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

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 $\Sigma 1, \Sigma 2, \Sigma 2.1, \Sigma 2.2$, 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 $s t r$ random variables representing the effect of the nested replicates. Assume that the str random variables $R(S T){(i j) k} \sim$ iid $\mathrm{N}1\left(0, \sigma{R(S T)}^2\right)$. Therefore, the str $\times 1$ random vector $\mathbf{Y} \sim \mathrm{N}{s t r}(\boldsymbol{\mu}, \mathbf{\Sigma})$ where the $s t r \times 1$ 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 $$\boldsymbol{\Sigma}=\sigma{R(S T)}^2\left[\mathbf{I}_s \otimes \mathbf{I}_t \otimes \mathbf{I}_r\right]$$

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

Example 4.5.2 Consider the same two-way layout as in Example 4.5.1 except now let $S$ and $T$ both be random factors. The model is
$$Y_{i j k}=\alpha+S_i+T_j+S T_{i j}+R(S T){(i j)_k}$$ where $\alpha$ is a constant representing the overall mean effect; $s_i$ are random variables representing the effect of the first random factor; $T_j$ are the random variables representing the second random factor; $S T{i j}$ are random variables representing the interaction between $S$ and $T$; and $R(S T){(i j) k}$ are random variable defined as in Example 4.5.1. Assume the $s$ random variables $S_i \sim$ iid $\mathrm{N}_1\left(0, \sigma_S^2\right)$; the $t$ random variables $T_j \sim$ iid $\mathrm{N}_1\left(0, \sigma_T^2\right)$; the $s t$ random variables $S T{i j} \sim$ iid $\mathrm{N}1\left(0, \sigma{S T}^2\right)$; and the $s t r$ random variables $R(S T){(i j) k} \sim$ iid $\mathrm{N}_1\left(0, \sigma{R(S T)}\right)$. Furthermore, assume that $\left{S_i, i=1, \ldots, s\right},\left{T_j, j=1, \ldots, t\right},\left{S T_{i j}, i=1, \ldots, s, j=1 \ldots, t\right}$, and $\left{R(S T){(i j) k}, i=1, \ldots, s, j=1, \ldots, t, k=1, \ldots, r\right}$ are mutually independent sets of random variables. Therefore, the str $\times 1$ random vector $\mathbf{Y} \sim \mathbf{N}{s t r}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ where the $s t r \times 1$ mean vector
$$\boldsymbol{\mu}=\mathrm{E}\left(Y_{111}, \ldots, Y_{11 r}, \ldots, Y_{s t 1}, \ldots, Y_{s t r}\right)^{\prime}=\alpha \mathbf{1}s \otimes \mathbf{1}_t \otimes \mathbf{1}_r$$ and, by the covariance algorithm, the str $\times$ str covariance matrix \begin{aligned} \Sigma=& \sigma_S^2\left[\mathbf{I}_s \otimes \mathbf{J}_t \otimes \mathbf{J}_r\right]+\sigma_T^2\left[\mathbf{J}_s \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right] \ &+\sigma{S T}^2\left[\mathbf{I}s \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right]+\sigma{R(S T)}^2\left[\mathbf{I}s \otimes \mathbf{I}_t \otimes \mathbf{I}_r\right] . \end{aligned} The sum of squares matrices are not dependent on whether the factors are fixed or random. Therefore, the sum of squares $\mathbf{Y}^{\prime} \mathbf{A}_m \mathbf{Y}$ for $m=1, \ldots, 5$ are the same as those given in Example 4.5.1. Furthermore, $\mathbf{A}_m \boldsymbol{\Sigma}=c_m \mathbf{A}_m$ for $m=1, \ldots, 5$ where \begin{aligned} &c_1=\operatorname{tr} \sigma_S^2+s r \sigma_T^2+r \sigma{S T}^2+\sigma_{R(S T)}^2 \ &c_2=\operatorname{tr} \sigma_S^2+r \sigma_{S T}^2+\sigma_{R(S T)}^2 \ &c_3=s r \sigma_T^2+r \sigma_{S T}^2+\sigma_{R(S T)}^2 \ &c_4=r \sigma_{S T}^2+\sigma_{R(S T)}^2 \ &c_5=\sigma_{R(S T)}^2 \end{aligned}

