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

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

Cited in various places such as Hilbe (1993b) and Francis, Green, and Payne (1993), GLMs are characterized by an expanded itemized list given by the following:
A random component for the response, $y$, which has the characteristic variance of a distribution that belongs to the exponential family.
A linear systematic component relating the linear predictor, $\eta=X \boldsymbol{\beta}$, to the product of the design matrix $X$ and the parameters $\boldsymbol{\beta}$.
A known monotonic, one-to-one, differentiable link function $g(\cdot)$ relating the linear predictor to the fitted values. Because the function is one-to-one, there is an inverse function relating the mean expected response, $E(y)=\mu$, to the linear predictor such that $\mu=g^{-1}(\eta)=E(y)$.
The variance may change with the covariates only as a function of the mean.
There is one IRLS algorithm that suffices to fit all members of the class.
Item 5 is of special interest. The traditional formulation of the theory certainly supposed that there was one algorithm that could fit all GLMs. We will see later how this was implemented. However, there have been extensions to this traditional viewpoint. Adjustments to the weight function have been added to match the usual Newton-Raphson algorithms more closely and so that more appropriate standard errors may be calculated for noncanonical link models. Such features as scaling and robust variance estimators have also been added to the basic algorithm. More importantly, sometimes a traditional GLM must be restructured and fit using a model-specific Newton-Raphson algorithm. Of course, one may simply define a GLM as a model requiring only the standard approach but doing so would severely limit the range of possible models. We prefer to think of a GLM as a model that is ultimately based on the probability function belonging to the exponential family of distributions, but with the proviso that this criterion may be relaxed to include quasilikelihood models as well as certain types of multinomial, truncated , censored , and inflated models. Most of the latter tvpe require a Newton-Raphson approach rather than the traditional IRLS algorithm.

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

The link function relates the mean $\mu=E(y)$ to the linear predictor $X \boldsymbol{\beta}$, and the variance function relates the variance as a function of the mean $V(y)=a(\phi) v(\mu)$, where $a(\phi)$ is the scale factor. For the Poisson, binomial, and negative binomial variance models, $a(\phi)=1$.
Breslow (1996) points out that the critical assumptions in the GLM framework may be stated as follows:
Statistical independence of the $n$ observations.
The variance function $v(\mu)$ is correctly specified.
$a(\phi)$ is correctly specified (1 for Poisson, binomial, and negative binomial).
The link function is correctly specified.
Explanatory variables are of the correct form.
There is no undue influence of the individual observations on the fit.
As a simple illustration, in table 2.1 we demonstrate the effect of the assumed variance function on the model and fitted values of a simple GLM.
Note: The models are all fit using the identity link, and the data consist of three observations $(y, x)={(1,1),(2,2),(9,3)}$. The fitted models are included in the last column.

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

Breslow(1996)指出，GLM框架中的关键假设可以表述如下:
$n$观测值的统计独立性。

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