Cox-Model-withTime-Dependent-Covariates说课材料

proc phreg; model time*status(0) = smoke; array tt{*} t1-t4; array zz{*} z1-z4; t1 = 2; t2 = 4; t3 = 5; t4 = 8; do i=1 to 4; if time=tt[i] then smoke=zz[i]; end; run;
• For example, blood pressure, disease complications, etc. • 2. An external or ancillary(辅助的) time-dependent
covariate is one whose path is generated exte然函数为：
i
L( )
n exp 1 X i1 2 X i2 m X im
i 1
SR(
t
exp
i)
1 X s1
2 X s2
m X sm
两边取自然对数
ln L(
)
n i 1
i
1 X i1
m X im
• 方法二：Identify patients that received a heart transplant
and those that did not; measure survival times for heart transplant patients as time from transplant to death, and for the patients that did not receive a heart transplant measure survival times as time from entry to the study until death; compare two groups.
ln
SR(
ti
)
exp
m
j 1
j
X
sj
求关于 j j 1,2, ,m 的一阶偏导数，并求其等于 0
（即
ln L( j
)
0
）的解，得到
j
的最大似然估计值。
对于固定协变量的比例风险模型
此偏似然函数应注意的地方：不同时刻的风险集，所用到的同一个体的 值是不同的！！
• 参数估计：MPLE（maximizing the log partial likelihood） • 参数的假设检验 ：Wald, score and likelihood ratio tests
• 利用时间依赖协变量构建风险率函数，但时依变量
的风险函数不一定能用于构建生存率函数（生存分 布）！
Cox 模型不直接考察生存函数 St与协变量的关系，而
是用风险率函数 ht 作为因变量，并假定：
ht, X h0 texp X h0 texp1 X1 2 X 2 m X m
Cox模型的 基本形式
• Time-varying regression coeffcients model
Logo
Examples of time dependent covariates
方法一
方法一的基本思想：Z --- Z(t) 如果Z为时依协变量，那么Z(t)为不随时间改变的变量 方法一的基本模型：
the history of the vector of the time-dependent covariates up to time t
time-dependent covariates for this purpose.
• 1. An internal time-dependent covariate is one where the
change of the covariate over time is related to the behavior of the individual.
covariates involving interactions between covariates and time in the standard Cox model.
• Proportional odds model (Bennett,1983).Still cannot
deal with crossing hazards.
献的个体
h0 t exp1 X s1 2 X s2 P X sm
SR(t i )
exp 1 X i1 2 X i2 p X im
exp1 X s1 2 X s2 P X sm
SR(t i )
将n个病人死亡的条件概率相乘
n
L
i1
exp 1 X i1 2 X i2 p X im exp 1 X s1 2 X s2 P X sm
For example
• For example, suppose we want to consider the effect of exposure to asbestos(石棉)
over time on mortality. A sample of workers in a factory where asbestos is made were monitored for a period of time and data were collected on survival and asbestos exposure. For the i th individual in the sample, the data could be summarized as
SR( ti)
第 i 个研究对象在 ti 时刻死亡的概率应当是两部分的乘积，一是患 者存活到 ti 时刻的概率（与 h0(t)有关），二是该暴露人群 Ri 中恰好第
i 个患者死亡的概率（qi）， L 忽略了前者，故称之为偏似然函数。
对于固定协变量的比例风险模型
有截尾值时，用 i 来表示数据类型：i 1，表示病人在 ti 时刻病
1 What should we use for the function
?
2 注意事项 (解释两种不同的使依变量)
• A cautionary note must be made when interpreting
hazard rates with time-dependent covariates, the hazard function with time-dependent covariates may NOT necessarily be used to construct survival distributions.
• Question: How do we evaluate the effectiveness of the
heart transplant?
早期的解决方案
• 方 法 一 ： Identify the patients that received a heart
transplant and those that did not; measure their survival times from the time they entered the study and compare the survival times between these two groups using, say, a log rank test.
• For example, levels of air pollution.
Logo
因此，只有对于外部协变量，才能计算生存率函数
3 回归参数的估计
COX 比例风险模型参数估计方法
h0 t exp 1 X i1 2 X i2 p X im
代表ti时刻 以后危险集 R(ti)中对似 然函数作贡
4 两个例子 （应用SAS）
• Example 1 ：吸烟状态（z1-z4）：1吸烟 0不吸
烟
SAS
• The above model can best
using the following SAS program:
• options ls=72 ps=72; • data smoking; • input time status z1-z4; • cards; • 211... • 4111.. • 51010. • 70101. • 811001 •;
Cox-Model-withTimeDependent-Covariates
COX model
• Cox比例风险回归模型（Cox’s proportional hazards
regression model），简称Cox回归模型。该模型由英 国统计学家D.R.Cox于1972年提出，主要用于肿瘤和 其它慢性病的预后分析，也可用于队列研究的病因 探索。
利用生存率函数S(t,X)与 风险函数h(t,X)的关系可
导出
St,Xexp0tht,Xdt
exp0th0texpXdtS0texp(X)
j
较好地解 决截尾值
的问题
反映了协变量X与生存函数的关系
对于固定协变量的比例风险模型
两种时依协变量
• It is useful to differentiate between internal and external
？
We assumed that the hazard rate at time t given the entire history of the covariates up to time t is only effected by the current values of the covariates at time t. This,of course, may or may not be true.
COX model 基本形式
Cox 模型是用风险率函数 ht 作为因变量，并假定：
ht, X h0 texp X h0 texp1 X1 2 X 2 m X m
生存时间
协变量
基准风险函数
称为具有协变量x的个体在t时刻的风险函数，表示生存时间已经达到t的个体在t时刻的瞬时风险率
COX model 优点
• Experiment: A group of patients are recruited that are
eligible for heart transplants. However, a heart has to become available and then the patients with the closest match receives this heart.
建立时间t与吸烟状态的关系。
Example 2: Heart-Transplant Data
心脏病人
是否心脏移植
结局（死亡）
• We want to evaluate whether patients receiving heart
transplants will benefit with increased survival.
• Comment: Patients that died early will not have the chance
to receive a heart transplant.(发生死亡时间早的人，更不 可能接受心脏移植) Thus the two groups being compared are selectively biased favoring the heart transplant patients. （存在选择偏倚）
exp 1 X1 ' X1 2 X2 ' X2 LmXm ' Xm
该比值保持一个恒定的比例，与时间t无关，称为比 例风险假定，简称PH假定。如果不满足PH假定呢？
Alternative methods when PH assumption is not valid
• Cox model with manufactured time-dependent
• 多因素分析方法（发现疾病结局影响因素） • 不考虑生存时间分布 • 利用截尾数据
COX model 假定
R Rh h'((tt))h h0 0((tt))e ex x p p 1 0)()(exp ) (
• 任意两个个体风险函数之比，即相对风险度RR或
风险比（risk ratio）
RR i h h'tth h 00tteex x p p11 X X 11' 2 2X X2 2' L L m mX Xm m'