Treatment Spring Late Summer Fall Mean = 1.33 Mean = 4.88 Mean = 3.

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1 The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only two groups t time. Scientists often find tht they need to compre the mens of three or more groups. The sttisticl hypothesis test used to compre the mens of three or more groups is the nlysis of vrince (ANOVA). ANOVA: n exmple ANOVA clcultions re tedious enough tht they re rrely done by hnd, but there is no better wy to relly understnd this prticulr sttisticl test. As n exmple, consider gin the issue of the role of controlled fire in pririe restortions. One prticulrly contentious issue mong restortion ecologists is the timing of pririe burns. Although nturl fires my primrily hve been sprked by lte-summer lightening strikes, most controlled burns re done during the spring or fll. The timing of burning my strongly influence the outcome of pririe restortions becuse burns done t different times of yer cn fvor drmticlly different plnt species. As scientists, you cn collect dt to help resolve such issues. For exmple, you could collect dt to nswer the following question: How does the timing of controlled burns influence the biomss of desirble pririe plnt species? An exmple of the dt you might collect is in Tble 2, below. Tble 2. Totl biomss (g) of Rudbecki hirt (Blck-eyed Susn) growing in ech of 15, 0.5 m 2 plots. One third of the plots were burned in the spring, lte summer, nd fll of 1998, respectively. All plots were smpled during summer Tretment Spring Lte Summer Fll Men = 1.33 Men = 4.88 Men = 3.52 Before we go ny further, we need to stte null nd lterntive hypotheses. For exmple, the null hypothesis could be H 0 : The timing of controlled burning does not influence biomss of Rudbecki hirt. While the lterntive hypothesis could red s follows: H A : The timing of controlled burns influences the biomss of Rudbecki hirt. ANOVA: clcultions To nlyze these dt using n ANOVA, you first need to clculte the grnd men. The grnd men is simply fncy term for the men of ll of your observtions, regrdless of group. The grnd men cn be clculted using the following formul: 46

2 ! 1 n "" n Where is the number of groups, n is the number of observtions within ech group, nd is single observtion. Applying this formul to our pririe burn dt, where = 3 nd n = 5, gives the following grnd men:! 1 $ 0.10 # 0.61#1.91# 2.99 #1.06 # 5.56 #...# 3.53 # 3.85 # 3.01# 2.13# 2.50 # 6.10%! Once you hve clculted the grnd men, you need to clculte the sum of squres mong groups (SS mong ). The SS mong is n estimte of the vrition mong your groups or, more precisely, n estimte of the devition of the group mens from the grnd men. The SS mong cn be clculted using the following formul: SS mong! n " $ & % 2 Where is the number of groups, n is the number of observtions within ech group, nd is the men of ech group. Applying this formul to our pririe burn dt, where = 3 nd n = 5, gives the following SS mong : SS mong! 5$ 3.52& 3.24% 2 # $ 1.33 & 3.24% 2 # $ 4.88 & 3.24% 2 %! The lst thing you need to clculte is the sum of squres within groups (SS within ). The SS within is n estimte of the vrition mong observtions within groups, or, more precisely, n estimte of the devition of the observtions within ech group from the group men. The SS within cn be clculted using the following formul: SS within! "" n $ &% 2 Where is the number of groups, n is the number of observtions within ech group, is n individul observtion, nd is the men of ech group. Applying this formul to our pririe burn dt, where = 3 nd n = 5, gives the following SS within : SS within! $ 0.10 &1.33% 2 # $ 0.61&1.33% 2 #...# (2.50 & 3.52) 2 # $ 6.10 & 3.52% 2! ANOVA: the F-vlue The test sttistic for n ANOVA is clled n F-vlue. The F-vlue for n AVOVA is clculted using the 47

