Sampling Designs• 1. Simple random sampling (SRS)Steps:– (1) Assign a single number to each element in the sampling frame. – (2) Use random numbers to select elements into the sample until the desired number of cases is obtained. • The method is not very different from winning a lottery.2. Systematic Sampling•Steps:– (1) Calculate the sampling interval as the ratio between population size and sample size, I = N/n. – (2) Arrange all elements in the population in an order. – (3) Select a case in the first intervalrandomly. – (4) Select every ithcase from this point.2. Systematic Sampling (continued) – Systematic Sampling is easier and simpler than SRS– The text warns of a danger to this method. What is it?I1st element, randomly chosenII II I3. Stratified Sampling• Stratified sampling is more complicated than SRS. The advantage is the guaranteed representativeness in some important characteristics.• For example, say that we want to select a sample of 100 individuals. Sex ratio in the sample is up to chance if we do SRS. We can guarantee the 50-50 split if we do stratified sampling:Stratified Sampling, Graphic Representation• Stratified sampling is often used to reduce the variability of a sample.SRSSRSPopulationSampleOversampling, Graphic Representation• Increasing the representation of a group in a sample. This is often done when groups are very different in size – e.g., race SRSSRSPopulationSample4. Cluster Sampling• Cluster sampling is desirable from an economic point of view: • It saves money, at the expense of lowering the quality of data. • e.g.: we are interested in studying students' experiences at the University of Wisconsin, the unit of analysis is the student. An economic way is to sample classes. Once a class is selected, everyone in the class is selected.Cluster Sampling, Graphic RepresentationCollection of selected elementsCluster Sampling, Loss of Efficiency• For cluster sampling to work well, we assume clusters do not differ radically from each other. • If this assumption is not true, => the sample has more variability than a sample obtained by SRS, resulting in inefficiency. • In general, we can only lose efficiency with cluster sampling.Sources of Variability in a Sample Statistic• 1. Population VariabilityAll elements in a population are inherently variant. • 2. Random SelectionPrecisely because elements in the population have different values on a variable, random selection is meaningful and necessary.Decomposing Variability• 1. ANOVA: Analysis of VarianceTotal variation =between-group variation + within-group variation.• Internal homogeneity => external heterogeneity => between-group variance is large (e.g., gender and height). • Internal heterogeneity => external homogeneity => between-group variance is small (e.g., gender and GPA).Effects of Stratification • Stratification reduces sampling variation.• Total variation - between-strata variation = within-strata variation. • The more heterogeneous are the strata externally (or equivalently, the more homogeneous internally), the greater the gain in precision arising from stratification.• Example of gun control law and region.Design Effect• The ratio of the variance of the estimator based on the complex design to the variance of the estimator based on simple random sampling of the same size. •D2(z) = V(z)/V(z0)• For stratified samples, D2≤ 1. That is, stratified samples cannot be less efficient than simple random samples. D2= 1 if strata do not differ from each other.Effects of Clustering• Clustering increases sampling variation. • For a cluster sample, the Design Effect (D2) ≥ 1. That is, cluster samples cannot be more efficient than simple random samples. D2= 1 if clusters do not differ from each other. • Example of cluster sampling of individuals based on state of residence.Cluster sampling vs. Stratification. • Since strata are all represented in the sample, it is advantageous if they are internally homogeneous (i.e., externally heterogeneous). • With cluster sampling, it is best when the clusters are internally heterogeneous in the characteristics being studied. • The cluster size also affects the design effect. The larger the cluster size, the less efficient is the sample (i.e., the higher the variance).Practical Matters• 1. Implicit Stratified SamplingCombining systematic sampling with stratified sampling. • 2. Multi-stage Cluster SamplingCombining cluster sampling at one level with cluster sampling at another level.Combination of Clustering and Stratification• You may have cluster sampling at a higher level and have stratification at a lower level. e.g., cluster sampling of counties first, and then stratification by race and sex second. • Or, you may have stratified sampling at a higher level and have cluster sampling at a lower level. e.g., stratify regions, and then cluster sampling of counties.Probability Proportionate to Size Sampling• 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 are selected. For individual i in cluster j (by the rule of conditional probability):• P(ij) = P(i|j) p(j) = (1/Nj) ×Nj/ N = 1/N, equal probability for the element.Disproportionate Sampling and Weighting• Disproportionate sampling usually is a kind of stratified sampling where strata have different sampling ratios. Example: oversampling of American Indians. • Weighting is necessary. Weight is usually the inverse of the sampling ratio (e.g., if blacks sampled at three times population representation we would weight each black respondent by 1/3).Alternative Ways of Sampling• 1. Convenience SamplingSoc. 357 class in 2009. • 2. Quota SamplingMatching overt characteristics. No random selection. – Problem: bias in other characteristics. • 3. Purposive sampling• 4. Snowball
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