揭秘生存曲线背后的统计学（上）_新药临床开发故事会_传送门
揭秘生存曲线背后的统计学（上）
新药临床开发故事会
点击上方“公众号”可以订阅哦！前言 新药临床开发故事会《深度解读诺华Midostaurin三期临床试验结果》一文写完发表两天后，作者碰巧翻阅了Robert Weinberg教授的著名教科书《The Biology of Cancer》第15章，其中他介绍了美国FDA在2011年批准史上首个免疫哨卡抑制剂ipilimumab上市这一里程碑。令人遗憾的是，他在解读FDA审批所依据的重要三期临床试验结果时也出现了作者前文所提及的常见误解：以为中位总生存期的差别是比较两条生存曲线的关键。生物统计学由于其中无法回避的数学内容而容易让以临床医学，实验生物学，或者药物化学为训练背景的制药业研发人员望而生畏。作者本人从事的是计算生物学研究，虽然在统计学的一些分支有不少实战经验和理论知识，但是对于临床生统这一颇有深度的领域还处在入门阶段。解读Midostaurin三期结果的文章虽然强调了风险比率值 (hazard ratio, HR)和概率值的重要性，但是限于篇幅和对数学公式的回避而只能在“生存分析海洋”的沙滩附近游泳。生存分析有三大支柱：(1)Kaplan-Meier曲线；(2)对数秩检验 (Log Rank Test)；(3)Cox回归模型。其中方法(1)和(2)背后的概念和运算除了与条件概率有关的定义很难科普之外(但可以诉诸直觉而绕过)，其余内容只需中学数学知识就能理解；方法(3)涉及的数学内容比较复杂，只能留待将来另文介绍。然而幸运的是，对于临床试验中常见的双样本分析而言(也就是只比较试验组和对照组的两条生存曲线，而不涉及分层分析或连续协变量)，对数秩方法既是给出差异显著性概率值的最常用手段，也能用来估算出与Cox回归法渐近等价(asymptotically equivalent)的HR数值。近年来中文出版界出现了包含数学公式的专业科普未必曲高和寡的可喜现象。在这些成功范例的鼓舞下，本文将从最基本的概念开始对“生存分析海洋”的两大水域大胆进行深度潜水式科普。这样的写作自然比大众科普要辛苦一些，然而无疑也让作者从边学边写的过程中不断享受到透彻理解后的愉快。【Weinberg教授的误读】《The Biology of Cancer》是癌症生物学领域不可多得的经典教科书，自2007年第一版发行以来，无数生物专业学生以及癌症研究人员从中获益。自2011年以来，CTLA-4抗体和PD-1抗体的先后成功使得癌症免疫疗法在全球制药界变得炙手可热。Weinberg教授自然要在2013年的第二版中增加不少关于免疫哨卡抑制剂 (immune checkpoint inhibitors)的内容，他在解释美国FDA为何批准CTLA-4抗体ipilimumab (商标名：Yervoy) 用于治疗已经扩散的黑色素瘤病人时写道：“试验组病人接受ipilimumab治疗后的中位总生存期是10个月，而对照组病人接受gp100多肽疫苗治疗后的中位总生存期是6个月”[1]。言下之意显然是说，这4个月的中位总生存期差异是FDA最后给绿灯放行的关键。在这里Weinberg教授对参考文献 [2]的摘要可谓“捡了芝麻，丢了西瓜”：他只关注了“横切”两条生存曲线后得到的中位总生存期的差异，而不知原文括号里的风险比率值 (HR = 0.66) 和概率值（P = 0.003）才是FDA专家委员会决策的更重要依据！要想真正理解Weinberg教授产生误读的原因，我们首先需要掌握生存分析的一些基本概念。生存分析中的基本概念事件(event)其具体定义取决于新药试验中事先指定的临床终点 (clinical endpoint)。如果终点是总体生存期(overall survival, OS)，那么事件就是患者的死亡。如果终点是无增恶生存期(progression-free survival, PFS)，那么事件就是患者病情的恶化(例如固体瘤增大或者白血病的血液指标恶化)或者死亡。生存分析能在生物医学以外的许多不同领域有用武之地 [3]，其关键就在于事件这个概念在定义上的灵活性。 生存时间 (survival time)指从病人开始临床试验(相对的时间零点)直到事件发生所经历的时间跨度。临床试验的病人招募通常是个持续的过程，不同病人的试验一般始于日历上不同的具体时间点，只有在数据分析时采用相对时间才能有同样的时间轴及其零点。对于一个临床试验的病人群体而言，个体病人的生存时间是一个随机变量，用大写的T表示。而生存曲线横坐标则对应各病人事件发生的时间点，它不是随机变量(而用做函数的自变量)，用小写的t表示。随机变量T一般不遵从正态分布。删失（censoring）指由于事件没有被观测到或者无法观测到，而导致生存时间无法精确记录的情况。其中最为常见的情形称为右删失(right censoring, 图一)，对这样的病人我们只知道其生存时间要大于从试验开始到删失发生的时间。出现右删失的情况一般有两种原因：(1)病人在某时间点上退出试验而失去随访信息；(2)病人在整个试验结束时事件还未发生。生存时间T的非正态分布以及删失情况的存在让传统的统计方法在分析这类数据时无用武之地，于是统计学家们殚精竭虑，直到1970年代才使生存分析这一方法体系趋于成熟。生存函数(survival function)又叫累积生存概率(cumulative survival probability)，其定义公式为S(t) = P(T >t)。该函数的意义就是生存时间大于时间点t的概率。显然当t=0时，S(t)=1，每个病人在各自的试验开始时都还没有事件发生。而随着t的增加，S(t)逐渐向0方向递减 (严格的说是不增)。理论上的T和t一般是连续变量，相应的生存函数S(t)的图像就是一条光滑的曲线(图二左)。而在临床试验的实践中，T和t都是离散变量，因此我们常用不增的阶梯曲线来描述估测出的生存函数(图二右)。图三可以帮助我们进一步理解光滑曲线和阶梯曲线之间的区别：红色数据点的纵坐标代表根据试验结果估算出的累积生存概率值，我们若想用光滑曲线来连接就需要对随机变量T的分布做出假设的参数拟合法，而曲线一般不宜正好经过所有的红点(那样会导致过度拟合而使得统计模型没有多大效用)；若用非参数的阶梯函数来连接，那么曲线简单而唯一确定！这一区别正是Kaplan-Meier非参数估计法能够成为生存分析领域一大支柱的关键。风险函数(hazard function)和风险比例(hazard ratio)定义自变量为时间的函数之动机是要研究其随时间的变化规律，也就离不开变化率这个重要概念而需要用到微分或差分的数学工具。生存函数S(t)描述的是组内尚未发生事件的累积病人比例，由于删失情况的存在，它与已发生事件的累积病人比例之间的数学关系不再简单。统计学家们通过实践发现，研究与病人发生事件的概率有关的风险函数能提取出更多蕴含在生存数据中的信息。风险函数(又称为风险率，hazard rate)的直接数学定义需要用到条件概率和极限两种概念的组合，不利于科普文章用非数学语言的解读。好在该函数还可以通过生存函数S(t)来间接定义：从定义式可以看出，h(t)基本就是S(t)对时间的导数除以S(t)。由于S(t)一般随时间递减，加上一个负号是可以让h(t)在取值上非负，从而在使用上更加方便。而h(t)在某一时间段内的积分，H(t)，则被称为累积风险函数 (cumulative hazard function)以上两个定义式为了简洁而假定时间t为连续变量，若t为离散变量时也有相对应的定义，我们只需记得微分的离散对应是差分，而积分的离散对应是求和。 