PART THREE SAMPLING AND EXPERIMENTAL DESIGN

Transcription

1 PART THREE SAMPLING AND EXPERIMENTAL DESIGN Ecologists are more ad more cofroted with the eed to do their work i the most efficiet maer. To achieve these desirable practical goals a ecologist must lear somethig about samplig theory ad experimetal desig. These two subjects are well covered i may statistical books such as Cochra (1977), Cox (1958), ad Mead (1988), but the termiology is ufortuately foreig to ecologists ad some traslatio is eeded. I the ext four chapters I discuss samplig ad experimetal desig from a ecological viewpoit. I emphasize methods ad desigs that seem particularly eeded i ecological research. All of this should be viewed as a prelimiary discussio that will direct studets to more comprehesive texts i samplig ad experimetal desig. Samplig ad experimetal desig are statistical jargo for the three most obvious questios that ca occur to a field ecologist: where should I take my samples, how should I collect the data i space ad time, ad how may samples should I try to take? Over the past 90 years statisticias have provided a great deal of practical advice o how to aswer these questios ad - for those that pla ahead - how ot to aswer these questios. Let us begi with the simplest questio of the three: how may samples should I take?

2 CHAPTER 7 SAMPLE SIZE DETERMINATION AND STATISTICAL POWER (Versio 4, 14 March 013) Page 7.1 SAMPLE SIZE FOR CONTINUOUS VARIABLES Meas From A Normal Distributio Compariso of Two Meas Variaces From A Normal Distributio DISCRETE VARIABLES Proportios ad Percetages Couts From A Poisso Distributio Couts From A Negative Biomial Distributio Couts From A Logormal Distributio SPECIALIZED ECOLOGICAL VARIABLES Mark-ad-Recapture Estimates Lie Trasect Estimates Distace Methods Chage-I-Ratio Methods STATISTICAL POWER ANALYSIS Estimatio of Effect Size for Cotiuous Variables Effect Size for Categorical Variables Power Aalysis Calculatios WHAT TO DO IF NOTHING WORKS SUMMARY SELECTED REFERENCES A recurret practical problem i ecological research projects is how much work do I eed to do? The statisticia traslates this questio ito the more easily aswered questio - how large a sample should I take? I spite of the close

3 Chapter 7 Page 77 appearace of these two questios, there is a great gap betwee them. The detailed objectives of the research program eed to be specified for a complete aswer to the first questio, ad a statisticia is ot very helpful at this stage. For example, you may wish to estimate the desity of rig-ecked pheasats o your study area. Give this overall objective, you must specify much more ecological detail before you see your statisticia. You must decide, for example, how ofte i each year is estimatio required, the type of samplig gear you will use, ad whether your study area cotais a closed populatio or is subject to extesive immigratio ad emigratio. All of this iformatio ad more must be used to help decide o the method of populatio estimatio you should apply. Do ot expect a statisticia to help you with these difficult decisios - go back ad read the previous three chapters! But the, whe you have decided, for example, to use the Peterse method of mark-recapture estimatio, it is useful to ask a statisticia how large a sample should I take? Throughout the last three chapters we have periodically come across this questio ad tried to provide a partial aswer. This chapter will draw together a series of geeral methods that oe ca apply to ay situatio, ad will guide you toward more complex methods whe ecessary. There is a excellet review of the sample size problem i populatio ecology by Eberhardt (1978) ad geeral surveys by Kraemer ad Thiema (1987), Mace (1964) ad Cochra (1977). We will start here with the simplest questio of statistical estimatio, ad determie how to determie the sample size ecessary to obtai a specified level of precisio. For example, we might ask how may fish we eed to weigh to obtai a mea weight with 95% cofidece limits of.5 grams. From this simple questio we will move later i the chapter to discuss the more complex questios of the sample sizes required for statistical iferece i which, for example, we wish to compare the size of fish i two or more differet lakes. To test hypotheses about statistical populatios we must clearly uderstad the cocept of statistical power. 7.1 SAMPLE SIZE FOR CONTINUOUS VARIABLES We will begi by cosiderig variables measured o the iterval or ratio scale which ca show a ormal distributio. How ca we decide how big a sample to take?

4 Chapter 7 Page Meas From A Normal Distributio If you eed to measure a sample to estimate the average legth of whitefish i a particular lake, you ca begi with a very simple statistical assumptio that legths will have a ormal, bell-shaped frequecy distributio. The two-sided cofidece iterval for a ormal variable is: x t s x t s Probability x x 1- where: t Studet's t value for - 1 degrees of freedom for / Variace of mea sx Stadard error of mea Sample size Sample size This cofidece iterval formulatio is usually writte i statistical shorthad as: x t s x The width of the cofidece iterval depeds o the t-value ad the stadard error. To proceed further some iformatio is required, ad sice the steps ivolved are geeral, I will describe them i some detail. Three geeral steps must be take to estimate sample size: Step 1. Decide what level of precisio you require. Do you wat your estimate of the mea to be accurate withi ±10% of the true populatio mea? Withi ±1% of the true mea? You must decide what the desired error limits will be for your sample. This is ot a statistical questio but a ecological oe, ad it will deped o what theory you are testig or what comparisos you will make with your data. The desired level of precisio ca be stated i two differet ways ad these ca be most cofusig if they are mixed up. You ca defie the absolute level of precisio desired. For example, you may wish the 95% cofidece iterval to be ±.8 mm or less for your whitefish. Alteratively, you ca defie the relative level of precisio desired. For example, you may wish the 95% cofidece iterval to be ± 6% of the mea or less. These two are related simply:

