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

#### Doug I. Jones

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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

$$\hat{E}i=n[\hat{S}(i)-\hat{S}(i+1)]$$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 $$\chi^2=\sum{i=0}^{k-1} \frac{\left(\hat{E}_i-d_i\right)^2}{\hat{E}_i},$$

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

$$D_n=\max t\left|\hat{S}(t)-S^o(t)\right|$$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

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