在新药临床试验中，两条不同生存曲线的背后隐藏着两个不同的风险函数：h1(t)和h2(t)。统计学家们通过研究许多生存数据后发现，如果试验组和对照组病人的随机分派做得足够好，在很多情况下就可以用一个近似成立的比例风险假设(proportional hazards)来简化生存数据的分析：在这里我们假设被比较的两个风险函数之比值是一个不随时间变化的常数，用HR(hazard ratio)来表示。制药业的常用惯例是：试验组的风险函数作为分子，而对照组的风险函数作为分母 [注意：某些论文和教科书可能会反过来]。如果试验组生存曲线总体在对照组之上，那么HR就在0到1之间取值，数值越小，两条曲线的差别就越大，也就意味着新药的相对疗效优势越大。需要指出的是，来自临床试验的生存数据有时也会有明显偏离比例风险假设的情况，其统计分析就需要对本文所介绍的方法做一些在数学上比较复杂的改进，这已经超出了本文的讨论范围。【揭秘Kaplan-Meier曲线】可以毫不夸张地说，近几十年来发表的临床癌症学会议报告和期刊论文中，几乎每篇都至少要包含一幅Kaplan-Meier生存曲线图。没有统计学背景的新药研发人员在专家的指引下通过多看虽能大致明白生存曲线的含义，但是对其背后的简单运算往往有一种不必要的神秘感。而统计学家们赋予Kaplan-Meier法的专业术语“乘积极限估计”(product limit estimator)更是让非专业人士望而生畏。若想揭开笼罩在Kaplan-Meier曲线上的神秘面纱，最佳途径似乎是“解剖麻雀法”：通过彻底展示一个非常简单实例的详细计算过程，我们就可以让非专业读者也能用手画出一条生存曲线。作者在此改编了文献 [4]中的一组只有10个病人的生存数据，用字母依次给病人编号后列表如下：其中第二列是时间点t,以月为单位。而第三列的病人状态只有两种取值：0代表右删失，1代表死亡事件。整个试验持续的最长相对时间是12个月，期间10人中有7人发生死亡事件。病人I在第7月出现删失(退出试验或失去随访)，我们对其生存时间的信息不完整，只知道大于7个月。而病人D在研究结束时依然存活。要从这张数据表格估计出生存曲线，我们首先需将各行按照时间点t从小到大排序，记为ti，其中下标i表示序号。由于发生死亡事件与删失的离散时间点一般不是均匀分布，这些点把时间轴分割成了长短不一的区间，每个ti值代表了区间的起点。然后我们把每个区间里发生的死亡事件累加，记为di。若同时出现了死亡和删失，我们将用独立的两行来表示该区间。例如第7月就对应两个ti值：t3 = 7，t4 = 7+，其中的"+"号代表删失。每个区间起点时的存活病人数记为ni，那么在该时间段的存活比例(surviving proportion)就是：pi=(ni–di)/ni。这样，我们通过整理和简单计算就得到了下表：为了使读者概念更加清晰，作者特地添加了第一行的时间零点。如果不存在删失，上面表格中每行的生存概率显然可以用一个简单的比值公式来估算：但是删失情况的出现会给这个思路带来困难。例如第8月结束时，病人F的死亡让组内存活人数下降为4人，用4/10 = 0.4来估算该时间段的生存概率显然不合理，它把第7月病人I的删失计为死亡，因此这里的0.4在统计学上被称为有偏估计量 (biased estimator)。要想找到生存概率的无偏估计量 (unbiased estimator)并非易事。Paul Meier和Edward Kaplan这两个普林斯顿大学数学系博士生在1940年代末跟随同一个导师(统计学巨匠John Tukey)，而且同在1951年完成博士论文答辩，两人在校期间居然互不相识 [5]。毕业离校各奔东西后，他们又在互不知晓的情况下独立研究了同样的问题，却殊途同归地找到了类似的答案。最后在Tukey教授和期刊编辑的协调下，花了4年左右的时间才将两篇风格迥异的手稿整合成一篇论文 [6]，于是著名的Kaplan-Meier乘积极限估计法终于在入难产后诞生。此文问世半个多世纪以来，被引用已达5万次左右，这个数字在任何一个自然科学领域都是惊人的！Kaplan和Meier基于条件概率的概念，经过一番并不简单的数学推导和探索，发现上面表格中pi的连乘积作为生存概率S(t)的非参数估计非常合适：需要注意的是，出现删失的时间点没有对应的pi值，这些项也就自动被排除在算式之外(本例中的p4,p7,p9都不存在)。这个连乘积公式也是Kaplan和Meier将他们的方法命名为“乘积极限估计”的主要原因，其实一点也不神秘。现在我们可用该公式来重新估算第8月时间段的生存概率：0.48这个数值显然要比把在前一时间段发生删失的病人I计为死亡而得出的0.4要来得合理 (也要比把病人I当成肯定在第8月依然存活而算出的0.5来得合理)。通过Kaplan-Meier法计算出的各时间段内的生存概率值放在上表的最右一列，将这些数值作为相应区间的纵坐标对时间作图，就可得到下面的生存曲线：图四只有6个数据点，有兴趣的读者完全可以用手工来完成。不过用专业软件来画(例如本文使用的R语言)可以更方便地在图中添加注解性内容，例如在纵坐标为0.5处用一条水平线来就横切该曲线，就发现本试验组的中位生存期为8个月。通过研究这个只有10个病人的简单实例，没有统计学背景的读者跟随作者一路走来，或许也能对Kaplan-Meier曲线有“会心不远”的感觉。【1】Weinberg, R.A. (2013) Chapter 15, The Biology of Cancer, 2nd Ed. Garland Science, Taylor & Francis Group, LLC.【2】Hodi, F.S. et al. (2010)New England Journal of Medicine 363: 711-723.【3】Linoff, G.S. & Berry, M.J.A. (2011) Chapter 10, Data Mining Techniques: For Marketing, Sales, and Customer Relationship Management, 3rd ed. Wiley Publisging, Inc.【4】Glantz, S.A. (2005) Chapter 11, Primer of Biostatistics, 6th ed. The McGraw-Hill Companies, Inc.【5】Marks, H.M. (2004) Clinical Trials 1: 131-138.【6】Kaplan, E.L. & Meier, P. (1958) Journal of the American Statistical Association 53: 457-481.原创作品，转载和拷贝请注明出自微信公众台“新药临床开发故事会” 公众号ID：biostocks新药临床开发故事会故事会 | 交流 | 分享
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　　1、生存分析表
　　Survival Analysis for TIME　　remission time（days）
　　对生存时间变量Time进行分析，其变量标签是remission time（days）。
　　Factor GROUP ＝ A疗法
　　Time　　Status　　Cumulative　　Standard　　Cumulative　　Number
