Designing Efficient Sampling Plans for Ecology and Conservation. Edward F. Connor Department of Biology San Francisco State University

Transcription

1 Designing Efficient Sampling Plans for Ecology and Conservation Edward F. Connor Department of Biology San Francisco State University

2 Issues in Study Design Sampling intensity (how much to sample) Choosing among different sampling designs (between and within-subjects designs) Determining sampling frequency (how often to sample) Determining which study design is most cost effective

3 Choosing a Sample Size Obtain a preliminary estimate of σ Conjecture the minimally interesting difference in treatment means, or effect size, δ Set the Type I error rate, α Set the desired power for the test (1- β) Calculate necessary sample size by approximation, web-based calculators, software, or simulation

4

5 Recommended Calculators for Sample Size Determination Web Based Author Location URL R. Lenth University of Iowa J. Bond UCLA Rollin Brant U Calgary M. Friendly York University Software Author Package Availability - W. Dupont & W. Plummer PS owersamplesize R. O Brien UnifyPow A. Buchner, F. Faul, & E. Erdfelder G*Power J. Hintze PASS (for purchase)

6 Choosing an Experimental Design ANOVA Designs Between Subjects* Within Subjects* Regression Designs Between Subject Mixed Model Designs Between Subject Within Subject Covariates

7 n s n s s b a x x b a = n s s r n s n s n s s s b a ab b a d x x d b a = = = Difference between Separate Groups and Paired t-test Separate Groups t-test Paired t-test 2 ) ( ) ( x a x b b a b a S x x t = μ μ 2 S d u d t = n 2cov ab 1) ( 2 = n df ( 1) = n df

8 70 60 Paired versus Separate Groups t-test δ/σ = k = 2 40 n 30 δ/σ = δ/σ = 1.0 δ/σ = 1.25 δ/σ = α = 0.05, β = r

9 Estimating flowering plant species richness in upland regions and seasonal stream channels in Freeman Meadow (Lakes Basin, California)

10 Preliminary Data Collection Meadow Example WS1 BS-O2 BS-I1 WS= Within Subjects BS = Between Subjects O = out of stream channel I = In stream channel BS-O1 BS-I2 WS2

11 Sample Size Calculations for Meadow Example Using PASS Within Subjects Between Subjects Power vs N with Mean0=0.5 Mean1=2.5 S=0.7 Alpha=0.05 T Test Power vs N1 with M1=0.5 M2=2.5 S1=1.8 S2=1.8 Alpha=0.05 N2=N1 1-Sided T Test Power Sample size with Power = 0.8 Power Sample size with Power = N N1 Number of measurements required = 6 Number of measurements required = 22

12 Difference between Within-subjects and Between-subjects ANOVA Between Subjects F = MS MS treat error dfs = k 1 k( n 1) MS error = var F = MS MS treat error dfs Within Subjects = k 1 ( n 1)( k 1) MS error = var cov cov r = MS error = var(1 r ) var

13 Within versus Between Subjects ANOVA δ/σ = 0.5 k = 3 ε = 1 k = 4 ε = 1 δ/σ = 0.5 n 10 5 δ/σ = 0.75 δ/σ = 1.0 δ/σ = 1.25 δ/σ = 1.5 δ/σ = 0.75 δ/σ = 1.0 δ/σ = 1.25 δ/σ = r r α = 0.05, β = 0.2

14 Advantage of Within-subjects Design For modest levels of correlation within a subject, within subjects designs will be more powerful than the equivalent between subjects design. When testing the same H o against the same H a (δ fixed), with the same α, β, within subjects designs are more efficient designs since σ will most likely be sufficiently smaller.

15 Northern Spotted Owl

16 Northern Spotted Owl Point Reyes National Seashore, Muir Woods and Golden Gate National Recreation Area Fledglings/nest Year

17 Spotted Owl Fecundity Sample Size Calculation Effect of varying sampling frequency Effect of Sampling Frequency (α = 0.05) Effect of Sampling Frequency (α = 0.20) % Annual Decline % Annual Decline Power Every Year (12) 3 - Years on, 3 - Years off (6) Every Other Year (6) 2 - Years on, 2 - Years off (6) 1 - Year on, 2 - Years off (4) 1 - Year on, 3 - Years off (3) Power Every Year (12) 3 - Years on, 3 - Years off (6) Every Other Year (6) 2 - Years on, 2 - Years off (6) 1 - Year on, 2 - Years off (4) 1 - Year on, 3 - Years off (3) n (Sample Size) n (Sample Size)

