如何在R中编写多参数对数似然函数

时间：2013-12-26 14:08:21

标签： r simulation inference weibull

我想估计以下问题的力量。我有兴趣比较两组都遵循Weibull分布。因此，A组有两个参数（形状par = a1，scale par = b1），两个参数有B组（a2，b2）。通过模拟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，b1 = b2 * 0.5;或者组之间的差异只是形状或尺度参数），应用log-似然比测试，以测试a1 = a2和b1 = b2（或者例如a1 = a1，当我们知道b1 = b2时），并估计测试的功效。

问题将是完整模型的对数似然性，以及如何在R中编码 a）有确切的数据， 和b）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是：

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]))))}

对于完整模型，当a1！= a2，b1！= b2时考虑到两种不同的观测方案，即当需要估计4个参数时（或者，如果有兴趣查看差异时）形状参数，3个参数必须估算）？

是否有可能估计它为不同的组购买构建两个对数似然并将它们加在一起（即 LL.full＆lt; -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，所有观察都是间隔删失的。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R代码以获取确切数据以说明问题。非常感谢你提前。

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

1 个答案:

答案 0 :(得分：4)

是的，您可以对两组的对数似然进行求和（如果它们是单独计算的）。就像你将观测矢量的对数似然加起来一样，每个观测都有不同的生成参数。

我更倾向于根据一个大矢量（即形状参数）来思考，该矢量包含根据协变量的结构（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器（一旦通过链接函数适当地变换）：设计矩阵的点积和回归系数的向量。

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些线性函数，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
    }

然后你的功率模拟可能如下所示：

## 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

在上面的示例中，标识链接可以正常工作，但对于更复杂的模型，可能需要进行某种转换。

并且您不必在对数似然函数中使用线性代数 - 显然，您可以以您认为合适的任何方式构造形状向量（只要您明确索引向量中的相应生成参数） par）。

区间删失数据

Weibull分布的累积分布函数 F（T）（R中的pweibull）给出了时间 T 之前的失败概率。所以， 如果观察是在 T [0] 和 T [1] 之间检查的间隔，那么对象在 T [0] 之间失败的概率和 T [1] 是 F（T [1]） - F（T [0]）： 在 T [1] 之前对象失败的概率减去 T [0] 之前失败的概率（ T [0]之间PDF的积分和 T [1] ）。 因为Weibull CDF已经在R中实现，所以修改上面的似然函数是微不足道的：

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，并且仅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)))
    }

测试两个函数是否提供相同的解决方案：

## 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)
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