5 Chapter 7 Page 79 Absolute error desired Percet relative error desired 100 x I will use the absolute level of precisio to begi but for some purposes as we proceed we will fid it easier to use relative error as a measure of precisio. A secod potetial source of cofusio ivolves how to specify the level of precisio you require. Some authors defie the level of precisio i terms of ± 1 stadard error of the mea. Others defie the level of precisio i terms of the width of the cofidece iterval, which for 95% cofidece gives approximately ± stadard errors of the mea. I this chapter I will always defie the desired level of precisio i terms of the width * of the cofidece iterval. If you use other statistical texts, be sure to check which defiitio of "desired precisio" they adopt. Step. Fid some equatio that coects sample size () with the desired precisio of the mea. This equatio will deped o the kid of samplig beig doe ad the type of variable beig measured. The purpose of this chapter is to give you these equatios. For the case of the mea from a ormal distributio we have the equatio: or, expadig: Desired absolute error = d t s x d t s rearragig: t s d (7.1) where: * Techically speakig, I should say the "half-width" of the cofidece iterval; but this cumbersome wordig is omitted. I use the term to idicate that i geeral we thik of cofidece itervals as a parameter ± width of the cofidece iterval.

6 Chapter 7 Page 80 = Sample size eeded to estimate the mea t = Studet's t value for - 1 degrees of freedom for the 1 - level of cofidece s = Stadard deviatio of variable d = Desired absolute error This is the formula to estimate sample size for a mea from a ormal distributio. Step (3) Estimate or guess the ukow parameters of the populatio that you eed to solve the equatio. I this simple case we eed to have a estimate of s, the stadard deviatio of the variable measured. How ca we get a estimate of the stadard deviatio? There are four ways you may get this for the variable you are studyig: (a) By previous samplig of a similar populatio: You may kow, for example, from work doe last year o the same whitefish i aother lake that the stadard deviatio was about 6.5 mm. (b) By the results of a pilot study: You may wish to sped a day or two samplig to get a estimate of the stadard deviatio i your populatio. (c) By guesswork: Ofte a experieced perso may have implicit kowledge of the amout of variability i a variable. From this type of iformatio o the rage of measuremets to be expected (maximum value - miimum value), you ca estimate the stadard deviatio of a measure that has a ormal distributio by the formulae give i Table 7.1. For example, if you kow that i a sample of about 50 fish, legth will vary from mm, from Table 7.1: Estimated stadard deviatio (0.)(rage) (0.)(55) = 1. mm (d) By two-stage samplig: If it is feasible for you to sample i two steps, you ca first take a sample of size 1 ad calculate from these measuremets a prelimiary estimate of the stadard deviatio. Cochra (1977 p. 79) shows that i the secod sample you eed to take additioal measuremets to make a total sample size of:

7 Chapter 7 Page 81 TABLE 7.1 ESTIMATION OF THE STANDARD DEVIATION (s) OF A VARIABLE FROM KNOWLEDGE OF THE RANGE (w) FOR SAMPLES OF VARIOUS SIZES a Sample size Coversio factor Sample size Coversio factor a Rage = maximum value - miimum value. Multiply the observed rage by the tabled values to obtai a ubiased estimate of the stadard deviatio. A ormal distributio is assumed. Source: Dixo ad Massey, 1983 ( ts1) 1 d 1 (7.) where: = Fial total sample size eeded 1 = Size of first sample take s1 = Stadard deviatio estimated from first sample take t = Studet's t with - 1 d.f for 1- level of cofidece d = Desired absolute error

8 Chapter 7 Page 8 Sice you are usig data from the first sample to estimate total sample size, there is a elemet of statistical circularity here, but the importat poit is to remember that these procedures are approximatios to help you to decide the appropriate sample size eeded. A example will illustrate the applicatio of these three steps with equatio (7.1). Assume you wish to measure whitefish to obtai a 95% cofidece iterval of ±.8 mm. From previous samplig it is kow that the stadard deviatio of legths is about 9.4 mm. Thus, from equatio (7.1): ts t(9.4) d.8 We immediately have aother problem because to look up a t -value we eed to kow the sample size, so we are trapped i a statistical Catch-. For those who desire precisio, equatio (7.1) ca be solved iteratively by trial ad error. I practice it is ever worth the effort because t-values for 95% cofidece limits are almost always aroud (uless is very small), ad we will use i equatio (7.1) the approximatio t = whe we eed 95% * cofidece limits. Thus: (9.4) The recommedatio is to measure about 45 fish. This techique for estimatig sample size ca also be used with relative measures of precisio (Eberhardt 1978). The most coveiet measure of relative variability is the coefficiet of variatio: CV s (7.3) x * If you eed to work i 90% cofidece itervals, you ca use t ~ 1.7. If you use 99% cofidece itervals, use t ~.7. These are approximatios.

9 Chapter 7 Page 83 where: s x = stadard deviatio = Observed mea If you kow the coefficiet of variatio for the variable you are measurig, you ca estimate the sample size eeded i the followig way. From the formulas give above: ts Desired relative error r x 100 x or, expadig: r ts x 100 rearragig: t s 100 x r or 100CV t r (7.4) We ca simplify this equatio by assumig for 95% cofidece limits that t = so this reduces to: 00CV r (7.5) where: r desired relative error (width of cofidece iterval as percetage) CV coefficiet of variatio (equatio 7.3) As a example, suppose you are coutig plakto samples ad kow that the coefficiet of variatio is about 0.70 for such data. If you wish to have ±5% i the relative precisio of your mea estimate, the:

