The real problems of contextual analysis concern the conceptualization of contextual effects, the kinds of data with which to estimate them, and the selection and implementation of appropriate statistical techniques. This paper focuses on detection; specifically, an approach to contextual analysis based on the estimation and interpretation of a covariance component model with hyperparameters. The restricted maximum likelihood/Bayes (REML/Bayes) estimation procedure described is based on Mason and Wong (forthcoming). This paper illustrates the specification, estimation, and interpretation of a covariance component model with hyperparameters, using an example concerning systematic variability across countries in the determination of individual fertility within countries. The purpose in developing this example is to make the approach accessible to others interested in contextual analysis, but not in the algebraic and statistical complexities that comprise formal presentations. Multilevel analysis using the covariance component framework is an advance in contextual analysis. Whether contextual effects exist for a given substantive problem is one matter; their estimation is another. Researchers may be hard pressed to find such effects unless they use efficient and appropriate estimation techniques in conjunction with appropriate theoretical reasoning. This illustration indicates that the REML/Bayes estimation procedure extracted results which otherwise would have been invisible. (Author/PN)