• 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
– (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 interval randomly. – (4) Select every ith case from this point.
2. Systematic Sampling (continued) I
1st element, randomly chosen
– Systematic ...view middle of the document...
• 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 Variability All elements in a population are inherently variant. • 2. Random Selection Precisely because elements in the population have different values on a variable, random selection is meaningful and necessary.
• 1. ANOVA: Analysis of Variance
Total 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.
• 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...