Sampling Designs Avery and Burkhart, Chapter 3 Source: J. Hollenbeck

Complete Enumeration (Census) 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

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

Sampling Designs Covered Today: Simple Random Sampling Systematic Sampling Stratified Sampling FOR 220 Aerial Photo Interpretation and Forest Measurements

Assumptions: Sampling Design I: Simple Random Sampling 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

Design: Sampling Design I: Simple Random Sampling 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 mean: Estimate the variance of individual values: Compute the coefficient of variation: FOR 220 Aerial Photo Interpretation and Forest Measurements

Sampling Design I: Simple Random Sampling Compute the standard error: OR (with replacement, or infinite population) (without replacement and from a finite population) Compute confidence limits: t = t statistic from table, 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

How many samples to take? 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 (withoout replacement, or finite population): Knowing the coefficient of variation and that error is expected to be within x % of the value of the mean, we use the following forumula n = where, n = required sample size, A = allowable error percent, t = t-value, CV = coefficient of variation, and N = population size

Sampling Design I: Simple Random Sampling Determine sample size (withoout replacement, or finite population): Example using 0.10 ac fixed radius plots in a ten acre stand Assume: We have calculated the CV and found it was 30% Our allowable error is +/- 10% of the mean Are using t-value of 2 N is the total number of potential plots that could be placed in the stand (i.e., population). In this case, N is stand size/plot size or 10/0.10 = 100 n = = 26 n =

Assumptions: Sampling Design II: Systematic Sampling 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

Design: Sampling Design II: Systematic Sampling A Grid Scheme is most common FOR 220 Aerial Photo Interpretation and Forest Measurements

Arguments: For: *** Against: Sampling Design II: Systematic Sampling 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

Arguments: For: Against: Sampling Design II: Systematic Sampling 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

Summary: Sampling Design II: Systematic Sampling 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

Assumptions: Sampling Design III: Stratified Sampling A population (i.e., tract) is subdivided into subpopulations (i.e., stands) of known sizes. A simple random sample (or systematic sample) is completed in each subpopulation (i.e., stand). Why? To obtain a more precise estimate of the tract mean. If the variation within a stand is small in relation to the tract 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 each stand. FOR 220 Aerial Photo Interpretation and Forest Measurements

Design: Sampling Design III: Stratified Sampling FOR 220 Aerial Photo Interpretation and Forest Measurements

Sampling Design III: Stratified Sampling Estimate the overall tract mean: Where: << Essentially a weighted average >> We often use area (acres) for stand units in natural resource applications; total stand volume may also be used. FOR 220 Aerial Photo Interpretation and Forest Measurements

Sampling Design III: Stratified Sampling Estimate the overall tract mean: Example: 76 acre tract Stand Acreage Volume (bdft per acre) 1 10 981.6 2 18 2017.9 3 48 3859.7 FOR 220 Aerial Photo Interpretation and Forest Measurements

Sampling Design III: Stratified Sampling Estimate the overall population standard error of the mean: 1. First compute variance within each strata Where: nh = total number of [tree] units sampled in strata h FOR 220 Aerial Photo Interpretation and Forest Measurements

(without replacement) Sampling Design III: Stratified Random Sampling Estimate the overall population standard error of the mean: 2. Second, compute population standard error of the mean (with replacement) OR (without replacement) Remember: N = [area] units, n = [tree] 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: FOR 220 Aerial Photo Interpretation and Forest Measurements

Sampling Design III: Stratified Sampling Calculating Sample Size for Stratified Random Sample: Specify the sampling error objective for tract as a whole Subdivide (stratify) the tract (i.e., population) into sampling components (i.e., stands). The purpose is to reduce the coefficient of variation within stands. Calculate coefficient of variation by stand and a weighted CV for the tract using preliminary cruise data. Calculate number of plots for the tract, then allocate by stand

Sampling Design III: Stratified Sampling Calculating Sample Size for Stratified Random Sample: An Illustrative Example

Sampling Design III: Stratified Sampling

Sampling Design III: Stratified Sampling

Sampling Design III: Stratified Sampling