统计代写|广义线性模型代写generalized linear model代考|STAT3030

2022年10月11日

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统计代写|广义线性模型代写generalized linear model代考|EXPECTED MEAN SQUARES

The mean square of an effect is the sum of squares of that effect divided by the corresponding degrees of freedom. For complete, balanced designs the mean

square is $\left[1 / \operatorname{tr}\left(\mathbf{A}_s\right)\right] \mathbf{Y}^{\prime} \mathbf{A}_s \mathbf{Y}$, since the degrees of freedom equal the $\operatorname{tr}\left(\mathbf{A}_s\right)$. The expected value of the mean square, usually called the expected mean square (EMS), is a function of the mean vector $\mu=\mathbf{E}(\mathbf{Y})$ and of the variance parameters in $\boldsymbol{\Sigma}=\operatorname{cov}(\mathbf{Y})$. The expected mean square indicates how the mean squares can be used to obtain unbiased estimates of functions of the variance parameters.

The expected mean square in complete, balanced designs is defined in the following theorem. The proof of the theorem is a direct result of Theorem 1.3.2.
Theorem 4.4.1 Let $\mathbf{Y}$ be an $n \times 1$ random vector associated with the observations of a complete, balanced factorial experiment with an $n \times 1$ mean vector $\mu=E(\mathbf{Y})$ and $n \times n$ covariance matrix $\mathbf{\Sigma}=\operatorname{cov}(\mathbf{Y})$. The expected mean square associated with the sum of squares $\mathbf{Y}^{\prime} \mathbf{A Y}$ is $\mathbf{E}\left{\left[1 / \operatorname{tr}\left(\mathbf{A}_s\right)\right] \mathbf{Y}^{\prime} \mathbf{A Y}\right}=\left[\operatorname{tr}(\mathbf{A} \mathbf{\Sigma})+\boldsymbol{\mu}^{\prime} \mathbf{A} \boldsymbol{\mu}\right] / \operatorname{tr}(\mathbf{A})$ where $\operatorname{tr}$ (A) equals the degrees of freedom associated with $\mathbf{Y}^{\prime} \mathbf{A} \mathbf{Y}$.

Example 4.4.1 Consider the experiment described in Examples 4.3.1 and 4.3.3 in which a finite model was assumed. The sums of squares due to the random effect $B$ and the fixed effect $T$ are $\mathbf{Y}^{\prime} \mathbf{A}2 \mathbf{Y}$ and $\mathbf{Y}^{\prime} \mathbf{A}_3 \mathbf{Y}$, respectively, where $\mathbf{A}_2=\left(\mathbf{1}_b-\frac{1}{b} \mathbf{J}_b\right) \otimes \frac{1}{t} \mathbf{J}_t \otimes \frac{1}{r} \mathbf{J}_r, \mathbf{A}_3=\frac{1}{b} \mathbf{J}_b \otimes\left(\mathbf{I}_t-\frac{1}{t} \mathbf{J}_t\right) \otimes \frac{1}{r} \mathbf{J}_r$, and the btr $\times 1$ random vector $\mathbf{Y}=\left(Y{111}, \ldots, Y_{11 r}, \ldots, Y_{b t 1}, \ldots, Y_{b t r}\right)^{\prime}$. The mean vector $\boldsymbol{\mu}=$ $\mathrm{E}(\mathbf{Y})=\mathrm{E}\left(Y_{111}, \ldots, Y_{11 r}, \ldots, Y_{b t 1}, \ldots, Y_{b t r}\right)^{\prime}=\mathbf{1}b \otimes\left(\mu_1, \ldots, \mu_t\right)^{\prime} \otimes \mathbf{1}_r$. Note that $\mathbf{A}_2 \boldsymbol{\Sigma}=\left[\operatorname{tr} \sigma_B^2+\sigma{R(B T)}^2\right] \mathbf{A}2$ and $\mathbf{A}_3 \Sigma=\left[r \sigma{B T}^2+\sigma_{R(B T)}^2\right] \mathbf{A}3$. Therefore, hy Theorem 4.4.1, the expected mean square of the random effect $B$ equals \begin{aligned} \mathrm{E}\left[\mathbf{Y}^{\prime} \mathbf{A}_2 \mathbf{Y} / \operatorname{tr}\left(\mathbf{A}_2\right)\right]=& {\left[r\left(\mathbf{A}_2 \Sigma\right)+\boldsymbol{\mu}^{\prime} \mathbf{A}_2 \mu\right] / \operatorname{tr}\left(\mathbf{A}_2\right)=\left[\operatorname{tr} \sigma_B^2+\sigma{R(B T)}^2\right] } \ &+\left{\left[\mathbf{1}b \otimes\left(\mu_1, \ldots, \mu_t\right)^{\prime} \otimes \mathbf{1}_r\right]^{\prime}\left[\left(\mathbf{I}_b-\frac{1}{b} \mathbf{J}_b\right) \otimes \frac{1}{t} \mathbf{J}_t \otimes \frac{1}{r} \mathbf{J}_r\right]\right.\ &\left.\times\left[\mathbf{1}_b \otimes\left(\mu_1, \ldots, \mu_t\right)^{\prime} \otimes \mathbf{1}_r\right] /(b-1)\right} \ =& \operatorname{tr} \sigma_B^2+\sigma{R(B T)}^2 \end{aligned}
and the expected mean square of the fixed factor $T$ equals

