1	Survival Analysis: Analysis of Right Censored Time to Event Data Scott S. Emerson, M.D., Ph.D. Professor of Biostatistics, University of Washington May 1, 2004
2	Course Structure Topics: First (and a half) Session: General Setting Censored data setting Estimation of survivor functions Survival analysis models Second session: Comparison of Two Samples Logrank statistic Nonproportional hazards Weighted logrank statistics Third session: Sequential Analysis Stopping rules Weighted logrank statistics in nonproportional hazards Reweighted statistics
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6	Overview Scientific Studies
7	Where Am I Going? I claim that Statistical analyses should be driven by the scientific questions they are meant to answer Before addressing limitations imposed by our sampling scheme (e.g., missing data, sequential analysis), we should review the extent to which our practice adheres to those goals
8	Fundamental Philosophy Statistics is about science. Science is about proving things to people. Other scientists Community at large
9	Scientific Studies A well designed study Discriminates between the most important, viable hypotheses “Discriminates” defined by what convinces your audience Is equally informative for all possible study results Binary search using prior probability of being true Also consider simplicity of experiments, time, cost
10	Scientific Questions Ultimately, scientific questions are most often concerned with investigating cause and effect E.g., in biomedical settings: What are the causes of disease? What are the effects of interventions?
11	Typical Inferential Setting In the studies considered here, we define Some “primary outcome” measurement A “response variable” in regression Groups that are homogeneous with respect to the level of some factor(s) Predictor of interest Effect modifiers Confounders Precision variables
12	Primary Outcome Measurement The primary outcome can be derived from more than one measured variable E.g., for repeated measurements made on the same experimental unit Contrast across repeated measurements Weighted average of repeated measurements E.g., for random process defined by longitudinal follow up of experimental units Contrast across time Weighted average over time Time until an event
13	Typical Scientific Hypotheses The specified level of some factor will cause outcome measurements that are
14	Causation vs Association Truly determining causation requires a suitable interventional study (experiment) Statistical analyses tell us about associations Associations in the presence of an appropriate experimental design allows us to infer causation But even then, we need to be circumspect in identifying the true mechanistic cause E.g., a treatment that causes headaches, and therefore aspirin use, may result in lower heart attack rates due entirely to the use of aspirin
15	First Statistical Refinement The group with the specified level of some factor will have outcome measurements that are
16	Deterministic Setting Conditions of scientific studies might make answering questions difficult even when study results are deterministic Difficulties in isolating specific causes E.g., isolating REM sleep from total sleep E.g., interactions between genetics and environment Difficulties in measuring potential effects E.g., measuring time to survival length of study competing risks
17	Can Statistics Help? Litmus Test # 1: If the scientific question cannot be answered by an experiment when outcomes are entirely deterministic, there is NO chance that statistics can be of any help.
18	Variation in Response There is, of course, usually variation in outcome measurements across repetitions of an experiment Variation can be due to Unmeasured (hidden) variables E.g., mix of etiologies, duration of disease, comorbid conditions, genetics when studying new cancer therapies Inherent randomness (as dictated by quantum theory)
19	Second Statistical Refinement The group with the specified level of some factor will tend to have outcome measurements that are
20	Refining Scientific Hypotheses In order to be able to perform analysis we must define “will tend to have” Probability model for response Nonparametric, semiparametric, parametric (Looking ahead: I am a big proponent of nonparametric interpretations of statistical analyses)
21	Ordering Probability Distributions In general, the space of all probability distributions is not totally ordered There are an infinite number of ways we can define a tendency toward a “larger” outcome This can be difficult to decide even when we have data on the entire population Ex: Is the highest paid occupation in the US the one with the higher mean? the higher median? the higher maximum? the higher proportion making $1M per year?
22	Can Statistics Help? Litmus Test # 2: If the scientific researcher cannot decide on an ordering of probability distributions which would be appropriate when measurements are available on the entire population, there is NO chance that statistics can be of any help.
23	Summary Measures Typically we order probability distributions on the basis of some summary measure Statistical hypotheses are then stated in terms of the summary measure Primary analysis based on detecting an effect on (most often) one summary measure Avoids pitfalls of multiple comparisons Especially important in a regulatory environment
24	Purposeful Vagueness What I call “summary measures”, others might call “parameters” “Parameters” suggests use of parametric and semiparametric statistical models I am generally against such analysis methods “Functionals” is probably the best word “Functional”= anything computed from a cdf But too much of a feeling of “statistical jargon”
25	Marginal Summary Measures Many times, statistical hypotheses are stated in terms of summary measures for univariate (marginal) distributions Means (arithmetic, geometric, harmonic, …) Medians (or other quantiles) Proportion exceeding some threshold Odds of exceeding some threshold Time averaged hazard function (instantaneous risk) …
26	Comparisons Across Groups Comparisons across groups then use differences or ratios Difference / ratio of means (arithmetic, geometric, …) Difference / ratio of proportion exceeding some threshold Difference / ratio of medians (or other quantiles) Ratio of odds of exceeding some threshold Ratio of hazard (averaged across time?) …
27	Joint Summary Measures Other times groups are compared using a summary measure for the joint distribution Median difference / ratio of paired observations Probability that a randomly chosen measurement from one population might exceed that from the other …
28	Looking Ahead: Transitivity The distinction between marginal versus joint summary measures impacts comparisons across studies Most often (always?) transitivity is not guaranteed unless comparisons can be defined using marginal distributions Intransitivity: Pairwise comparisons might suggest A > B, and B > C, but C > A
29	Can Statisticians Help? While I claim that the choice of the definition for “tends to be larger” is primarily a scientific issue, statisticians do usually play an important role Statisticians do explain how different summary measures capture key features of a probability distribution
30	Overview Choice of Summary Measure for Inference
31	Where Am I Going? I have claimed that We usually address scientific questions using summary measures of probability distributions I now claim that Selection of a summary measure is best based on scientific criteria Relevance to this course: With censored data, we often choose summary measures that are not the logical choice from a scientific standpoint
32	Hypothetical Example: Setting Consider survival with a particular treatment used in renal dialysis patients Extract data from registry of dialysis patients To ensure quality, only use data after 1995 Incident cases in 1995: Follow-up 1995 – 2002 (8 years) Prevalent cases in 1995: Data from 1995 - 2002 Incident in 1994: Information about 2^nd – 9^th year Incident in 1993: Information about 3^rd – 10^th year … Incident in 1988: Information about 8^th – 15^th year
33	Hypothetical Example: Analysis Methods to account for censoring/truncation Descriptive statistics using Kaplan-Meier Options for inference Parametric models Weibull, lognormal, etc. Semiparametric models Proportional hazards, etc. Nonparametric Weighted rank tests: logrank, Wilcoxon, etc. Comparison of Kaplan-Meier estimates
34	Hypothetical Example: KM Curves
35	Who Wants To Be A Millionaire? Proportional hazards analysis estimates a Treatment : Control hazard ratio of A: 2.07 (logrank P = .0018) B: 1.13 (logrank P = .0018) C: 0.87 (logrank P = .0018) D: 0.48 (logrank P = .0018) Lifelines: 50-50? Ask the audience? Call a friend?
