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# 统计代写|多尺度模型代写Multilevel Models代考|UV9253 Maximum likelihood estimation using iterative generalised least squares

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## 统计代写|多尺度模型代写Multilevel Models代考|Maximum likelihood estimation using iterative generalised least squares

The iterative generalised least squares (IGLS) algorithm forms the basis for many of the developments in later chapters and we now summarise the main features. Appendix $2.1$ sets out the details.
Consider the simple 2-level variance components model
$$y_{i j}=\beta_0+\beta_1 x_{i j}+u_{0 j}+e_{0 i j}, \quad \operatorname{var}\left(e_{0 i j}\right)=\sigma_{e 0}^2, \quad \operatorname{var}\left(u_{0 j}\right)=\sigma_{u 0}^2$$
Suppose that we knew the values of the variances, and so could construct immediately the block-diagonal matrix $V_2$, which we will refer to simply as $V$. We can then apply the usual Generalised Least Squares (GLS) estimation procedure to obtain the estimator for the fixed coefficients, namely
$$\hat{\beta}=\left(X^T V^{-1} X\right)^{-1} X^T V^{-1} Y$$
where in this case
$$X=\left(\begin{array}{cc} 1 & x_{11} \ 1 & x_{21} \ \vdots & \vdots \ 1 & x_{n_m m} \end{array}\right) \quad Y=\left(\begin{array}{c} y_{11} \ y_{12} \ \vdots \ y_{n_m m} \end{array}\right)$$
with $m$ level 2 units and $n_j$ level 1 units in the $j$-th level 2 unit. Since we are assuming that the residuals have normal distributions, (2.8) also yields maximum likelihood estimates.

## 统计代写|多尺度模型代写Multilevel Models代考|Marginal models and generalised estimating equations

IIt is worth emphasising the distinction between multilevel models, sometimes referred to in this context as ‘subject specific’ models, and so-called ‘marginal’ models such as the GEE model (Zeger et al., 1988; Liang et al., 1992). When dealing with hierarchical data these latter models typically start with a formulation for the covariance structure, for example, but not necessarily, based upon a multilevel structure, and aim to provide estimates with acceptable properties only for the fixed parameters in the model, treating the existence of any random parameters as a necessary ‘nuisance’. More specifically, the estimation procedures used in marginal models are known to have useful asymptotic properties in the case where the exact form of the random structure is unknown.

If interest lies only in the fixed parameters, marginal models can be useful since they give directly unbiased estimates for these parameters. Even here, however, they may be inefficient if they utilise a covariance structure that is substantially incorrect. They are, however, generally more robust than multilevel models to serious misspecification of the covariance structure (Heagerty and Zeger, 2000). Fundamentally, however, marginal models address different research questions. From a multilevel perspective, the failure explicitly to model the covariance structure of complex data is to ignore information about variability that, potentially, is as important as knowledge of the average or fixed effects. Thus, in a repeated measures growth study, knowledge of how individual growth rates vary, possibly differentially according to say demographic factors, will be important information and in Chapter 5 we will show how such information can be used to provide efficient predictions in the case of human growth.

Also, when we discuss multilevel models for discrete response data in Chapter 4 we will show how to obtain estimates for population or subpopulation means equivalent to those obtained from marginal models. In the case of normal response linear multilevel models, GEE and multilevel models lead to the same fixed coefficient estimates. For a further discussion of the limitations of marginal models, see the paper by Lindsey and Lambert (1998).

## 统计代写|多尺度模型代写MULTILEVEL MODELS代考|MAXIMUM LIKELIHOOD ESTIMATION USING ITERATIVE GENERALISED LEAST SQUARES

$$y_{i j}=\beta_0+\beta_1 x_{i j}+u_{0 j}+e_{0 i j}, \quad \operatorname{var}\left(e_{0 i j}\right)=\sigma_{e 0}^2, \quad \operatorname{var}\left(u_{0 j}\right)=\sigma_{u 0}^2$$

$$\hat{\beta}=\left(X^T V^{-1} X\right)^{-1} X^T V^{-1} Y$$

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