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|ALGORITHM APPLICATIONS

$$Y_{i j k}=\mu_{i j}+R(S T)(i j) k$$

$\boldsymbol{\mu}=\mathrm{E}\left(Y 111, \ldots, Y_{11 r}, \ldots, Y_{s t 1}, \ldots, Y_{s t r}\right)^{\prime} \quad=\left(\mu_{11} \mathbf{1} r, \ldots, \mu s t \mathbf{1} r\right)^{\prime}=\left(\mu 11, \ldots, \mu_{s t}\right)^{\prime} \otimes \mathbf{1} r$

$$\boldsymbol{\Sigma}=\sigma R(S T)^2\left[\mathbf{I}_s \otimes \mathbf{I}_t \otimes \mathbf{I}_r\right]$$

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

$$Y_{i j k}=\alpha+S_i+T_j+S T_{i j}+R(S T)(i j)k$$ 在哪里 $\alpha$ 是代表整体平均效应的常数； $s_i$ 是代表第一个随机因素影响的随机变量； $T_j$ 是代表第二个随机 因子的随机变量；STij是代表之间相互作用的随机变量 $S$ 和 $T$; 和 $R(S T)(i j) k$ 是如示例 $4.5 .1$ 中定义的 随机变量。假设 $s$ 随机变量 $S_i \sim$ 独立同居 $\mathrm{N}_1\left(0, \sigma_S^2\right)$; 这 $t$ 随机变量 $T_j \sim$ 独立同居 $\mathrm{N}_1\left(0, \sigma_T^2\right) ;$ 这 $s t$ 随机 变量 $S T i j \sim$ 独立同居N1 $\left(0, \sigma S T^2\right)$; 和str随机变量 $R(S T)(i j) k \sim$ 独立同居 $\mathrm{N}_1(0, \sigma R(S T))$. 此外， 向量 $\mathbf{Y} \sim \mathbf{N} \operatorname{str}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ 在哪里str $\times 1$ 平均向量 $$\boldsymbol{\mu}=\mathrm{E}\left(Y{111}, \ldots, Y_{11 r}, \ldots, Y_{s t 1}, \ldots, Y_{s t r}\right)^{\prime}=\alpha \mathbf{1} s \otimes \mathbf{1}t \otimes \mathbf{1}_r$$ 并且，通过协方差算法，str $\times \operatorname{str}$ 协方差矩阵 $$\Sigma=\sigma_S^2\left[\mathbf{I}_s \otimes \mathbf{J}_t \otimes \mathbf{J}_r\right]+\sigma_T^2\left[\mathbf{J}_s \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right] \quad+\sigma S T^2\left[\mathbf{I} s \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right]+\sigma R(S T)^2\left[\mathbf{I} s \otimes \mathbf{I}_t \otimes \mathbf{I}_r\right] .$$ 平方和矩阵不取决于因子是固定的还是随机的。因此，平方和 $\mathbf{Y}^{\prime} \mathbf{A}_m \mathbf{Y}$ 为了 $m=1, \ldots, 5$ 与示例 4.5.1 中给出的相同。此外， $\mathbf{A}_m \boldsymbol{\Sigma}=c_m \mathbf{A}_m$ 为了 $m=1, \ldots, 5$ 在哪里 $$c_1=\operatorname{tr} \sigma_S^2+s r \sigma_T^2+r \sigma S T^2+\sigma{R(S T)}^2 \quad c_2=\operatorname{tr} \sigma_S^2+r \sigma_{S T}^2+\sigma_{R(S T)}^2 c_3=s r \sigma_T^2+r \sigma_{S T}^2+\sigma_{R(S T)}^2$$

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