Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may of a certai orgaism are i a certai place, or at a certai time. Iformatio o the abudace of a orgaism, or group of orgaisms is fudametal to most questios i ecology. However, we ca rarely do a complete cesus of the orgaisms i the area of iterest because of limitatios to time or research fuds. Therefore, we usually have to estimate the abudace of orgaisms by samplig them, or coutig a subset of the populatio of iterest. For example, suppose you wated to kow how may slugs there were i the forests o Mt. Moosilauke. It would take a lifetime to cout them all, but you could estimate their abudace by coutig all the slugs i carefully chose smaller areas o the moutai. Accuracy vs. Precisio. Obviously, we would like our method for samplig the populatio to produce a good estimate. A "good" estimate should maximize both precisio ad accuracy. I everyday Eglish we ofte use these terms iterchageably, but i sciece, they have differet meaigs. Accuracy refers to how close to the true mea (µ) our estimate is. That is, if we somehow could kow the true umber of slugs residig o Mt. Moosilauke we could compare our estimate to it ad fid out how accurate we are. Obviously, we would like our estimate to be as close to the true value as possible. I additio, we would like to avoid ay bias i our estimate. A estimate would be biased if it cosistetly over- or uder-estimated the true mea. Bias may arise i may ways, but oe frequet source is by the selectio of sample plots that are oradom with respect to the abudace of the target orgaism. For example, if we looked for slugs at Moosilauke oly i suy, dry ope fields, our estimate would probably be much lower tha the true abudace. Radom samplig avoids this source of bias. A radom sample is oe where every potetial sample plot withi the study area sample has a exactly equal chace of beig chose for samplig. Radom samplig is ot the same as haphazard samplig. True radom samplig usually requires the use a radom umber table (available i some books), or a radom umber geerator (such as is cotaied i some calculators, most spreadsheets, ad some other software packages). I additio to obtaiig a accurate, ubiased sample, we are also cocered with the precisio of our estimates. Precisio refers to the repeatability of our estimates of the true sample mea. If we were to estimate slug abudace may times ad got early the same estimate each time, we would say that our estimate was very precise. Note that it is possible to have accuracy without precisio ad vice versa. Sokal ad Rohlf (1981) wrote: "a biased but sesitive scale might yield iaccurate but precise weight. By chace, a isesitive scale might result i a accurate readig, which would however be imprecise, sice a repeated measuremet would be ulikely to yield a equally accurate weight." If measuremets are ubiased, precisio will lead to accuracy. I this exercise, we are cocered maily with precisio. Overview We will use the computer to simulate ad sample populatios of two plat species, virtual beech trees (represetig Fagus gradifolia) ad virtual hobblebush (represetig Viburum alifolium). The objective is to develop a ituitio for the issues ivolved whe estimatig the size of a populatio with quadrat samplig. Quadrat samplig is based o measuremet of replicated sample uits referred to as quadrats or plots (sometimes trasects or relevés). This method is appropriate for estimatig the abudace of plats ad other orgaisms that are sufficietly sedetary that we ca usually sample plots faster tha idividuals move betwee plots. This approach allows estimatio of absolute desity (umber of idividuals per uit area withi the study site). Our challege is to idetify samplig strategies that will provide satisfactory precisio with miimum sample effort. Some of the factors that affect precisio are: 1) Measuremet error. I the real world, it is importat to cout orgaisms carefully ad lay out plots accurately for good estimates of desity. This is ot a cocer here however, because the computer will be layig out the plots ad coutig the plats. 2) Total area sampled. I geeral, the more area sampled, the more precise the estimates will be, but at the expese of additioal samplig effort. 3) Dispersio of the populatio. Whether the populatio teds to be aggregated, evely spaced, or radomly dispersed ca affect precisio. Note that the dispersio patter of the same populatio may be differet at differet spatial scales (e.g., 1 x 1 m plots vs 100 x 100 m plots). 4) Size ad shape of quadrats. The size ad shape of the plots ca affect samplig precisio. Ofte, the optimal plot size ad shape will deped o the dispersio patter of the populatio. We will explore the role of some of these factors i ifluecig estimates of absolute desity.