10 Chapter 7 Page 84 (00)(0.70) 5 = 31.4 ad you require a sample size of about 31 samples to get a cofidece iterval of approximately 5% of the mea. Eberhardt (1978) has draw together estimates of the coefficiet of variatio from published ecological data (Table 7.). Clearly it is a oversimplificatio to assume that all differet samplig regios will have a costat coefficiet of variatio for ay ecological measuremet. But the poit is that as a approximate guide ad i the absece of more detailed backgroud data it will give you a geeral idea of the variatio you might expect with differet kids of samplig. Table 7. ad equatio (7.5) ca supply a rough guide to the samplig itesity you should provide. TABLE 7. COEFFICIENTS OF VARIATION OBSERVED IN A VARIETY OF POPULATION SAMPLING TECHNIQUES TO ESTIMATE POPULATION SIZE a Aquatic orgaisms Group of orgaisms Coefficiet of variatio Plakto 0.70 Bethic orgaisms Surber sampler, couts 0.60 Surber sampler, biomass or volume 0.80 Grab samples or cores 0.40 Shellfish 0.40 Fish 0.50 to.00 Terrestrial orgaisms Roadside couts 0.80 Call couts 0.70 Trasects (o foot) 0.50 to.00 Fecal pellet couts 1.00 a Average values complied by Eberhardt (1978a). All of the above formulas assume that the populatio is very large relative to the umber sampled. If you are measurig more tha 5-10% of the whole populatio, you

11 Chapter 7 Page 85 do ot eed to take so large a sample. The procedure is to calculate the sample size from equatio (7.1) or (7.5) above ad the correct it with the followig fiite populatio correctio: * 1 / N (7.6) where: * Estimated sample size required for fiite populatio of size N Estimated sample size from (7.1) ad (7.5) for ifiite populatio N Total size of fiite populatio For example, if we kow that the whitefish populatio described above (page 8) totals oly 50 idividuals, to get a 95% cofidece iterval of ±.8 mm, we first estimate = 45.1 as above, ad the: * / N = 38. fish Note that the required sample size is always less whe we use the fiite populatio correctio, so that less effort is required to sample a fiite populatio. These equatios to estimate sample size are derived from the ormal distributio. What happes if the variable we are measurig does ot have a ormal, bell-shaped distributio? Fortuately, it does ot matter much because of the Cetral Limit Theorem. The Cetral Limit Theorem states that, as sample size icreases, the meas of samples draw from a populatio with ay shape of distributio will approach the ormal distributio *. I practice this theorem meas that with large sample sizes ( > 30) we do ot have to worry about the assumptio of a ormal distributio. This theorem is amog the most importat practical fidigs of theoretical statistics because it meas that you do ot eed to worry if you are measurig a variable that has a skewed distributio - you ca still use the approach * Assumig the variace is fiite for the distributio.

12 Chapter 7 Page 86 outlied above (Sokal ad Rohlf 1995). But if the distributio is strogly skewed, you should be cautious ad apply these formulas coservatively Compariso of Two Meas Plaig experimets which ivolve comparisos of two or more meas is more complicated tha decidig o a sigle sample size. You may, for example, wish to compare whitefish legths i two differet lakes. After the data are collected, you will evaluate how differet the lakes are i this parameter. How ca you decide beforehad how big a sample to use? To do this you must first decide o the smallest differece you wish to be able to detect: d (7.7) A B where: d A B Smallest differece you wish to detect Mea value for populatio A Mea value for populatio B For example, you may decide you wish to pick up a differece of 8 mm or more i fish legths betwee lakes A ad B. This differece d must be expressed i uits of the stadard deviatio of the variable beig measured: D d (7.8) s where: D d s Stadardized smallest differece you wish to detect Smallest differece you wish to detect Stadard deviatio of variable measured The stadard deviatio is assumed to be the same for both populatios A ad B, ad is estimated from previous kowledge or guesswork. Note that the size of this differece is dictated by your maagemet objectives or the ecological hypothesis you are testig.

13 Chapter 7 Page 87 The secod decisio you must make is the probability you will tolerate for makig a Type I or a Type II error. These are usually called ad : Probability of rejectig the ull hypothesis of o differece whe i fact it is true (Type I error) Probability of acceptig the ull hypothesis whe i fact it is false ad the meas really do differ (Type II error) By covetio is ofte set to 0.05, but of course it ca be set to ay other value depedig o the cosequeces of makig a Type I mistake. The probability is less well kow to ecologists ad yet is critical. It is related to the "power" of the statistical test: Power 1 (7.9) Whe is very small the test is said to be very powerful, which meas that you will ot make Type II errors very ofte. Ecologists are less familiar with because it is fixed i ay typical study after you have specified, measured a particular sample size, ad specified the alterative hypothesis. But if you have some choice i decidig o what sample size to take, you ca specify idepedetly of. Sokal ad Rohlf (01, pg. 167) ad Zar (010, pg. 81) have a good discussio of the problem of specifyig ad. Give that you have made these decisios, you ca ow estimate the sample size eeded either by solvig the equatios give i Mace (1964 page 39) or by the use of Table 7.3 from Davies (1956). I most cases the table is more coveiet. For example, suppose for your compariso of fish legths you have: d s A B = 8 mm 9.4 mm (from previous studies) ad you decide you wish = 0.01 ad = 0.05 ad are coductig a two-tailed test. Calculate the stadardized differece D from equatio (7.8): d D s 8 = =

14 Chapter 7 Page 88 Table 7.3 NUMBER OF SAMPLES REQUIRED FOR A t-test OF DIFFERENCE BETWEEN THE MEANS OF TWO POPULATIONS a Two-sided test Level of t-test = 0.01 = 0.0 = 0.05 = 0.1 = V A L U E O F D

15 Chapter 7 Page a a The etries i the table show the umber of observatios eeded to test for differeces of specified size (D) betwee two meas at fixed levels of ad. Etries i the table show the sample size eeded i each of two samples of equal size. Source: Davies (1956), Table E.1.