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

The sum of squares and covariance matrix algorithms are now applied to a series of complete, balanced factorial experiments. As mentioned in Section 4.3, finite and infinite model covariance structures are reparameterizations of each other. Therefore, the choice between the finite and infinite model is somewhat arbitrary. For the remainder of this text, the finite model is assumed. Therefore, unless specifically stated, subsequent covariance matrices for complete, balanced factorial experiments will always be constructed with Rules $\Sigma 1, \Sigma 2, \Sigma 2.1, \Sigma 2$. , and $\Sigma 2.3$.
Example 4.5.1 Consider a two-way cross classification where the first two factors $S$ and $T$ are fixed with $i=1, \ldots, s$ and $j=1, \ldots, t$ levels and the third factor is a set of $k=1, \ldots, r$ random replicates nested in the $s t$ combinations of the first two factors. Let $Y_{i j k}$ be a random variable representing the $k^{\text {th }}$ replicate observation in the $i j^{\text {th }}$ combination of the two fixed factors. Let the $s t r \times 1$ random vector $\mathbf{Y}=\left(Y_{111}, \ldots, Y_{11 r}, \ldots, Y_{s t 1}, \ldots, Y_{s t r}\right)^{\prime}$. The model is
$$Y_{i j k}=\mu_{i j}+R(S T){(i j) k}$$ where $\mu{i j}$ are constants representing the mean effect of the $i j^{\text {th }}$ combination of the two fixed factors and $R(S T){(i j) k}$ are str random variables representing the effect of the nested replicates. Assume that the str random variables $R(S T){(i j) k} \sim$ iid $\mathbf{N}1\left(0, \sigma{R(S T)}^2\right)$. Therefore, the $s t r \times 1$ random vector $\mathbf{Y} \sim \mathbf{N}{s t r}(\mu, \mathbf{\Sigma})$ where the str $\times 1$ mean vector \begin{aligned} \boldsymbol{\mu} &=\mathrm{E}\left(Y{111}, \ldots, Y_{11 r}, \ldots, Y_{s t 1}, \ldots, Y_{s t r}\right)^{\prime} \ &=\left(\mu_{11} \mathbf{1}r, \ldots, \mu{s t} \mathbf{1}r\right)^{\prime} \ &=\left(\mu{11}, \ldots, \mu_{s t}\right)^{\prime} \otimes \mathbf{1}r \end{aligned} and the $\operatorname{str} \times \operatorname{str}$ covariance matrix $$\boldsymbol{\Sigma}=\sigma{R(S T)}^2\left[\mathbf{I}_s \otimes \mathbf{I}_t \otimes \mathbf{I}_r\right]$$

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|平方和和协方差矩阵算法

$$\Sigma=\sigma_B^2\left[\mathbf{I} b \otimes \mathbf{J}_t \otimes \mathbf{J}_r\right] \quad+\sigma B T^2\left[\mathbf{I} b \otimes\left(\mathbf{I}_t-\frac{1}{t} \mathbf{J}_t\right) \otimes \mathbf{J}_r\right]+\sigma R(B T)^2\left[\mathbf{I}_b\right.$$

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

$\Sigma=\sigma_B^2 \quad\left[\mathbf{I} b \otimes \mathbf{J}t \otimes \mathbf{J}_r\right] \quad+\sigma B T^2 \quad\left[\mathbf{I} b \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right] \quad+\sigma R(B T)^2 \quad\left[\mathbf{I} b \otimes \mathbf{I}_t \otimes\right.$ 屈管有限和无限模型以不同的方式激发并产生不同的协方差结构，但从代数上讲，这两个协 方差结构只是彼此的简单重新参数化。为了说明这一点，考虑前面的实验 $b$ 随机块， $t$ 固定治 疗，和 $r$ 每个处理随机重复。首先，用以下等价形式重写有限模型的协方差矩阵。 $\Sigma=\left(\sigma_B^2-\frac{1}{t} \sigma B T^2\right) \quad\left[\mathbf{I} b \otimes \mathbf{J}_t \otimes \mathbf{J}_r\right] \quad+\sigma B T^2 \quad\left[\mathbf{I} b \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right]+\sigma R(B T)^2$ 现在要区分有限和无限模型中的参数，重命名方差参数 $\sigma_B^2, \sigma B T^2$ ，和 $\sigma{R(B T)}^2$ 在无限模型

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MATLAB代写

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