1. Lecture 6 Forestry 3218 Forest Mensuration II Lecture 6 Double Sampling Cluster Sampling Sampling for Discrete Variables Avery and Burkhart, Chapter 3

2. Lecture 6 Forestry 3218 Double Sampling (two-phase sampling) Double sampling with regression and ratio estimator Double sampling for stratification

3. Lecture 6 Forestry 3218 Double Sampling with Regression and Ratio Estimators Remember: regression and ratio estimators require known Take a large sample in which x alone is measured – allow a good estimate of Establish a regression or ratio relationship between paired x and y

4. Lecture 6 Forestry 3218 Double Sampling with Regression Estimate of the population mean of y

5. Lecture 6 Forestry 3218 Double sampling with regression vs. regression estimation Complete enumeration of x vs. a large sample of it Both gain precision from using regression estimators

6. Lecture 6 Forestry 3218 Double Sampling With Ratio where

7. Lecture 6 Forestry 3218 Double Sampling for Stratification Recall: stratified random sampling requires that the strata size ( N h ) be known in advance of sampling Double sampling for stratification applies when – N h is not known, but can be estimated by sampling

8. Lecture 6 Forestry 3218 Double Sampling for Stratification 1. Estimate N h using a large sample 2. Estimate overall population mean How is this different from that in stratified random sampling?

9. Lecture 6 Forestry 3218 Cluster Sampling A practical problem – A forester needs to estimate average seedling heights or root collar of a nursery. Seedlings are grown on benches, blocks, or clusters of styrofoam How are you going to sample?

10. Lecture 6 Forestry 3218 Cluster Sampling A cluster sample is a sample in which each sampling unit is a collection, or cluster, of elements Reasons 1. A list of elements is not available, but a list of clusters is 2. Even when a list of elements is available, it is more economical to randomly select clusters than individual elements

11. Lecture 6 Forestry 3218 Cluster Sampling We need to know: – How many clusters in the population ( N ) – How many clusters selected ( n ), often by simple random sampling – How many elements in a cluster ( m ) – Measured value for sampled elements ( y ij ), e.g., seedling height Estimation of population mean

12. Lecture 6 Forestry 3218 Two-stage Sampling What if there are too many elements in a cluster? For examples, – You want to know seedling dry weight of the previous example

13. Lecture 6 Forestry 3218 Sampling for Discrete Variables For qualitative attributes such as dead or alive, deciduous or evergreen – binomial distribution Species composition – multinomial distribution

14. Lecture 6 Forestry 3218 Sampling for Discrete Variables Estimate proportion Estimate standard error of the proportion Estimate confidence interval

15. Lecture 6 Forestry 3218 Sampling for Discrete Variables Use Cluster Sampling for Attributes – recall how we calculate mean, variance, and standard error of the mean for simple random sampling

16. Lecture 6 Forestry 3218 Relative Efficiencies of Sampling Plans Measure by cost or time with the same level of accuracy (not precision, why?) When samples are unbiased, standard error of mean can serve as a measure of accuracy Most efficient plan is: min { (standard error) 2 ×cost (time) } Remember: The objective of sampling design is to obtain a specified amount of information about a population parameter at minimum cost