r - 如何在 R 中编写多参数对数似然函数

Question

我想估计以下问题的威力。我有兴趣比较两个都遵循威 bool 分布的组。所以，A 组有两个参数(shape par = a1，scale par = b1)，B 组有两个参数(a2, b2)。通过模拟来自感兴趣分布的随机变量(例如，假设不同的尺度和形状参数，即 a1=1.5*a2 和 b1=b2*0.5；或者组之间的差异仅在于形状或尺度参数)，应用 log-似然比检验以检验是否 a1=a2 和 b1=b2(或例如 a1=a1，当我们知道 b1=b2 时)，并估计检验的功效。

问题是完整模型的对数似然是什么，以及如何在 R 中对其进行编码
a) 有准确的数据，
b) 对于区间删失数据？

也就是说，对于简化模型(当 a1=a2,b1=b2 时)精确和区间删失数据的对数似然为:

LL.reduced.exact <- function(par,data){sum(log(dweibull(data,shape=par[1],scale=par[2])))};
LL.reduced.interval.censored<-function(par, data.lower, data.upper) {sum(log((1-pweibull(data.lower, par[1], par[2])) – (1-pweibull(data.upper, par[1],par[2]))))}

完整模型是什么，当 a1!=a2, b1!=b2 时，考虑到两种不同的观察方案，即当必须估计 4 个参数时(或者，如果有兴趣查看形状参数的差异，必须估计 3 个参数)？

是否可以估计它为不同的组构建两个对数似然并将其相加(即 LL.full<-LL.group1+LL.group2 )？

关于区间删失数据的对数似然，删失是非信息性的，所有观察都是区间删失的。任何更好的想法如何执行此任务将不胜感激。

请在下面找到确切数据的 R 代码来说明问题。非常感谢您提前。

R Code:    
# n (sample size) = 500
# sim (number of simulations) = 1000
# alpha  = .05
# Parameters of Weibull distributions: 
   #group 1: a1=1, b1=20
   #group 2: a2=1*1.5 b2=b1

n=500
sim=1000
alpha=.05
a1=1
b1=20
a2=a1*1.5
b2=b1
#OR: a1=1, b1=20, a2=a1*1.5, b2=b1*0.5 

# the main question is how to build this log-likelihood model, when a1!=a2, and b1=b2
# (or a1!=a2, and b1!=b2)
LL.full<-????? 
LL.reduced <- function(par,data){sum(log(dweibull(data,shape=par[1],scale=par[2])))}

LR.test<-function(red,full,df) {
lrt<-(-2)*(red-full)
pvalue<-1-pchisq(lrt,df)
return(data.frame(lrt,pvalue))
}

rejections<-NULL

for (i in 1:sim) {

RV1<-rweibull (n, a1, b1)
RV2<-rweibull (n, a2, b2)
RV.Total<-c(RV1, RV2)

par.start<-c(1, 15)

mle.full<- ????????????  
mle.reduced<-optim(par.start, LL, data=RV.Total, control=list(fnscale=-1))

LL.full<-????? 
LL.reduced<-mle.reduced$value

LRT<-LR.test(LL.reduced, LL.full, 1)

rejections1<-ifelse(LRT$pvalue<alpha,1,0)
rejections<-c(rejections, rejections1)
}

table(rejections)
sum(table(rejections)[[2]])/sim   # estimated power

Answer 1

是的，您可以将两组的对数似然相加(如果它们是单独计算的)。就像您对观测向量的对数似然求和一样，每个观测都有不同的生成参数。

我更喜欢考虑一个大向量(即形状参数)，它包含根据协变量结构(即组成员资格)而变化的值。在线性模型上下文中，该向量可以等于线性预测变量(一旦被链接函数适当转换):设计矩阵和回归系数向量的点积。

这是一个(非功能化的)示例:

## setup true values
nobs = 50 ## number of observations
a1 = 1  ## shape for first group
b1 = 2  ## scale parameter for both groups
beta = c(a1, a1 * 1.5)  ## vector of linear coefficients (group shapes)

## model matrix for full, null models
mm_full = cbind(grp1 = rep(c(1,0), each = nobs), grp2 = rep(c(0,1), each = nobs))
mm_null = cbind(grp1 = rep(1, nobs*2))

## shape parameter vector for the full, null models
shapes_full = mm_full %*% beta ## different shape parameters by group (full model)
shapes_null = mm_null %*% beta[1] ## same shape parameter for all obs
scales = rep(b1, length(shapes_full)) ## scale parameters the same for both groups

## simulate response from full model
response = rweibull(length(shapes_full), shapes_full, scales)

## the log likelihood for the full, null models:
LL_full = sum(dweibull(response, shapes_full, scales, log = T)) 
LL_null = sum(dweibull(response, shapes_null, scales, log = T)) 

