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# 统计代写|广义线性模型代考GENERALIZED LINEAR MODEL代考|MAT22006 The model η = β0 + β1×1 + β2×2

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The linear model $\eta=\beta_{0}+\beta_{1} x+\beta_{2} x_{2}=\boldsymbol{f}^{\top}(\boldsymbol{x}) \boldsymbol{\beta}$ may be transformed to the canonical form $\eta=z_{1}+z_{2}$ by defining $z_{1}=\beta_{0}+\beta_{1} x_{1}$ and $z_{2}=\beta_{2} x_{2}$. This is equivalent to
$$\left[\begin{array}{c} 1 \ z_{1} \ z_{2} \end{array}\right]=\left[\begin{array}{ccc} 1 & 0 & 0 \ \beta_{0} & \beta_{1} & 0 \ 0 & 0 & \beta_{2} \end{array}\right]\left[\begin{array}{c} 1 \ x_{1} \ x_{2} \end{array}\right]$$
or
$$\boldsymbol{f}(\boldsymbol{z})=\boldsymbol{B} \boldsymbol{f}(\boldsymbol{x})$$
While globally D-optimal designs were successfully found in Section $4.4$ using the canonical form of the linear predictor, this cannot be achieved for the case of two or more explanatory variables. Although no constraints were placed on the value of $z$ in Section $4.4$ for the logit link, a constraint effectively occurred through the model weight $\omega=\pi(1-\pi)$ becoming very small as $\pi$ became close to 0 or 1 . Similarly, constraints occurred through the behaviour of the model weights for the probit and complementary log-log links. However, in the $\left(z_{1}, z_{2}\right)$ plane considered in this section, the entire length of the line $\eta=z_{1}+z_{2}=0$ gives $\pi=0.5$ for the logit and probit links, and $\pi=1-\exp (-1)=0.6321$ for the complementary log-log link (from (4.2), (4.4) and (4.6), respectively). The model weight is constant along the entire line, so no constraining occurs.
There is nothing “special” about the line $z_{1}+z_{2}=0$. For a given value of $c$, the model weight $\omega(\boldsymbol{z})$ takes a constant value along the line $\eta=z_{1}+z_{2}=c$. So in order to restrict the values of the explanatory variables $x_{1}$ and $x_{2}$ to “reasonable” values, constraints must be placed on them.

It has become standard to investigate locally optimal designs for values of $x_{1}$ and $x_{2}$ in the region $\mathcal{X}=\left{\left(x_{1}, x_{2}\right):-1 \leq x_{1} \leq 1,-1 \leq x_{2} \leq 1\right}$. This is not really restrictive. If an explanatory variable, $w$, lies between $a$ and $b(>a)$, the transformation
$$x=\frac{2 w-(a+b)}{b-a}$$
gives an explanatory variable $x$ that lies between $-1$ and 1 . Rewriting the linear predictor $\eta$ in terms of variables $x_{i}$ instead of variables $w_{i}$ necessitates a change to the parameter vector $\boldsymbol{\beta}$. Once the optimal design has been determined in terms of $x$, transform back to $w$ using
$$w=\frac{1}{2}[(a+b)+(b-a) x]$$
This is illustrated in Example $4.5 .5$.

## 统计代写|广义线性模型代考GENERALIZED LINEAR MODEL代考|The logit link

Consider a situation where one seeks a locally D-optimal design on the region $\mathcal{X}=\left{\left(x_{1}, x_{2}\right):-1 \leq x_{1} \leq 1,-1 \leq x_{2} \leq 1\right}$ using the logit link and the
THE MODEL $\eta=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}$ parameter vector $\boldsymbol{\beta}=(1,1,1)^{\top}$. Remember that this is the assumed parameter vector. One purpose of using the design is to estimate the actual value of $\boldsymbol{\beta}$.
The method is a straightforward extension of that demonstrated earlier in Sub-section 4.4.1. Note first that there are now $p=3$ parameters, so that the required number of support points, $s$, will lie between $p=3$ and $p(p+1) / 2=6$. The values of $x_{1}$ and $x_{2}$ are constrained to lie between $-1$ and 1 , and the design weights must satisfy $0<\delta_{i}<1(i=1, \ldots, s)$ and $\delta_{1}+\cdots+\delta_{s}=1$. If constrOptim is used, these constraints must be written in the form $\boldsymbol{C v}-\boldsymbol{u} \geq \mathbf{0}$; see page 34 . Note here that there are two constraints on each of the $s$ values of $x_{1}$, two constraints on each of the $s$ values of $x_{2}$, two constraints on each of the $s$ individual values of $\delta_{i}$, and then one remaining constraint $\delta_{1}+\cdots+\delta_{s}=1$; i.e., a total of $6 s+1$ constraints. Denote by $x_{i j}$ the value of $x_{j}$ at the $i$ th support point $(i=1, \ldots, s ; j=1,2)$.

Let $\boldsymbol{x}{1}=\left(x{11}, x_{21}, \ldots, x_{s 1}\right)^{\top}, \boldsymbol{x}{2}=\left(x{12}, x_{22}, \ldots, x_{s 2}\right)^{\top}$ and $\boldsymbol{\delta}=\left(\delta_{1}, \ldots, \delta_{s}\right)^{\top}$. Then the constraints are written as

## 统计代写|广义线性模型代考 GENERALIZED LINEAR MODEL代 考|Preliminary comments

$$\boldsymbol{f}(\boldsymbol{z})=\boldsymbol{B} \boldsymbol{f}(\boldsymbol{x})$$

䢒条线没有什么“特别”的地方 $z_{1}+z_{2}=0$. 对于给定的值 $c$, 模型权重 $\omega(\boldsymbol{z})$ 沿线取一个常数 值 $\eta=z_{1}+z_{2}=c$. 所以为了限制解释变量的值 $x_{1}$ 和 $x_{2}$ 为了 “合理”的价值观，必须对其施 加约束。

\left 的分隔符缺失或无法识别
. 这并不是真正的限制。如果是解释变

$$x=\frac{2 w-(a+b)}{b-a}$$

$$w=\frac{1}{2}[(a+b)+(b-a) x]$$

## 统计代写|广义线性模型代考 GENERALIZED LINEAR MODEL代 考|The logit link

\left 的分隔符缺失或无法识别 使用 logit 链接和
THE MODEL $\eta=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}$ 参数向量 $\boldsymbol{\beta}=(1,1,1)^{\top}$. 请记住，这是假定的参数向 量。使用该设计的目的之一是估计实际值 $\boldsymbol{\beta}$.

## MATLAB代写

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