Glossary

4.4 Analysis and estimation in stratified sampling

3 min read•august 9, 2024

Stratified sampling divides a population into subgroups, or strata, before sampling. This method ensures representation across all subgroups and can improve precision. The notes cover key concepts, implementation steps, and allocation methods for stratified sampling.

Analysis and estimation in stratified sampling involve calculating weighted means and variances from each . The notes explain how to compute stratified estimates, assess precision, and understand the benefits of stratification in improving estimation accuracy and sample representativeness.

Stratification and Sampling

Understanding Stratification Concepts

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  • Stratum represents a homogeneous within a population sharing common characteristics
  • Stratification divides a heterogeneous population into distinct, non-overlapping subgroups (age groups, income levels)
  • Stratified random sampling selects samples independently from each stratum, ensuring representation across all subgroups
  • Post-stratification applies stratification after data collection to improve estimation accuracy
    • Adjusts for unequal selection probabilities
    • Corrects for non-response

Implementing Stratified Sampling

  • Identify relevant stratification variables based on research objectives (geographic regions, educational levels)
  • Determine optimal number of strata balancing precision and cost
  • Allocate sample sizes to each stratum using appropriate allocation methods
  • Conduct simple random sampling within each stratum
  • Combine data from all strata for analysis and estimation

Allocation Methods

Proportional Allocation

  • Allocates sample sizes proportionally to stratum sizes in the population
  • Calculation: nh=n×NhNn_h = n \times \frac{N_h}{N}
    • n_h: sample size for stratum h
    • n: total sample size
    • N_h: population size of stratum h
    • N: total population size
  • Ensures representation of larger strata in the sample
  • Simple to implement and understand
  • Optimal when variances are equal across strata

Optimal Allocation

  • Allocates sample sizes based on stratum size and variability
  • Neyman allocation formula: nh=n×NhSh∑NhShn_h = n \times \frac{N_h S_h}{\sum N_h S_h}
    • S_h: standard deviation of the variable of interest in stratum h
  • Minimizes overall of the estimator
  • Assigns larger samples to strata with higher variability or larger sizes
  • Requires prior knowledge or estimates of stratum variances
  • More complex to implement than proportional allocation

Stratified Estimates

Calculating Stratified Mean and Variance

  • Stratified mean combines weighted means from each stratum
    • Formula: yˉst=∑h=1HWhyˉh\bar{y}_{st} = \sum_{h=1}^{H} W_h \bar{y}_h
    • W_h: stratum weight (N_h / N)
    • ȳ_h: sample mean of stratum h
  • Stratified variance measures variability of the stratified estimator
    • Formula: V(yˉst)=∑h=1HWh2sh2nhV(\bar{y}_{st}) = \sum_{h=1}^{H} W_h^2 \frac{s_h^2}{n_h}
    • s_h^2: sample variance of stratum h
    • n_h: sample size of stratum h

Assessing Precision and Confidence

  • Stratified quantifies the precision of the stratified mean estimate
    • Calculated as the square root of the stratified variance
  • provides a range of plausible values for the population parameter
    • Formula: yˉst±tα/2,df×SE(yˉst)\bar{y}_{st} \pm t_{\alpha/2, df} \times SE(\bar{y}_{st})
    • t_α/2, df: critical value from t-distribution
    • SE(ȳ_st): stratified standard error
  • Narrower confidence intervals indicate higher precision of estimates

Benefits of Stratification

Improving Estimation Precision

  • Gain in precision reduces sampling error compared to simple random sampling
    • Achieved by accounting for between-strata variability
    • Leads to smaller standard errors and narrower confidence intervals
  • Stratification effect measures the relative efficiency of stratified sampling
    • Calculated as the ratio of variances: deff=V(yˉSRS)V(yˉst)\text{deff} = \frac{V(\bar{y}_{SRS})}{V(\bar{y}_{st})}
    • Values greater than 1 indicate improved precision through stratification

Enhancing Sample Representativeness

  • Ensures representation of important subgroups in the sample
  • Allows for separate analysis of individual strata
  • Improves overall population estimates by combining stratum-level information
  • Facilitates comparisons between different strata
  • Reduces the impact of outliers on overall estimates

Key Terms to Review (19)

