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For other models we can be more explicit. In cases where explicit, closed-form solutions exist for the maximizing values, an oft-successful method for finding those solutions is to differentiate the likelihood or restricted likelihood and set the derivatives equal to zero. From the resulting equations, solutions for the parameter symbols might well be thought of as ML or REML estimators, in which case they would be denoted by a "hat" or tilde over the parameter symbol. However, this is not a fail-safe method because, in some cases, solutions to these estimating equations may not be in the parameter space.

For total sample sizes N < 50 the differences can be appreciable. 2. d. Confidence intervals Confidence intervals for /Zf, //j — ^, and cr2 are easily derived and widely available (Snedecor and Cochran, 1989, Chapters 5 and 6). For completeness we list them here. In doing so we emphasize that these intervals are for fixed effects models only, not for mixed models. 1. 1: Critical values based on ^"-distribution (dashed lines) and ^-distributions (solid lines) plotted versus N for m = 10 (top set of lines), 5 (middle set of lines), and 2 (bottom set of lines).

Yij — E[yij|ai] may not even take on integer values. Thus we would have an awkward-to-deal-with distribution. - iv. Variance and conditional expected values To obtain var(y) we will often use the formula which relates variance to conditional expected values in order to partition variability: 12 CHAPTER 1. 7), this gives the usual components of variance breakdown: A similar formula holds for covariances: a derivation of which is to be found in Searle et al. (1992, p. 462). , the covariance between every pair of observations in the same class; and s2a/(s2a+s2) is the intra-class correlation coefficient.

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