ANALYZING ECOLOGICAL DATA

Transcription

1 Geeral Ecology (BIO 60) Aalyzig Ecological Data Sacrameto State ANALYZING ECOLOGICAL DATA Let Start With a Eample Whe coductig ecological eperimet, we would like to kow whether a eperimetal treatmet had a effect o ome variable. A a imple but itructive eample, uppoe we wat to kow whether a ew formulatio of fertilizer icreae plat growth over that of the old fertilizer formula. To tet thi, we might meaure the growth repoe (let ay height) of two et of plat, each of which i grow o oe of the two fertilizer. Let imagie that we grow 0 plat o the old fertilizer ad 0 plat o the ew fertilizer; the height of each idividual plat ad the mea for each fertilizer are give i the table below. Plat Height (m) Old Fertilizer New Fertilizer Mea height:.40 Mea height:.9 A you ca ee, the calculated mea height of plat grow o the ew fertilizer wa greater tha that of plat grow o the old fertilizer. But before we jump to the cocluio that the ew fertilizer i better tha the old, let take a cloer look at the raw data. You ll otice that there quite a bit of variatio i height amog idividual plat grow o the old fertilizer ad alo amog thoe grow o the ew fertilizer. For eample, oe of the plat grow o the old fertilizer grew to.0 m. Therefore, thi plat actually grew taller tha the mea height of plat grow o the ew fertilizer; ideed, it alo grew taller tha may of the other plat grow o the ew fertilizer. So how ca we claim that the ew fertilizer i i fact better if ometime it i ad ometime it ot? Of coure, may of the plat grow o the ew fertilizer were taller tha thoe grow o the old fertilizer, but ot all. Likewie, may of the plat grow o the old fertilizer were horter tha thoe grow o the ew fertilizer, but ot all. To look at thee data aother way, Figure how the plat height above a poit o a graph. Plat height (m) 4 Fig.. Idividual height (m) meauremet of plat grow i old or ew fertilizer. Data are quite variable (lot of pread amog the poit). 0 Old New The dotted lie paig through each et of data i the figure idicate the mea plat height for each et of data. Plotted i thi way, it i eay to ee that the poit (i.e., plat height) overlap betwee the old ad ew fertilizer ad that thi i due to radom variatio i plat height. Coequetly, though the calculated mea of plat height do differ betwee old ad ew fertilizer, our cocluio about whether mea plat height differ betwee the two fertilizer deped o how much variability there i

2 Geeral Ecology (BIO 60) Aalyzig Ecological Data Sacrameto State i the data. The more variable our data, the le cofidet we will be that the mea are i fact differet. To make thi poit, let look at two ew et of plat height data that have the ame mea a i Figure but are much le variable. A i Figure, Figure how the height of idividual plat grow o the old ad ew fertilizer but thi time the data are ot early a variable a i Figure. Plat height (m) 4 Fig.. Idividual height (m) meauremet of plat grow i old or ew fertilizer. Data have the ame mea but are le variable tha i Figure (i.e., little pread amog the poit). 0 Old New If you had a choice betwee uig the data i Figure or Figure to determie whether the old or ew fertilizer differed i their effect o plat height growth, which data would you feel mot cofidet about? Give that the data i Figure i much le variable tha the data i Figure, our mea i Figure are actually more precie tha thoe i Figure. A a reult, we are more cofidet that the mea i Figure differ from oe aother tha we are that the mea i Figure differ. Thu, the variability of our data i what i truly critical whe makig cocluio about whether or ot real differece actually eit. A cietit who are taked with beig objective whe makig uch cocluio, thi i where we mut tur to tatitical approache. A it tur out, the mea i Figure do ot differ tatitically from oe aother whe they are compared uig a objective tatitical tet, wherea the mea i Figure do. If you were to coclude that imply becaue the calculated mea were differet i the Figure data, the you would have made a icorrect cocluio. Statitic i a approach that miimize the rik of makig thi type of mitake. Sigificace Tet If we wih to kow whether two mea are tatitically differet from oe aother, we mut coduct a igificace tet. A igificace tet i a tatitical tet that tet the ull hypothei. The actual ull hypothei that i ued i differet depedig o the particular tatitical tet, but i geeral a ull hypothei i oe i which there i o differece betwee or effect of ome kid of eperimetal treatmet. Regardle of the type of igificace tet ued, a P- value i alway give. A P- value i the probability (hece, the letter P) that ay differece betwee treatmet or eperimetal effect could have arie by chace aloe. What the reultig P- value tell u i eetially the probability that our ull hypothei i true. A probability i epreed a a decimal value; for eample, P = 0. i equivalet to 0% probability ad P = 0.0 equal a % probability. A low P- value (P < 0.0) idicate that there i a very low probability that the ull hypothei (i.e., o effect) i true; i other word, that there wa a tatitically igificat effect or differece i your eperimet. By cotrat, a high P- value (P 0.0) idicate that there i a high probability that the ull hypothei i true; i other word, there wa o tatitically igificat effect or differece i your eperimet. Whe coductig igificace tet, we mut decide at what P- value we accept or reject our ull hypothei. Typically, cietit chooe a % igificace (α) level (that i: P < 0.0) a the threhold for what cotitute tatitical igificace. For mot purpoe, thi balace the rik of makig a error i

3 Geeral Ecology (BIO 60) Aalyzig Ecological Data Sacrameto State your cocluio, either by rejectig a true ull hypothei or acceptig a fale ull hypothei. The ame of thee error are Type I ad Type II error, repectively. If the threhold P- value i et too high, the you are more likely to reject a true ull hypothei, thereby icreaig the rik of coductig a Type I error. Coductig a Type I error reult i cocludig that there i a igificat differece whe oe eit. However, if the threhold P- value i et too low, the you are more likely to accept a fale ull hypothei, thereby icreaig the rik of coductig a Type II error. Coductig a Type II error reult i the cocluio that there i o igificat differece whe a differece actually eit. Thu, the choice of igificace level will affect the relative rik of coductig thee two type of error. Clearly, coductig a Type I error i a more eriou problem i cietific reearch. Hece, igificace level are rarely et higher tha 0.0. The table below illutrate the poible outcome of cocluio that ca be made from a tatitical tet: Cocluio Null Hypothei Accept Reject True Correct deciio Type I error Fale Type II error Correct deciio There are two tet that are commoly ued to tet the differece amog ample. The firt i the t- tet. A t- tet compare (ad oly ) ample mea. The ecod i Aalyi of Variace (ANOVA). A ANOVA tet the differece betwee more tha mea, but it doe o by comparig ample variace. The t- tet The t- tet i a tatitical tet that allow u to compare two ample mea. It i called a t- tet becaue we calculate a tet tatitic called a t- value. The t- value i calculated baed o the differece betwee the two mea ad the variatio i the data. If the differece betwee two mea i large, the it i likely that the two mea are differet. However, a decribed i the fertilizer eample above, we mut alo coider variability of the data. If variatio i the data i low, the it i more likely that ay differece i the mea will be detected i the tet. The ize (or magitude) of the t- value i idicative of how differet our ample mea are. A large t- value idicate that the ample mea are igificatly differet, wherea a mall t- value idicate o igificat differece betwee the mea. For eample, we might wat to compare the deity of a marh gra, called Spartia, betwee two differet marhe. The t- tet provide a ubiaed way of decidig whether ay oberved differece i the mea deity of Spartia betwee the two marhe i real or imply due to chace. A with all tatitical tet, the t- tet tet the ull hypothei. I thi cae, the t- tet tet the hypothei that there i o igificat differece betwee the mea; i other word, that the mea are equal ( = ). The tet relie o a ditributio of t- value called the t- ditributio, which i a ditributio that give the probability of gettig a particular t- value. Becaue the t- value reflect the magitude of the differece betwee the mea, t = 0 if the mea are idetical (a rare occurrece). Sice the t- ditributio i baed o the ull hypothei (i.e., that the mea do ot differ), it ha a mea

4 Geeral Ecology (BIO 60) Aalyzig Ecological Data Sacrameto State of zero. The greater the differece betwee the mea (aumig low variace), the higher the t- value will be. The higher the t- value, the le probable it i that the mea are the ame. The t- ditributio wa developed to take accout of the fact that ample ize () i mall i mot practical applicatio ad, therefore, require a differet ditributio tha the ormal ditributio. Ideed, the t- ditributio i imply a modified form of the ormal ditributio, which if you remember ha certai propertie that allow u to aig probabilitie of occurrece for our data. For eample, data poit that fall farther away from the ceter (i.e., the mea) of a ormal curve are le likely (le probable) to occur tha thoe that lad cloer to the ceter of the curve. Thi alo applie to the t- ditributio. The t- tet relie o thi property whe it compare two mea. To ay that two mea are tatitically differet, we ue the % igificace level (P < 0.0). That i, the differece betwee the mea mut be great eough uch that it i improbable that we would get uch a large differece betwee the mea if, i fact, they were the ame. Said aother way, uig the % igificace level, there i a % chace that we would get a t- value outide of the % level if the mea were the ame. The t- ditributio ad it % (.%) probability regio are how below:.% probability.% probability The aumptio of the tet are: ) The data are ditributed ormally that i, the frequecy ditributio of the data form a ormal (bell- haped) curve; ad ) The variace of the two ample beig compared are approimately equal. I a t- tet, the t- value i calculated baed o the differece i the ample mea ( ad ) ad the tadard error of the differece betwee ample mea ( ): It tur out that: t = 0 t-ditributio = + Thi i becaue there i a mathematical relatiohip that equate the tadard error of the differece i the mea with the um of the tadard error of the two variable. Subtitutig thi for i the equatio for t, we get: t = + 4

5 Geeral Ecology (BIO 60) Aalyzig Ecological Data Sacrameto State where the i value are the variace of the idividual variable. Thi equatio ca be rearraged a follow: t = + where equal the pooled tadard deviatio of both ample: = i= i i= ( i ) + i i= + The calculated t- value will lie omewhere alog the t- ditributio ad therefore will idicate how likely it would be for that value of t to occur by chace. If t i large (toward the tail of the curve), the probability of the mea beig the ame i low. By covetio, we et the cut- off poit for the mea to be igificatly differet at a probability of le tha % (0.0) o the curve; coequetly, we ay that there i le tha a % probability that the mea are differet due to chace aloe. Said aother way, there i greater tha a 9% chace that the mea are actually differet from oe aother. By cotrat, if t i mall (toward the ceter or mea of the curve), the probability of the mea beig the ame i high. We ue the ame cut- off but talk about it i a differet way. Whe t i mall, we would ay that there i a 9% probability that our mea do ot differ by chace. The value of t above which the mea are coidered igificatly differet i called the critical t- value ad it ca be looked up i a t- table (called Critical value of the t ditributio ; Appedi D). Thi critical t value i determied by the igificace level (α) (i thi cae % or 0.0) ad the degree of freedom df = ( + - ). A eample of a t- tet. For the Spartia eample above, let ay that we collect plat cout from five m plot i each of the two marhe. We get the followig data: i= ( ) Plot # Marh A Marh B i

6 Geeral Ecology (BIO 60) Aalyzig Ecological Data Sacrameto State t = Uig our calculatio equatio for the pooled tadard deviatio (): + = ( 99.6) ( 66.) Therefore: = ad which give, = t = = ( 0.4) = 8. The critical t- value at the 0.0 igificace level with 8 degree of freedom (df) i.06 (i.e., t (0.0, 8) =.06). Becaue our calculated t- value (t = 8.) i higher tha the critical t- value at the 0.0 igificace level, the probability (P) that the mea are the ame i le tha % (P < 0.0). Thu, we coclude that the mea of marh A ad B are igificatly differet from oe aother. Whe evaluatig tatitical igificace, you mut alway report the igificace level ued ad the P- value i your reult. Aalyi of Variace (ANOVA) Aalyi of variace (ANOVA) i a powerful tatitical techique that i ofte ued i the aalyi of reult from more comple eperimet. The advatage of uig thi method i that more tha two et of data ca be eamied at the ame time. There are may differet type of ANOVA, but they all ue the ame baic method of comparig the variace betwee differet treatmet with the variace withi treatmet. The implet type of ANOVA i a completely radomized deig. Thi i whe the treatmet are radomly applied to ay particular eperimetal uit. That i, each plot or eperimetal uit ha a equal chace of havig ay of the treatmet applied to it. The geeral priciple for three treatmet i decribed below. Withi each treatmet there are i eperimetal uit from which meauremet were take. Although there ca be ay umber of treatmet, it i preferable (i.e., reult i fewer headache!) to keep the umber of treatmet low for eae of iterpretatio. Treatmet 4 7 Treatmet 6 9 Treatmet

7 Geeral Ecology (BIO 60) Aalyzig Ecological Data Sacrameto State Sice the ANOVA i a variace method, the compario of thee three treatmet relie o comparig the amout of variace betwee each treatmet compared to the variace withi each treatmet. The idea here i that there i goig to be ome variability i the eperimet. But, for the treatmet to differ, there mut be ubtatially more variability amog the treatmet tha withi them. To ee how thi work, let look at a eample imilar to that above uig three Spartia marhe. The data are below: Plot # Marh A Marh B Marh C Sum ( ) ( ) ( ) SS. SS 8. SS To do thi, we calculate what i called the total mea quare. Thi repreet the variace of all the data collectively. The total mea quare i the partitioed (divided up) ito it compoet part: Total variace = Total um of quare Total degree of freedom The total um of quare i partitioed ito it compoet part a follow: Total SS = Betwee treatmet SS + Withi treatmet SS k ( ) k g = ( ) k i g + ( i ) Or 7

8 Geeral Ecology (BIO 60) Aalyzig Ecological Data k k = k k k + k k Sacrameto State Eterig the value i the: ( 4.) = ( 4.) = Now we mut calculate the total degree of freedom: Total v = Betwee v + Withi v k = k + k( ) = + ( ) 4 = + We ummarize the reult or our ANOVA i the followig tadard table: Source v SS MS Total Betwee (Treatmet) Withi (Error) where MS tad for the mea quare. To calculate the mea quare, imply divide the SS value by their correpodig degree of freedom (v). No total MS value i calculated becaue we do ot ue it i determiig whether our treatmet actually differed from oe aother. To evaluate whether our treatmet differ from oe aother, we compare the Betwee (Treatmet) variatio to the Withi (Error) variatio. If our treatmet variatio i igificatly greater tha our error variatio, our treatmet differ. The tatitic we calculate to tell u thi i a F tatitic. The F tatitic i aalogou to the t tatitic we ued i our t- tet. Oce we calculate our F tatitic, we compare it to a critical F value that i determied by the Treatmet ad Error degree of freedom ad our igificace level (α), which i ormally 0.0. I the tet above, we calculate our F value a follow: F (0.0,, ) = Treatmet MS = 9.9 = Error MS.08 F (0.0,, ) = 6. 8

9 Geeral Ecology (BIO 60) Aalyzig Ecological Data Sacrameto State To determie the critical F value, we ue the Error degree of freedom. Lookig up our F value i a table uig 0.0 a our igificace level, Treatmet v = ad Error v =, our critical F value i.0. Sice i much higher tha our critical F value, our treatmet ca be aid to differ igificatly. Note: Eve if our calculated F value wa., we would till ay that our treatmet differ igificatly. By the ame toke, if our calculated F value wa.09, the we would have to ay that our treatmet did ot differ igificatly. A fial ote i that ormally we do ot had calculate our F value or geerate our critical F value i uch a way. That what computer are for. Whe a ANOVA i ru o a computer, you will get all thi ifo, plu a P value (a eact igificace level of the tet). I thi tet, our P value would be P < That i, it i highly igificat! Thi tet aloe, however, doe ot determie which of our treatmet differ from each other. Aother tet called Tukey tet will allow u to ditiguih betwee each of the idividual treatmet with repect to the other. We will dicu ad ue thi tet later. 9