## likelihood ratio test
LR_test = function(LL_null, LL_full, df) {
    LR = -2 * (LL_null - LL_full) ## test statistic
    pchisq(LR, df = df, ncp = 0, lower = F) ## probability of test statistic under central chi-sq distribution
    }
LR_test(LL_null, LL_full, 1) ## 1 degrees freedom (1 parameter added)

要编写一个对数似然函数来查找 Weibull 模型的 MLE，其中形状参数是协变量的某个线性函数，您可以使用相同的方法:

## (negative) log-likelihood function
LL_weibull = function(par, data, mm, inv_link_fun = function(.) .){
    P = ncol(mm) ## number of regression coefficients
    N = nrow(mm) ## number of observations
    shapes = inv_link_fun(mm %*% par[1:P]) ## shape vector (possibly transformed)
    scales = rep(par[P+1], N) ## scale vector
    -sum(dweibull(data, shape = shapes, scale = scales, log = T)) ## negative log likelihood
    }

那么您的电源模拟可能如下所示:

## function to simulate data, perform LRT
weibull_sim = function(true_shapes, true_scales, mm_full, mm_null){
    ## simulate response
    response = rweibull(length(true_shapes), true_shapes, true_scales)

    ## find MLE
    mle_full = optim(par = rep(1, ncol(mm_full)+1), fn = LL_weibull, data = response, mm = mm_full) 
    mle_null = optim(par = rep(1, ncol(mm_null)+1), fn = LL_weibull, data = response, mm = mm_null)

    ## likelihood ratio test
    df = ncol(mm_full) - ncol(mm_null)
    return(LR_test(-mle_null$value, -mle_full$value, df))
    }

## run simulations
nsim = 1000
pvals = sapply(1:nsim, function(.) weibull_sim(shapes_full, scales, mm_full, mm_null) )

## calculate power
alpha = 0.05
power = sum(pvals < alpha) / nsim

身份链接在上面的示例中工作正常，但对于更复杂的模型，可能需要某种转换。

并且您不必在对数似然函数中使用线性代数——显然，您可以以任何您认为合适的方式构造形状向量(只要您在向量 par 中明确索引适当的生成参数) )。

区间删失数据

Weibull 分布(R 中的 pweibull)的累积分布函数 F(T) 给出了时间 T 之前的失效概率。所以，
如果观察在时间 T[0] 和 T[1] 之间进行间隔删失，则对象在 T[0] 和 T[1] 之间失败的概率为 F(T[1]) - F(T[0]) :
对象在 T[1] 之前失败的概率减去它在 T[0] 之前失败的概率(T[0] 和 T[1] 之间的 PDF 积分)。
因为 Weibull CDF 已经在 R 中实现，所以修改上面的似然函数很简单:

LL_ic_weibull <- function(par, data, mm){
    ## 'data' has two columns, left and right times of censoring interval
    P = ncol(mm) ## number of regression coefficients
    shapes = mm %*% par[1:P]
    scales = par[P+1]
    -sum(log(pweibull(data[,2], shape = shapes, scale = scales) - pweibull(data[,1], shape = shapes, scale = scales)))
    }

或者，如果您不想使用模型矩阵等，而只是限制自己按组索引形状参数向量，则可以执行以下操作:

LL_ic_weibull2 <- function(par, data, nobs){
    ## 'data' has two columns, left and right times of censoring interval
    ## 'nobs' is a vector that contains the num. observations for each group (grp1, grp2, ...)
    P = length(nobs) ## number of regression coefficients
    shapes = rep(par[1:P], nobs)
    scales = par[P+1]
    -sum(log(pweibull(data[,2], shape = shapes, scale = scales) - pweibull(data[,1], shape = shapes, scale = scales)))
    }

测试两个函数是否给出相同的解决方案:

## generate intervals from simulated response (above)
left = ifelse(response - 0.2 < 0, 0, response - 0.2)
right = response + 0.2
response_ic = cbind(left, right)

## find MLE w/ first LL function (model matrix)
mle_ic_full = optim(par = c(1,1,3), fn = LL_ic_weibull, data = response_ic, mm = mm_full)
mle_ic_null = optim(par = c(1,3), fn = LL_ic_weibull, data = response_ic, mm = mm_null)

## find MLE w/ second LL function (groups only)
nobs_per_group = apply(mm_full, 2, sum) ## just contains number of observations per group
nobs_one_group = nrow(mm_null) ## one group so only one value
mle_ic_full2 = optim(par = c(1,1,3), fn = LL_ic_weibull2, data = response_ic, nobs = nobs_per_group)
mle_ic_null2 = optim(par = c(1,3), fn = LL_ic_weibull2, data = response_ic, nobs = nobs_one_group)