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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|广义线性模型代写generalized linear model代考|Selecting a good fit covariance structure using SAS

Recently, it has been possible to fit an analysis of variance model (3.3) based on the restricted maximum likelihood estimation (REML, see Appendix B, Section B.2.1 for the reason why the REML is used) and also select a model for covariance structure that is a good fit to the repeated measurements using some information criterion such as AIC (Akaike information criterion, 1974) or BIC (Schwarz’s Bayesian information criterion, 1981),
\begin{aligned} &A I C(\mathrm{REML})=-2 \text { Res Log Likelihood }+2 p \ &B I C(\mathrm{REML})=-2 \text { Res Log Likelihood }+p \log N \end{aligned}
where “Res Log Likelihood” (shown in SAS outputs) denotes the value of the restricted log-likelihood function and $p$ denotes the number of parameters in the covariance structure model. The model that minimizes $\mathrm{AIC}$ or $\mathrm{BIC}$ is preferred. If $\mathrm{AIC}$ or $\mathrm{BIC}$ is close, then the simpler model is usually considered preferable.

It should be noted that, if the REML estimator has been used, the above $\mathrm{AIC}$ and BIC can be used for comparing models with different covariance structure only if both models have the same set of fixed effects parameters. For comparing models with different sets of fixed effects parameters, one should consider the following AIC and BIC based on the maximum likelihood (ML) estimation,
\begin{aligned} A I C(\mathrm{ML}) &=-2 \log \text { Likelihood }+2(p+q) \ B I C(\mathrm{ML}) &=-2 \log \text { Likelihood }+(p+q) \log N \end{aligned}
where “Log Likelihood” denotes the value of the log-likelihood function and $q$ denotes the number of fixed-effects parameters to be estimated.

## 统计代写|广义线性模型代写generalized linear model代考|Heterogeneous covariance

So far, homogeneous covariance structure is assumed for all the treatment groups, i.e., $\boldsymbol{\Sigma}_k=\boldsymbol{\Sigma}$. In this section, to check the homogeneity assumption, we shall consider the analysis of variance model with heterogeneous covariance. To do this in PROC MIXED, we have only to add the option group = group to the REPEATED statement, where the former group is the SAS statement and the latter group is a numeric factor denoting the treatment group. Then the modified REPEATED statement will be
repeated / type $=\mathrm{cs}$ subject $=$ id r rcorr group $=$ group
In this case, the variable group must be declared as a numeric factor in the CLASS statement. Now we shall fit the two models, CS and UN, to the Rat Data. The respective sets of SAS programs appear in Program 3.2.

In the CS model, two variance estimates, $\hat{\sigma}_B^2$ and $\hat{\sigma}_E^2$, are shown in the table labeled “Covariance Parameter Estimates” by treatment group. You can see that the difference between groups is small for both variances. In the unstructured model, the covariance matrix is shown by treatment group in the table labeled “Covariance Parameter Estimates” in the form of $\operatorname{UN}\left(j_1, j_2\right)$. Here also, we can observe small differences between groups. When we observe the change of AICs from the homogeneous model to the heterogeneous model, we have $144.8 \rightarrow 148.4$ for the CS model and $108.3 \rightarrow 111.2$ for the UN model, indicating that the homogeneous models are preferred to the heterogeneous ones.

# 广义线性模型代考

## 统计代写|广义线性模型代写generalized linear model代考|Selecting a good fit covariance structure using SAS

$$A I C(\mathrm{REML})=-2 \text { Res Log Likelihood }+2 p \quad B I C(\mathrm{REML})=-2 \text { Res Log Likelihood }+p \log N$$
$\mathrm{AIC}$ 或者 $\mathrm{BIC}$ 接近，则通常认为更简单的模型更可取。

$$A I C(\mathrm{ML})=-2 \log \text { Likelihood }+2(p+q) B I C(\mathrm{ML})=-2 \log \text { Likelihood }+(p+q) \log N$$

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## MATLAB代写

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