# 统计代写|线性回归分析代写linear regression analysis代考|Generalized Additive Models

## 统计代写|线性回归分析代写linear regression analysis代考|Generalized Additive Models

There are many alternatives to the binomial and Poisson regression GLMs. Alternatives to the binomial GLM of Definition 13.6 include the discriminant function model of Definition 13.7, the quasi-binomial model, the binomial generalized additive model (GAM), and the beta-binomial model of Definition 13.2 .

Alternatives to the Poisson GLM of Definition 13.11 include the quasiPoisson model, the Poisson GAM, and the negative binomial regression model of Definition 13.3. Other alternatives include the zero truncated Poisson model, the zero truncated negative binomial model, the hurdle or zero inflated Poisson model, the hurdle or zero inflated negative binomial model, the hurdle or zero inflated additive Poisson model, and the hurdle or zero inflated additive negative binomial model. See Zuur et al. (2009), Simonoff (2003), and Hilbe (2011).

Many of these models can be visualized with response plots. An interesting research project would be to make response plots for these models, adding the conditional mean function and lowess to the plot. Also make OD plots to check whether the model handled overdispersion. This section will examine several of the above models, especially GAMs.

Definition 13.18. In a $1 D$ regression, $Y$ is independent of $\boldsymbol{x}$ given the sufficient predictor $S P=h(\boldsymbol{x})$ where $S P=\alpha+\boldsymbol{\beta}^T \boldsymbol{x}$ for a GLM. In a generalized additive model, $Y$ is independent of $\boldsymbol{x}=\left(x_1, \ldots, x_p\right)^T$ given the additive predictor $A P=\alpha+\sum_{j=1}^p S_j\left(x_j\right)$ for some (usually unknown) functions $S_j$. The estimated sufficient predictor $\mathrm{ESP}=\hat{h}(\boldsymbol{x})$ and $\mathrm{ESP}=\hat{\alpha}+\hat{\boldsymbol{\beta}}^T \boldsymbol{x}$ for a GLM. The estimated additive predictor $\mathrm{EAP}=\hat{\alpha}+\sum_{j=1}^p \hat{S}_j\left(\boldsymbol{x}_j\right)$. An ESPresponse plot is a plot of ESP versus $Y$ while an EAP-response plot is a plot of EAP versus $Y$.

## 统计代写|线性回归分析代写linear regression analysis代考|Response Plots

It is well known that the residual plot of $E S P$ or $E A P$ versus the residuals (on the vertical axis) is useful for checking the model, but there are several other plots using the ESP that can be generalized to a GAM by replacing the $E S P$ by the $E A P$. The response plots are used to visualize the $1 \mathrm{D}$ regression model or GAM in the background of the data. For 1D regression, a response plot is the plot of the $E S P$ versus the response $Y$ with the estimated model conditional mean function and a scatterplot smoother often added as visual aids. Note that the response plot is used to visualize $Y \mid S P$ while for the additive error regression model, a residual plot of the ESP versus the residual is used to visualize $e \mid S P$. For a GAM, these two plots replace the ESP by the EAP . Assume that the ESP or EAP takes on many values.

Suppose the zero mean constant variance errors $e_1, \ldots, e_n$ are iid from a unimodal distribution that is not highly skewed. For additive error regression, see Definition 13.1i), the estimated mean function is the identity line with unit slope and zero intercept. If the sample size $n$ is large, then the plotted points should scatter about the identity line and the residual $=0$ line in an evenly populated band for the response and residual plots, with no other pattern. To avoid overfitting, assume $n \geq 10 d$ where $d$ is the model degrees of freedom. Hence $d=p$ for multiple linear regression with OLS.

If $Z_i=Y_i / m_i$, then the conditional distribution $Z_i \mid \boldsymbol{x}_i$ of the binomial GAM can be visualized with a response plot of the EAP versus $Z_i$ with the estimated mean function of the $Z_i, \hat{E}(Z \mid A P)=\frac{\exp (E A P)}{1+\exp (E A P)}$, and a scatterplot smoother added to the plot as a visual aids. Instead of adding a lowess curve to the plot, consider the following alternative. Divide the EAP into $J$ slices with approximately the same number of cases in each slice. Then compute $\hat{\rho}_s=\sum_s Y_i / \sum_s m_i$ where the sum is over the cases in slice $s$. Then plot the resulting step function. For binary data the step function is simply the sample proportion in each slice. The response plot for the beta-binomial GAM is similar.

# 线性回归代考

## 统计代写|线性回归分析代写linear regression analysis代考|Generalized Additive Models

$\mathrm{EAP}=\hat{\alpha}+\sum_{j=1}^p \hat{S}_j\left(\boldsymbol{x}_j\right)$. ESPresponse 图是 ESP 与 $Y$ 而 EAP 响应图是 EAP 与 $Y$.

## 统计代写|线性回归分析代写linear regression analysis代考|Response Plots

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