By Piotr Garbaczewski, Marek Wolf, Aleksander Weron

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Extra resources for Chaos - The Interplay Between Stochastic and Deterministic Behaviour

Example text

10. The 95% confidence region is the ellipse 8. 3(β1 + 0. 36) 2 + 0. 22 (β1 + 0. 36)(β 2 − 0. 72 ) + 6. 7 (β 2 − 0. 72 ) 2 = 5. 99 is the 5% point of the χ 22 distribution. 10 displays this region. 11 95% confidence region for coefficients of Social Class and Gender In some situations we may be interested in separate confidence intervals for all possible linear functions involving a subset of q parameters or q linearly independent functions of the parameters, while maintaining a fixed probability that all the intervals include the population value of these functions of the parameters.

When we are judging the fixed effects, a simple comparison of the estimate with its standard error is usually adequate. Because the model adjusts for the earlier maths score we can interpret the social class and gender differences is in terms of the relative progress of girls versus boys or non-manual versus manual children. 2 shows the effects when 8-year maths score is removed from the model and the interpretation is now in terms of the actual differences found at 11 years. Note that the level 1 and level 2 variances are increased, reflecting the importance of the earlier score as a predictor, and the intra-school correlation is slightly reduced.

4. 11. As is clear, apart from some of the extreme intervals, each interval overlaps with most of the other intervals. 45. This will give a similar display but with intervals 25% wider. In reality the complete set of schools typically will be compared in overlapping subsets of different sizes, and a value for c can be determined by averaging over all such possibilities. 5 3 1 Conf. 12 are useful for conveying the inherent uncertainty associated with estimates for individual level 2 (or higher) units, where the number of level 1 units per higher level unit is not large.