修改glm函数以采用R中用户指定的链接函数

In glm在 R 中，默认链接函数为Gamma家人是inverse,identity and log。现在对于我的特定问题，我需要使用伽玛回归和响应Y以及修改后的链接函数，其形式为log(E(Y)-1))。于是我考虑修改一些glmR 中的相关函数。有几个可能相关的函数，我正在为以前有过这方面经验的人寻求帮助。

例如，函数Gamma定义为

function (link = "inverse") 
{
  linktemp <- substitute(link)
  if (!is.character(linktemp)) 
    linktemp <- deparse(linktemp)
  okLinks <- c("inverse", "log", "identity")
  if (linktemp %in% okLinks) 
    stats <- make.link(linktemp)
  else if (is.character(link)) 
    stats <- make.link(link)
  else {
    if (inherits(link, "link-glm")) {
      stats <- link
      if (!is.null(stats$name)) 
        linktemp <- stats$name
    }
    else {
      stop(gettextf("link \"%s\" not available for gamma family; available links are %s", 
                    linktemp, paste(sQuote(okLinks), collapse = ", ")), 
           domain = NA)
    }
  }
  variance <- function(mu) mu^2
  validmu <- function(mu) all(mu > 0)
  dev.resids <- function(y, mu, wt) -2 * wt * (log(ifelse(y == 
                                                            0, 1, y/mu)) - (y - mu)/mu)
  aic <- function(y, n, mu, wt, dev) {
    n <- sum(wt)
    disp <- dev/n
    -2 * sum(dgamma(y, 1/disp, scale = mu * disp, log = TRUE) * 
               wt) + 2
  }
  initialize <- expression({
    if (any(y <= 0)) stop("non-positive values not allowed for the 'gamma' family")
    n <- rep.int(1, nobs)
    mustart <- y
  })
  simfun <- function(object, nsim) {
    wts <- object$prior.weights
    if (any(wts != 1)) 
      message("using weights as shape parameters")
    ftd <- fitted(object)
    shape <- MASS::gamma.shape(object)$alpha * wts
    rgamma(nsim * length(ftd), shape = shape, rate = shape/ftd)
  }
  structure(list(family = "Gamma", link = linktemp, linkfun = stats$linkfun, 
                 linkinv = stats$linkinv, variance = variance, dev.resids = dev.resids, 
                 aic = aic, mu.eta = stats$mu.eta, initialize = initialize, 
                 validmu = validmu, valideta = stats$valideta, simulate = simfun), 
            class = "family")
}

另外，为了使用命令glm(y ~ log(mu), family = Gamma(link = MyLink))，我是否还需要修改glm.fit功能？谢谢你！

更新和新问题

根据@Ben Bolker的评论，我们需要编写一个新的链接函数，名为vlog（以真实姓名"log(exp(y)-1)"）。我发现make.link函数可能负责这种修改。它被定义为

function (link) 
{
  switch(link, logit = {
    linkfun <- function(mu) .Call(C_logit_link, mu)
    linkinv <- function(eta) .Call(C_logit_linkinv, eta)
    mu.eta <- function(eta) .Call(C_logit_mu_eta, eta)
    valideta <- function(eta) TRUE
  }, 

  ...

  }, log = {
    linkfun <- function(mu) log(mu)
    linkinv <- function(eta) pmax(exp(eta), .Machine$double.eps)
    mu.eta <- function(eta) pmax(exp(eta), .Machine$double.eps)
    valideta <- function(eta) TRUE
  }, 

  ...

  structure(list(linkfun = linkfun, linkinv = linkinv, mu.eta = mu.eta, 
                 valideta = valideta, name = link), class = "link-glm")
}

我的问题是: 如果我们愿意的话永久添加此链接功能vlog to glm，这样在每个 R 会话中，我们可以使用glm(y~x,family=Gamma(link="log(exp(y)-1)"))直接，我们应该使用fix(make.link)然后添加定义vlog到它的身体？或者fix()只能在当前的 R 会话中执行此操作吗？再次感谢！