1. Double click o the Ecobeaker ico to ope it. The Lab 2. Load the "Samplig" situatio file by choosig Ope i the File meu. You should see four widows ope o the scree: (1) Species Grid showig brow ad gree dots. This is a aerial view of our forest, 100 m o each side. The brow dots represet beech trees ad the gree dots represet hobblebush; (2) Samplig Parameters is where we specify the type ad umber of quadrats we would like to sample. They are automatically placed radomly; (3) Total Populatio shows the umber of beech ad hobblebush preset o the grid; (4) Cotrol Pael is used to ru the simulatio ad tell the computer whe to sample. I this exercise we wo t be usig STOP, GO, or RESET. If you accidetally hit GO or RESET, thus chagig the species abudaces, you will have to reload the situatio file. 3. First, ru a sample to familiarize yourself with the procedure. Specify plot size ad umber i the Samplig Parameters widow. Use 5 x 5 m plots, ad sample =20 of them. After you eter the parameters i the widow, press Chage. Now go to the Cotrol Pael widow ad press sample. Oe by oe, a plot will appear o the scree i a radomly selected locatio, ad you will be told how may beech ad how may hobblebush were i that plot. 4. For this exercise, you will eed to calculate the mea, stadard deviatio, stadard error, ad 95% cofidece itervals for your samples (see Appedix). So before we begi, you will eed to specify how you will receive the data. You could copy the results by had as the samplig is performed, but it is easier to save the data ito a file that ca be opeed by Excel. To do so: Choose Samplig... from the Setup meu. Press the Set Save File butto to specify where you would like to save the file. You should create a separate file for each samplig ru. So each time you chage the samplig parameters, chage the file ame BEFORE you press Sample. Whe you wat to do the calculatios o this data, import the data file ito Excel as "Tab-delimited text". To speed up the samplig, tur off the dialog boxes put up durig the samplig rus. Choose 'Samplig...' from the Setup meu agai. Press the Advaced Stuff butto. Set verbosity to No Feedback. 5. Begi the data collectio by samplig beech with a small plot size, 5 x 5 m. Assume that our fudig for this study is limited, ad we ca oly sample a total of 500 m 2, so set = 20. As before, set the samplig parameters i the appropriate widow, ad press 'Chage'. Now, sample the populatio (press 'Sample') ad calculate the mea desity of the plots (see Appedix 1). To facilitate comparisos with subsequet samplig usig differet plot sizes, covert your raw data to idividuals / m 2. At this plot size (5X5m= 25m 2, you should divide each sample cout by 25; do this i Excel with a equatio i the adjoiig your raw data (see Appedix 2 for a recommeded structure for your Excel worksheet). Record the mea o the aswer sheet i the Results table, o the appropriate lie. Be sure that they are i the correct uits. This is your estimate of the true mea desity. If you sampled the populatio agai with the same plot size, how close do you thik the ext estimate would be? We ca estimate the precisio of the sample based o oly oe ru i order to fid out how variable our estimates would be if we sampled the populatio may times. Calculate the stadard deviatio (SD), stadard error (SE), ad a 95% cofidece iterval (CI) (see Appedix). Record your calculatios i the Results table. 6. You have just received a research grat, ad ca therefore icrease our total area sampled 4-fold to 2000 m 2. Cosider what will happe to our estimate of (1) the mea, (2) the SD, ad (3) the CI. The chage the umber of plots sampled () i the samplig parameters widow to 80. Record data for BOTH beech ad hobblebush. Sample, ad calculate the mea, SD, SE, ad CI for beech, as before. Eter your results i the Results table, i the appropriate space. Agai, be sure that they are i the correct uits. 7. Now cosider how dispersio would affect our estimates of the mea, ad our precisio. First, look at the patter of beech i our forest. Would you describe the dispersio as aggregated, evely spaced, or radom? Aswer at lie 1 o the aswer sheet. A simple calculatio ca give us a quatitative estimate of the degree of aggregatio i a populatio. This statistic (δ) is simply the variace divided by the mea (see Appedix). A value of 1 idicates radom dispersio, values less tha 1 idicate eve-spacig, ad values greater tha 1 idicate aggregatio. (I this case, the calculatios should be based o the raw data uits of idividuals / sample plot; see Appedix 2). Calculate this statistic for beech (5 x 5 plots, = 80) ad evaluate how it
