An Extension of Kish's Formula for Design Effects to Two- and Three-Stage Designs with Stratification

We extend the model-based justification of Kish's 1987 formula for design effects proposed by Gabler, Haeder and Lahiri (1999) to two -stage designs with stratification at the first stage, and unstratified three-stage designs. In each case this approach gives rise to a formula for the overall design effect, in terms of the survey weights and the population intra-cluster correlations at each stage. The proposed formulae give rise to informative interpretations of design effects from different sources. When analyzing the data from a completed survey, the formulae can be used to estimate the intra-cluster correlation at each stage. In planning for future surveys the formulae can be used to predict the design effect due to weighting and clustering at each stage. The application of the proposed method to data from the 2012 Programme for the International Student Assessment (PISA) is also presented.