还有一件事：我意识到也许另一个功能需要修改。这是Gamma， 定义为

function (link = "inverse") 
{
  linktemp <- substitute(link)
  if (!is.character(linktemp)) 
    linktemp <- deparse(linktemp)
  okLinks <- c("inverse", "log", "identity")
  if (linktemp %in% okLinks) 
    stats <- make.link(linktemp)
  else if (is.character(link)) 
    stats <- make.link(link)
  else {
    if (inherits(link, "link-glm")) {
      stats <- link
      if (!is.null(stats$name)) 
        linktemp <- stats$name
    }
    else {
      stop(gettextf("link \"%s\" not available for gamma family; available links are %s", 
                    linktemp, paste(sQuote(okLinks), collapse = ", ")), 
           domain = NA)
    }
  }
  variance <- function(mu) mu^2
  validmu <- function(mu) all(mu > 0)
  dev.resids <- function(y, mu, wt) -2 * wt * (log(ifelse(y == 
                                                            0, 1, y/mu)) - (y - mu)/mu)
  aic <- function(y, n, mu, wt, dev) {
    n <- sum(wt)
    disp <- dev/n
    -2 * sum(dgamma(y, 1/disp, scale = mu * disp, log = TRUE) * 
               wt) + 2
  }
  initialize <- expression({
    if (any(y <= 0)) stop("non-positive values not allowed for the 'gamma' family")
    n <- rep.int(1, nobs)
    mustart <- y
  })
  simfun <- function(object, nsim) {
    wts <- object$prior.weights
    if (any(wts != 1)) 
      message("using weights as shape parameters")
    ftd <- fitted(object)
    shape <- MASS::gamma.shape(object)$alpha * wts
    rgamma(nsim * length(ftd), shape = shape, rate = shape/ftd)
  }
  structure(list(family = "Gamma", link = linktemp, linkfun = stats$linkfun, 
                 linkinv = stats$linkinv, variance = variance, dev.resids = dev.resids, 
                 aic = aic, mu.eta = stats$mu.eta, initialize = initialize, 
                 validmu = validmu, valideta = stats$valideta, simulate = simfun), 
            class = "family")
}

我认为我们还需要修改

okLinks <- c("inverse", "log", "identity")

to

okLinks <- c("inverse", "log", "identity", "log(exp(y)-1)")

?

我基本上遵循示例的形式?family显示用户指定的表单链接qlogis(mu^(1/days)).

我们想要一个以下形式的链接eta = log(exp(y)-1)（所以反向链接是y=log(exp(eta)+1), and mu.eta = dy/d(eta) = 1/(1+exp(-eta))

vlog <- function() {
    ## link
    linkfun <- function(y) log(exp(y)-1)
    ## inverse link
    linkinv <- function(eta)  log(exp(eta)+1)
    ## derivative of invlink wrt eta
    mu.eta <- function(eta) { 1/(exp(-eta) + 1) }
    valideta <- function(eta) TRUE
    link <- "log(exp(y)-1)"
    structure(list(linkfun = linkfun, linkinv = linkinv,
                   mu.eta = mu.eta, valideta = valideta, 
                   name = link),
              class = "link-glm")
}

基本检查：

vv <- vlog()
vv$linkfun(vv$linkinv(27))  ## check invertibility
library("numDeriv")
all.equal(grad(vv$linkinv,2),vv$mu.eta(2))  ## check derivative

Example:

set.seed(101)
n <- 1000                       
x <- runif(n)
sh <- 2                        
y <- rgamma(n,scale=vv$linkinv(2+3*x)/sh,shape=sh)
glm(y~x,family=Gamma(link=vv))                       
## 
## Call:  glm(formula = y ~ x, family = Gamma(link = vv))
## 
## Coefficients:
## (Intercept)            x  
##       1.956        3.083  
## 
## Degrees of Freedom: 999 Total (i.e. Null);  998 Residual
## Null Deviance:       642.2 
## Residual Deviance: 581.8     AIC: 4268 
##