# 统计代写|广义线性模型代写generalized linear model代考|The power family

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## 统计代写|广义线性模型代写generalized linear model代考|The power family

The power function may be extended so that it is regarded as a family of distributions. That is, we may model data as a power model, like a gamma or logit model. The difference is that the power distribution is not directly based on a probability distribution, that is, a member of the exponential family of distributions.
Although the log-likelihood and deviance functions were derived directly from the exponential family form of a probability distribution, one can derive a working quasideviance from the variance function. See chapter 17 on quasilikelihood for more details. The quasideviance for the power family is
$$\frac{2 y}{(1-a)\left(y^{1-a}-\mu^{1-a}\right)}-\frac{2}{(2-a)\left(y^{2-a}-\mu^{2-a}\right)}$$
One can then create a power algorithm that uses the above deviance function with the power link, the power inverse link, the derivative of the power link, and the power variance functions. This is an example of a quasideviance model; one that is an extended member of the GLM family.

One use of power links is to assess optimal models within a GLM distribution family. One can use the link function to check various statistics resulting from changes to $a$ in the power link. This gives a wide variety of possible links for each continuous family.

However, we may need to use a general power link model to assess models between families. Of course, we may simply create a series of models using a continuum of links within each of the Gaussian $(a=1)$, Poisson $(a=0)$, gamma ( $a=-1$ ), and inverse Gaussian $(a=-2$ ) families. If all other aspects of the models are the same, then the model having the lower deviance value and bestfitting residuals will be the model of choice. However, we are thereby giving wide leniency to the notion of “aspects”.

## 统计代写|广义线性模型代写generalized linear model代考|The binomial–logit family

Except for the canonical Gaussian model, binomial models are used more often than any other member of the GLM family.
Binomial regression models are used in analyses having discrete (corresponding to the number of successes for $k$ trials ) or grouped responses. As shown in table 9.1 , models of binary responses are based on the Bernoulli $(n=1$ ) or binomial distribution. The Bernoulli distribution is a degenerate case of the binomial, where the number of trials is equal to 1 .
Table 9.1: Binomial regression models
\begin{tabular}{ll}
Response & Model \
\hline Binary: ${0,1}$ & $\operatorname{Bernoulli}=\operatorname{Binomial}(1)$ \
Proportional: ${0,1, \ldots, k}$ & $\operatorname{Binomial}(k)$
\end{tabular}
Proportional-response data involve two variables corresponding to the number of successes, $y_i$, of a population of $k_i$ trials , where both variables are indexed by $i$ because grouped data do not require that the number of trials be the same for each observation. Hence, a binomial model can be considered as a weighted binary response model or simply as a two-part response relating the proportion of successes to trials. We discuss the relationship between these two methods of dealing with discrete data. Binary data can be modeled as grouped, but the reverse is not always possible.

Binary responses take the form of 0 or 1 , with 1 typically representing a success or some other positive result, whereas 0 is taken as a failure or some negative result. Responses may also take the form of $1 / 2, a / b$, or a similar pair of values. However, values other than 0 and 1 are translated to a $0 / 1$ format by the binary regression algorithm before estimation. The reason for this will be apparent when we describe the probability function. Also in Stata, a 0 indicates failure and any other nonzero nonmissing value represents a success in Bernoulli models.

# 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|The power family

$$\frac{2 y}{(1-a)\left(y^{1-a}-\mu^{1-a}\right)}-\frac{2}{(2-a)\left(y^{2-a}-\mu^{2-a}\right)}$$

## 统计代写|广义线性模型代写generalized linear model代考|The binomial–logit family

\begin{tabular}{ll}
Response & Model \hline Binary: ${0,1}$ & $\operatorname{Bernoulli}=\operatorname{Binomial}(1)$ \Proportional:${0,1, \ldots, k}$ & $\operatorname{Binomial}(k)$
\end{tabular}

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