18 Trade off between Sampling Intensity and Frequency Achieve approximately equal power using fewer sites sampled more years Or more sites sampled fewer years For Spotted Owl more cost effective to sample more sites fewer years (20/yr x 12 yrs = 240 surveys, 25/yr x 4 yrs = 100 surveys

19 In Search of the Optimal Quadrat Size and Shape What is Optimal? Sampling variance Sampling time/replicate Transit time/replicate Minimizes bias

20 Field effort = (p x n) + (d x t) p = processing time per quadrat n = sample size (number of quadrats) d = total distance traveled between quadrats t = rate of movement between quadrats

21 Field Transit rate through vegetation ( m/sec) Processing time for each size quadrat t p Computer Estimate σ from simulated population Use σ to calculate estimate of required sample size For estimated n determine shortest distance between quadrats n d

22 Field effort = (p x n) + (d x t) p = processing time per quadrat n = sample size (number of quadrats) d = total distance traveled between quadrats t = rate of movement between quadrats

23 Field Using a within-subjects design (n = 4) we estimated the time required to set-up and sample a variety of quadrat sizes and shapes (p) We estimated the time required to walk a variety of distances through vegetation on a fixed bearing (t)

24 Time Costs for Different Size and Shape Quadrats Carmen Valley, CA

25 Computer We estimated the between quadrat variance in plant density for each size and shaped quadrat for each simulated plant population We used the estimated variance to calculate the sample size (n) required to place a (1 - α) confidence interval on the mean population density that was no greater than ±0.3 x μ

26 Computer Simulated plant populations with a mean density of 0.8 plants/m 2 in a 50 x 100 m region with different spatial structures Aggregation Random = 1 Low aggregation = 5 High aggregation = 10

27 Computer Gradient No Gradient Weak Gradient Strong Gradient

28 Shape - Rectangularity Length/Width Square = x 50m = 200 Rotation 0 o along gradient 90 o perpendicular to gradient

29 From Zar p ) 2, / (1 2 d t s n n = α α μ α α = + 1 ) ( ) 1 2, / (1 1) 2, / (1 n s t x n s t x P n n 2 2 1) 2, / ( x t s n n = α

30 Conditional Probability of having the correct width interval (W), given that it is a 1 - α interval (V) (Jiroutek et al 2003) { ( ) δ ( θ )} Pr( W V) = Pr U L L U = Pr V 1 α Where ( ) { } Pr W ( ) Pr IV V ( ) { } Pr W = U L δ ( ) ( ) Let U and L indicate the upper and lower confidence interval bounds, respectively Let δ indicate the desired Confidence Interval width

31 Pr( ) x Φ Φ 0 { ( ) ( )} 1 1 x; v ( ) 1 2 x W V c x c x dx f e (1 α) Where Φ () indicates the CDF of a standard normal variate c 1 f xv indicates a central chi-square density function with ( ) x 2 ; e = F F crit crit v e v e degrees of freedom x 2 v δ = 4σ Fcrit N e 1 2 indicates the critical value from a central F distribution with 1 numerator and v e denominator degrees of freedom ve = N r where N indicates the estimated sample size and r = 2 for a two-sided test

32 Effort (hrs) x 1m 0.25 x 5m x 1m 0.25 x 2m 1 x 1m 4 x 2 2 x 5 10 Area Nominal v/m = 1 Gradient = 0 Rotation = 0 4 x 5 2 x 10 4 x x 25m 0.5 x 50m x 50m Rectangularity (L/W)

33 x 1m Nominal v/m = 5 Gradient = 0 Rotation = Effort (hrs) x 1m Area 4 x 10m x 25m 4 x 25m 2 x 50m 4 x 50m Rectangularity (L/W)

34 Nominal v/m = 10 Gradient = 0 Rotation = 0 Effort (hrs) Area x 50m Rectangularity (L/W)

35 25 Nominal v/m =1 Gradient = steep Rotation = x 1m Effort (hrs) x 50m Area x 50m 1 x 50m 2 x 50m 4 x 50m Rectangularity (L/W)

36 25 Nominal v/m = 1 Gradient = steep Rotation = Effort (hrs) x 5m Area Rectangularity (L/W)

37 25 Nominal v/m = 1 Gradient = steep Rotation = Effort (hrs) x 5m Area Rectangularity (L/W)

38 Summary Preliminary data is essential to designing cost effective sampling programs Within-subject and reduced-frequency sampling designs are generally more cost effective Field data combined with computer simulations can provide concrete guidance on sampling methods and study design

39 Think and calculate before you go to the field

40 Acknowledgements Lucas Bohnett Anastasia Chavez Jen Carah Numerous students from Biology Field Methods in Ecology