　　Survival　　Error　　Events　　Remaining
　　4　　恶化　　．9600　　．0392　　1　　24
　　5　　恶化　　．9200　　．0543　　2　　23
　　9　　恶化　　．8800　　．0650　　3　　22
　　10　　恶化　　．8400　　．0733　　4　　21
　　11　　恶化　　．8000　　．0800　　5　　20
　　12　　恶化　　．7600　　．0854　　6　　19
　　13　　恶化　　．7200　　．0898　　7　　18
　　20　　删失　　7　　17
　　23　　恶化　　．6776　　．0940　　8　　16
　　28　　恶化　　9　　15
　　28　　恶化　　10　　14
　　28　　恶化　　．5506　　．1010　　11　　13
　　29　　恶化　　．5082　　．1017　　12　　12
　　31　　恶化　　．4659　　．1017　　13　　11
　　32　　恶化　　．4235　　．1009　　14　　10
　　37　　恶化　　．3812　　．0993　　15　　9
　　41　　恶化　　16　　8
　　41　　恶化　　．2965　　．0936　　17　　7
　　57　　恶化　　．2541　　．0893　　18　　6
　　62　　恶化　　．2118　　．0838　　19　　5
　　74　　恶化　　．1694　　．0770　　20　　4
　　100　　恶化　　．1271　　．0684　　21　　3
　　139　　恶化　　．0847　　．0572　　22　　2
　　258　　删失　　22　　1
　　269　　删失　　22　　0
　　Number of Cases：25　　Censored：3　　（12．00%）Events：22
　　Survival Time　　Standard Error　 95% Confidence Interval
　　Mean：57　　15　　（28．86 ）
　　（Limited to　　269 ）
　　Median：31　　3　　（25．37 ）说明：
　　限于篇幅原因，此处仅列出A治疗组的结果。
　　Time：观察时间。
　　Status：生存状态。
　　Cumulative Survival：累积生存率。
　　Standard Error：累积生存率的标准差。
　　Cumulative Events：累计死亡数。
　　Number remaining：组中剩余人数，即在时间Time的暴露人数。
　　2、生存时间估计
　　Survival Analysis for TIME　　remission time（days）Factor GROUP ＝ A疗法
　　Survival Time　　Standard Error　 95% Confidence Interval
　　Mean：57　　15　　（28．86 ）
　　（Limited to　　269 ）
　　Median：31　　3　　（25．37 ）Factor GROUP ＝ B疗法
　　Survival Time　　Standard Error　 95% Confidence Interval
　　Mean：112　　20　　（72．152 ）
　　（Limited to　　245 ）
　　Median：99　　24　　（52．146 ）Factor GROUP ＝ C疗法
　　Survival Time　　Standard Error　 95% Confidence Interval
　　Mean：95　　19　　（58．132 ）
　　（Limited to　　219 ）
　　Median：40　　11　　（18．62 ）Total　　Number　　Number　　Percent
　　Events　　Censored　　Censored
　　GROUP　　A疗法　　25　　22　　3　　12.00
　　GROUP　　B疗法　　19　　15　　4　　21.05
　　GROUP　　C疗法　　22　　15　　7　　31.82
　　Overall　　66　　52　　14　　21．21说明：
　　Mean是生存时间的算术均数。
　　&Limit to 269&表示A疗法组的最长生存时间为219天。
　　Median为中位生存时间，即生存率为50%所对应的生存时间。A、B、C疗法的中位生存时间分别为31、99、40。
　　A、B、C疗法组中位生存时间的95%可信区间分别为（25，37）、（52，146）、（18，62）。
　　A、B、C疗法的删失例数分别为3、4、7，删失率分别为12%、21.05%、31.82%。
　　3、水平间的整体比较
　　Test Statistics for Equality of Survival Distributions for GROUP
　　Statistic　　df　　Significance
　　Log Rank　　4．31　　2　　．1158
　　Breslow　　3．67　　2　　．1595
　　Tarone-Ware　　4．35　　2　　．1137说明：
　　3种疗法的生存时间差异无显著性意义，3个检验统计量的P值均大于0.1。在实际分析中，当各组的总体水平比较无统计学意义时，不宜再进行两两比较，此处仅是为了演示一下。
　　4、水平间的两两比较
　　Log Rank Statistic and （Significance）
　　Factor　　1　　2
　　2　　3.65
　　（．0561）
　　3　　2．84　　．03
　　（．0917）（．8677）
　　Breslow Statistic and （Significance）
　　Factor　　1　　2
　　2　　3.23
　　（．0722）
　　3　　1．77　　．07
　　（．1832）（．7967）
　　Tarone-Ware Statistic and （Significance）
　　Factor　　1　　2
　　2　　3.85
　　（．0498）
　　3　　2．26　　．07
　　（．1324）（．7981）说明：
　　3种检验方法两两比较差异均无显著性意义。括号外数值为检验统计量，括号内数值为P值。
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更多Joe Man的产品CRAN Task View: Survival Analysis
CRAN Task View: Survival Analysis
Maintainer:Arthur Allignol and Aurelien Latouche
Contact:arthur.allignol at uni-ulm.de
Survival analysis, also called event history analysis in social science,
or reliability analysis in engineering, deals with time until occurrence
of an event of interest. However, this failure time may not be observed
within the relevant time period, producing so-called censored observations.
