Class 12

1. Class 12 The Logic of Sampling

2. Class Outline • Capture-Recapture Method • Sample Representation • Probability Sampling • Nonprobability Sampling

3. Fun Example: How Do Scientists Estimate the Number of Pandas in a Closed Region? • The capture-recapture method • You first capture n1 pandas, mark them, and release them. • You then capture n2 pandas and see how many of them had been captured before. • Intuition: the higher the proportion in both captures, the smaller the population size.

4. The Capture-Recapture Method The calculation of the total number of pandas assumes that capture is random. Specifically, the probability of being captured at time 1 is independent of that at time 2.

5. Probability Sampling Each element in the population has a known probability of selection. A sampling frame is used. Avoids researcher’s biases in element selection. Nonprobability Sampling The probability of selection is unknown. Typically no sampling frame is used. The selection process may be biased. Probability and Nonprobability Sampling

6. Sample Representation Who do you want to generalize to? Theoretical population Elements not captured by sampling frame Population represented by the sampling frame Who can you get access to? Target sample Non-response Who’s in your study? Actual sample

7. Sample Size and Population Homogeneity • The larger the sample, the more confidence we can have in the sample’s representativeness. • The more homogeneous the population, the more confidence we can have in the representativeness of a sample of any particular size. • The fraction of the total population that a sample contains does not affect the sample’s representativeness.

8. Simple Random Sampling (SRS) Steps: • Assign a unique number to each element in the sampling frame. • Use random numbers to select elements into the sample until the desired number of cases is obtained. • Use a table of random numbers • Use computers to generate random numbers.

9. Systematic Sampling Steps: • Calculate the sampling interval as the ratio between population size and sample size, I = N/n. • Arrange all elements in the population in an order. • Select an element in the first interval randomly. • Select every Ith element from this point.

10. Systematic Sampling I I I I I I 1st element is randomly chosen • Systematic sampling is easier than simple random sampling (SRS). • But there is a potential danger. What is it?

11. Stratified Sampling Steps • Divide the population into subpopulations (called strata), say, males and females. • Among males, we select cases randomly using SRS or systematic sampling; among females, we also select cases randomly. • The resulting sample will guarantee the desired numbers of males and females. We could use any other characteristics, such as region, ethnicity, age, or education to define the strata. Stratified sampling guarantees representativeness in these characteristics.

12. Stratified Sampling Population Stratum 1 Stratum 2 SRS SRS Sample

13. Advantage and Disadvantage of Stratified Sampling • Stratified sampling ensures representativeness in the stratifying variables by decreasing the probable sampling error. • Stratified sampling is advantageous when strata are vastly different from each other for the variables of interest. • It requires some knowledge about the elements in the sampling frame, which may be unavailable. E.g., in order to stratify the sample by education, we must know the education of each individual in the sampling frame.

14. Stratified Sampling: Oversample • Oversampling increases the representation of a particular group in a sample. Population SRS SRS Sample

15. Multistage Cluster Sampling • A cluster is a natural aggregate of elements of the population. • Steps • List all clusters. • Draw a random sample of clusters. • List all elements in the selected clusters. • Draw a random sample of elements from each cluster. • Multistage cluster sampling: • states  cities schools  students

16. Multistage Cluster Sampling Selected clusters

17. Advantage and Disadvantage of Cluster Sampling • Cluster sampling is desirable from an economic point of view. • It saves money but lowers the quality (representativeness) of data. • Cluster sampling is not a good sampling choice if the clusters are very different from each other. • The trade-off between the number of clusters and the number of elements selected within clusters.

18. Probability Proportionate to Size • A type of cluster sampling where a cluster's probability of being selected is proportional to its size. That is, the larger a cluster, the higher its probability of being selected. • Within each cluster, a fixed number of cases is selected. For element i in cluster j (by the rule of conditional probability): • Equal probability for each element

19. Nonprobability Sampling 1. Reliance on available subjects • Only justified if less risky sampling methods are not possible. • Researchers must exercise great caution in generalizing from their data when this method is used.

20. Nonprobability Sampling 2. Purposive or judgmental sampling • Selecting a sample on the basis of knowledge of a population, its elements, and the purpose of the study. • Often used when field researchers are interested in studying cases that don’t fit into regular patterns of attitudes and behaviors

21. Nonprobability Sampling 3. Snowball sampling • Appropriate when members of a population are difficult to locate (homeless, migrant workers, undocumented immigrants). • Researcher collects data on members she can locate, then asks those individuals to help locate other members of that population.

22. Nonprobability Sampling 4. Quota sampling • Begins with a matrix of the target population. • Data is collected from people with the characteristics of a given cell. • Each group is assigned a weight appropriate to their portion of the total population. • When the elements are properly weighted, the data should provide a representation of the total population.