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# 统计代写|广义线性模型代写Generalized linear model代考|Criterion measures

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## 统计代写|广义线性模型代写Generalized linear model代考|Criterion measures

In the following definitions,
\begin{aligned} p & =\text { number of predictors } \ n & =\text { number of observations } \ L\left(M_k\right) & =\text { likelihood for model } k \ \mathcal{L}\left(M_k\right) & =\text { log likelihood for model } k \ D\left(M_k\right) & =\text { deviance of model } k \ G^2\left(M_k\right) & =\text { likelihood-ratio test of model } k \end{aligned}
Next, we provide the formulas for two criterion measures useful for model comparison. They include terms based on the log likelihood along with a penalty term based on the number of parameters in the model. In this way, the criterion measures seek to balance our competing desires for finding the best model (in terms of maximizing the likelihood) with model parsimony (including only those terms that significantly contribute to the model).
We introduce the two main model selection criterion measures below; also see Hilbe (1994, 2011) for more considerations.
AIC
The Akaike ( $\underline{1973})$ information criterion may be used to compare competing nested or nonnested models. The information criterion is a measure of the information lost in using the associated model. The goal is to find the model that has the lowest loss of information. In that sense, lower values of the criterion are indicative of a preferable model. Furthermore, a difference of greater than 2 indicates a marked preference for the model with the smaller criterion measure. When the measure is defined without scaling by the sample size, marked preference is called when there is a difference greater than two. The (scaled) formula is given by
$$\mathrm{AIC}=-2 \mathcal{L}\left(M_k\right)+2 p$$

## 统计代写|广义线性模型代写Generalized linear model代考|The interpretation of $R_2$ in linear regression

One of the most prevalent model measures is $R^2$. This statistic is usually discussed in introductory linear regression along with various ad hoc rules on its interpretation. A fortunate student is taught that there are many ways to interpret this statistic and that these interpretations have been generalized to areas outside linear regression.
For the linear regression model, we can define the $R^2$ measure in the following ways:
\begin{aligned} n & =\text { number of observations } \ p & =\text { number of predictors } \ M_\alpha & =\text { model with only an intercept } \ M_\beta & =\text { model with intercept and predictors } \end{aligned}
Percentage variance explained
The most popular interpretation is the percentage variance explained, where it can be shown that the $R^2$ statistic is equal to the ratio of the variance of the fitted values and to the total variance of the fitted values.
\begin{aligned} \text { RSS } & =\text { residual sum of squares }=\sum_{i=1}^n\left(y_i-\widehat{y}i\right)^2 \ \text { TSS } & =\text { total sum of squares }=\sum{i=1}^n\left(y_i-\bar{y}\right)^2 \ R^2 & =\frac{\text { TSS }- \text { RSS }}{\text { TSS }}=1-\frac{\text { RSS }}{\text { TSS }}=1-\frac{\sum_{i=1}^n\left(y_i-\widehat{y}i\right)^2}{\sum{i=1}^n\left(y_i-\bar{y}\right)^2} \end{aligned}

## 统计代写|广义线性模型代写Generalized linear model代考|Criterion measures

\begin{aligned} p & =\text { number of predictors } \ n & =\text { number of observations } \ L\left(M_k\right) & =\text { likelihood for model } k \ \mathcal{L}\left(M_k\right) & =\text { log likelihood for model } k \ D\left(M_k\right) & =\text { deviance of model } k \ G^2\left(M_k\right) & =\text { likelihood-ratio test of model } k \end{aligned}

aic
Akaike ($\underline{1973})$)信息标准可用于比较相互竞争的嵌套模型或非嵌套模型。信息标准是对使用相关模型时丢失的信息的度量。目标是找到信息损失最小的模型。从这个意义上说，较低的标准值表示较好的模型。此外，大于2的差异表明对具有较小标准度量的模型有明显的偏好。当测量的定义没有按样本大小进行缩放时，当差异大于2时，称为标记偏好。(缩放后的)公式由
$$\mathrm{AIC}=-2 \mathcal{L}\left(M_k\right)+2 p$$

## 统计代写|广义线性模型代写Generalized linear model代考|The interpretation of $R_2$ in linear regression

\begin{aligned} n & =\text { number of observations } \ p & =\text { number of predictors } \ M_\alpha & =\text { model with only an intercept } \ M_\beta & =\text { model with intercept and predictors } \end{aligned}

\begin{aligned} \text { RSS } & =\text { residual sum of squares }=\sum_{i=1}^n\left(y_i-\widehat{y}i\right)^2 \ \text { TSS } & =\text { total sum of squares }=\sum{i=1}^n\left(y_i-\bar{y}\right)^2 \ R^2 & =\frac{\text { TSS }- \text { RSS }}{\text { TSS }}=1-\frac{\text { RSS }}{\text { TSS }}=1-\frac{\sum_{i=1}^n\left(y_i-\widehat{y}i\right)^2}{\sum{i=1}^n\left(y_i-\bar{y}\right)^2} \end{aligned}

## MATLAB代写

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