11.2 Stratified sampling

Stratified sampling is a powerful statistical technique that divides a population into subgroups before sampling. This method ensures representation of key subgroups and can increase precision in population estimates. It's particularly useful when studying diverse populations or when certain subgroups are of special interest.

The process involves defining strata, allocating sample sizes, and selecting samples within each stratum. Different allocation methods, such as proportional or optimal allocation, can be used depending on study goals. Stratified sampling offers advantages in precision and representation, but requires careful planning and consideration of potential biases.

Definition of stratified sampling

• Divides population into non-overlapping subgroups (strata) based on specific characteristics
• Selects samples independently from each stratum using probability sampling methods
• Combines stratum samples to form overall sample representative of entire population

Purpose and advantages

• Increases precision of population estimates by reducing sampling error
• Ensures representation of important subgroups that might be missed in simple random sampling
• Allows for separate analysis of individual strata to compare group differences

Population stratification process

Identifying strata characteristics

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• Selects variables strongly related to the study's outcome of interest
• Considers demographic factors (age, gender, income) or geographic regions
• Ensures mutually exclusive and collectively exhaustive strata
• Aims for homogeneity within strata and heterogeneity between strata

Determining optimal strata number

• Balances increased precision with added complexity and cost
• Uses statistical methods (Cumulative Square Root Frequency method)
• Considers practical constraints (budget, time, available resources)
• Typically ranges from 3 to 6 strata in most applications

Sample allocation methods

Proportional allocation

• Allocates sample size to each stratum proportional to stratum size in population
• Calculation: $n_h = n \times (N_h / N)$ where $n_h$ is stratum sample size, $n$ is total sample size, $N_h$ is stratum population size, and $N$ is total population size
• Maintains population proportions in the sample
• Simple to implement and understand

Optimal allocation

• Allocates sample size based on stratum size and variability
• Aims to minimize overall sampling variance for a given total sample size
• Requires knowledge of within-stratum variances
• Formula: $n_h = n \times \frac{N_h S_h}{\sum_{i=1}^{L} N_i S_i}$ where $S_h$ is the standard deviation of the variable of interest in stratum $h$

Neyman allocation

• Special case of optimal allocation when cost per unit is constant across strata
• Allocates larger samples to strata with higher variability or larger sizes
• Calculation: $n_h = n \times \frac{N_h S_h}{\sum_{i=1}^{L} N_i S_i}$ (same as optimal allocation formula)
• Provides minimum variance for a fixed total sample size

Stratified random sampling procedure

Within-stratum sampling techniques

• Employs simple random sampling within each stratum
• Uses systematic sampling for ordered lists within strata
• Applies probability proportional to size (PPS) sampling for unequal selection probabilities
• Ensures independence between samples from different strata

Sample size determination

• Considers desired precision level (margin of error)
• Accounts for expected response rate and budget constraints
• Uses power analysis for hypothesis testing scenarios
• Adjusts for finite population correction in small populations

Statistical properties

Variance estimation

• Calculates within-stratum variances separately
• Combines stratum variances using appropriate weighting
• Formula for stratified sample variance: $V(\bar{y}_{st}) = \sum_{h=1}^{L} W_h^2 \frac{s_h^2}{n_h}$ where $W_h$ is the stratum weight and $s_h^2$ is the sample variance in stratum $h$
• Provides more precise estimates compared to simple random sampling

Precision vs simple random sampling

• Offers increased precision when strata are homogeneous
• Reduces standard error of estimates for given sample size
• Quantifies improvement using design effect (DEFF) measure
• Achieves greater efficiency in population parameter estimation

Bias considerations

Selection bias in strata

• Occurs when strata are not properly defined or identified
• Results from incomplete or inaccurate sampling frames within strata
• Mitigated by careful stratification variable selection and frame development
• Requires thorough understanding of population characteristics

Non-response bias effects

• Varies across strata due to different response rates
• Impacts representativeness of final sample
• Addressed through weighting adjustments or imputation techniques
• Requires analysis of non-response patterns within each stratum

Stratified sampling applications

Market research examples

• Customer satisfaction surveys stratified by product lines
• Brand awareness studies stratified by geographic regions
• Consumer behavior analysis stratified by age groups and income levels

Environmental studies cases

• Water quality assessment stratified by river sections
• Air pollution monitoring stratified by urban vs rural areas
• Wildlife population estimates stratified by habitat types

Limitations and challenges

Stratum boundary issues

• Difficulty in defining clear boundaries between strata
• Overlapping characteristics leading to ambiguous stratum assignment
• Potential for misclassification of population units
• Requires careful consideration of stratification variables and cutoff points

Small stratum problems

• Insufficient sample sizes in some strata for reliable estimates
• Increased variability of estimates for small strata
• Potential need for collapsing or combining small strata
• Trade-off between maintaining stratum identity and achieving adequate precision

Analysis of stratified data

Weighted estimators

• Uses stratum weights to calculate population estimates
• Formula for stratified mean: $\bar{y}_{st} = \sum_{h=1}^{L} W_h \bar{y}_h$ where $W_h$ is the stratum weight and $\bar{y}_h$ is the sample mean in stratum $h$
• Applies weights in regression analysis and other statistical procedures
• Ensures proper representation of population structure in final estimates

Confidence interval construction

• Accounts for stratified design in interval calculations
• Uses stratified variance estimates for more accurate intervals
• Formula: $CI = \bar{y}_{st} \pm t_{\alpha/2, df} \times \sqrt{V(\bar{y}_{st})}$ where $t_{\alpha/2, df}$ is the t-value for desired confidence level
• Provides narrower intervals compared to simple random sampling

Stratified sampling vs other methods

Cluster sampling comparison

• Stratified sampling selects units from all strata, cluster sampling selects entire clusters
• Stratified sampling generally more precise than cluster sampling
• Cluster sampling more cost-effective for geographically dispersed populations
• Stratified sampling requires more information about population characteristics

Multistage sampling differences

• Stratified sampling involves one stage of selection within strata
• Multistage sampling uses multiple levels of sampling units
• Stratified sampling offers more control over sample composition
• Multistage sampling more suitable for complex, hierarchical populations

Software tools for stratified sampling

• Statistical packages (R, SAS, SPSS) with built-in stratified sampling functions
• Specialized survey software (Qualtrics, SurveyMonkey) for online stratified surveys
• GIS tools (ArcGIS, QGIS) for spatial stratification in environmental studies
• Custom programming languages (Python, MATLAB) for complex sampling designs

Ethical considerations in stratification

• Potential for reinforcing stereotypes or discrimination through stratification variables
• Privacy concerns when using sensitive characteristics for stratification
• Balancing representativeness with individual rights and protections
• Ensuring transparency in reporting stratification methods and limitations