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

#### Doug I. Jones

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

The probit model was first used in bioassay or quantal analysis. Typically, the probability of death was measured against the log of some toxin (for example, an insecticide). Whether death occurred depended on the tolerance the subject had to the toxic agent. Subjects with a lower tolerance were more likely to die. Tolerance was assumed to be distributed normally, hence the use of the probit transform, $\mu=\Phi(x \boldsymbol{\beta})$. Researchers still use probit for bioassay analysis, and they often use it to model other data situations.
The probit link function is the inverse of $\Phi$, the cumulative normal distribution defined as
$$\mu=\int_{-\infty}^\eta \phi(u) d u$$
where $\phi(u)$ is the standard normal density function
$$\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{1}{2} u^2\right)$$
Using a probit regression model on binary or grouped binomial data typically results in output similar to that from logistic regression. Logistic regression is usually preferred because of the wide variety of fit statistics associated with the model. Moreover, exponentiated probit coefficients cannot be interpreted as odds ratios, but they can be interpreted as odds ratios in logistic models (see section 10.6.2). However, if normality is involved in the linear relationship, as it often is in bioassay, then probit may be the appropriate model. It may also be used when the researcher is not interested in odds but rather in prediction or classification. Then if the deviance of a probit model is significantly lower than that of the corresponding logit model, the former is preferred. This dictum holds when comparing any of the links within the binomial family.

## 统计代写|广义线性模型代写generalized linear model代考|The clog-log and log-log models

Clog-log and log-log links are asymmetrically sigmoidal. For clog-log models, the upper part of the sigmoid is more elongated or stretched out than the logit or probit. Log-log models are based on the converse. The bottom of the sigmoid is elongated or skewed to the left.

We provide the link, inverse link, and derivative of the link for both clog-log and log-log models in table 10.2. As we showed for the probit model, we can change logit or probit to clog-log or log-log by replacing the respective functions in the GLM-binomial algorithm.
Figure 10.3 shows the asymmetry of the clog-log and log-log links. We investigate the consequences of this asymmetry on the fitted values and discuss the relationship of the coefficient estimates from models using these links.

We may wish to change the focus of modeling from the positive to the negative outcomes. Assume that we have a binary outcome variable $y$ and the reverse outcome variable $z=1-y$. We have a set of covariates $X$ for our model. If we use a symmetric link, then we can fit a binary model to either
outcome. The resulting coefficients will differ in sign for the two models and the predicted probabilities (fitted values) will be complementary. They are complementary in that the fitted probabilities for the model on $y$ plus the fitted probabilities for the model on $z$ add to one. See our previous warning note on the sign change of the coefficients in section 9.2.

We say that $\operatorname{probit}(y)$ and $\operatorname{probit}(z)$ form a complementary pair and that logit( $y)$ and $\operatorname{logit}(z)$ form a complementary pair. Although the equivalence of the link in the complementary pair is true with symmetric links, it is not true for the asymmetric links. The log-log and clog-log links form a complementary pair. This means that $\log \log (y)$ and $\operatorname{cog} \log (z)$ form a complementary pair and that $\log \log (z)$ and cloglog $(y)$ form a complementary pair.

# 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 probit model

probit链接函数是$\Phi$的倒数，累积正态分布定义为
$$\mu=\int_{-\infty}^\eta \phi(u) d u$$

$$\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{1}{2} u^2\right)$$

## 统计代写|广义线性模型代写generalized linear model代考|The clog-log and log-log models

log-log和log-log链接是非对称的s型。对于log-log模型，s型曲线的上半部分比logit或probit更拉长或伸展。对数-对数模型基于相反的情况。乙状结肠底部拉长或向左倾斜。

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