compares with your visual estimate. Now examie the hobblebush populatio. How would you describe its dispersio patter (3 o aswer sheet)? Calculate the mea, variace, ad dispersio statistic (δ) for hobblebush (Eter the mea i the results table. Eter G o lie 4). How would you expect the precisio for samplig hobblebush to compare with that of beech? Calculate SD, SE ad CI for hobblebush (Eter them i the results table) Compare the SD, SE, ad CI for beech ad hobblebush. Which oe was more precise? Why? (lie 5). 8. Examie the effect of chagig plot size ad shape o our estimates of the mea desity ad o precisio. We ll cocer ourselves oly with hobblebush i this part of the exercise. First, test what happes whe we chage plot size keepig everythig else costat. Chage the samplig parameters so that you are usig a 20 x 20 m plot, keepig total area costat at 2000 m 2 (=5). Now sample the populatio. Be sure to covert your data to the proper uits to facilitate compariso with the earlier samplig. Calculate the mea, SD, SE, ad CI (Eter i the results table). Did the precisio icrease or decrease compared to the 5 x 5m plots (=80)? Briefly explai why (lie 6) Now chage the plot shape, keepig ad total area sampled costat. Chage the plot size to be 4 x 100. Leave at 5. Sample. Covert to idividuals / m 2. Calculate the mea, SD, SE, ad CI (Eter i the Results table). How do these compare with the values from the 20 x 20 plots? If they are differet, explai why (lie 7) OPTIONAL: Sample beech for plot sizes of 20 x 20 ad 4 x 100. Calculate mea, etc. ad compare for other samples of beech. How did your estimates of the mea ad precisio chage with plot size ad shape? How does this compare with hobblebush? Why? (lie 7 - optioal) 9. I this exercise we have a advatage because we ca kow the true desity of our populatios. Calculate the true desity (µ) of each species usig the values i the 'Total populatio' widow. Note that the total area of our forest is 10,000 m 2. Eter these values i the Results table. Which samplig scheme, of all you have tried so far, came closest to estimatig the true meas? (lie 8) Was the true mea icluded i all of the CI's? If ot, explai why. (lie 9) 10. Based o your pilot samplig of these virtual plat populatios, suggest a optimal plot size ad shape for hobblebush.
Appedix 1: a primer o the statistical descriptio of populatios Problem: How to estimate the populatio mea from a sample (e.g., How ca we estimate the mea desity of beech i a forest where we sampled = 20 quadrats that were each 5 x 5 m?) There are two importat compoets to estimatig the mea. First is the estimate itself, ad secod is the variability associated with that estimate. You are probably familiar with the first compoet; it is simply the average. 1. To fid the average desity of a sample of plots, add the umber of idividuals foud i each plot ad divide by the umber of plots. This is ow the average umber of idividuals per plot ad ca be expressed as a absolute desity (e.g., 4.2 beech / 25 m 2 ). This sample mea ( x ) represets a estimate of the true populatio mea (µ). x = x i i=1 Where: x i = each sample observatio (idividuals / plot) = sample size (umber of plots) I Excel, the mea ca be calculated as: =AVERAGE(umber1, umber2...) or =AVERAGE(firstcell referece:lastcell referece) 2. The stadard deviatio is oe measure of variability amog the populatio of plots i the study area. The sample stadard deviatio (SD) represets a estimate of the true populatio stadard deviatio (σ). SD the average amout by which a sample differs from the sample mea. (I a ormal distributio, 68% of the populatio lies withi ± 1 SD of the mea, ad 96% lies withi ± 2 SD's of the mea). Thus the SD is a measure of the variability i the populatio. SD = i=1 ( x i x ) 2 1 where: x i = each sample observatio x = sample mea = sample size I Excel, the stadard deviatio ca be calculated as: =STDEV(umber1, umber2...) or =STDEV(firstcell referece:lastcell referece) 3. The stadard error (SE)is a measure of the repeatability of the populatio estimate ( x ). SE the average amout by which a sample mea ( x ) differs from the true populatio mea (µ). (Give a ormal distributio ad radom, idepedet samples, RIWKHpossible sample meas lie withi ± 1 SE of the true, ad OLHVZLWKLQ SE's of the mea). Thus the SE is a measure of the variability that could be expected of repeated samples from a populatio. The SE is a estimate of the SD aroud the x 's that would be obtaied from repeated samples of size from the study populatio. It estimates the precisio of a samplig scheme.
SE = SD where: SD = sample stadard deviatio = sample size I Excel, calculate SE by referecig the cell holdig the SD ad divide by the square root of the sample size: e.g., = C6/(^0.5), where C6 is the cell holdig the SD, ad = sample size. 4. The precisio of a estimate of the sample mea ca be expressed i terms of a cofidece iterval (CI). A cofidece iterval represets the rage of values that ca be assumed to cotai the true mea with a certai (specified) probability. For example, suppose we wated to fid out the rage of values that would cotai the true mea with a probability of 0.95. I other words, if we were to take 100 radom samples of our populatio, 95 of the calculated CI s based o those samples would cotai the true populatio mea. This would be called a 95% cofidece iterval. Other values, (e.g. 99%, 99.9%) may be used, but 95% is most commo. The higher the percetage, the wider the cofidece iterval, thus it becomes a trade-off betwee a rage arrow eough to be meaigful, ad precise eough to be useful. The cofidece iterval is expressed as a lower ad upper limit (L 1 ad L 2, respectively), ad is calculated as follows: L 1 = x t α[ 1] SE ad L 2 = x + t α[ 1] SE where L 1 = lower cofidece limit L 2 = upper cofidece limit x = sample mea t = the t statistic, from statistical table or Excel α = the probability associated with the critical t - value. α = 1-P where P is the desired cofidece of the iterval (e.g. for a 95% CI, α = 0. 05) = the sample size SE = stadard error To calculate cofidece limits i Excel, be sure to take advatage of previously calculated values for the mea ad the stadard deviatio. Note that Excel uses the SD i its calculatio, ad NOT the SE. To calculate a CI i Excel, the lower limit is =(mea) - TINV(alpha, -1) Â6(. The upper limit is idetical, except the TINV( )Â6( term is added to the mea, ot subtracted. To calculate these limits, substitute cell umbers or calculated values for the variables i italics. 3. A simple statistic for describig the spatial dispersio of a populatio is simply the variace (= SD 2) ) divided by the mea. It is deoted here as δ, delta. δ = SD2 x Values of δ greater tha 1 idicate aggregatio, while values less tha 1 idicate a uiform dispersio. A value of 1 idicates radom dispersio.
Appedix 2: Example spreadsheet Samplig 20 quadrats, each 5 by 5 at time step 365 Idividuals / sample plot Idividuals / m2 X_pos Y_pos Hobblebush Beech Hobblebush Beech 52 78 0 1 0 0.04 62 39 0 1 0 0.04 87 94 0 0 0 0 93 8 0 0 0 0 48 74 0 0 0 0 34 53 4 0 0.16 0 74 11 0 0 0 0 13 30 0 2 0 0.08 74 81 0 0 0 0 59 79 0 2 0 0.08 48 74 0 0 0 0 34 53 4 0 0.16 0 74 11 0 0 0 0 13 30 0 1 0 0.04 72 78 0 0 0 0 91 13 0 1 0 0.04 6 19 1 0 0.04 0 26 31 0 1 0 0.04 93 52 0 0 0 0 94 77 0 1 0 0.04 Mea 0.450 0.500 0.0180 0.0200 SD 1.234 0.688 0.0494 0.0275 N 20 20 SE 0.0110 0.0062 CI(lower) 0.0101 0.0112 CI(upper) 0.0396 0.0321 δ 3.39 0.95