MechanismsofMitosisDataAnalysis

Youarenowreadytostatisticallyanalyzethedatayourteamhascollected.

I.MitoticIndexandMitoticPhaseIndicesYourmitoticcellcountscompriseasurveyofthenumberofdifferentstagesofmitosisinyourtwopopulations(treatmentandcontrol).Youcountedmitoticcellsin8treatmentand8controlroots,andthencalculatedaMitoticIndex(M)foreachsample.

M=nm/N

nm=thenumberofmitoticcellsinthesampleN=thetotalnumberofcellscountedinthesample.

You also calculated aMitotic Phase Index (Mxxx) for each phase ofmitosis you found in yourtreatmentandcontrolgroups.

Mphase=nphase/nm

nphase=#ofcellsin[mitoticphase]inthesamplenm=totalnumberofmitoticcellsinthesample

UsetheindicesyourecordedinthetabletemplatesduringSessions3and4forthedataanalysisdescribedinthefollowingsection.II.ApplyingaStatisticalTesttoYourMitoticIndicesYourmitotic indicesareordinal, non-parametric data thatarenotdistributedalonganormalcurve.Thenon-parametricMann-WhitneyUtestisappropriateforthistypeofdata.

TheMannWhitneyUtestmeasuresthedegreeofoverlapbetweentwosetsofdatathatcanberanked(i.e.,putinorderofascendingvalues).

• largeoverlapmeansnosignificantdifferencebetweenyourpopulationso failtorejectthenullhypothesis

• small(5%orless)overlapmeansasignificantdifferencebetweenyourpopulations

o rejectyournullhypothesis.Non-parametrictestfortwosamples:Mann-WhitneyUTheMann-Whitneytestallowstheinvestigator(you)tocompareyourtwocellpopulationswithoutassumingthatyourMitoticIndexvaluesarenormallydistributed.TheMann-WhitneyUdoeshaveitsrules.Forthistesttobeappropriate:

• Youmustbecomparingtworandom,independentsamples(treatment&control)• Themeasurements(MitoticIndices)shouldbeordinal• Notwomeasurementsshouldhaveexactlythesamevalue

o (thoughwecandealwith“ties”inawaythatwillbeexplainedshortly)

1.Stateyournullandalternativehypotheses.

Ho:

HA:

Example: Ho:ThereisnodifferenceintheranksofMitoticIndices(M)betweenmeristematiccellsin anoniontreatedwithaqueoustrifluralinandanoniontreatedwithplainwater.

HA:ThereisadifferenceintheranksofMitoticIndices(M)betweenmeristematiccellsin anoniontreatedwithaqueoustrifluralinandanoniontreatedwithplainwater.

2.Statethesignificancelevel(tobecomparedtoa,0.05)requiredtorejectHo. Thisistypicallyaprobabilityvalue(P)of<0.05

3.RankyourMitoticIndicesfromsmallesttolargestinatable Notewhichindexcamefromwhichpopulationofcells(TreatmentorControl).

Example:• Table1shows16(imaginary)MIfromtreatment(T)andcontrol(C)onionroottips.• Table2showsthevaluesrankedandlabeledbypopulation.

Table1.MitoticIndicesfor Table2.RankedMitoticIndicestreatmentandcontrolroottips Notetiedvaluesinblue.Sample#

Mtreatment

Mcontrol Rank RankedMvalues

CellPopulation

1 0.20 0.55 1 0.10 T2 0.25 0.60 2 0.15 T3 0.45 0.65 3 0.20 T4 0.35 0.80 4 0.25 T5 0.15 0.35 5 0.35 T6 0.10 0.75 6 0.35 C7 0.55 0.70 7 0.40 T8 0.40 0.85 8 0.45 T 9 0.55 T 10 0.55 C 11 0.60 C 12 0.65 C 13 0.70 C 14 0.75 C 15 0.80 C 16 0.85 C

4.Assignpointstoeachrankedvalue(seeTable3):• Each“treatment”rankgetsonepointforevery“control”rankthatappearsbelowit.• Every“control”valuegetsonepointforevery“treatment”valuethatappearsbelowit.• Forexample,thefirstvalue,0.10(T)has8Controlvaluesbelowit,soitgets8points.• Value10(C)has3Treatmentvaluesbelowit,soitgets3points.• Tiedvaluessplitthesumoftheirpoints.Forexample:

o Rank5(0.35)has8pointso Rank6(0.35)has3pointso 8+3=11o Eachrankgetshalfof11,or5.5

Table3.PointsassignedtorankedMvaluesinTreatmentandControlonioncellpopulations.(example)Tiedvaluessplittheirtotalpointsequally.Rank RankedM

valuesCellpopulation

Points

1 0.10 T 82 0.15 T 83 0.20 T 84 0.25 T 85 0.35 T 8à5.56 0.35 C 3à5.57 0.40 T 78 0.45 T 79 0.55 T 7à3.510 0.55 C 0à3.511 0.60 C 012 0.65 C 013 0.70 C 014 0.75 C 015 0.80 C 016 0.85 C 05.CalculateaUstatisticforeachcategorybyaddingthepointsforeachcellpopulation.

