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.
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. 16363 7995 4115 6659 1860 51577 7169 1668 6734 83419 3940 0741 6611 30863 34081 04607 7550 04707 68653 43338 87116 46480 36041 74787 96575 5386 41704 15194 5679 49971 4649 70175 95444 76761 35816 55837 5166 96367 47481 1886 4563 93595 10650 1053 3874 78957 9886 86154 9108 6511 1180 46755 3955 9634 79516 6343 10341 946 39179 6900 7709 8594 5839 87636 3944 8048 74996 19481 66351 06377 9784 37041 91136 8588 4857 5199 10838 73647 61668 7778 37789 53166 58454 41833 71188 6689 0481 706 70331 17899 51875 7753 86078 134 9391 038 8676 71131 5968 80300 57614 9357 55946 89789 1838 55 70653 37393 83339 77436 38773 50081 99 3649 1546 31504 5341 74610 190 91666 4947 4463 4554 58705 358 0055 9473 96090 70086 3184 4361 19557 8186 4164 9387 39381 99737 07 47305 58768 7531 46704 14361 6560 3884 4983 67787 94116 5161 38414 84319 50978 01803 19969 56564 88407 79466 4667 4145 91616 6537 86449 4668 54459 9583 1610 1366 570 11775 5778 1965 0699 01513 8836 98785 7116 61859 53933 31861 3445 6648 79998 13670 6490 1531 54748 9354 94888 73377 74669 69746 84505 80039 39369 59470 6716 4646 59 78911 19953 93147 1379 1761 35869 99366 4313 90418 61337 1460 56876 7499 677 91075 91137 11354 64755 33913 9611 4338 9594 0116 0949 66586 74661 94809 1476 78894 75333 4419 59181 4333 3314 37576 47110 07 7384 79877 46516 75488 873 04478 8499 68978 693 9885 7677 94407 79 81553 4619 11806 019 3881 64814 475 86915 49195 8851 71306 8407 94648 17014 38834 9108 97443 01973 61484 6615 41741 99773 1110 15931 0910 33874 84738 08786 85704 9851 9175 76784 9431 1884 817 35496 66678 75891 76086 97310 4480 07655 40183 556 095 84493 6368 774 87341 57773 1578 6833 64858 69010 31483 37740 46141 981 53313 6641 33150 50417 43686 87116 96176 99746 95971 496 8590 3537 1789 67766 189 14555 416 15937 8438 3787 68455 18433 94419 79448 7174 5031 09557 7967 78776 51468 8608 9116 1344 4605 36451 756 9497 5871 1333 6943 30157 38766 88959 4817 419 1986 59749 96376 54314 34511 11377 15936 14015 79795 71608 45585 48448 86156 44586 3414 681 95516 51596 6139 86385 13587 54399 7148 3594 93594 5565 755 8033 9176 35457 71737 639 51060 59755 7465 8637 39716 4508 36755 7577 74799 1169 31143 754 78596 17961 16698 6913 93388 89813 581 18803 39039 9764 4619 97375 38838 1698 96671 5799 16061 69740 56814 6334 35363 54903 90979 6079 83056 44946 1769 1764 68 6350 4863 5366 7367 7757 1694 4410 7388
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 = 0.05. 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 31 19 50 o 8 50 Total 53 47 = 100 To calculat G: 1) a = 31 l 31 + l + 19 l 19 + 8 l 8 = 33.7 ) b = 50 l 50 + 50 l 50 + 53 l 53 +47 l 47 = 78.6 3) c = 100 l 100 = 460.5 4) G = (33.7-78.6 +460.5) = 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 3.841. 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.
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. 1981. 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 3 6 3 4 f 4 5 f 5 6 f 6 7+ f 7 4 4 5 10 6 70 7 5040 total N=Σf =
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 = 0.70. Frqucy (# trials) # galls/laf obsrvd (f) (= # lavs) 0 145 = 0. 7 xpctd (f^) dviatio from xpctd (f- f^) 01 = 99.8 + 1 ( ) 01 = 69.9 - = (0.7) 0.7 1 3 8 4 7 5 4 6 0 7+ 3 = 4.5-3 6 = 5.7 + 4 4 = 1.0 + 5 10 = 0.1 + 6 70 = 0 7 5040 = 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:
Frqucy (# trials) # galls/laf obsrvd (f) (= # lavs) 0 145 = 0. 7 xpctd (f^) dviatio from xpctd (f- f^) 01 = 99.8 + 1 ( ) 01 = 69.9 - = (0.7) 0.7 1 3+ = 4.5-3 6 = 5.7 + 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.