Fundamentals of Sampling
Sampling provides methods for selecting subsets from populations to make inferences about the whole. Proper sampling enables valid conclusions while reducing data collection costs and time. Understanding sampling methods is essential for designing studies and interpreting data.
The goal of sampling is to obtain a representative subset that reflects population characteristics. Poor sampling leads to biased estimates regardless of analysis sophistication. Even big data cannot correct fundamental sampling problems.
Sampling theory bridges descriptive and inferential statistics. It provides the foundation for understanding how sample statistics relate to population parameters. This connection enables valid inference from samples to populations.
Simple Random Sampling
Simple random sampling provides the foundation for sampling theory. Each member of the population has an equal chance of selection. This approach is conceptually straightforward and provides certain theoretical properties.
Random Number Generation
Modern sampling uses random number generators to select units. Pseudo-random number generators produce deterministic sequences appearing random. They are sufficient for most practical purposes. True random number generators use physical processes for additional randomness.
Random selection can use tables of random numbers, random number functions in software, or systematic approaches. The key is ensuring each unit has equal selection probability.
The sample should be drawn randomly even when systematic or convenience sampling seems easier. Convenience sampling introduces selection bias that undermines inference.
Properties of Simple Random Samples
Simple random samples are unbiased estimators of population parameters. The expected value of the sample mean equals the population mean. The expected value of other statistics generally equals population values.
Variability of simple random sample estimates depends on sample size and population variability. Larger samples provide more precise estimates. Less variable populations yield more precise estimates.
The sampling distribution describes how statistics vary across samples. Understanding sampling distributions enables inference about population parameters from sample statistics.
Stratified Sampling
Stratified sampling divides the population into homogeneous subgroups (strata) and samples within each stratum. This approach can improve precision when strata differ in the variable of interest.
Stratification Process
Strata should be internally homogeneous and externally different. Typical stratification uses demographic characteristics like age, gender, or geography. The goal is groups with different values on the outcome variable.
Proportional allocation samples proportionally across strata based on their sizes. This approach is efficient when stratum variances are similar. Disproportional allocation samples more from high-variance strata when they differ.
Post-stratification adjusts for known population proportions. This can improve estimates when strata sizes in the sample differ from population proportions.
Advantages and Considerations
Stratified sampling often provides more precise estimates than simple random sampling, especially when strata means differ substantially. It ensures representation of small subgroups that might be missed in simple random sampling.
The approach requires knowledge of population stratum composition. It also requires sample within each stratum. Stratification might increase design complexity but often improves efficiency.
Analytic methods should account for stratification when analyzing data. Weighted analyses using stratum information preserve efficiency gains from the design.
Cluster Sampling
Cluster sampling groups population units into clusters and samples clusters rather than individual units. This approach is useful when population units are naturally grouped or when no sampling frame exists.
Cluster Design
Clusters should be internally heterogeneous (like mini-populations) and externally similar to each other. School classes, neighborhoods, or hospitals might serve as clusters. The choice depends on the population structure and data collection practicality.
One-stage cluster sampling selects all units within selected clusters. Two-stage cluster sampling selects units within selected clusters, then samples within those. Multi-stage designs can be cost-effective for large populations.
Cluster sampling is often less efficient than simple random sampling when clusters are similar to each other. The intraclass correlation describes how similar units within clusters are.
Applications
Cluster sampling is common in household surveys, where blocks or municipalities are clusters and households within them are sampled. It is also used in medical research sampling patients within clinics.
The approach reduces travel costs when clusters are geographically concentrated. It simplifies sampling frame construction by listing clusters rather than individuals.
Analysis should account for cluster structure. Standard errors computed ignoring clustering underestimate uncertainty. Cluster-robust standard errors or hierarchical models account for within-cluster correlation.
Systematic Sampling
Systemistic sampling selects units at regular intervals from an ordered list. After a random start, every kth unit is selected, where k equals population size divided by desired sample size.
Implementation
First, order the population (by any variable, but random ordering is typical). Then, select a random starting point between 1 and k. Finally, select every kth unit after the starting point.
Systematic sampling is simple to implement in the field. It spreads the sample evenly across the population. It can be nearly as efficient as simple random sampling when there is no cyclic pattern.
Cyclic patterns in the ordering can create bias if the cycle matches the sampling interval. Checking for periodicity in the variable of interest is important.
Random Start and Interval
The random start determines which unit is selected first. Different random starts produce different samples. The interval k is determined by sample size requirements.
For population N and desired sample n, k ≈ N/n. This should be close to integer. If not, some randomization in the interval selection might be needed.
Systematic sampling with a random start is equivalent to simple random sampling if there is no pattern in the ordering. This equivalence is often used for analysis.
Ratio and Regression Estimation
Ratio and regression estimation use auxiliary information to improve precision when strong relationships exist between the variable of interest and auxiliary variables.
Ratio Estimation
Ratio estimators use the relationship between the variable of interest and a related auxiliary variable. For each unit, we observe both variables. The ratio of population totals is estimated using the sample ratio.
