如果你也在 怎样代写生存模型Survival Models这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。生存模型Survival Models在许多可用于分析事件时间数据的模型中，有4个是最突出的:Kaplan Meier模型、指数模型、Weibull模型和Cox比例风险模型。

生存模型Survival Models精算师和其他应用数学家使用预测人类或其他实体(有生命或无生命)生存模式的模型，并经常使用这些模型作为相当重要的财务计算的基础。具体来说，精算师使用这些模型来计算与个人人寿保险单、养老金计划和收入损失保险相关的财务价值。人口统计学家和其他社会科学家使用生存模型对该模型适用的人口的未来构成做出预测。

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统计代写|生存模型代写survival model代考|Grouped Times of Death

Suppose, in the grouped times of death situation of Section 8.2.2, the parameters of the chosen parametric model have been estimated by some procedure such as maximum likelihood. We are then hypothesizing that the fitted model, $\hat{S}(t)$, is a good representation of the real $S(t)$, and we wish to test this hypothesis. Let
$$
\hat{E}i=n[\hat{S}(i)-\hat{S}(i+1)] $$ represent the expected number of deaths in $(i, i+1]$ according to the hypothesized model. Recall that $d_i$ is the number observed in $(i, i+1]$ from our sample. The quantity $$ \chi^2=\sum{i=0}^{k-1} \frac{\left(\hat{E}_i-d_i\right)^2}{\hat{E}_i},
$$

has approximately a $\chi^2$ distribution with $k-1-r$ degrees of freedom, where $r$ is the number of parameters in $S(t)$ which were estimated from the data. If the hypothesized model is fully specified including its parameters (i.e., if the parameters are not estimated from the data), then the approximate $\chi^2$ statistic defined by (8.41) has $k-1$ degrees of freedom.

统计代写|生存模型代写survival model代考|Exact Times of Death

In this case we wish to compare our $\hat{S}(t)$, which is hypothesized as a representation of $S(t)$, with the observed $S^o(t)$. To do so we need a measure of the departure of $\hat{S}(t)$ from $S^o(t)$.

A fairly simple departure measure is the Kolmogorov-Smirnov statistic, defined as
$$
D_n=\max t\left|\hat{S}(t)-S^o(t)\right| $$ the largest absolute deviation between $\hat{S}(t)$ and $S^o(t)$ to be found over the domain of $t$. The subscript of $D_n$ reminds us that this measure depends on the sample size $n$. We then calculate $$ Y=\sqrt{n} \cdot D_n $$ as the actual departure measure to be tested. For small $n$, tables of critical values of $y$ for various significance levels can be found in some books of tables or textbooks (see, for example, Hollander and Wolfe [38]). As $n \rightarrow \infty$, the probabilities can be calculated from $$ \operatorname{Pr}(Y>y)=2 \sum{j=1}^{\infty}(-1)^{j+1} e^{-2 j^2 y^2}
$$
so, in effect, (8.45) gives approximate probabilities for large values of $n$.

CONCOMITANT VARIABLES IN PARAMETRIC MODELS

Throughout most of this text we have only considered survival models that were a function of chronological age, $S(x)$, or those that were a function of time since some initial event, $S(t)$. In both cases the model was univariate.
Many cases arise in which survival probabilities are a function of two or more variables, such as those used for insurance premium calculations which depend on age at issue as well as time since issue. This leads us to the select survival model $S(t ; x)$ defined in Section 1.2.1. The tabular form of $S(t ; x)$ was described in Section 3.6, and illustrated in Table 3.5. There we saw that the concomitant variable age at issue, denoted $[x]$, was taken into account by having a separate $S(t ; x)$ for that value of $x$. In other words, each row in a select table constitutes a separate univariate model (varying with $t$ only); age at selection is reflected by choice of the appropriate row. For this reason we say that the concomitant variable has been taken into account through separation.

Similarly, suppose we consider the survival of cancer patients as a function of time since diagnosis. We might believe that type of cancer, sex of the patient, and type of treatment all affect survival, so we would estimate a separate $S(t)$ for each type/sex/treatment combination. Again $S(t)$ is univariate, with concomitant variables taken into account by separation.

生存模型代考

统计代写|生存模型代写survival model代考|Grouped Times of Death

假设，在第8.2.2节的分组死亡时间情况下，所选参数模型的参数已通过最大似然等程序进行估计。然后，我们假设拟合的模型$\hat{S}(t)$很好地代表了真实的$S(t)$，我们希望测试这个假设。让
$$
\hat{E}i=n[\hat{S}(i)-\hat{S}(i+1)] $$表示根据假设模型$(i, i+1]$的预期死亡人数。回想一下，$d_i$是我们样本中$(i, i+1]$中观察到的数字。数量 $$ \chi^2=\sum{i=0}^{k-1} \frac{\left(\hat{E}_i-d_i\right)^2}{\hat{E}_i},
$$

近似为$\chi^2$分布，自由度为$k-1-r$，其中$r$为$S(t)$中由数据估计的参数个数。如果假设模型是完全指定的，包括它的参数(即，如果参数不是从数据中估计出来的)，那么近似的$\chi^2$统计量由(8.41)定义，其自由度为$k-1$。

统计代写|生存模型代写survival model代考|Exact Times of Death

在这种情况下，我们希望将我们的$\hat{S}(t)$(假设为$S(t)$的表示)与观察到的$S^o(t)$进行比较。为此，我们需要测量$\hat{S}(t)$与$S^o(t)$的偏离程度。

一个相当简单的偏离度量是Kolmogorov-Smirnov统计量，定义为
$$
D_n=\max t\left|\hat{S}(t)-S^o(t)\right| $$在$t$的范围内，$\hat{S}(t)$和$S^o(t)$之间的最大绝对偏差。$D_n$的下标提醒我们，这一措施取决于样本量$n$。然后，我们计算$$ Y=\sqrt{n} \cdot D_n $$作为要测试的实际偏离度量。对于较小的$n$，可以在一些表格或教科书中找到各种显著性水平的$y$临界值表(例如，参见Hollander和Wolfe[38])。如$n \rightarrow \infty$，概率可以从$$ \operatorname{Pr}(Y>y)=2 \sum{j=1}^{\infty}(-1)^{j+1} e^{-2 j^2 y^2}
$$计算
因此，实际上，(8.45)给出了$n$较大值的近似概率。

在这篇文章的大部分内容中，我们只考虑了作为实际年龄函数的生存模式，$S(x)$，或者那些自某些初始事件以来作为时间函数的生存模式，$S(t)$。在这两种情况下，模型都是单变量的。
在许多情况下，生存概率是两个或两个以上变量的函数，例如用于保险费计算的那些变量，它取决于问题的年龄以及问题后的时间。这就引出了1.2.1节中定义的选择生存模型$S(t ; x)$。$S(t ; x)$的表格形式已在第3.6节中描述，并在表3.5中说明。在那里，我们看到伴随的变量年龄，表示为$[x]$，通过对$x$的值使用单独的$S(t ; x)$来考虑。换句话说，选择表中的每一行都构成一个单独的单变量模型(仅随$t$变化);选择时的年龄通过选择适当的行来反映。由于这个原因，我们说伴随变量已经通过分离考虑进去了。

类似地，假设我们将癌症患者的生存率视为自诊断以来时间的函数。我们可能会认为癌症的类型、患者的性别和治疗的类型都会影响生存率，因此我们会对每种类型/性别/治疗组合分别估计一个$S(t)$。同样，$S(t)$是单变量，通过分离考虑了伴随变量。

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