This task view aims at presenting the useful R packages for the analysis
of time to event data.
Please let the
something is inaccurate or missing.
Standard Survival Analysis
Estimation of the Survival Distribution
Kaplan-Meier:
function from the
computes the Kaplan-Meier estimator for truncated and/or censored data.
(replacement of the Design package)
proposes a modified version of the
package implements a fast algorithm and some features
not included in
Various confidence intervals and confidence bands for the Kaplan-Meier estimator
are implemented in the
of package
the Kaplan-Meier estimator.
package includes a function to compute the Kaplan-Meier
estimator for left-censored data.
provides a weighted
Kaplan-Meier estimator.
estimates the
survival curve for each level of categorical variables with
missing data.
kaplan-meier
computes the Kaplan-Meier estimator from
histogram data.
package permits to compute a
weighted Kaplan-Meier estimate.
function in
plots the survival function using a
variant of the Kaplan-Meier estimator in a hospitalisation risk
package computes
presmoothed estimates of the main quantities used for
right-censored data, i.e., survival, hazard and density functions.
package permits to compute the Kaplan-Meier
estimator following Pollock et al. (1998). The
package provides several functions for computing confidence
intervals of the survival distribution (e.g., beta product
confidence procedure). The
package offers
various length-bias corrections to survival curve
estimation. Non-Parametric confidance bands for the Kaplan-Meier
estimator can be computed using the
package implements the Kaplan-Meier estimator
with constraints. The
package allows landmark
estimation and testing of survival
probabilities. The
package computes the
original and modified jackknife estimates of Kaplan-Meier
estimators. The
package permits to estimate a
survival distribution in the presence of dependent left-truncation
and right-censoring. The
package provides
methods for estimating the conditional survival function for
ordered multivariate failure time data. The
implements the generalised Turnbull estimator proposed by Dehghan
and Duchesne for estimating the conditional survival function with
interval-censored data.
Non-Parametric maximum likelihood estimation (NPMLE):
package provides several ways to compute the NPMLE
of the survival distribution for various censoring and truncation
can also be used to compute the MLE for interval-censored data.
permits to compute the NPMLE of the cumulative
distribution function for left- and right-censored data.
function in package
computes the NPMLE for interval-censored data.
package implements several algorithms
permitting to analyse possibly doubly truncated survival
computes the NPMLE of a survival function
for general interval-censored data.
Parametric:
permits to fit an univariate distribution by maximum
likelihood. Data can be interval censored.
package provides routines for fitting
models in the vitality family of mortality models.
Hazard Estimation
package permits
to estimate the hazard function through kernel methods for right-censored data.
epi.insthaz
function from
the instantaneous hazard from the Kaplan-Meier estimator.
to estimate the hazard function using splines.
package aims at estimating the hazard function for interval
censored data.
package provides non-parametric smoothing
of the hazard through B-splines.
function in
compares survival curves using the Fleming-Harrington G-rho family of test.
implements this class of tests for left-censored
implements a permutation version of the
logrank test and a version of the logrank that adjusts for
covariates.
implements the shift-algorithm by Streitberg and Roehmel for
computing exact conditional p-values and quantiles, possibly for censored data.
package implements
the logrank test reformulated as a linear rank test.
package performs tests using maximally selected
rank statistics.
package implements logrank and Wilcoxon type tests
for interval-censored data.
Three generalised logrank tests and a score test for interval-censored data
are implemented in the
compares 2 hazard ratios.
implements a two stage procedure for comparing
hazard functions.
package proposes to test the equality of
two survival distributions based on the Gini index.
package offers several tests based on the
Fleming-Harrington class for comparing surival curves with right-
and interval-censored data.
package provides a logrank test for which
aggregated data can be used as input.
The short term and long term hazard ratio model for two samples
survival data can be found in the
implements a nonparametric two-sample
procedure for comparing the median survival time.
package performs two-sample comparison
of the restricted mean survival time
package permits to compare two samples
with censored data using empirical likelihood ratio tests.
Regression Modelling
Cox model:
function in
package fits the Cox model.
package and
package propose some extensions to the
The package
implements the Firth's penalised maximum likelihood bias reduction
method for the Cox model.
An implementation of weighted
estimation in Cox regression can be found in
package proposes a robust implementation
of the Cox model.
in package
fits Cox models
with possibly time-varying effects.
permits to fit Cox models with multiple fractional
polynomial. The
fits Cox models for
covariates with missing data.
A Cox model model can be fitted to
data from complex survey design using the
function in
fits Cox models using a weighted partial likelihood for nested
case-control studies.
package implements
Pan's (2000) multiple imputation approach to Cox models for
interval censored data.
package fits Cox
models for interval-censored data through an EM algorithm.
package fits time-varying coefficient
models for interval censored and right censored survival data
using a Bayesian Cox model, a spline based Cox model or a
transformation model.
package computes
the Cox proportional hazards model with shape constrained hazard
functions.
package implements the Cox
model using an active set algorithm for dummy variables of ordered
factors. The
package fits Cox models using
maximum penalised likelihood and provide a non parametric smooth
estimate of the baseline hazard function. A Cox model with
piecewise constant hazards can be fitted using the
package. The
allows nonparametric estimation of
an isotonic covariate effect for proportional hazards
model. The
package implements several models
for interval-censored data, e.g., Cox, proportional odds, and
accelerated failure time models. A Cox type Self-Exciting
Intensity model can be fitted to right-censored data using
package. The
methods for estimation of proportional hazards models with
intermittently observed longitudinal
covariates. The
package provides routines to fit
the Cox model with left-truncated data using augmented information
from the marginal of the truncation times.
function in
goodness-of-fit methods for the Cox proportional hazards model.
The proportionality assumption can be checked using
function in
package calculates concordance probability
estimate for the Cox model, as does the
function in
coxphQuantile
the latter package draws a quantile curve of the survival
distribution as a function of covariates.
package computes simultaneous tests and confidence intervals for
the Cox model and other parametric survival
models. The
package permits to obtain
least-squares means (and contrasts thereof) from linear models. In
particular, it provides support for
functions.
package on Bioconductor proposes a resampling based multiple
hypothesis testing that can be applied to the Cox model.
coefficients of Cox regression models using a Wald test with a
sandwich estimator of variance can be done using
permits to plot visualisation of the relative importance of
covariates in a proportional hazards
model. The
package provides hazard ratio
curves that allows for nonlinear relationship between predictor
and survival. The
package permits to compute the
unadjusted/adjusted attributable fraction function from a Cox
proportional hazards model. The
package proposes
tools to check the proportional hazards assumption using a
standardised score process. The
package implements
empirical likelihood analysis for the Cox Model and Yang-Prentice
(2005) Model.
Parametric Proportional Hazards Model:
) fits a parametric
proportional hazards model.
packages implement a proportional hazards
model with a parametric baseline hazard. The
translates an AFT model to a proportional
hazards form. The
package includes
function that fits a hazard regression
model, using splines to model the baseline hazard. Hazards can be,
but not necessarily, proportional. The
implements the model of Royston and Parmar (2002). The model uses
natural cubic splines for the baseline survival function, and
proportional hazards, proportional odds or probit functions for
regression. The
package allows
estimation of a Weibull Regression for a right-censored endpoint,
one interval-censored covariate, and an arbitrary number of
non-censored covariates.
Accelerated Failure Time (AFT) Models:
function in package
fit an accelerated failure time model.
A modified version of
is implemented in the
function). It permits to use some of the
functionalities.
package also
proposes an implementation of the AFT model (function
An AFT model with an error distribution
assumed to be a mixture of G-splines is implemented in the
proposes the front end of the
function for
left-censored data.
A least-square principled implementation of
the AFT model can be found in the
package implements the
Simulation-Extrapolation algorithm for the AFT model, that can be
used when covariates are subject to measurement error.
version of the accelerated failure time model can be found in
package fits
AFT models for interval censored data. The
package implements both rank-based estimates and least square
estimates (via generalised estimating equations) to the AFT
model. An alternative weighting scheme for parameter estimation in
the AFT model is proposed in the
package. The
package implements elastic net
regularisation for the AFT model.
Additive Models:
fit the additive hazards model of Aalen in
respectively.
also proposes an implementation
of the Cox-Aalen model (that can also be used to perform the Lin,
Wei and Ying (1994) goodness-of-fit for Cox regression models) and
the partly parametric additive risk model of McKeague and
Sasieni. A version of the Cox-Aalen model for interval censored
data is available in the
package. The
package fits shape-restricted
additive hazards models. The
package contains
tools to fit additive hazards model to random sampling, two-phase
sampling and two-phase sampling with auxiliary information.
Buckley-James Models:
function in
compute the
Buckley-James model, though the latter does it without
an intercept term. The
package fits the Buckley-James
model with high-dimensional covariates (L2 boosting, regression
trees and boosted MARS, elastic net).
Other models:
Functions like
can fit other types of models depending on the chosen
distribution,
, a tobit model.
package provides the
function, which is a
wrapper of
to fit the tobit model.
implementation of the tobit model for cross-sectional data and
panel data can be found in the
package provides implementation of the
proportional odds model and of the proportional excess hazards
model. The
package fits the inverse Gaussian
distribution to survival data. The model is based on describing
time to event as the barrier hitting time of a Wiener process,
where drift towards the barrier has been randomized with a
Gaussian distribution. The
package computes the
pseudo-observation for modelling the survival function based on
the Kaplan-Meier estimator and the restricted
package dose the same for the
restricted mean survival time.
parametric time-to-event models, in which any parametric
distribution can be used to model the survival probability, and
where one of the parameters is a linear function of covariates.
function in package
a multiplicative relative risk and an additive excess risk model
for interval-censored data.
package can fit
vector generalised linear and additive models for censored data.
package implements the generalised
additive model for location, scale and shape that can be fitted to
censored data.
locfit.censor
produces local regression estimates.
function included in the
package implements a conditional quantile regression model for
censored data.
package fits shared parameter
models for the joint modelling of a longitudinal response and
event times.
The temporal process regression model is implemented
Aster models, which combine
aspects of generalized linear models and Cox models, are
implemented in the
package implements conditional
logistic regression for survival data as an alternative to the Cox
model when hazards are non-proportional.
extension of the
package, fits latent variable models
for censored outcomes via a probit link
formulation. The
package implements Markov
beta and gamma processes for modelling the hazard ratio for
discrete failure time data. The
packages proposes some model-free contrast comparison measures
such as difference/ratio of cumulative hazards, quantiles and
restricted mean. The
package provides link-based
survival models that extend the Royston-Parmar models, a family of
flexible parametric models. The
implements a unified estimation procedure for the analysis of
censored data using linear transformation
models. The
package fits a flexible parametric
regression model to possibly right-censored, left-truncated
fits the generalized odds rate hazards
model to interval-censored data while
generalized odds rate mixture cure model to interval-censored
package permits to fit a threshold
regression model for interval-censored data based on the
first-hitting-time of a boundary by the sample path of a Wiener
diffusion process. The
package fits
semiparametric promotion time cure models with possibly
mis-measured covariates. The
implements semiparametric cure rate estimators for interval
censored data. The
package permits to fit
semiparametric proportional hazards and accelerated failure time
mixture cure models.
Multistate Models
General Multistate Models:
function from package
can be fitted for any
transition of a multistate model. It can also be used for
comparing two transition hazards, using correspondence between
multistate models and time-dependent covariates. Besides, all the
regression methods presented above can be used for multistate
models as long as they allow for left-truncation.
package provides convenient functions for
estimating and plotting the cumulative transition hazards in any
multistate model, possibly subject to right-censoring and
left-truncation. The
package estimates and plots transition
probabilities for any multistate models. It can also estimate the
variance of the Aalen-Johansen estimator, and handles
left-truncated data. The
package provides non-parametric estimation for
multistate models subject to right-censoring (possibly
state-dependent) and left-truncation. The
package permits to estimate hazards and probabilities, possibly
depending on covariates, and to obtain prediction probabilities in
the context of competing risks and multistate models.
package contains functions for fitting general
continuous-time Markov and hidden Markov multistate models to
longitudinal data. Transition rates and output processes can be
modelled in terms of covariates. The
provides utilities to facilitate the modelling of longitudinal
data under a multistate framework using the
package.The
package can be used to fit
semi-Markov multistate models in continuous time.
distribution of the waiting times can be chosen between the
exponential, the Weibull and exponentiated Weibull distributions.
Non-parametric estimates in illness-death models and other three
state models can be obtained with package
package permits to
estimate transition probabilities of an illness-death model or
three-state progressive model. The
extends the
package to estimation in the
mulstistate model framework, while the
proposes L1 penalised estimation. The
package permits to fit Cox models to the progressive illness-death
model observed under right-censored survival times and interval-
or right-censored progression times. The
package fits proportional hazards models for the illness-death model
with possibly interval-censored data for transition toward the
transient state. Left-truncated and right-censored data are also
allowed. The model is either parametric (Weibull) or
semi-parametric with M-splines approximation of the baseline
intensities. The
package implement the estimator
of Una-Alvarez and Meira-Machado (2015) for non-Markov
illness-death models.
package implements Lexis objects as a way to
represent, manipulate and summarise data from multistate models.
package, based on
permits to draw Lexis diagrams. The
package is
intended for analysing state or event sequences that describe life
courses. The
package permits to describe and
analyse life histories following a multistate perspective on the
life course.
computes the expected numbers of
individuals in specified age classes or life stages given
survivorship probabilities from a transition matrix.
Competing risks:
The package
estimates the cumulative incidence functions, but they can be
compared in more than two samples.
The package also implements
the Fine and Gray model for regressing the subdistribution hazard
of a competing risk.
extends the
package to
stratified and clustered data.
performs a Kaplan-Meier multiple imputation to recover missing
potential censoring information from competing risks events,
permitting to use standard right-censored methods to analyse
cumulative incidence functions. The
implements stepwise covariate selection for the Fine and Gray
model. Package
computes pseudo observations for
modelling competing risks based on the cumulative incidence
functions.
does flexible regression modelling for
competing risks data based on the on the
inverse-probability-censoring-weights and direct binomial
regression approach.
implements risk regression for competing
risks data, along with other extensions of existing packages
useful for survival analysis and competing risks data.
package estimates the conditional probability
of a competing event, aka., the conditional cumulative
incidence. It also implements a proportional-odds model using
either the temporal process regression or the pseudo-value
approaches.
survfit) and
can also be used
to estimate the cumulative incidence function.
package estimates event-specific incidence
rates, rate ratios, event-specific incidence proportions and
cumulative incidence functions.
package implements the semi-parametric mixture model for competing
risks data. The
package implements a
proportional subdistribution hazards model with adjustment for
covariate-dependent censoring. The
implements Pan's (2000) multiple imputation approach to the Fine
and Gray model for interval censored data. The
package provides graphical and analytical approaches for checking
the assumptions of the Fine and Gray model. The
package permits to perform Bayesian, and non-Bayesian,
cause-specific competing risks analysis for parametric and
non-parametric survival functions. The
provides some methods for competing risks data. Estimation,
testing and regression modeling of subdistribution functions in
the competing risks setting using quantile regressions can be had
Recurrent event data:
package can be used to analyse recurrent event
function of the
fits the Anderson-Gill model for recurrent events, model that can
also be fitted with the
package. The latter
also permits to fit joint frailty models for joint modelling of
recurrent events and a terminal event.
package proposes implementations of several models for recurrent
events data, such as the Pe?a-Strawderman-Hollander,
Wang-Chang estimators, and MLE estimation under a Gamma Frailty
package implements the conditional
GEE for recurrent event gap times.
package implements weighted logrank type tests for recurrent
events. The
package provides function to fit gamma
frailty model with either a piecewise constant or a spline as the
baseline rate function for recurrent event data, as well as some
miscellaneous functions for recurrent event data. Several
regression models for recurrent event data are implemented in
Relative Survival
package proposes several functions to deal
with relative survival data. For example,
computes a relative
survival curve.
fits an additive model and
fits the Cox model of Andersen et al. for relative survival, while
fits a Cox model in transformed time.
package permits to fit relative survival models like
the proportional excess and additive excess models.
package allows fitting an hazard regression
model using different shapes for the baseline hazard. The model
can be used in the relative survival setting (excess mortality
hazard) as well as in the overall survival setting (overall
mortality hazard).
package implements the models of Remontet
et al. (2007) and Mahboubi et al. (2011).
package implements methods for
population-based survival analysis, like the proportional hazard
relative survival model and the join point relative survival model.
package computes relative survival,
absolute excess risk and standardized mortality ratio based on
French death rates.
package permits to fit multiplicative
regression models for relative survival.
allows for estimation of EdererII and Pohar
Perme relative / net survival as well as standardized mortality
package implements time-dependent ROC curves
and extensions to relative survival.
Random Effect Models
Frailties:
Frailty terms can be added in
functions in package
. A mixed-effects Cox model is implemented in
package fits the Clayton-Oakes-Glidden
package fits fully parametric frailty
models via maximisation of the marginal likelihood.
package fits proportional hazards models
with a shared Gamma frailty to right-censored and/or
left-truncated data using a penalised likelihood on the hazard
function. The package also fits additive and nested frailty models
that can be used for, e.g., meta-analysis and for hierarchically
clustered data (with 2 levels of clustering), respectively.
proportional hazards model with mixed effects can be fitted using
package fits a
linear mixed-effects model for left-censored data. The Cox model
using h-likelihood estimation for the frailty terms can be fitted
package implements a linear mixed effects model for censored data
with Student-t or normal distributions. The
package simulates and fits semiparametric shared frailty models
under a wide range of frailty distributions. The
package implements parametric frailty models by maximum marginal
likelihood. The
package provides a
regularisation approach for Cox frailty models through
penalisation.
enables modelling of the
excess hazard regression model with time-dependent and/or
non-linear effect(s) and a random effect defined at the cluster
Joint modelling of time-to-event and longitudinal
package allows the analysis
of repeated measurements and time-to-event data via joint random
effects models. The
package performs Cox
regression and dynamic prediction under the joint frailty-copula
model between tumour progression and death for
meta-analysis.
fits semiparametric
regression model for longitudinal responses and a semiparametric
transformation model for time-to-event data.
Multivariate Survival
Multivariate survival refers to the analysis of unit, e.g., the
survival of twins or a family. To analyse such data, we can estimate
the joint distribution of the survival times
Joint modelling:
can estimate bivariate
survival data subject to interval censoring.
package implements various statistical models
for multivariate event history data, e.g., multivariate cumulative
incidence models, bivariate random effects probit models,
Clayton-Oakes model.
package constructs trees for multivariate
survival data using marginal and frailty models.
package permits to estimate correlation
coefficients with associated confidence limits for bivariate,
partially censored survival times.
Bayesian Models
package proposes an implementation of a bivariate
AFT model.
The package
computes a Bayesian model averaging for
Cox proportional hazards models.
function in
fits a Bayesian
semi-parametric AFT model.
LDDPsurvival
in the same package
fits a Linear Dependent Dirichlet Process Mixture of survival models.
performs an MCMC estimation
of normal mixtures for censored data.
A MCMC for Gaussian linear regression with left-, right- or interval-censored
data can be fitted using the
package estimates the hazard function from censored
data in a Bayesian framework.
weibullregpost
function in
the log posterior density for a Weibull proportional-odds regression model.
fits generalised linear mixed models using MCMC
to right-, left- and interval censored data.
package aims at drawing inference on
age-specific mortality from capture-recapture/recovery data when
some or all records have missing information on times of birth
and death. Covariates can also be included in the model.
package performs joint modelling of
longitudinal and time-to-event data under a bayesian approach.
Bayesian parametric and semi-parametric estimation for
semi-competing risks data is available via the
package implements penalized
semi-parametric Bayesian Cox models with elastic net, fused lasso and
group lasso priors.
package fits a Bayesian parametric
proportional hazards model for which events have been geo-located.
package implements Bayesian clustering
using a Dirichlet process mixture model to censored responses.
package provides Bayesian model fitting
for several survival models including spatial copula, linear
dependent Dirichlet process mixture model, anova Dirichlet process
mixture model, proportional hazards model and marginal spatial
proportional hazards model.
package implements non-parametric
survival analysis techniques using a prior near-ignorant Dirichlet
packages permits to fit Bayesian
semiparametric regression survival models (proportional hazards
model, proportional odds model, and probit model) to
interval-censored time-to-event data
package fits a piecewise
exponential hazard to survival data using a Hierarchical Bayesian
High-Dimensional Data
Recursive partitioning:
implements CART-like trees that can be used with
censored outcomes.
package implements recursive partitioning for survival
can perform logic regression.
implements K-adaptive partitioning and recursive
partitioning algorithms for censored survival data.
package implements trees and bagged trees
for discrete-times survival data. The
provides recursive partition algorithms designed for fitting
survival tree with left-truncated and right censored
implements a bootstrap aggregated
version of the k-nearest neighbors survival probability prediction
Random forest:
implements
bagging for survival data.
fits random forest to survival data, while a variant of the random
forest is implemented in
. A faster implementation
can be found in package
. An alternative
algorithm for random forests is implemented in
Regularised and shrinkage methods:
package provides procedures for fitting the
entire lasso or elastic-net regularization path for Cox models.
package implements a L1 regularised Cox
proportional hazards model.
An L1 and L2 penalised Cox models are
available in
computes a nearest shrunken centroid for survival gene expression
A high dimensional Cox model using univariate shrinkage is
available in
implements the lassoed principal components method.
package implements the LASSO and elastic net
estimator for the additive risk model.
package implements the Lasso and elastic-net penalized Cox's
regression using the cockail algorithm.
package permits to fit Cox models with a combination of lasso and
group lasso regularisation.
fits Cox models
with penalized ridge-type (ridge, dynamic and weighted dynamic)
partial likelihood. The
package implements 9
types of penalised Cox regression methods and provides methods for
model validation, calibration, comparison, and nomogram
visualisation. Another implementation of regularised Cox models
can be found in
. A penalised version of the Fine
and Gray model can be found
package implements
cyclic coordinate descent for the Cox proportional hazards model.
Gradient boosting for the Cox model is implemented in the
package includes a generic gradient boosting algorithm
for the construction of prognostic and diagnostic models for right-censored data.
implements permutation-based testing procedure to test
the additional predictive value of high-dimensional data. It is based on
provides routines for fitting the Cox proportional hazards model
and the Fine and Gray model by likelihood based boosting.
package implements
the supervised principal components for survival data.
package can construct index models for survival
outcomes, that is, construct scores based on a training dataset.
package fits Cox proportional hazards
model using the compound covariate method.
provides partial least squares regression and various techniques
for fitting Cox models in high dimensionnal
settings. The
package implements feature selection
algorithms based on subsampling and averaging linear models
obtained from the Lasso algorithm for predicting survival risk.
Predictions and Prediction Performance
package provides utilities to plot prediction error
curves for several survival models
implements prediction error techniques which can
be computed in a parallelised way. Useful for high-dimensional
package permits to estimate time-dependent
ROC curves and time-dependent AUC with censored data, possibly
with competing risks.
computes time-dependent ROC curves and time-dependent AUC from
censored data using Kaplan-Meier or Akritas's nearest neighbour estimation method
(Cumulative sensitivity and dynamic specificity).
can be used to compute time-dependent ROC curve
from censored survival data using nonparametric weight
adjustments.
implements time-dependent ROC curves,
AUC and integrated AUC of Heagerty and Zheng (Biometrics, 2005).
Various time-dependent true/false positive rates and
Cumulative/Dynamic AUC are implemented in the
package provides several functions to
assess and compare the performance of survival models.
C-statistics for risk prediction models with censored survival
data can be computed via the
package implements the integrated
discrimination improvement index and the category-less net
reclassification index for comparing competing risks prediction
package provides functions for
estimating the AUC, TPR(c), FPR(c), PPV(c), and NPV(c) for
survival data.
package provides functions for the
estimation of the prediction accuracy in a unified survival AUC
package permits to compare C indices
with right-censored survival outcomes
package provide tools to estimate the
average positive predictive values and the AUC for risk scores or
Power Analysis
package proposes power calculation for weighted
Log-Rank tests in cure rate models.
permits to calculate sample size based on
proportional hazards mixture cure models.
package provides power and sample size
calculation for survival analysis (with a focus towards
epidemiological studies).
Power analysis and sample size calculation for SNP association
studies with time-to-event outcomes can be done using
Simulation
package permits to generate data wih one
binary time-dependent covariate and data stemming from a
progressive illness-death model.
package permits the user to simulate
complex survival data, in which event and censoring times could be
conditional on an user-specified list of (possibly time-dependent)
covariates.
package proposes some functions for
simulating complex event history data.
package also permits to simulate and analyse
multistate models. The package allows for a general specification
of the transition hazard functions, for non-Markov models and
for dependencies on the history.
package provides functions for simulating
complex multistate models data with possibly nonlinear baseline
hazards and nonlinear covariate effects.
package implements tools for simulating and
plotting quantities of interest estimated from proportional
hazards models.
package permits to simulate simple and
complex survival data such as recurrent event data and competing
package provides routines for performing
continuous-time microsimulation for population projection. The
basis for the microsimulation are a multistate model, Markov or
non-Markov, for which the transition intensities are specified, as
well as an initial cohort.
package permits to simulate data with a
dichotomous time-dependent exposure.
package can be used to simulate
univariate and semi-competing risks data given covariates and
piecewise exponential baseline hazards.
This section tries to list some specialised plot functions that might be
useful in the context of event history analysis.
package proposes
functions for plotting survival curves with the at risk table aligned to
the x axis.
extends this to the competing risks
function in
to draw the states and transitions that characterize a multistate
package provides many plot functions for
representing multistate data, in particular Lexis diagrams.
package provide multistate-type graphics
for competing risks, in which the thickness of the transition
arrows from the initial event to each competing event describes
the particular amount of every incidence rate.
generates time-to-event outcomes for
families that habour genetic mutation under different sampling
designs and estimates the penetrance functions for family data
with ascertainment correction.
Miscellaneous
package contains the
ggsurvplot
for drawing survival curves with
the 'number at risk' table. Other functions are also available for
visual examinations of cox model assumptions.
package multiple imputation
methods for dealing with informative censoring.
provides data transformations, estimation
utilities, predictive evaluation measures and simulation functions for
discrete time survival analysis.
is the companion package to &Dynamic Prediction
in Clinical Survival Analysis&.
proposes the
function that
implements several types of bootstrap techniques for right-censored data.
package estimates the current
cumulative incidence and the current leukaemia free survival function.
package provides functions for performing meta-analyses
of gene expression data and to predict patients' survival and risk assessment.
provides tools for individual patient data meta-analysis, mixed-level meta-analysis with patient
level data and mulivariate survival estimates for aggregate studies.
package includes the data sets from Klein
and Moeschberger (1997).
Some supplementary data sets and
functions can be found in the
that accompanies Aitkin et al. (2009),
that accompanies Davidson (2003)
that accompanies Maindonald, J.H. and Braun,
W.J. () also contain survival data sets.
package permits to construct, validate and
calibrate nomograms stemming from complex right-censored survey
package compute the MLE of a density
(log-concave) possibly for interval censored data.
package fits parametric
Transform-both-sides models used in reliability analysis
package implements algorithms to detect outliers
based on quantile regression for censored data.
package implements an EM algorithm
to estimate the relative case fatality ratio between two groups.
package proposes a fully efficient sieve
maximum likelihood method to estimate genotype-specific distribution
of time-to-event outcomes under a nonparametric model
power and sample size calculation based on the difference in
restricted mean survival times can be performed using
package allows for the estimation of
multivariate average hazard ratios as defined by Kalbfleisch and
provides miscellaneous routines to help in
the analysis of right-censored survival data.
Accompanying data sets to the book
Applied Survival Analysis
can be found in package
CRAN packages:
Related links:
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