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# 统计代写|时间序列分析代写Time-Series Analysis代考|Random effects and mixed effects models

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## 统计代写|时间序列分析代写Time-Series Analysis代考|Random effects and mixed effects models

The model in Eq. (7.14) is a special case of the fixed effects model with two factors. When the levels of these factors are randomly selected, and we want to generalize the result from analysis to a much larger population, then the model becomes a random effects model. The model becomes a mixed effects model when some factors are random and some are fixed. For example, in Model (7.14), if treatments are randomly assigned, we have
$$Z_{i, j, t}=\mu+\alpha_i+\beta_t+\gamma_{i, t}+e_{i, j, t},$$
where $\alpha_i$ is random, i.i.d. $N\left(0, \sigma_a^2\right)$, independent of $e_{i, j, t}$ The analysis of variance table for the fixed effects model, the random effects model, and the mixed effects model are the same as Table 7.7, but when a model contains a random factor like Eq. (7.30), it is important to note the following:

1. The variance of $Z_{i, j, t}$ is no longer equal to the variance of $e_{i, j, t}$. Instead, if we also assume that $\operatorname{Var}\left(\alpha_i\right)=\sigma_a^2$, we have
$$\operatorname{Var}\left(Z_{i, j, t}\right)=\operatorname{Var}\left(\alpha_i\right)+\operatorname{Var}\left(e_{i, j, t}\right)=\sigma_\alpha^2+\sigma^2$$
2. The model in (7.31) can also be written as
$$Z_{i, j, t}=\mu+\beta_t+\gamma_{i, t}+\varepsilon_{i, j, t},$$
where the $\varepsilon_{i, j, t}$ are now i.i.d. $N\left(0, \sigma_{\varepsilon}^2\right), \sigma_{\varepsilon}^2=\sigma_\alpha^2+\sigma^2$.
3. The Expected Mean Squares (EMS) for treatment is $\sigma^2+n \sigma_\alpha^2$. Hence, we can estimate the variance of the random treatment term using
$$\hat{\sigma}_\alpha^2=\frac{\text { Mean Squares for Treatment }-s^2}{n},$$
where $s^2$ is the residual mean square error.
4. The null and alternative hypotheses for the random treatment in Model (7.30) are now
\begin{aligned} & H_0: \sigma_\alpha^2=0, \ & H_a: \sigma_\alpha^2>0 \end{aligned}

## 统计代写|时间序列分析代写Time-Series Analysis代考|Nested random effects model

In some applications, subjects are randomly selected from a population. For example, in agricultural studies, where researchers want to compare the effects of three different fertilizers in terms of the yield of a certain product such as tomatoes. In this case, “subjects” refers to plots of land. By realizing the effects of land, and more importantly, being interested in the effects of fertilizers on a wide variety of plots, the researchers may randomly select a certain number of plots of land from a population of plots when assigning fertilizers within their experiments. In such a case, we will consider the following nested random effects model:
$$Z_{i, j, t}=\mu+\alpha_i+\theta_{j(i)}+\beta_t+\gamma_{i, t}+e_{i, j, t}$$
where $\alpha_i, \beta_t$, and $\gamma_{i, t}$ are fixed effects defined in Eq. (7.14), but $\theta_{j(i)}$ is a random effect for subject $j$ associated with treatment $i$. We assume that the $\theta_{j(i)}$ are i.i.d. $N\left(0, \sigma_\theta^2\right)$, which are independent of $e_{i, j, t}$. The variance of $Z_{i, j, t}$ is no longer equal to the variance of $e_{i, j, t}$ It becomes the sum of the variances of $\theta_{j(i)}$ and $e_{i, j, t}$. Hence, the variance-covariance matrix of $\mathbf{Z}{i, j}=\left[Z{i, j, 1}, \ldots, Z_{i, j, p}\right]^{\prime}$ becomes
$$\sigma_\theta^2 \mathbf{H}+\mathbf{\Sigma}$$
where $\mathbf{H}$ is a matrix of ones.

Equivalently, we can rewrite the model in Eq. (7.34) as
$$Z_{i, j, t}=\mu+\alpha_i+\beta_t+\gamma_{i, t}+\varepsilon_{i, j, t},$$
where $\varepsilon_{i, j, t}=\theta_{j(i)}+e_{i, j, t}$. If the $e_{i, j, t}$ are i.i.d. $N\left(0, \sigma^2\right)$, then it can be shown that the variance and covariance of $\varepsilon_{i, j, t}$ and hence $Z_{i, j, t}$ will follow the structure of common symmetry given in Eq. (7.22) of Section 7.3.2.

The analysis of variance table for this nested random effects model is now modified as given in Table 7.8.

## 间序列分析代写Time-Series Analysis代考|Random effects and mixed effects models

$$Z_{i, j, t}=\mu+\alpha_i+\beta_t+\gamma_{i, t}+e_{i, j, t},$$

$Z_{i, j, t}$的方差不再等于$e_{i, j, t}$的方差。相反，如果我们也假设$\operatorname{Var}\left(\alpha_i\right)=\sigma_a^2$，我们有
$$\operatorname{Var}\left(Z_{i, j, t}\right)=\operatorname{Var}\left(\alpha_i\right)+\operatorname{Var}\left(e_{i, j, t}\right)=\sigma_\alpha^2+\sigma^2$$

$$Z_{i, j, t}=\mu+\beta_t+\gamma_{i, t}+\varepsilon_{i, j, t},$$

$$\hat{\sigma}_\alpha^2=\frac{\text { Mean Squares for Treatment }-s^2}{n},$$

\begin{aligned} & H_0: \sigma_\alpha^2=0, \ & H_a: \sigma_\alpha^2>0 \end{aligned}

## 统计代写|时间序列分析代写Time-Series Analysis代考|Nested random effects model

$$Z_{i, j, t}=\mu+\alpha_i+\theta_{j(i)}+\beta_t+\gamma_{i, t}+e_{i, j, t}$$

$$\sigma_\theta^2 \mathbf{H}+\mathbf{\Sigma}$$

$$Z_{i, j, t}=\mu+\alpha_i+\beta_t+\gamma_{i, t}+\varepsilon_{i, j, t},$$

## MATLAB代写

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