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FOR 220 Aerial Photo Interpretation and Forest Measurements Lecture 4 Sampling Designs Avery and Burkhart, Chapter 3 Why/Why Not Randomly Select Samples? We will discuss several sampling designs in this lecture • Most statistical methods assume simple random sampling was used. • Sampling methods, however, will vary depending on the objectives of the survey, the nature of the population being sampled, and prior information about the population being sampled. FOR 220 Aerial Photo Interpretation and Forest Measurements Why Not Measure Everything? Complete Enumeration (Census) Measure every feature of interest. The result is a highly accurate description of the population. Drawbacks: Only viable with small populations. Only cost-effective with high-valued features. FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Frame Sampling Frame The list of all possible units that might be drawn in a sample. • Developing a reliable frame may be difficult (e.g., campgrounds) • Access How is a sampling frame different than a population? FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Designs Sampling Design The method of selecting [non-overlapping] sample units to be included in a sample. FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Designs Sampling Designs Covered Today: 1. Simple Random Sampling 2. Systematic Sampling 3. Stratified Random Sampling FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling Assumptions: Every possible combination of sampling units has an equal and independent chance of being selected. The selection of a particular unit to be sampled is not influenced by the other units that have been selected or will be selected. Samples are either chosen with replacement or without replacement. FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling With and without replacement? Why worry about replacement? Small populations – samples may “remove” considerable portion. Removes “independence” N 12 n -6 5 FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling Design: FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling We use the familiar equations for estimating common statistics for the population Estimate the population x mean: x n Estimate the variance of 2 x 2 x 2 / n s individual values: n 1 Compute the coefficient CV s of variation: x FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling Compute the standard error: s2 s2 N n SE SE n N OR n (with replacement, or (without replacement and infinite population) from a finite population) Compute confidence limits: t = t statistic from table, 95% CI x t SE determined by degrees of freedom (n-1). In general, it is safe to use: t = 2.0 for small n (n < 30), or t = 1.96 for large n (n > 30) if you do not have access to a t-table. FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling Finite population term: N n N = Population size n = sample size N s2 N n SE N Standard error (without replacement n and from a finite population) FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling How many samples to take? • Sample size should be statistically and practically efficient. • Enough sample units should be measured to obtain the desired level of precision (no more, no less). FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling Determine sample size (with replacement, or infinite population) Knowing the standard deviation and the desired 2 t s objective (e.g., to be within xxx board feet per acre) n we can calculate the necessary minimum sample size. E Stein’s Formula Example: s = 2000 board feet / acre 2 2 (2000) t = 2 (assuming a 95% confidence interval) 500 64 n E = want to be within +/- 500 board feet / acre (E must be in same units as std. deviation) FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling Determine sample size (with replacement, or infinite population): Knowing the coefficient of variation and the 2 t CV desired objective (e.g., error is expected to be n A within x % of the value of the mean) Example: CV = 30% t = 2 (assuming a 95% confidence 2 2 (30) interval) 5 144 n A = we want the allowable error to be within +/- 5% of the mean (A is expressed as a percentage of the mean) FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design I: Simple Random Sampling How many samples to take? • How do we know variance in population before we sample it?!?! FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design II: Systematic Sampling Assumptions: The initial sampling unit is randomly selected or established on the ground. All other sample units are spaced at uniform intervals throughout the area sampled. Sampling units are easy to locate. Sampling units appear to be representative of an area. FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design II: Systematic Sampling Design: A Grid Scheme is most common FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design II: Systematic Sampling Arguments: For: Regular spacing of sample units may yield efficient estimates of populations under certain conditions. *** Against: Accuracy of population estimates can be low if there is periodic or cyclic variation inherent in the population. FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design II: Systematic Sampling Arguments: For: There is no practical alternative to assuming that populations are distributed in a random order across the landscape. Against: Simple random sampling statistical techniques can’t logically be applied to a systematic design unless populations are assumed to be randomly distributed across the landscape. FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design II: Systematic Sampling Summary: We can (and often do) use systematic sampling to obtain estimates about the mean of populations. When an objective, numerical statement of precision is required, however, it should be viewed as an approximation of the precision of the sampling effort. (i.e. 95% confidence intervals) Use formulas presented for simple random sampling, and where appropriate, use the “without replacement” variations of those equations (if sampling from a small population), otherwise