如何在R中编码多参数对数似然函数

是的，您可以将两组的对数似然相加（如果它们是分别计算的）。就像您将观察向量的对数似然相加一样，其中每个观察具有不同的生成参数。

我更愿意考虑一个大向量（即形状参数），该向量包含根据协变量的结构（即组成员）而变化的值。在线性模型上下文中，此向量可以等于线性预测变量（一旦通过链接函数进行了适当的转换）：设计矩阵的点积与回归系数的向量。

这是一个（未功能化的）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以找到Weibull模型的MLE，其中形状参数是协变量的某些线性函数，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，您的功耗模拟可能如下所示：

## 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，但是对于更复杂的模型，可能需要某种转换。

而且，您不必在对数似然函数中使用线性代数-显然，您可以按照您认为合适的任何方式构造形状的向量（只要您在向量中明确索引了适当的生成参数par）。

间隔检查的数据

威布尔分布（在R中）的累积分布函数F（Tpweibull）给出了在时间T之前发生故障的概率。因此，如果观察是在时间T [0]和T [1]之间进行间隔检查的，则对象在T [0]和T [1]之间失效的概率为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（PDF在T [0]和T [1]之间的积分）。由于Weibull CDF已在R中实现，因此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，而只是限制自己只能按组索引形状参数矢量，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)