APPENDIX: STATISTICAL TOOLS

Transcription

1 I. Nots o radom samplig Why do you d to sampl radomly? APPENDI: STATISTICAL TOOLS I ordr to masur som valu o a populatio of orgaisms, you usually caot masur all orgaisms, so you sampl a subst of th populatio. I ordr for this subst to b rprstativ of th populatio as a whol, it caot b slctd subjctivly from th populatio. That is, thr should ot b ay bias about which particular orgaisms ar slctd to masur. Th way i which w avoid bias is by takig radom sampls. This mas that ach idividual to b sampld is chos radomly (or by chac) from th populatio. Most statistical tsts that you us assum that you hav sampld radomly. Cosidr a xampl of how oradom samplig could affct your rsults. Suppos you ar tstig th hypothsis that plats growig i acidic soils hav lowr growth rats tha thos growig i utral soils. You grow plats from sd, th trasfr th sdligs to th two soil typs. You subcosciously choos mor vigorous sdligs to trasfr to th utral soil, ad lss vigorous sdligs (prhaps thos with som hrbivor damag) to trasfr to th acidic soils. If you obsrv a diffrc i growth rats btw ths two groups, you caot b sur that it was du to soil diffrcs. If you had radomly chos which sdlig would b trasplatd to th two soil typs, you could b mor crtai that ay diffrc i growth was du to th soil. Ways to sampl radomly Radom samplig ca b difficult i fild biology. I may othr situatios, you ca assig th idividuals umbrs, th us a radom umbrs tabl to dcid which tratmt ach idividual would rciv. You somtims must b crativ i dvisig ways to sampl radomly, or at last avoid bias. Exampl: If you d to radomly sampl a laf or brach from a tr, stad with your back to th tr ad rach back, samplig th first laf or brach you happ to touch. Aothr optio is to pick a brach, th choos which laf you will sampl by lookig up a umbr i a radom umbrs tabl (clos your ys ad put your figr somplac o th pag). If th radom umbr is 4, choos th fourth laf from th brach tip. You ca xprimt with othr mthods. Th importat thig is to ot look at th lavs, ad do't choos o bcaus it has fw or may galls.

2 II. Radom Numbrs Tabl Etr th tabl at radom (.g., poit to it with a pcil). Procd horizotally or vrtically from that poit to obtai as may digits or umbrs as dd

3 III. Tsts of Idpdc usig th G-statistic: A two-way tst of idpdc is commoly usd i cology for situatios i which w wat to tst whthr two diffrt charactristics or coditios occur idpdtly of o aothr. (A altrativ, th Chi Squar tst, is mor commoly usd, but G is quit simpl to calculat ad mor closly follows a actual chi-squar distributio. A advatag of th chi squar tst is that studts must calculat xpctd frqucis, so thy s for ach cll i th tabl how closly th obsrvd valu matchs th xpctd.) Th calculatios for a -way tst of idpdc follow: 1) a = Σ (f l f) for cll frqucis ) b = Σ (f l f) for row ad colum totals 3) c = l 4) G = (a - b+c) 5) Compar G with th critical valu of Χ. I a x tabl, thr is (-1)(-1) =1 dgr of frdom. Us α = If G < critical valu, accpt H o ; if G critical valu, rjct H o. Exampl: A cologist wishs to kow whthr lavs that hav sawfly galls ar mor suscptibl to hrbivory by laf-chwig iscts tha ar lavs without galls. Th cologist rcordd prsc or absc of laf chwig damag o 50 lavs with galls ad 50 lavs without galls. Hr ar th rsults (hypothtical data): Galls o laf chwig damag o laf ys o Total ys o 8 50 Total = 100 To calculat G: 1) a = 31 l 31 + l + 19 l l 8 = 33.7 ) b = 50 l l l l 47 = ) c = 100 l 100 = ) G = ( ) = 3.4 5) I th tabl of critical valus of th chi squar distributio, th critical valu of G for 1 d.f. ad α = 0.05 is Th cologist accpts th ull hypothsis that chwig damag occurs idpdtly of gall prsc o willow lavs at this sit ad cocluds that th obsrvd diffrcs wr small ough to hav rsultd from chac.

4 IV. Tstig a Poisso Distributio For cological vts that occur rlativly rarly ad idpdtly of othr vts of th sam typ, th frqucy of vts should follow a Poisso distributio. For xampl, adult fmal sawflis may oviposit o oly a small fractio of th lavs of a willow tr. W could tst whthr sawfly galls occur idpdtly of whthr thr ar othr galls o th sam laf by comparig th frqucy distributio of galls to a Poisso distributio. H o : Th umbr of galls pr laf o willows follows a Poisso distributio. H a : th umbr of galls pr laf o willows dos ot follow a Poisso distributio. Gral tst procdur: Compar obsrvd frqucy data with xpctd frqucis (Sokal, R. R., ad F. J. Rohlf Biomtry. W.H. Frma, Sa Fracisco, sctio 5.3) Frqucy (# trials) #vts/trial obsrvd (f) 0 f 0 xpctd (f^) dviatio from xpctd (f- f^) 1 f 1 ( ) f 3 f f 4 5 f 5 6 f 6 7+ f total N=Σf =

5 N is th sampl siz (umbr of lavs sampld), (avrag umbr of galls pr laf) = total umbr of galls/total umbr of lavs. For calculatios of xpctd frqucis, s xampl data, blow. Exampl (from ral class data, 1998): From a sampl of 01 lavs, w foud 140 galls. So = Frqucy (# trials) # galls/laf obsrvd (f) (= # lavs) = 0. 7 xpctd (f^) dviatio from xpctd (f- f^) 01 = ( ) 01 = = (0.7) = = = = = = 0 + Total =Σf = 01 Not that it is bst to lump classs with xpctd frqucis lss tha 5. I this xampl, w would combi lavs with 3 or mor galls, givig th tabl o th followig pag:

6 Frqucy (# trials) # galls/laf obsrvd (f) (= # lavs) = 0. 7 xpctd (f^) dviatio from xpctd (f- f^) 01 = ( ) 01 = = (0.7) = = Total =Σf = 01 Th data do ot fit th Poisso distributio vry wll. Sic th umbr of lavs with just o gall is much smallr tha xpctd (ad th umbr with 3 or mor galls gratr), w ca coclud that fmal sawflis do ot avoid ovipositig o lavs that alrady hav 1 gall.