use the normal SRS statistical techniques. FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design III: Stratified Random Sampling Assumptions: A population is subdivided into subpopulations of known sizes. A simple random sample of at least two units is drawn from each subpopulation. Why? To obtain a more precise estimate of the population mean. If the variation within a subpopulation is small in relation to the total population variance, the estimate of the population mean will be considerably more precise than a simple random sample of the same size. Why? To obtain an estimate of the resources within the subpopulations. FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design III: Stratified Random Sampling Design: Simple random samples within 3 strata FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design III: Stratified Random Sampling Estimate the overall population mean: Where: L L = number of strata Nh y h y st h 1 Nh = total number of [area] units in strata h N = total number of [area] units in all strata N << Essentially a weighted average >> We often use area for strata units (acres, hectares) in natural resource applications FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design III: Stratified Random Sampling Estimate the overall population mean: Example: Strata acres mean dbh 1 10 12.2 2 12 31.6 3 7 20.1 (10)(12.2) (12)(31.6) (7)(20.1) y st 22.1 inches 29 FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design III: Stratified Random Sampling Estimate the overall population standard error of the mean: 1. First compute variance within each strata x h x h 2 sh 2 nh 1 Where: nh = total number of [tree or plot] units sampled in strata h 2 x h 2 x h 2 / nh sh nh 1 FOR 220 Aerial Photo Interpretation and Forest Measurements Avery and Burkhart formula Sampling Design III: Stratified Random Sampling Estimate the overall population standard error of the mean: 2. Second, compute population standard error of the mean 1 L Nh 2 sh 2 SE st 2 (with replacement) N h 1 nh OR 1 L Nh sh Nh nh 2 2 (without replacement) SE st 2 N h1 nh Nh Remember: N = [area] units, n = [tree or plot] units FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design III: Stratified Random Sampling Estimate the overall population standard error of the mean: 3. Compute confidence intervals if desired Estimated confidence intervals: y st t SE st FOR 220 Aerial Photo Interpretation and Forest Measurements Sampling Design III: Stratified Random Sampling Stratum (h) Stratum Size ac. (N) Stratum Mean Example: Mean 1 20 43 2 40 80 3 70 27 4 60 56 5 50 63 (20 43) (40 80 ) (70 27 ) (60 56 ) (50 63) yst 51 .9 (20 40 70 60 50 ) 2 yst N yh h 4 N 5 h 3 1 Sampling Design III: Stratified Random Sampling Example: Standard Error Stratum (h) Sample Size (n) Stratum Size ac. (N) Stratum Mean Stratum s2 1 10 20 43 285 2 25 40 80 472 3 30 70 27 295 4 30 60 56 304 5 25 50 63 387 1 20 2 285 40 2 472 70 2 295 60 2 304 50 2 387 SEst 2 240 10 25 30 30 25 SEst 2.86 N h sh 2 2 1 2 SEst ( N h ) nh Sampling Design III: Stratified Random Sampling Example: Confidence Interval Remember: Mean = 51.9 95 %CI yst SEst SE = 2.86 Overall n > 30, t = 1.96 95%CI 51.9 1.96 2.86 95%CI 51.9 5.6 Sampling Design III: Stratified Random Sampling Minimum sample size – modified Stein’s formula: t s 2 2 N t Nh s 2 2 n 2 n h AE N AE 2 2 Here, N = Total N (N1 + N2 + …Nn) AE = Allowable Error Example: Estimate Average Age of Campers: 28 41 27 27 29 Complete enumeration: (Census) 32 35 26 29 23 45 63 38 Mean age = 35.95 years 38 46 19 31 41 56 35 25 19 19 55 Shower 48 26 25 22 47 27 33 54 56 26 32 42 28 58 48 39 FOR 220 Aerial Photo Interpretation and Forest Measurements Example: Estimate Average Age of Campers: Simple 28 41 27 27 29 Random Sampling: 32 35 26 29 23 n = 10 45 63 38 38 46 19 31 41 56 35 Average age = 25 19 19 55 38.4 7.6 years Shower 48 26 25 30.8 to 46.0 22 47 27 33 54 56 26 32 42 28 58 48 39 True average = 35.95 FOR 220 Aerial Photo Interpretation and Forest Measurements Example: Estimate Average Age of Campers: Simple 28 41 27 27 29 Random Sampling: 32 35 26 29 23 n = 10 45 63 38 38 46 19 31 41 56 35 Average age = 25 19 19 55 40.2 8.5 years Shower 48 26 25 22 47 31.7 to 48.7 27 33 54 56 26 32 42 28 58 48 39 True average = 35.95 FOR 220 Aerial Photo Interpretation and Forest Measurements Example: Estimate Average Age of Campers: Systematic 27 28 41 27 29 Sampling: 32 35 26 n = 10 29 23 45 63 38 38 46 19 31 41 56 35 Average age = 25 19 19 55 37.7 8.0 years Shower 48 26 25 22 47 29.7 to 45.7 27 33 54 56 26 32 42 28 58 48 39 True average = 35.95 FOR 220 Aerial Photo Interpretation and Forest Measurements Example: Estimate Average Age of Campers: Every 4th camper Systematic 27 28 41 27 29 Sampling: 32 35 26 n = 10 29 23 45 63 38 38 46 19 31 41 56 Average age = 35 25 19 19 37.6 8.2 years 55 Shower 48 29.4 to 45.8 26 25 22 47 27 33 54 56 26 32 42 28 58 48 39 True average = 35.95 FOR 220 Aerial Photo Interpretation and Forest Measurements Example: Estimate Average Age of Campers: Systematic 27 28 41 27 29 Sampling: 32 35 26 n = 15 29 23 45 63 38 38 46 19 31 41 56 Average age = 35 25 19 19 38.5 5.9 years 55 Shower 48 32.6 to 44.4 26 25 22 47 27 33 54 56 26 32 42 28 58 48 39 True average = 35.95 FOR 220 Aerial Photo Interpretation and Forest Measurements Class Example: Timber Cruise Stand ~ 32 acres Lets use plots and a systematic sampling scheme typical for a timber cruise FOR 220 Aerial Photo Interpretation and Forest Measurements Class Example: Timber Cruise 32 plots cruise lines # # 5 chains apart # # # # 4 chains # # # between plots # # # # # # # 32 acres = 320 square chains 320/16plots = 2 ac. represented by each plot 2 acres = 20 square chains Plot spacing: (5 x 4 = 20) Randomize location of first plot FOR 220 Aerial Photo Interpretation and Forest Measurements Class Example: Lab 2 1/10 acre plots (37.2 ft radius) # # # # # # # # # # # # # # # # FOR 220 Aerial Photo Interpretation and Forest Measurements What happened here? FOR 220 Aerial Photo Interpretation and Forest Measurements What is this? What is the red “stuff”? FOR 220 Aerial Photo Interpretation and Forest Measurements