# Chapter 5: Linearization and sample reuse in variance estimation

In Chapter 5 of the textbook Practical Methods for Design and Analysis of Complex Surveys, various approximate techniques for variance and covariance estimation of nonlinear estimators are introduced. Variance estimation is needed to obtain standard error estimates of sample means and proportions for the total population and, more importantly, for various subpopulations. In modelling procedures, variance estimates of estimated model coefficients, such as regression coefficients, are needed for proper test statistics. The linearization method is used as the basic approximation method. Alternative methods include balanced half-samples, jackknife and bootstrap, which are based on sample reuse techniques. The variance approximation methods are demonstrated for the Mini-Finland Health Survey, providing a complex analytical survey in which stratified cluster sampling is used with regional stratification and two regional sample clusters per stratum. A more complex sampling design is provided by the Occupational Health Care Survey (OHC), to be used for covariance matrix estimation of a vector of nonlinear estimators. The sampling design of the OHC Survey is a combination of stratified one-stage and two-stage sampling with industrial establishments as clusters.

In Training Key 145, the linearization method is demonstrated by reproducing the results of Example 5.1. The Mini-Finland Health Survey data set is used. Two different response variables are used: Chronic morbidity, providing an example of a binary variable, and systolic blood pressure, providing an example of a continuous variable.

In Training Key 158, the jackknife technique is demonstrated by reproducing the results of Example 5.3.

In Training Key 162, the bootstrap technique is demonstrated by reproducing the results of Example 5.4. An option is provided for the examination of the distribution of bootstrap estimates with varying number of generated bootstrap samples.

NOTE: Instructions for the use of Training Keys are given in the Instructions section.