Better representation: Better representation refers to the increased accuracy and fairness in reflecting the characteristics of a population within a sample. This concept emphasizes the importance of capturing diversity, ensuring that all subgroups are appropriately included, particularly in stratified sampling methods where the population is divided into distinct strata or groups. By achieving better representation, researchers can make more reliable inferences about the overall population based on sample data.
Bias: Bias refers to a systematic error that leads to an inaccurate representation of a population in sampling or survey results. It can occur in various forms, affecting the validity and reliability of research findings. Understanding bias is crucial as it influences sampling designs, estimation processes, and ultimately the interpretation of data.
Confidence Interval: A confidence interval is a range of values, derived from a data set, that is likely to contain the true population parameter with a specified level of confidence, often expressed as a percentage. It provides an estimate of uncertainty around a sample statistic, allowing researchers to make inferences about the larger population from which the sample was drawn.
Disproportional Stratified Sampling: Disproportional stratified sampling is a technique where the sample sizes from different strata (subgroups) do not reflect their proportions in the population. Instead, certain strata may be over-sampled or under-sampled to ensure adequate representation of specific groups or to improve the precision of estimates for those groups. This method helps to address the needs of analysis by allowing researchers to focus on particular segments of a population that are of interest, making it vital for effective analysis and estimation.
Increased Precision: Increased precision refers to the enhancement of the accuracy and reliability of estimates derived from a sample, particularly through stratification methods. It ensures that the sample closely represents the entire population by minimizing variability within subgroups, allowing for more accurate generalizations. Achieving increased precision is crucial for effective analysis and estimation as it leads to reduced sampling error and more trustworthy results.
Interval Estimate: An interval estimate is a range of values that is used to estimate an unknown population parameter, providing both a lower and upper bound for that parameter. This concept is important in statistical analysis as it reflects the uncertainty inherent in sampling and allows researchers to convey the precision of their estimates. Interval estimates are often associated with confidence intervals, which indicate the likelihood that the true parameter lies within the specified range.
Point Estimate: A point estimate is a single value or statistic that serves as a best guess or approximation of an unknown population parameter. It provides a simple way to summarize data and make inferences about a larger group based on a sample. Point estimates are commonly used in various sampling methods, providing a foundation for further statistical analysis and decision-making.
Proportional Stratified Sampling: Proportional stratified sampling is a sampling method where the population is divided into distinct subgroups, or strata, and samples are drawn from each stratum in proportion to its size relative to the entire population. This approach ensures that each subgroup is adequately represented in the final sample, which enhances the accuracy of estimates and analysis. By using this technique, researchers can better understand the characteristics of different segments within a population and improve the reliability of their findings.
R: In statistics, 'r' typically represents the correlation coefficient, a numerical measure of the strength and direction of a linear relationship between two variables. It plays a vital role in various analytical techniques, helping to quantify how closely related different sets of data are. Understanding 'r' can be crucial when interpreting results from stratified sampling, managing missing data, performing imputation methods, and employing propensity score techniques.
Sample Size Determination: Sample size determination is the process of calculating the number of observations or replicates needed in a study to achieve reliable and valid results. It ensures that the sample is large enough to accurately reflect the population, providing sufficient data for estimation and inference while balancing resources and time constraints.
Sampling frame: A sampling frame is a list or database from which a sample is drawn for a study, serving as the foundation for selecting participants. It connects to the overall effectiveness of different sampling methods and is crucial for ensuring that every individual in the population has a known chance of being selected, thus minimizing bias and increasing representativeness.
SAS: SAS stands for Statistical Analysis System, a powerful software suite used for advanced analytics, business intelligence, data management, and predictive analytics. It plays a crucial role in various statistical methodologies, enhancing the analysis of complex data sets and improving estimation techniques across different sampling strategies.
SPSS: SPSS, which stands for Statistical Package for the Social Sciences, is a powerful software tool used for statistical analysis and data management. It provides users with a user-friendly interface to perform complex statistical analyses, making it an essential resource for researchers and analysts in various fields, including social sciences, health sciences, and marketing. The software's capabilities extend to various analyses, such as stratified sampling analysis, multivariate techniques, and methods for managing missing data, allowing researchers to gain valuable insights from their data.
Standard Error: Standard error refers to the measure of the amount of variability or dispersion in a sample statistic, typically the mean, from the true population parameter. It provides insights into how much sample means might vary from the actual population mean, making it crucial for understanding the reliability of estimates derived from sample data.
Stratified Mean Formula: The stratified mean formula is a method used to calculate the overall mean of a population that has been divided into distinct subgroups or strata. This formula takes into account the individual means of each stratum, weighted by their respective sizes in the total population. By using this approach, the stratified mean provides a more accurate estimate of the population mean compared to using a simple random sample, especially when there are significant differences between strata.
Stratum: A stratum is a subset of a population that shares a specific characteristic, which is used in stratified sampling to ensure representation across different segments. Each stratum is formed based on key attributes like age, income, or education level, helping to provide a more accurate reflection of the population. This division allows for tailored sampling methods that enhance the precision of estimates and analyses.
Subgroup: A subgroup is a smaller group derived from a larger population, often defined by specific characteristics or criteria. In the context of stratified sampling, subgroups allow researchers to ensure that various segments of the population are adequately represented, leading to more accurate and reliable estimates. Understanding subgroups is crucial for analyzing the differences between these segments and making informed decisions based on the data collected.
Variance: Variance is a statistical measure that indicates the degree to which data points in a set differ from the mean of that set. It helps in understanding the spread or dispersion of the data, which is crucial when analyzing how different groups or strata behave within a larger population. Variance plays a significant role in estimating parameters and understanding data quality, especially when dealing with survey data and missing values.
Weighted Average Formula: The weighted average formula is a statistical calculation that determines the average of a set of values, where each value has a specific weight that reflects its importance or frequency in the dataset. This formula is particularly useful in stratified sampling, as it allows for the estimation of population parameters by accounting for the different sizes and contributions of each stratum within the sample.
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