16 Chapter 7 Page 90 From Table 7.3 i the secod colum, we read 51 This meas that you should measure 51 fish from lake A ad 51 fish from lake B to achieve your stated goal. Note from Table 7.3 that the smaller the ad the smaller the, ad the smaller the stadardized distace D that you wish to detect, the larger the sample size required. This table thus restates the geeral priciple i statistics that if you wish to gai i precisio you must icrease sample size. The approach used i Table 7.3 ca be geeralized for the compariso of 3, 4, 5... meas from differet populatios, ad cosequetly this table ca be useful for plaig relatively complex experimets. Kastebaum et al. (1970) give a more detailed discussio of sample sizes required whe several meas are to be compared. If Table 7.3 does ot cover your particular eeds, you ca use the approximate equatio suggested by Sedecor ad Cochra (1967, pg. 113) for a compariso of two meas: ( z z) s (7.10) d where: Sample size required from each of the two populatios z Stadard ormal deviate for level of probability ( z.05 is 1.96; z.01 is.576) z Stadard ormal deviate for the probability of a type II error s (see table below) Variace of measuremets (kow or guessed) d A B Smallest differece betwee meas you wish to detect with probability 1 - Equatio (7.10) is oly approximate because we are usig z-values i place of the more proper t-values, but this is a mior error whe > 50. The z values are

17 Chapter 7 Page 91 obtaied from tables of the stadard ormal deviate (z). A few examples are (Eberhardt 1978): Type II error Power Two-tailed ( ) (1 - ) z For example, if you wished to detect a differece i fish legths i the above example of 4 mm, the: d 4 mm A B s (9.4) mm (from previous studies) ad = 0.01 ad = Thus from equatio (7.10): ( ) (9,4) fish 4 Note that Table 7.3 does ot cover this particular combiatio of d ad s. A alterative approach is to use the tables i Kastebaum et al. (1970) to estimate the required sample size. Figure 7.1 is derived from these tables. I this case the meas are expressed as a ratio you wish to detect: k A (7.11) B where: A = Larger mea B = Smaller mea k = Ratio you wish to detect betwee the meas If you have a rough idea of the coefficiet of variatio i your data (cf. Table 7.), Figure 7.1 ca be used i plaig a study ivolvig the compariso of two or more

18 Chapter 7 Page 9 meas. Note that the sample size give i Figure 7.1 is required for each of the two populatios beig compared Sample Size C = 0.50 C = 0.75 C = 1.00 C = Ratio of Meas, k Figure 7.1 Sample sizes required to detect a sigificat differece betwee two meas expressed as a ratio (k) with = 0.05 ad = 0.0, plotted for differet coefficiets of variatio (C) that are typical of may populatio techiques. (From Eberhardt, 1978b.) Variaces From A Normal Distributio I some cases a ecologist may wish to estimate the variace of a give measuremet to a specified level of precisio. How ca you estimate how large a sample you should take? First, you must specify the allowable limits for the cofidece limits o the variace. This is most easily expressed as a percetage (± 5%) or as a proportio (± 0.5). Secod, you must choose the probability ( ) of the cofidece iterval ot icludig the true variace. The, if sample sizes are ot too small ( > 30), from Mace (1964 p. 57):

19 Chapter 7 Page z (7.1) where: Approximate sample size required to estimate the variace Allowable limits of error (expressed as a proportio) of the variace I equatio (7.1) substitute z = 1.96 for = 0.05; if you wish to have = 0.01 use z =.58. Other costats may be chose for other values from tables of the stadard ormal distributio. For example, suppose that you wish to estimate the variace of a measuremet with allowable limits for the cofidece limits of ±35% ad you wish to use = The from equatio (7.1): so you would eed to take about 60 samples. Ufortuately, this procedure is applicable oly to variaces from a ormal distributio, ad is quite sesitive to departures from ormality. 7. DISCRETE VARIABLES Couts of the umbers of plats i a quadrat or the umbers of eggs i a est differ from cotiuous variables i their statistical properties. The frequecy distributio of couts will ofte be described by either the biomial distributio, the Poisso distributio, or the egative biomial distributio (Elliott 1977). The samplig properties of these distributios differ, ad thus we require a differet approach to estimatig sample sizes eeded for couts.

20 Chapter 7 Page Proportios ad Percetages Proportios like the sex ratio or the fractio of juveiles i a populatio are described statistically by the biomial distributio. All the orgaisms are classified ito classes, ad the distributio has oly two parameters: p = Proportio of x types i the populatio q = 1 - p = Proportio of y types i the populatio We must specify a margi of error (d) that is acceptable i our estimate of p, ad the probability ( ) of ot achievig this margi of error. If sample size is above 0, we ca use the ormal approximatio to the cofidece iterval: pˆ t s P ˆ (7.13) where: pˆ t Observed proportio Value of Studet's t-distributio for -1 degrees of freedom s Stadard error of pˆ pq ˆ ˆ / Pˆ Thus the desired margi of error is: d t s t Pˆ pq ˆˆ Solvig for, the sample size required is: t ˆˆ pq (7.14) d where: d Sample size eeded for estimatig the proportio Desired margi of error i our estimate p ad the other parameters are defied above. As a first approximatio for =.05 we ca use t =.0. We eed to have a approximate value of p to use i this equatio. Prior iformatio, or a guess, should be used, with the oly rule-of-thumb that whe i doubt pick a value of p closer to 0.5 tha you guess - this will make your aswer coservative.

