英文:
maximum likelihood estimation of parameters following polynomial logistic regression
问题
这是数据集，使用frair库，数据为gammarus。
我想估计参数p0、p1、p2和p3的值，公式如下：
*NA/No= exp(P0+ P1*density+ P2*density^2+P3*density^3)/(1+exp(P0+P1*density+ P2*density^2+P3*density^3))*，其中Na是被捕食的猎物数量，No是提供的猎物数量。
英文:
this is the datset, library(frair), data=gammarus
i want to estimate the parameters p0, p1 p2 and p3 formula is
*NA/No= exp(P0+ P1*density+ P2*density^2+P3*density^3)/(1+exp(P0+P1*density+ P2*density^2+P3*density^3))*, where Na is prey eaten and No is prey offered
答案1
得分: 0
## 设置```Rlibrary(dplyr)library(frair)d <- gammarus %>% mutate(y = eaten/(eaten + alive))
步骤 1：回归
你可以使用 lm（线性模型）函数从方程 中估算系数：
lm(y ~ density, data = d)
步骤 2：多项式回归
如果想要使用多项式函数形式，你可以使用 poly 函数。第一个参数是变量，第二个是多项式的次数，然后你必须指定你想要原始多项式还是正交多项式。在我们的情况下，它将是一个原始多项式，可以查看此帖子获取更多细节。
你可以通过将密度替换为三次原始密度多项式来估算 中的四个系数：
lm(y ~ poly(density, 3, raw = T), data = d)
步骤 3：逻辑回归
最后一步是从线性的 转换为逻辑的
。为此，你将需要
glm 函数（广义线性模型），并且你必须指定你想要的是 logit（而不是例如 probit，参见此帖子）规范，使用 family = binomial(link = "logit")。
glm(y ~ poly(density, 3, raw = T), data = d, family = binomial(link = "logit"))
<details><summary>英文:</summary>## Setup
library(dplyr)
library(frair)
d <- gammarus %>% mutate(y = eaten/(eaten + alive))
## Step 1: RegressionYou can estimate the coefficients from an equation  with the `lm` (linear model) function:
lm(y ~ density, data = d)
## Step 2: Polynomial regressionTo have a polynomial functional form instead, you can use the `poly` function. The first argument is the variable, the second is the degree of the polynomial, and you must then specify whether you want a raw or an orthogonal polynomial. In our case it would be a raw polynomial, check [this post](https://stats.stackexchange.com/questions/258307/raw-or-orthogonal-polynomial-regression) for more detail.You can estimate the four coefficients from  by replacing density with a third degree raw polynomial of density:
lm(y ~ poly(density, 3, raw = T), data = d)
## Step 3: Logistic regressionThe final step is to switch from the linear  to the logistic . For this you would need the `glm` function (generalized linear model) and you must specify that you want a logit (and not a probit for instance, cf. [this post](https://stats.stackexchange.com/questions/20523/difference-between-logit-and-probit-models)) specification with `family = binomial(link = "logit")`.
glm(y ~ poly(density, 3, raw = T), data = d, family = binomial(link = "logit"))
</details>
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