whataresomecausesofthereproducibilitycrisis?

• Commonproblemsinplots?

Today

1. Terminology2. SummarizingData3. CentralLimitTheorem4. CovarianceandCorrelation

PopulationsandSamples

• PopulationsSomesortofgroupofsomething- couldbeanything• UndergraduatesatTexasA&M• JewelbeetlesinArizona• Strainoffliesinthelab

• Samples• Asubsetofindividualsdrawnfromapopulation

Whatisthepopulation?

Wewantedtoexamineanyassociationbetweentheseverityofinjuries,andtheheightfromwhichcatsfallinhigh-risebuildings.

IntheperiodbetweenJanuary1,1998andDecember12,2001attheClinicofSurgery,OrthopedicsandOphthalmologyoftheVeterinaryFaculty,119catsweretreatedafterafallorjumpfromabalconyorwindow,wheretheownerssawthefall,orwheretherewasareasonablesuspicionthatafallhadoccurred.Onlythosecatsthatfellfromthesecondorhigherstorieswereincluded.Theownersbroughtthecatsfortreatmentwithinvaryingperiodsoftimeafterthefall(from30mintooveramonth).

Vnuk,etal."Felinehigh-risesyndrome:119cases(1998–2001). JournalofFelineMedicine&Surgery 6.5(2004):305-312.

SamplingConsiderations

Targetpopulation• Needtosamplearepresentativepopulation• AsampleofpeoplefromCollegeStation,forinstance,would

probablynotberepresentativeofNewYorkers

SamplingError• Chancealonewillcauseyoursampletodepartfromthe

population

Parameter,estimates,samplingconsiderations

Parameter:Population-levelvariableswearetryingtoestimate

EstimateorStatistic:Thevalueoftheparameterinferredfromthesample

Bias:Ifsomethingaboutthesamplingprocedurecausesthesampletosystematicallymisrepresentthepopulation.

Precision:Howtightlygroupedaretheestimates?

AccuracyvsPrecision

•Precision is a measure of spread

•Accuracy is a measure of bias

RandomSampling

1. Every unit in a population should have an equal chance of being sampled.

2. The selection of units must be independent.

3. Lots of ways of being non-random…

Yourbigideashouldbeahypothesis

Astatisticalhypothesisisaspecificclaimaboutapopulationparameter

CaloricrestrictionincreasesthelifespanofDrosophilamelanogaster.

Therateofevolutioninwinglessspeciesishigherthanwingedspecies.

Pesticideexposurecausesfeminizationofamphibianmales.

RepetitiveDNAcontentishigherinvenomousthannonvenomousreptiles

Data

VariablesThecharacteristicsthatdifferamongindividuals

DataThemeasurementsofvariablestakenforasampleofindividuals

CategoricalVariablesIndividualsareinqualitativecategories

Data

NumericalVariablesIndividualsvaryonaquantitativescale

OrdinalThecategoriescanbeordered

NominalThecategorieshavenoinherentorder

ContinuousvsDiscrete

Continuousvariablesavariablethathasaninfinitenumberofpossiblevalues

Discretevariablesavariablethathasafinitenumberofpossiblevalues

ExplanatoryandResponseVariables

Experimentalvsobservationalstudies

• Doescaloricrestrictionincreaselifespaninmice?• Isglobalwarmingcausedbyhumanactivities?• Doessmokingcauselungcancerinhumans?• Doesparasiteinfectionreducematingsuccessofbeetles?• Doesoxytocinaffectsexualattractioninhumans?• Dosexchromosomesincreasetherateofspeciation?

Whyshouldwesummarizedata?

• Manydatasetsaresimplytoobigtolookatallvaluesandformanimpression?

• Ourimpressionsofsmalldatasetsareoftenmisledbyourtendencytolookforpatterns.

Typicalsummarystatistics

• Mean: Sumoftheobservationsdividedbythenumberofobservations

• Median: Themiddleobservationinasetofdata

• Variance: Theaveragesquareddeviationfromthemean

• StandardDeviation: Thesquarerootofthevariance

Meanandvariance

BoxPlot

InterquartileRange

BoxPlot

Rangeofalldata

BoxPlot

Datapointsmorethan1.5Interquartilerangesawayfromthemedian

BoxPlot

Symbolsforsamplesandpopulations

SamplesversusPopulationsThemeanorstandarddeviationstatisticyoucalculatefromyoursampleisanestimateofthepopulationparameter.

ParameterSymbols:μ :populationmeanσ :populationstandarddeviation

StatisticSymbols:𝑌" :samplemeans :samplesstandarddeviation

Forasampleofapopulation

Themeanisjust:𝑌" = ∑ %&'&()*

Thestandarddeviationis𝑠 = 𝑠,�

Where𝑠, orthevarianceis:𝑠, = ∑ %&/%" 0'&()*/1

Centrallimittheorem

• Imaginethatwesamplefromthesamepopulationmanytimes,sowehaveabunchofdifferent,independentsamples.

• Eachsamplewillhaveamean,butthemeanswillbedifferentduetochance Inprinciple,wecoulddrawahistogramofthesemeans.

• Ingeneral,youonlyhaveonesamplefromagivenpopulation,however,sowhatcanyouinferaboutthedistributionofthemeansfromyoursample?

• TheCentralLimitTheoremstatesthatregardlessoftheunderlyingpopulationdistributionofthevariableofinterest,thedistributionofthepopulationofmeanswillberoughlynormal.

Centrallimittheorem

Yourestimateofthesamplemeanisanestimateofthemeanofthisdistributionofmeans(thatis,it’syourbestestimateofthepopulationmean).

Thehypotheticaldistributionofsamplemeanshasastandarddeviationequaltosdividedbythesquarerootofn.

𝑆𝐸%" =4*�

Wecallthisstandarddeviationthestandarderrorofthemean(SEM).

Centrallimittheorem

Letstrythat

Estimatingwithuncertainty

Errorbars

• Errorbarscanbeausefulwaytoshowuncertaintywhenit’snotpossibletoshowtheactualdatapoints.

• Usually,theyrepresent1SEorthe95%CI,butnotalways.

• THEFIGURELEGENDSHOULDINDICATEWHATTHEERRORBARSREPRESENT!

ForThursday

1. Readchapters3WS

Bringlaptoptoclass!

HeathBlackmonBSBW309Acoleoguy@gmail.com

Yourturn

• Letsdemonstratethatthemeansofsamplesfromanexponentialdistributionarenormallydistributed.

Youwillneed:rexp, hist, mean

Mightuse:for, sample

BoxPlot