21 Chapter 7 Page 95 As a example, suppose you wish to estimate the sex ratio of a deer populatio. You expect p to be about 0.40 ad you would like to estimate p withi a error limit of ±0.0 with = From equatio (7.14): (.0 )(0.40)(1 0.40) 400 deer (0.0) Give this estimate of, we ca recycle to equatio (7.14) with a more precise value of t = 1.96, to get a better estimate of = 305. So you must classify 305 deer to achieve this level of precisio. If you wish to have = 0.01 (t =.576) you must classify 3981 deer, while if you will permit = 0.10 ( t = 1.645) you must classify oly 164 deer. You ca also use two-stage samplig (page 000) to estimate the proportio p (Cochra 1977, pg. 79). Take a first sample of size, ad calculate a prelimiary estimate of ˆp. The i the secod sample take additioal samples to make a total sample size of: pˆ qˆ 3 8pˆ qˆ 1 3pˆ qˆ pˆˆ 1q1 1 (7.15) where: = Total fial sample size eeded to estimate proportio pˆ 1 Proportio of x types i first sample qˆ p1 d / t (for =.05, d / 3.48; for =.01, d / 6.64) d Desired margi of error i ˆp If you are samplig a fiite populatio of size N, you may the correct your estimated sample size by equatio (7.6) usig the fiite populatio correctio to reduce the actual sample size eeded. For example, suppose the deer populatio is kow to be oly about 1500 aimals. From equatio (7.6): * / N / deer so a much smaller sample is required i this example if there is a fiite populatio.

22 Chapter 7 Page 96 Wildlife maagers ofte measure attributes of wildlife populatios i a slightly differet way with "populatio ratios". These are expressed, for example, as faws/100 does, or males/100 females *. For ratios of this type, Czaplewski et al. (1983) have preseted a useful series of charts ad graphs to determie how large a sample size you eed to attai a certai level of precisio. I all cases they have assumed samplig without replacemet. Figure 7. shows their recommeded sample sizes for a rage of total populatio sizes ad for 90% cofidece itervals. Note that if the total populatio is below aimals, the fiite populatio correctio (eq. 7.6) reduces the required sample size cosiderably. If populatio size exceeds 1,000, for all practical purposes it ca be cosidered ifiite. Oe example will illustrate the use of Figure 7.. Assume you have a deer populatio of about 3000 aimals, ad you expect a sex ratio of 70 males per 100 females. From Figure 7., you should classify at least 50 deer ad at most 1350 deer **. I the first case, with = 50, you will achieve a estimate of the populatio ratio of approximately ± 9 per 100 (i.e. 61 males/100 females to 79 males per 100 females). I the secod case, with = 1350, you will achieve a estimate of the populatio ratio of approximately ± 5 per 100. The geeral message is that to achieve a high precisio i estimatig ratios, you eed to take large samples. * The more covetioal otatio is as a proportio or percetage; e.g., if 70 males per 100 females, the 70/(100+70) or 0.41, or more usually 41% males i the populatio. * * Note that you ca get the same result usig equatio (7.14).

23 Chapter 7 Page 97 Precisio correspodig Populatio ratio to curve 0:100 40:100 50:100 70: :100 A 4:100 6:100 7:100 9:100 13:100 B 3:100 4:100 5:100 7:100 10:100 C :100 3:100 4:100 5:100 7:100 Sample Size Maximum recommeded sample size C B A 500 Miimum recommeded sample size Populatio Size Figure 7. Recommeded sample sizes for estimatig a populatio ratio of two sex or age classes i populatios of fiite size. Three curves are show for the three levels of precisio give at the top of the graph. Precisio is specified here as a 90% cofidece iterval. A fiite populatio correctio is icluded i these curves, ad samplig without replacemet is assumed. (From Czaplewski et al., 1983.) 7.. Couts From A Poisso Distributio Sample size estimatio is very simple for ay variable which ca be described by the Poisso distributio i which the variace equals the mea (Chapter 4). From this it follows that: CV = s s x x x x

24 Chapter 7 Page 98 or CV 1 x Thus from equatio (7.5) assumig = 0.05: 00CV 00 1 = r r x (7.16) where: = Sample size required for a Poisso variable r = Desired relative error (as percetage) CV = Coeeficiet of variatio = 1/ x For example, if you are coutig eggs i starlig ests ad kow that these couts fit a Poisso distributio ad that the mea is about 6.0, the if you wish to estimate this mea with precisio of ± 5% (width of cofidece iterval), you have: ests Equatio (7.16) ca be simplified for the ormal rage of relative errors as follows: For For For 10% precisio 5% precisio 50% precisio 400 x 64 x 16 x Note that these are all expressed i terms of the width of the cofidece iterval ad all are appropriate oly for variables that fit a Poisso distributio Couts From A Negative Biomial Distributio Sice may aimal ad plats have a clumped patter, quadrat couts will ofte be described by the egative biomial distributio istead of the Poisso (Chapter 4). To estimate the sample size for a series of couts which fit a egative biomial, you must kow two variables ad decide two more: 1. Mea value you expect i the data ( x )

25 Chapter 7 Page 99. Negative biomial expoet (k) 3. Desired level of error (r) s a percetage 4. Probability ( ) of ot achievig your desired level of error The level of error is set as a expected cofidece iterval, for example as ± 0% of the mea. Sice the variace of the egative biomial is give by: s x x k we ca substitute this expected variace ito equatio (7.1) ad do some algebra to derive a estimated sample size: t r x k (7.17) where: = Sample size required for a egative biomial variable t Studet's t value for - 1 degrees of freedom for probability x k r Estimated mea of couts Estimated egative biomial expoet Desired level of error (percet) For most practical purposes we ca assume t =.0 (for 95% cofidece limits). As a example, usig the data from Box 4. (page 000) o the black-bea aphid with x = 3.46, ˆk =.65, assume we would like to have cofidece limits of ± 15% of the mea: quadrats (stems) As with the Poisso distributio, we ca simplify equatio (7.17) for the ormal rage of relative errors used i ecological research:

26 Chapter 7 Page For 10% precisio 400 x k 1 1 For 5% precisio 64 x k 1 1 For 50% precisio 16 x k By comparig these formulas with those developed above for the Poisso, you ca see the extra samplig required to attai a give cofidece iterval with the egative biomial. For example, if you had erroeously assumed that the black bea aphids had a Poisso distributio with mea 3.46, you would estimate for 95% cofidece limits of ±15% that you would require a sample size of 51 quadrats, rather tha the correct value of 118 quadrats calculated above. This is a vivid illustratio of why the tests outlied i Chapter 4 are so critical i ecological samplig ad the desig of experimets Couts From A Logormal Distributio May ecological variables are ot well described by the ormal distributio because they represet rates ad are logormal i distributio (see Chapter 15, page 000). By trasformig logormal data with a logarithmic trasform, the trasformed data become more early a ormal distributio. The questio we address here is how to determie the sample size eeded for estimatig the mea of a variable that has a logormal distributio. We proceed by trasformig the origial data as follows: Y log X (7.18) where Y = trasformed data X = origial data Note that the base of logarithms used does ot matter, so that log e or log 10 ca be used as coveiet. Next, with the Y as data, calculate the usual mea ad variace of Y with the covetioal formulas. Olsso (005) showed that the best cofidece iterval for Y is give by:

27 Chapter 7 Page 301 s s s 4 Y t 1 (7.19) where s = variace of Y Covert these cofidece limits to the origial measuremet scale by takig ati-logs. To use this relatioship to produce a estimate of desired sample size oce you have a estimate of the mea Y ad the variace of Y for your data, decide o the width of cofidece iterval desired ad by trial-ad-error i Excel or similar program eter sample sizes ad determie whether or ot they satisfy your desired width of the cofidece iterval. Box 7.1 provides a example of this empirical approach to sample size determiatio for a logormal variable. Box 7.1 ESTIMATING SAMPLE SIZE FOR A VARIABLE THAT HAS A LOGNORMAL DISTRIBUTION A forest ecologist i a pilot study measured the umber of tree seedligs i 0 plots i a regeeratig forest, ad wishes to kow if 0 plots are sufficiet for obtaiig estimates of mea desity withi a lower cofidece limit about 5% of the mea. She measured the followig umbers from 0 quadrat couts: Sample quadrat Observed cout. Log e (observed # cout)

28 Chapter 7 Page Sum Mea Variace Estimate the 95% cofidece limits from equatio (7.19): 4 s s s Y t (0.9808) or to These log values ca be coverted back to the origial cout scale by ati-logs to give a mea of 44.1 with 95% cofidece itervals of to 430.9: Note that these are always asymmetric cofidece itervals (- 43%, +76%). Give these pilot data, you ca ow determie what size of sample you require to achieve a lower 95% cofidece iterval that is 5% of the mea. Solve equatio (7.19) iteratively for a hypothetical rage of sample sizes, for example = 50: 4 s s s Y t (0.9808) or to which gives 95% cofidece itervals of to 34.8, or 9% to +41%. Repeat these calculatios several times util you coverge to a = 71 which gives: 95% cofidece limits: to , or i the origial uits 183. to 35.4, or i percetages -5% to + 33% of the estimated mea. Further samplig from this or similar forest sites should thus cosider sample sizes of approximately 70 quadrats..

29 Chapter 7 Page SPECIALIZED ECOLOGICAL VARIABLES Some variables that ecologists estimate, like populatio desity, are basically cotiuous variables that are estimated i idirect ways. For example, we might use a lie trasect techique (page 000) to estimate populatio desity. Such derived variables are more difficult to aalyze i order to decide i advace how big a sample you eed. But some approximate techiques are available to help you pla a field study Mark-ad-Recapture Estimates I Chapter we aalyzed the simplest mark-recapture method (the Peterse method) i cosiderable detail, ad we icluded the Robso ad Regier (1964) charts (Figures.3,.4, page 000) for estimatig the umber of aimals that eed to be marked ad recaptured to achieve a give level of accuracy i the estimate of populatio size. A alterative procedure is to use the coefficiet of variatio of the estimated populatio size to estimate the required sample size directly. Seber (198 p. 60) gives the coefficiet of variatio of the Peterse populatio estimate as: CV( Nˆ ) 1 1 (7.0) R MC / N where: R M C Nˆ Expected umber of marked aimals to be caught i the secod Peterse sample Number of aimals marked ad released i the first Peterse sample Total umber of aimals caught i the secod Peterse sample Estimated populatio size This formula ca be used i two ways. First, if you have started a Peterse sample ad thus kow the value of M, ad you ca guess the approximate populatio size, you ca determie C for ay desired level of the coefficiet of variatio. For example, if you have marked ad released 00 fish (M) ad you thik the populatio is about 3000 (N), ad you wish the coefficiet of variatio of the estimated populatio size to

