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Hierarchical linear models provide a conceptual orientation and a flexible set of analytic techniques for studying psychological change in repeated measures studies. The researcher first formulates a model for individual change over time, with each individual's development characterized by a unique set of parameters. These parameters are then viewed as varying randomly over the population of persons. We illustrate this approach with data on attitudes toward deviance during adolescence (Raudenbush & Chan, 1992), indicating how one may assess the psychometric properties of an instrument for studying change, compare the adequacy of linear and curvilinear growth models, control for time invariant and time-varying covariates, and link overlapping cohorts of data. The results suggest that prodeviant attitudes characteristically increase during early adolescence, achieving a peak between 17 and 18 years of age. The typical trajectories for male and female adolescents have the same shape, although female adolescents tend to be less deviant than male adolescents at each age. We briefly consider the statistical power of tests of cohort differences at the points where they overlap.

Accelerated longitudinal designs enable researchers to study individual development over a long interval of the life course by gathering data during a comparatively short interval of time. Such designs also create possibilities not available in standard panel designs for separating developmental effects from cohort and period effects. However, these designs confront the investigator with a special set of inferential challenges and introduce complexity into statistical analysis. In this article the authors employ a hierarchical linear model to illustrate the application of growth curve analysis to data from an accelerated longitudinal design. The goal is to construct a picture of the development of attitudes toward deviance from ages 11 to 18 by linking data from two cohorts of the National Youth Survey, each observed for only 5 years. The example illustrates how the analyst may control for time-varying and time-invariant covariates and test for cohort effects and cohort-by-age interactions. Interesting features of growth include an inflection point (age at which the rate of increase in prodeviant attitude begins to slow down) and a peak age (age of maximally prodeviant attitude).

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We have identified incomplete particle vaporization and non-linear detector response as the major factors that cause the non-linearity of SP-ICP-MS measurements at high particle mass. The contribution of incomplete vaporization to the deviation from the linearity of the ICP-MS intensity is estimated using a mathematical model of particle vaporization. The non-linear detector response in the pulse-counting mode is due to the overlapping of the analyte ions at the detector within the 40 ns dead time of the electron multiplier. The overlap can be severe because of the relatively short pulse duration of 300 ms of the ion plumes of the discrete sample particles. The non-linear detector response has been modeled using Poisson statistics. We have also examined the applicability of calibration methods based on the standard particles, discrete standard solution droplets, and continuous nebulization of standard solution. The standard-solution calibration method requires linear calibration curves. The method may also suffer from errors due to the difference in the sensitivity of the standard solution and sample particles.Calibration using standard particles and discrete standard solution droplets do not have these limitations.