Autoregressive

An autoregressive correlation structure indicates that two observations taken close in time (or space) within an individual tend to be more closely correlated than two observations taken far apart in the same individual. Formally, p^ = 1 and pjkij / increases in value as the absolute difference between j and k falls. As a specific example, a first-order autoregressive (AR-1) correlation structure specifies that pjk = pli~k] where p is the correlation when | j — k\ = 1.

This is used in balanced data sets. No assumption is made about the relative magnitude of the correlation between any two pairs of observations. Formally, pj} = 1 and Pjku^k] is free to take

All correlation coefficients are fixed by the user rather than being estimated from the data. Formally, pn = 1 and pjk[j ¿ k] can take any value between — 1 and -I-1, but this value is fixed prior to the analysis rather than being estimated from the data.

We must first thank Patricia Cahill RN who supervized every peak flow measurement; without her it would have been impossible to conduct the research upon which our practical example was based. We would also like to thank all of the boys who participated in the study for their time, effort and patience. We thank the statisticians, epidemiologists and clinical scientists who commented on earlier versions of the manuscript. We gratefully acknowledge the teaching and advice provided by the Multilevel Models Project Team at the Institute of Education, London. This research was supported in part by grants from the National Health and Medical Research

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