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

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

Kempthorne (1952) provides the EMSs for $T, R T, S, S T$, and $R S(T)$ in his Table 19.2. Kempthorne’s EMSs are the same as the EMSs given here, although Kempthorne uses different notation. Kempthorne’s $\sigma^2, t_j, \sigma_s^2, s_k$, and $(t s){j k}$ are equivalent to our $s \sigma{R T}^2,\left(\bar{\mu}{j .}-\bar{\mu}{. .}\right), \sigma_{R S(T)}^2,\left(\bar{\mu}{. k}-\bar{\mu}{. .}\right)$, and $\left(\bar{\mu}{i j}-\bar{\mu}{j .}-\bar{\mu}{. k}+\bar{\mu}{. .}\right)$, respectively.

Example 4.5.4 Consider an experiment where bst experimental units are divided into $b$ homogeneous sets of $s t$ units. Within each of the $b$ sets (or, equivalently, within each of the $b$ random blocks $B$ ) the $s t$ units are randomly assigned to the $s t$ combinations of the two fixed factors $S$ and $T$ where factor $S$ has $s$ levels and $T$ has $t$ levels. This is a classic random block design where the fixed treatments are identified by two fixed factors. Let $Y_{i j k}$ be the random variable representing the observation in the $k^{\text {th }}$ level of factor $T$, the $j^{\text {th }}$ level of factor $S$, and the $i^{\text {th }}$ block for $i=1, \ldots, b, j=1, \ldots, s$, and $k=1, \ldots, t$. This experiment is characterized by the model
$$Y_{i j k}=\mu_{j k}+B_i+R_{i j k}$$
where $\mu_{j k}$ are $s t$ constants representing the mean effect of the $j k^{\text {th }}$ combination of factors $S$ and $T$; the $B_i$ are random variables representing the random effect of blocks; and the $R_{i j k}$ are random variables representing the random residual or remainder. It is our intention to write a covariance matrix that contains two variance components, one associated with the variance of the random variables $B_i$ and one associated with the variance of the random variables $R_{i j k}$. However, to use the covariance algorithm, we must first rewrite the variables $R_{i j k}$ in terms of the factor letters $B, S$, and $T$. Note that $R_{i j k}$ can be equivalently written as
$$R_{i j k}=B S_{i j}+B T_{i k}+B S T_{i j k}$$
where $B S_{i j}, B T_{i k}$, and $B S T_{i j k}$ are random variables representing the interaction of $B$ with $S, B$ with $T$, and $B$ with $S T$, respectively. We could proceed from here by putting the last two equations together to produce a model
$$Y_{i j k}=\mu_{j k}+B_i+B S_{i j}+B T_{i k}+B S T_{i j k} .$$
However, the last model has four sets of random variables and would thus require a definition with four, not two, random components.

## 统计代写|广义线性模型代写generalized linear model代考|ORDINARY LEAST-SQUARES ESTIMATION

We begin with a simple example. An engineer wants to relate the fuel consumption of a new type of automobile to the speed of the vehicle and the grade of the road traveled. He has a fleet of $n$ vehicles. Each vehicl0e is assigned to operate at a constant speed (in miles per hour) on a specific grade (in percent grade) and the fuel consumption (in $\mathrm{ml} / \mathrm{sec}$ ) is recorded. The engineer believes that the expected fuel consumption is a linear function of the speed of the vehicle and the speed of the vehicle times the grade of the road. Let $Y_i$ be a random variable that represents the observed fuel consumption of the $i^{\text {th }}$ vehicle, operating at a fixed speed, on a road with a constant grade. Let $x_{i 1}$ represent the speed of the $i^{\text {th }}$ vehicle and let $x_{i 2}$ represent the speed times the grade of the $i^{\text {th }}$ vehicle. The expected fuel consumption of the $i^{\text {th }}$ vehicle can be represented by
$$\mathrm{E}\left(Y_i\right)=\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}$$
where $\beta_0, \beta_1$, and $\beta_2$ are unknown parameters. Due to qualities intrinsic to each vehicle, the observed fuel consumptions differ somewhat from the expected fuel consumptions. Therefore, the observed fuel consumption of the $i^{\text {th }}$ vehicle is represented by
$$Y_i=\mathrm{E}\left(Y_i\right)+E_i$$
or
$$Y_i=\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}+E_i$$
where $E_i$ is a random variable representing the difference between the observed fuel consumption and the expected fuel consumption of the $i^{\text {th }}$ vehicle. An example data set for this fuel, speed, grade experiment is provided in Table 5.1.1. In a more general setting consider a problem where the expected value of a random variable $Y_i$ is assumed to be a linear combination of $p-1$ different variables $x_{i 1}, x_{i 2}, \ldots, x_{i, p-1}$. That is,
$$\mathrm{E}\left(Y_i\right)=\beta_0+\beta_1 x_{i 1}+\cdots+\beta_{p-1} x_{i, p-1} .$$
Adding a component of error, $E_i$, to represent the difference between the observed value of $Y_i$ and the expected value of $Y_i$ we obtain
$$Y_i=\beta_0+\beta_1 x_{i 1}+\cdots+\beta_{p-1} x_{i, p-1}+E_i .$$

# 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|Factorial Experiments

Kempthorne (1952) 为 $T, R T, S, S T$ ，和 $R S(T)$ 在他的表 $19.2$ 中。Kempthorne 的 EMS 与此处给出的 EMS 相同，尽管 Kempthorne 使用不同的符号。肯普索恩 $\sigma^2, t_j, \sigma_s^2, s_k$ ，和 $(t s) j k$ 相当于我们的 $s \sigma R T^2,(\bar{\mu} j .-\bar{\mu} .),. \sigma_{R S(T)}^2,(\bar{\mu} . k-\bar{\mu} .$.$) ， 和 (\bar{\mu} i j-\bar{\mu} j .-\bar{\mu} . k+\bar{\mu} .$.$) ，分别。$

$$Y_{i j k}=\mu_{j k}+B_i+R_{i j k}$$

$$R_{i j k}=B S_{i j}+B T_{i k}+B S T_{i j k}$$

$$Y_{i j k}=\mu_{j k}+B_i+B S_{i j}+B T_{i k}+B S T_{i j k} .$$

## 统计代写|广义线性模型代写generalized linear model代考|ORDINARY LEAST-SQUARES ESTIMATION

$$\mathrm{E}\left(Y_i\right)=\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}$$

$$Y_i=\mathrm{E}\left(Y_i\right)+E_i$$

$$Y_i=\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}+E_i$$

$$\mathrm{E}\left(Y_i\right)=\beta_0+\beta_1 x_{i 1}+\cdots+\beta_{p-1} x_{i, p-1} .$$

$$Y_i=\beta_0+\beta_1 x_{i 1}+\cdots+\beta_{p-1} x_{i, p-1}+E_i .$$

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