1. Simulation for Examining Margin of Error and Sample Size: Binomial Proportions Acknowledgements to Mandy Kauffman (WEST, Inc.) for photos and ‘background’ slides…simulation exercise adapts pedagogy of Trumbo,Suess, and Okumura (2005)

2. Background – Bovine brucellosis Bacterial disease –History in US –Elk, bison, cattle (humans) –Cattle  wildlife –Causes abortions –Environmental contamination –Potential transmission to cattle $$$$ Management implications 2

3. Harsh winters + development  elk starving, commingling with cattle 23 supplemental winter elk feedgrounds created 22 WGFD 1 USFWS Up 84% of elk use feedgrounds Low winter mortality Costly 22 % seroprevalence on feedgrounds 3.7% elsewhere Background - Elk Feedgrounds 3 Preble 1911

4. Background – Management Management strategies 1.Maintain cattle/elk separation -hazing elk-fencing haystacks -elk feedgrounds 2. ↓ likelihood of exposed cattle experiencing abortions (RB51) 3.↓ seroprevalence in elk -T&S-low density feeding -elk vaccination

5. Background: Management Despite ongoing management: –Recent cases in cattle/bison traced back to elk –Affected area expanding Limited $$ available for management –No clear scientifically sound method –Need for economic evaluation of available management strategies Groups 1 & 2 already evaluated/underway Evaluation of Group 3 strategies still needed –How to assess sero-prevalence of brucellosis in elk on feedgrounds…how many elk to sample?

6. A Beginning R Code: samp <- sample(0:1,25,rep=T,prob=c(0.78,0.22)) samp Let’s start with simulating brucellosis diagnosis from 25 randomly sampled elk…assume prevalence is 0.22…goal to estimate prevalence within 5%...how many elk needed? Issue with the assumption of random here?

7. Generate a Profile Let’s observe the incidence rate for a variety of sample sizes...keep in mind that this profile doesn’t display independent samples R Code: n <- 25 NumElk <- 1:n p.bruc <- c(0.78,0.22) x <- sample(0:1,n,rep=T,prob=p.bruc) run.tot.pos <- cumsum(x) Proportion <- run.tot.pos/NumElk tabresults <- round(cbind(NumElk,x,run.tot.pos,Proportion),3) tabresults

8. Generating a Profile R Code: plot(NumElk,Proportion,type="l", ylim= c(0,1)) abline(h=0.22,col="green",lwd=2) abline(h=0.17,col="blue",lwd=2,lty=3) abline(h=0.27,col="blue",lwd=2,lty=3) Running the simulation and corresponding plots several times will provide differing versions of the profile on the next page… variation in profiles displays the instability in our statistic

10. Generating Multiple Profiles R Code: set.seed(11); n <- 25; numsamps <- 20 plot(0,pch=" ", xlim=c(0,n), ylim = c(0.1,1.0), xlab = "NumElk", ylab="Proportion") #Loop below will produce a different profile for each of the specified #numsamps… for(i in 1:numsamps){ x <- cumsum(sample(0:1,n,rep=T,prob=p.bruc)) / (1:n) lines(1:n,x) } abline(h=0.22,col="green",lwd=2) abline(h=0.17,col="blue",lwd=2,lty=3) abline(h=0.27,col="blue",lwd=2,lty=3)

12. How about 263 elk?

13. Power is a concept that is often difficult to teach, regardless of the level of the course…probably because hypothesis testing is such a strange beast! Many practical applications involve evaluating sampling protocols in terms of the ability to detect change over a period of time Simulation often quite effective for determining power associated with a particular design Simulation for Determining Power

14. Cobble bars are an important feature for many streams…home of native and non-native plants and many species of birds nest in these regions What is the proportion of ‘woody vegetation’ cover on cobblebars in the Great Smokey Mountains? Monitoring Woody Vegetation in Cobble Bars

15. - 22 cobble bars exist in the BISO area…can afford to sample 9 of them -GRTS selection of sampling units - rotating panel monitoring schedule

16. transect Point intercept counts on transect

23. As a Single Function

24. The code to the left computes as we change the effect size (i.e. the per annum change)

25. Just as we would expect, the power increases with larger trends