30 Chapter 7 Page 304 be about ±5% (correspodig to a level of precisio of ±50% * ), we have from equatio (7.18): CV( Nˆ ) = MC / N 00 C / 300 Solvig for C, we obtai: 3000 C = = 40 fish (00)(0.5) so we should capture about 40 fish i our secod Peterse sample ad we would expect i these 40 fish to get about 16 marked idividuals (R). This is oly a crude approximatio because equatio (7.18) assumes large sample sizes for R ad to be coservative you should probably take a somewhat larger sample tha 40 fish. A secod use for this equatio (7.18) is to get a rough estimate of the umber of marked idividuals (R) that should be obtaied i the secod Peterse Sample. I this case samplig might cotiue util a prearraged umber of recaptures is obtaied (Eberhardt 1978). For example, suppose i the fish example above you wish to have a coefficiet of variatio of ( ˆN ) of 0.05 (5%). From equatio (7.18): CV( Nˆ ) 0.05 Solvig for R: 1 R (0.05) 1 R 400 fish If you wish to obtai such a high degree of accuracy i a Peterse estimate you must recapture 400 marked fish, so clearly you must mark ad release more tha 400 i the first Peterse sample. These examples assume a large populatio so that the fiite populatio correctio is egligible. * Note that the coefficiet of variatio for populatio estimates is equal to approximately oe-half the relative level of precisio (r) defied above for = 0.05 (page 46).

31 Chapter 7 Page 305 The Schabel method of populatio estimatio utilizes a series of samples (e.g. Table. page 000) ad ca be readily adapted to a predetermied level of precisio. The coefficiet of variatio for a Schabel estimateof populatio size is, from Seber (198, p. 190): CV( Nˆ ) 1 R t (7.1) where: CV( Nˆ ) Expected coefficiet of variatio for Schabel estimate of R populatio size Number of marked idividuals caught i sample t t This formula is similar to equatio (7.18). It ca be used as follows to decide whe to stop samplig i a Schabel experimet. Cosider the data i Table.1 (page 000). After Jue 11 (sample 10) there was a total of 13 recaptures ( Rt ). Thus: ˆ 1 CV( N) ad thus the 95% cofidece limits would be about twice 0.77 or ±55% of the estimated populatio size ˆN. By the ed of this experimet o Jue 15 there were 4 recaptures, ad thus ˆ 1 CV( N) so the 95% cofidece limits would be approximately ±40% of ˆN. If you wated to reduce the cofidece iterval to approximately ±0% of ˆN you would have to cotiue samplig util Rt = 100. These estimates of sample size are oly a rough guide because they igore fiite populatio correctios, ad equatio (7.19) assumes a ormal approximatio that is ot etirely appropriate whe sample sizes are small. The importat poit is that they provide some guidace i plaig a markig experimet usig the Schabel method. There are o simple methods available to estimate sample sizes eeded to attai a give level of precisio with the

32 Chapter 7 Page 306 Jolly-Seber model. At preset the oly approach possible is to use a simulatio model of the samplig process to help pla experimets. Program MARK (White 008, Cooch ad White 010) allows user iputs of certai samplig rates i a Jolly- Seber model ad will simulate the estimatio so you ca see the levels of precisio obtaied. Roff (1973) ad Pollock et al. (1990) have emphasized that samplig itesity must be very high to attai good levels of precisio with Jolly-Seber estimatio Lie Trasect Estimates If you wish to desig a lie trasect survey, you eed to determie i advace what legth of trasect (L) or what sample size () you eed to attai a specified level of precisio. We discussed i Chapter 5 the geeral methods of estimatio used i lie trasect surveys ad i aerial surveys. How ca we estimate the sample size we require i lie trasect surveys? If we measure the radial distaces (r i ) to each aimal see (see Figure 5. page 000), Eberhardt (1978a) has show that the coefficiet of variatio of the desity estimate is give by: 1 ˆ 1 CV( D) 1 CV ri (7.) where: CV( Dˆ ) Coefficiet of variatio of the lie trasect desity estimate D of equatio (5.) Sample size CV 1/ ri (Coefficiet of variatio of the reciprocals of the radial distaces) There are two possibilities for usig this equatio. The coefficiet of variatio of the reciprocals of the radial distaces ca be estimated empirically i a pilot study, ad this value used to solve equatio (7.). Alteratively, Eberhardt (1978) ad Seber (198, p. 31) suggest that the coefficiet of variatio of the reciprocals of radial distaces is ofte i the rage of 1-3, so that equatio (7.0) ca be simplified to:

33 Chapter 7 Page 307 CV( Dˆ ) = b (7.3) where b = a costat (typically 1-4) For ay particular study, you could assume b to be i the rage of 1-4 ad be coservative by estimatig it to be relatively high. For example, suppose that you wish to have a coefficiet of variatio of desity of about ±10% so that the 95% cofidece iterval is about ±0% of the populatio size. A pilot study has determied that b =. From equatio (7.3): ˆ CV( D) (assumes b ) 0.10 or 00 Alteratively, if the perpedicular distaces (x i ) are used i a lie trasect estimator (such as equatio 5.7), Eberhardt (1978) suggests that the approximate coefficiet of variatio is: ˆ 4 CV( D) (assumes b 4) (7.4) which is exactly double that give above for radial distaces. Oe alterative strategy to use with lie trasect data is two-step samplig i which the first sample is used as a pilot survey. Burham et al. (1980, pp ) discuss the details of plaig a lie trasect study by two-stage samplig. They provide a rule-of-thumb equatio to predict the legth of a lie trasect you would eed to achieve a give precisio: Lˆ b 1 1 L CV Dˆ (7.5) where:

34 Chapter 7 Page 308 Lˆ Legth of total survey lie required b A costat betwee 1.5 ad 4 (recommed b 3) CV( Dˆ ) Desired coefficiet of variatio i the fial desity estimate L1 Legth of pilot survey lie = Number of aimals see o pilot survey lie 1 There is some disagreemet about the value of b i ecological lie trasect studies (Eberhardt 1978a, Burham et al. 1980). It may be betwee 1.5 ad 4 but is more likely to be about 3, so Burham et al. (1980, p. 36) recommed usig b = 3 to be coservative or b =.5 to be more liberal. The ratio (L 1 / 1 ) could be kow from earlier studies ad will vary greatly from study to study. For example, i surveys of duck ests it could be 10 ests/km. If the ratio (L 1 / 1 ) is ot kow already, you ca estimate it from a pilot study. Equatio (7.5) ca also be used backwards to estimate what the coefficiet of variatio i desity will be if you ca sample a lie of legth L (due to fiacial costraits or time). To illustrate, cosider the data i Box 5.1 (page 000) as data from a pilot lie trasect survey. From equatio (7.5), assumig you wish to have a coefficiet of variatio i the fial desity of ±10% (so that a 95% cofidece iterval would be about ±0%), ad give that you have see 1 aimals i a lie trasect of 10 km ad wish to be coservative so assume that b = 3: Lˆ = 3 10 (0.10) 1 = 50 km Cosequetly you should pla a total trasect of 50 km if you wish to achieve this level of precisio. But suppose that you oly have moey to sample 60 km of trasect. From equatio (7.3) with a coservative estimate of b = 3: ˆ 3 10 L 60 CV D ˆ 1 Solvig for CV( ˆD ): CV D ˆ = 0.04

35 Chapter 7 Page 309 Thus you could achieve ±0% precisio (or a 95% cofidece iterval of ±40%) i your fial desity estimate if you could do oly 60 km of trasect i this particular example Distace Methods There has bee relatively little aalysis doe of the sample size requiremets for populatio desity estimates that use plotless samplig methods. Seber (198 p. 43) suggests that the coefficiet of variatio of desity estimates from distace samplig will be approximately: CV Nˆ sr 1 (7.6) where: CV Nˆ Coefficiet of variatio for plotless samplig estimate of s r populatio desity Number of radom poits or radom orgaisms from which distaces are measured Number of measuremets made from each poit (i.e., if oly earest eighbor r 1) Thus i samplig with plotless methods you ca achieve the same level of precisio with (say) 100 radom poits at which you measure oly earest-eighbors, as with 50 radom poits at which you measure the distace to the first ad secod earesteighbors. For example, if you wish to achieve a coefficiet of variatio of ±0% for populatio desity estimated from distace samplig (which will give a 95% cofidece iterval of about ±40%), ad you are measurig oly earest eighbors (r = 1), you have from equatio (7.6): CV Nˆ 0.0 s 1 Solvig for s: s = 7 radom poits

36 Chapter 7 Page 310 As with all estimates of populatio size, this should be take as a approximate guidelie to achieve the desired precisio. If you are lucky, this sample size will give you better precisio tha you expect, but if you are ulucky, you may ot achieve the precisio specified Chage-I-Ratio Methods To estimate populatio desity from the chage-i-ratio estimator discussed previously i Chapter 3 (page 000), you should cosult Paulik ad Robso (1969) ad Figures 3.1 ad 3.. These graphs will allow you to select the required sample size directly without the eed for ay computatios. 7.4 STATISTICAL POWER ANALYSIS Much of the material preseted i this chapter has bee discussed by statisticias uder the rubric of power aalysis. Statistical power ca be viewed most clearly i the classical decisio tree ivolved i Neyma-Pearso statistical iferece (Hurlbert ad Lmobardi 009): Decisio State of real world Do ot reject ull hypothesis Reject the ull hypothesis Null hypothesis is actually true Null hypothesis is actually false Possibly correct decisio but susped judgmet (probability = 1-) Possibly a Type II error (probability = ) Type I error (probability = ) Correct decisio (probability = (1-) = power We will discuss i Chapter 15 the problems of the ull hypothesis. Suffice to say for the preset that the ull hypothesis eed ot be o effect, ad there ca be a whole set of alterative hypotheses that are a rage of values (e.g. that fish legths differ by more tha 10 mm, more tha 0 mm, etc.). Most ecologists worry about, the probability of a Type I error, but there is abudat evidece ow that we should worry just as much or more about, the probability of a Type II error (Peterma 1990, Fairweather 1991).

37 Chapter 7 Page 311 Power aalysis ca be carried out before you begi your study (a priori or prospective power aalysis) or after you have fiished (retrospective power aalysis). Here we discuss a priori power aalysis as it is used for the plaig of experimets. Thomas (1997) discusses retrospective power aalysis. The key poit you should remember is that there are four variables affectig ay statistical iferece: Sample size Probability of a Type I error Probability of a Type II error Magitude of the effect = effect size These four variables are itercoected ad oce ay 3 of them are fixed, the fourth is automatically determied. Looked at from aother perspective, give ay 3 of these you ca determie the fourth. We used this relatioship previously whe we discussed how to determie sample size (page 78). We ca ow broade the discussio ad i particular discuss effect size, a key item i all ecological research Estimatio of Effect Size for Cotiuous Variables Effect size is a geeral term that is iterpreted i differet ways by ecologists ad statisticias. To a ecologist the effect size is typically the differece i meas betwee two populatios. Fish i lake A are.5 kg heavier tha fish i lake B. But statisticias prefer to covert these biological effect sizes to stadardized effect sizes so they ca discuss the geeral case. So for a ecologist the first thig you eed to lear is how to traslate biological effect sizes ito statistical effect sizes. The most commo situatio ivolves the meas of two populatios which we would like to test for equality. Oe measure of effect size is the stadardized effect size, defied as: