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

#### Doug I. Jones

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

For ML estimation of a model, the log-likelihood function (rather than the deviance) plays a paramount role in estimation. It is the log-likelihood function that is maximized, hence the term “maximum likelihood”.

We can illustrate the log-likelihood function for a particular link function (parameterization) by substituting the inverse link function $g^{-1}(\eta)$ for $\mu$. For example, the gamma canonical link (reciprocal function) is
$$\mu=\frac{1}{x \beta}$$
For each instance of a $\mu$ in the gamma $\mathcal{L}$ function, we replace $\mu$ with the inverse link of the linear predictor $1 / x \boldsymbol{\beta}$.
\begin{aligned} \mathcal{L} & =\sum_{i=1}^n\left{\frac{y_i / \mu_i-\left(-\ln \mu_i\right)}{-\phi}+\frac{1-\phi}{\phi} \ln y_i-\frac{\ln \phi}{\phi}-\ln \Gamma\left(\frac{1}{\phi}\right)\right} \ & =\sum_{i=1}^n\left{\frac{y_i x_i \boldsymbol{\beta}-\ln \left(x_i \boldsymbol{\beta}\right)}{-\phi}+\frac{1-\phi}{\phi} \ln y_i-\frac{\ln \phi}{\phi}-\ln \Gamma\left(\frac{1}{\phi}\right)\right} \end{aligned}

## 统计代写|广义线性模型代写generalized linear model代考|Log-gamma models

We mentioned before that the reciprocal link estimates the rate per unit of the model response, given a specific set of explanatory variables or predictors. The log-linked gamma represents the log-rate of the response. This model specification is identical to exponential regression. Such a specification, of course, estimates data with a negative exponential decline. However, unlike the exponential models found in survival analysis, we cannot use the log-gamma model with censored data. We see, though, that uncensored exponential models can be fit with GLM specifications. We leave that to the end of this chapter.
The log-gamma model, like its reciprocal counterpart, is used with data in which the response is greater than 0 . Examples can be found in nearly every discipline. For instance, in health analysis, length of stay (LOS) can generally be estimated using log-gamma regression because stays are always constrained to be positive. LOS data are generally estimated using Poisson or negative binomial regression because the elements of LOS are discrete. However, when there are many LOS elements – that is, many different LOS values — many researchers find the gamma or inverse Gaussian models to be acceptable and preferable.

Before GLM, data that are now estimated using log-gamma techniques were generally estimated using Gaussian regression with a log-transformed response. Although the results are usually similar between the two methods, the loggamma technique, which requires no external transformation, is easier to interpret and comes with a set of residuals with which to evaluate the worth of the model. Hence, the log-gamma technique is finding increased use among researchers who once used Gaussian techniques.

The IRLS algorithm for the log-gamma model is the same as that for the canonical-link model except that the link and inverse link become $\ln (\mu)$ and $\exp (\eta)$, respectively, and the derivative of $g$ is now $1 / \mu$. The ease with which we can change between models is one of the marked beauties of GLM. However, because the log link is not canonical, the IRLS and modified ML algorithms will give different standard errors. But, except in extreme cases, differences in standard errors are usually minimal. Except perhaps when working with small datasets, the method of estimation used generally makes little inferential difference.

# 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|ML estimation

$$\mu=\frac{1}{x \beta}$$

\begin{aligned} \mathcal{L} & =\sum_{i=1}^n\left{\frac{y_i / \mu_i-\left(-\ln \mu_i\right)}{-\phi}+\frac{1-\phi}{\phi} \ln y_i-\frac{\ln \phi}{\phi}-\ln \Gamma\left(\frac{1}{\phi}\right)\right} \ & =\sum_{i=1}^n\left{\frac{y_i x_i \boldsymbol{\beta}-\ln \left(x_i \boldsymbol{\beta}\right)}{-\phi}+\frac{1-\phi}{\phi} \ln y_i-\frac{\ln \phi}{\phi}-\ln \Gamma\left(\frac{1}{\phi}\right)\right} \end{aligned}

## 统计代写|广义线性模型代写generalized linear model代考|Log-gamma models

log-gamma模型与其对等的对应模型一样，用于响应大于0的数据。几乎在每个学科中都可以找到这样的例子。例如，在健康分析中，通常可以使用log-gamma回归来估计住院时间(LOS)，因为住院时间总是被限制为正的。由于LOS的元素是离散的，因此通常使用泊松或负二项回归来估计LOS数据。然而，当存在许多LOS元素时，即有许多不同的LOS值时，许多研究人员发现伽马或逆高斯模型是可接受的和优选的。

log-gamma模型的IRLS算法与标准链接模型的IRLS算法相同，只是链接和反向链接分别变成$\ln (\mu)$和$\exp (\eta)$, $g$的导数现在变成$1 / \mu$。我们可以轻松地在模型之间切换，这是GLM的显著优点之一。然而，由于日志链接不是规范的，IRLS和修改后的ML算法会给出不同的标准误差。但是，除极端情况外，标准误差的差异通常很小。除了在处理小数据集时，所使用的估计方法通常不会产生推论差异。

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