Utreatment=8+8+8+8+5.5+7+7+3.5=55Ucontrol=5.5+3.5+0+0+0+0+0+0=9

YourUstatisticisthesmallerofthesetwovalues.IntheimaginaryexampleourUvalueis9.ThelowertheUvalue,thegreaterthedifferencebetweenthetwogroupsbeingcompared.(Forexample,ifnoneoftheMvaluesoverlapped,theUvaluewouldbezero.)III.Criticalvaluesfornon-parametricstatisticsWehavedefinedoursignificancelevel(a)as0.05.Thisimplies:

• atruenullhypothesiswillberejectedonly5%ofthetime• afalsenullhypothesiswillberejected95%ofthetime

…ifthePvalueobtainedfromyourdataislessthanorequalto0.05.Acritical value of a statistic (e.g.,Mann-WhitneyU) is thevalue associatedwith a significancelevellessthanorequaltoa.(Weareusingthetraditionalvalueofa,0.05.)CriticalvaluesfortheMann-WhitneyUstatistic(atdifferentsamplesizes)areshowninTable4.

Inthepreviousimaginaryexample,treatmentandcontrolgroupswith8sampleseach,acriticalvalueof13isrequiredforrejectionofthenullhypothesis.TheMannWhitneyUstatisticof9isfarlowerthanthiscut-offvalue.Thismeansthereisverylittleoverlapbetweenthetwopopulations(theyaresignificantlydifferent).Thenullhypothesisisrejected.

Table4. CriticalvaluesfortheMann-WhitneyUstatistic. Findthevaluethatcorrespondstothesamplesizes(8and8)ofyourtwocellpopulations.IfyourUvalueissmallerthanthatshowninthetable,thenthereislessthan5%chancethatthedifferencebetweenyourtwocellpopulationsis due to chance. If your U value is smaller than the one shown in this table, reject your nullhypothesis.IfyourUvalueislargerthanthatshowninthetable,failtorejectyournullhypothesis.

IV.GraphicRepresentationofyourDataTablesofnumericaldataareimportant,buttheyarenotalwaysthebestwaytopresentyourdatatoanaudience.Astheoldsayinggoes,“Apictureisworthathousandwords.”Themosteffectivewaytopresentyourexperimentalresults,wheneverpossible,iswithafigure.

A.MitosisRawData Asimplebargraphcanbeusedtorepresenttheproportionofcellsinyoursamplethatyoufoundineachstageofmitosis.AnexamplecanbeseeninFigure1.Inabargraph,categoriesmaybeplacedinanyorder,anddonotnecessarilyrepresentacontinuum.

Figure1.Abargraphshowingahypotheticaldistributionofcellsineachstageofmitosisinastudypopulationofcells.

Don’tconfuseabargraphwithahistogram.Ahistogramdepictscontinuousdata.AnexampleofahistogramisshowninFigure2.

Figure2.Ahistogramshowingahypotheticaldistributionofcellsofdifferentdiameterinapopulationofcells.Notethateachbaronthehistogramrepresentsaspecificsubsetofarangeofcontinuousnumericaldatathatoccurinasetorder.

Figures,unliketables,havetheirlegendsplacedunderneath.Alwaysuseproperformatforfiguresandtablesinyourwork.B.VisualizingMann-WhitneyUresultsBecausetheMann-WhitneyUprovidesameasureofhowgreattheoverlapisbetweentwogroupsbeingcompared,abox plot isagoodway torepresentyourMann-WhitneyUresults. Theboxgraphcanbecreatedtoshowthemedianofeachgroup,therangeofvalues,andtheiroverlap.AnexampleofaboxplotisshowninFigure3,withakeyandexplanationinFigure4.

Figure3.Sampleboxplotshowingoverlapofmitoticindexvaluesfortwopopulationsofcells.

Figure4. Theblackbarinthecenterofeachpopulation’svaluesrepresentsthemedian.TheInterquartileRange(IQR)includes50%ofthevalues,andisborderedonthebottomby the 25th percentile and on the top by the 75th percentile. The range is the regionbetweentheminimumandmaximumvalues. Thestarrepresentsadatapointthatisanoutlier.