3 following formul: F! '( SS mong *( )( &1 +( '( SS within *( )( (n &1) +( Where is the number of groups nd n is the number of observtions within ech group. Applying this formul to our pririe burn dt, where = 3 nd n = 5, gives the following F-vlue: F! $ % $ %! If you look t the formul for the F-vlue, you cn see tht it is essentilly the rtio of the vrition mong groups to the vrition within groups. If the vrition mong groups is reltively lrge compred to the vrition within groups, then the F-vlue will be reltively lrge. A reltively lrge F-vlue suggests tht the vrition mong groups is lrgely cused by given vrible or experimentl mnipultion (in our exmple, the timing of burning), rther thn chnce vrition. If the vrition mong groups is similr to the vrition within groups, the F-vlue will be reltively smll. A reltively smll F- vlue suggests tht the difference mong groups is lrgely due to chnce nturl vrition nd mesurement error, rther thn to given vrible or mnipultion. This interprettion of n F-vlue should sound fmilir to you, becuse it is very similr to the interprettion of t-sttistic discussed bove. ANOVA: EXCEL output For dt set of ny size, n ANOVA is extremely tedious to clculte by hnd. As such, you will be using EXCEL to do ANOVAs (see below). The EXCEL ANOVA output for our pririe burn study is inserted below: ANOVA Source of SS Df MS F P-vlue F crit Vrition Between Groups Within Groups Totl By custom, the results of n ANOVA re expressed in the bove formt, which not surprisingly is clled n ANOVA tble. As you cn see, this tble contins the (now fmilir to you) estimtes of the SS mong, SS within, nd F-vlue (the vlues in this tble re not identicl to the vlues clculted erlier in this hndout becuse of rounding error in the hnd clcultions). It lso contins the ssocited P-vlue. To review, P-vlue rnges from 0 to 1, nd is the probbility of clculting given test sttistic ssuming tht the mens of your groups re identicl. The lrger the test sttistic is, the lower the chnce (P) tht tht n observed difference mong groups is due to chnce environmentl vrition, nd the greter the chnce tht difference hs biologicl custion. With smple size of 15 observtions, the probbility tht the differences we see mong spring, lte summer, nd fll burns re due to chnce lone is This vlue is bolded in the EXCEL output. Tht is, the p-vlue equls We cn therefore reject the null hypothesis tht burning seson hs no effect on R. hirt biomss, nd ccept the lterntive 48

4 hypothesis tht it burning seson does ffect R. hirt biomss. If the p-vlue for our F-vlue hd been >0.05, then we would hve ccepted tht null hypothesis. Note tht the ANOVA does not tell you nything bout pirwise differences between groups, only if there re overll differences mong ll groups. There re procedures tht cn be done fter n ANOVA tht test for pirwise differences between groups, but they re beyond the scope of this hndbook. How to do n ANOVA in EXCEL 1. Enter your dt so tht ech group is in seprte column. Lbel your columns ppropritely. For exmple, in our pririe burn exmple, the columns could be lbeled "spring", "lte summer", nd "fll". How to do n ANOVA in Minitb 1. Enter your dt so tht one lbeled column contins the grouping vrible (e.g., for the pririe burn exmple, the entries in this column could be spring, summer, nd fll ) nd second lbeled column contins the dependent vrible. 2. Click on Tools > Dt nlysis > Anov: single fctor. If the Dt nlysis module is not vilble in the Tools menu, click on Add-ins nd instll the Anlysis ToolPk. 2. Select Stt > ANOVA > generl liner model. 3. In the dilog window, plce the dependent vrible in the response box. The dependent vrible is wht you mesure. In this cse biomss is the dependent vrible. Plce the grouping vrible in the model box. The grouping vrible is the sme s the independent vrible or tretment, so in this exmple it would be seson. 3. Input the rnge of your dt by highlighting ll of your dt (including lbels). Click Lbels, then click OK. The ANOVA output should pper on the screen 4. Click OK to view the ANOVA output. 49

5 Exmple problems Crry out ANOVAs for the following dt sets: 1. Three different methods were used to mesure dissolved oxygen (mg/kg) content of lke wter. (Dt from JH Zr's Biosttisticl Anlysis, 4 th ed.) Method Number of eggs lid per femle per dy for femles from ech of three lines of Drosophili melnogster. The RS nd SS lines were selected for resistnce nd susceptibility to DDT. The NS line is the nonselected control. (Dt from RR Sokl nd FJ Rohlf's Biometry, 3 rd ed.) Line RS SS NS Strontium concentrtions (mg/ml) in three different bodies of wter. (Dt from JH Zr's Biosttisticl Anlysis, 4 th ed.) Gryson's Pond Bever Lke Angler's Cove