The approach is particularly valuable when the auxiliary variable is strongly correlated with the outcome. The correlation improves precision. It requires knowing auxiliary values for the population.
Examples include using population to estimate area, or using previous period values to estimate current period values. The technique uses the known relationship to improve estimates.
Regression Estimation
Regression estimators use linear relationships with auxiliary variables to improve estimates. They model the relationship between outcome and auxiliary variables and use the model to adjust estimates.
The regression estimator is more flexible than ratio estimators, allowing multiple auxiliary variables and different functional forms. It can improve precision substantially when relationships are strong.
The auxiliary variable information must be known for population totals or means. This might come from census data, administrative records, or other sources.
Sample Size Determination
Determining appropriate sample sizes balances precision requirements against costs. This involves specifying desired precision, estimating population variability, and considering budget constraints.
Precision-Based Sample Size
Sample size for estimating means depends on desired confidence level, acceptable margin of error, and population standard deviation. Larger confidence levels, smaller margins of error, and larger variability require larger samples.
For proportions, the variance depends on the proportion itself. Using p = 0.5 provides the most conservative (largest) sample size. This assumes maximum variability.
These calculations assume simple random sampling. Adjustments are needed for complex sampling designs like cluster or stratified sampling.
Budget and Cost Considerations
Practical sample size depends on available resources. The cost per observation times desired sample size must fit the budget. Costs vary by data collection mode and population.
Cost-effective designs balance precision and cost. Stratified sampling might achieve required precision more cheaply. Cluster sampling might reduce travel costs.
Pilot studies help estimate variability and costs before full study implementation. This information improves sample size calculations.
Non-Probability Sampling
Non-probability sampling selects units without known selection probabilities. While convenient, this approach complicates inference because selection bias is difficult to quantify.
Convenience Sampling
Convenience sampling selects readily available units. It is quick and inexpensive but might produce biased estimates. The direction and magnitude of bias depend on how available units differ from the target population.
Generalizing from convenience samples requires strong assumptions about representativeness. These assumptions are often not justifiable. Results should be interpreted cautiously.
Examples include surveying students available in a particular class or using volunteers. These samples rarely represent broader populations.
Purposive Sampling
Purposive sampling selects units based on specific characteristics. It might select cases representing particular viewpoints, extreme cases, or typical cases. The selection is based on judgment rather than random selection.
This approach is appropriate for qualitative research or specific methodological purposes. It is not appropriate for generalizing to populations.
Judgement sampling selects cases based on expert judgment about representativeness. Snowball sampling starts with a few cases who recruit additional cases. These approaches are useful for hard-to-reach populations.
Complex Survey Designs
Real surveys often combine multiple design features. Complex designs require specialized analysis methods to produce valid estimates and appropriate measures of uncertainty.
Design Features
Complex surveys might stratify, cluster, and sample with unequal probabilities in combination. They might include multiple stages of selection and various adjustment mechanisms.
Weighting accounts for differential selection probabilities and nonresponse. Survey weights should be used in analysis to produce unbiased estimates. Weights can be substantial.
Analysis methods should reflect the design structure. Standard errors computed ignoring design features can be seriously underestimated.
Analysis of Complex Survey Data
Specialized software handles complex survey designs. SAS, Stata, R (with survey package), and SPSS have procedures for survey analysis. These implement design-based inference accounting for stratification, clustering, and weighting.
Analysis options include Taylor series linearization or replicate methods for variance estimation. Replicate methods like jackknife or bootstrap can handle complex designs.
Many standard statistical procedures have survey-specific versions. T-tests, regressions, and other methods have survey-appropriate alternatives. Using these procedures produces correct inference.
Small Area Estimation
Small area estimation produces estimates for geographic areas or domains with small sample sizes. These areas might have too few observations for direct reliable estimates.
Direct and Model-Based Estimates
Direct estimates use only data from the specific area. They might be unreliable for small areas with few observations. Precision varies across areas.
Model-based estimates borrow information from similar areas using statistical models. Fay-Herriot models combine direct and synthetic estimates. Spatial models borrow information from neighboring areas.
Model-based estimates are more precise but depend on model assumptions. Checking model fit and evaluating predictions is important.
Applications
Small area estimation is used for county-level estimates in health surveys, school district estimates in educational assessments, and neighborhood estimates in demographic studies.
Federal statistical programs often produce small area estimates. The Census Bureau, National Center for Health Statistics, and other agencies provide these estimates.
Model-based estimates should include measures of uncertainty. Users should understand the distinction between direct and model-based estimates.
Key Takeaways
- Simple random sampling provides the foundation for sampling theory with known properties
- Stratified sampling improves precision by sampling within homogeneous groups
- Cluster sampling is useful for geographically dispersed populations or when no frame exists
- Sample size determination balances precision requirements against cost constraints
- Non-probability sampling complicates inference due to unknown selection probabilities
- Complex survey designs require specialized analysis methods accounting for design features