STRATIFIED SAMPLING STRATIFIED SAMPLING 1 Stratification The elements

STRATIFIED SAMPLING

STRATIFIED SAMPLING 1. Stratification: The elements in the population are divided into layers/groups/ strata based on their values on one/several auxiliary variables. The strata must be nonoverlapping and together constitute the whole population. 2. Sampling within strata: Samples are selected independently from each stratum. Different selection methods can be used in different strata.

Ex. Stratification of individuals by age group Stratum Age group 1 17 or younger 2 18 -24 3 25 -34 4 35 -44 5 45 -54 6 55 -64 7 65 or older

Ex. Regional stratification Stratum 1: Northern Sweden Stratum 2: Mid. Sweden Stratum 3: Southern Sweden

Ex. Stratification of individuals by age group and region Stratum Age group Region 1 17 or younger Northern 2 17 or younger Mid 3 17 or younger Southern 4 18 -24 Northern 5 18 -24 Mid 6 18 -24 Southern etc.

WHY STRATIFY? • Gain in precision. If the strata are more homogenous with respect to the study variable(s) than the population as a whole, the precision of the estimates will improve. • Strata = domains of study. Precision requirements of estimates for certain subpopulations/domains can be assured by using domains as strata.

WHY STRATIFY? , cont’d • Practical reasons. For instance nonresponse rates, method of measurement and the quality of auxiliary information may differ between subpopulations, and can be efficiently handled by stratification. • Administrative reasons. The survey organization may be divided into geographical districts that makes it natural to let each district be a stratum.

ESTIMATION Assume a population divided into H strata of sizes. Independently, a sample of size nh is selected from each stratum. = y-value for element j in stratum h = population total for stratum h = sample mean for stratum h

ESTIMATION OF A TOTAL Assume: SRS within all strata. 9

ESTIMATION OF A TOTAL Assume: SRS within all strata. In general: What is the variance of this estimator? 10

VARIANCE OF THE ESTIMATOR OF A TOTAL Principle: Add the variances of the estimators for each stratum. A legitimate approach since samples are selected independently from each stratum. Remember: if X, Y are independent random variables. 11

VARIANCE OF THE ESTIMATOR OF A TOTAL, cont’d Result: One term per stratum Finite population correction (one per stratum!) where 12

ESTIMATION OF THE VARIANCE OF THE ESTIMATOR OF A TOTAL Principle: Estimate what’s unknown in the variance formula. where 13

ESTIMATORS FOR A MEAN Note: Start from the estimators for a total!

ESTIMATORS FOR A MEAN, cont’d Note: Start from the estimators for a total!

ESTIMATORS FOR A PROPORTION Note: Like the estimators for a mean, only with y a 0/1 -variable!

IMPORTANT DESIGN CHOICES IN STRATIFIED SAMPLING • Stratification variable(s) • Number of strata • Sample size in each stratum (allocation) • Sampling design in each stratum • Estimator for each stratum 17