36	Who Wants To Be A Millionaire? Proportional hazards analysis estimates a Treatment : Control hazard ratio of B: 1.13 (logrank P = .0018) C: 0.87 (logrank P = .0018) Lifelines: 50-50? Ask the audience? Call a friend?
37	Who Wants To Be A Millionaire? How could you have known this? In PH, the standard error of log hazard ratio estimates is approximately 2 divided by the square root of the number of events. A P value of .0018 corresponds to \| Z \| = 3.13 log(2.07) = -log(0.48) is approximately 0.7 3 x 2 / .7 is about 8.4 Number of deaths would be about 72 We had 5000+ subjects with survival estimated down to 30%
38	Criteria for Summary Measure We choose some summary measure of the probability distribution according to the following criteria (in order of importance) Scientifically (clinically) relevant Also reflects current state of knowledge Is likely to vary across levels of the factor of interest Ability to detect variety of changes Statistical precision Only relevant if all other things are equal
39	Example of Scientific Issues E.g., Is the arithmetic mean’s sensitivity to outliers desirable or undesirable? Do we want to detect better infant mortality? Does making one person immortal make up for killing others prematurely? E.g., Is the scientific importance of a difference in distribution best measured by the proportion exceeding some threshold? Is an increase in survival time only important if the patient eventually makes it out of intensive care?
40	Common Practice The overwhelming majority of statistical inference is based on means Means of continuous random variables t test, linear regression Proportions (means of binary random variables) chi square test (t test) Rates (means) for count data Poisson analyses
41	Use of the Mean Rationale Scientific relevance Measure of “central tendency” or “location” Related to totals, e.g. total health care costs Plausibility that it would differ across groups Sensitive to many patterns of differences in distributions (especially in tails of distributions) Statistical properties Distributional theory known Optimal (most precise) for many distributions (Ease of interpretation?)
42	When Not to Use the Mean Lack of scientific relevance The mean is not defined for nominal data The mean is sensitive to differences that occur only in the tail of the distribution E.g., increasing the jackpot in Lotto makes one person richer, but most people still lose Small differences may not be of scientific interest Extend life expectancy by 24 hours Decrease average cholesterol in patients with familial hypercholesterolemia by 20 mg/dl
43	When Not to Use the Mean Intervention unlikely to affect the mean Sometimes we are interested in controlling variability E.g., thermostats are designed to maintain house temperature within a certain range E.g., control of blood glucose in diabetics? (This is not typically a major criterion for avoiding the mean: It is rare that the mean is not affected by an intervention.)
44	When Not to Use the Mean Statistical criteria In the presence of heavy tails (outliers) the mean is not estimated with high precision asymptotic distributional theory may not yet hold When adjusting for covariates, it may be unreasonable to expect the mean to show constant differences across subgroups Especially invoked with binary data (we most often use the odds instead)
45	Comments on Statistical Criteria Many of the reasons used to justify other tests are based on misconceptions The validity of t tests does NOT depend heavily upon normally distributed data Modern computation allows exact small sample inference for means in same manner as used for other tests The statistical theory used to demonstrate inefficiency of the mean is most often based on unreasonable (and sometimes untestable) assumptions
46	Example: Wilcoxon Rank Sum Test Common teaching: A nonparametric alternative to the t test Not too bad against normal data Better than t test when data have heavy tails (Some texts refer to it as a test of medians)
47	More Accurate Guidelines In general, the t test and the Wilcoxon are not testing the same summary measure Wilcoxon test statistic based on Pr(X > Y) Null distribution is a permutation test Wrong size as a test of Pr(X > Y) = ½ (unless a semi-parametric model holds on some scale) (this can be fixed by modifying the null variance) Inconsistent test of F(t) = G(t) An infinite sample size may not detect the alternative (And the Wilcoxon is not transitive)
48	More Accurate Guidelines Efficiency theory derived when a shift model holds for some monotonic transformation If propensity to outliers is different between groups, the t test may be better even with heavy tails
49	Comments In any case, the decision regarding which parameter to use as the basis for inference sould be made prior to performing any analysis directly related to the question of interest Basing decisions regarding choice of analysis method on the observed data will tend to inflate the type I error Decrease our confidence in our statistical conclusions
50	Overview Impact of Censored Data
51	Where Am I Going? I claim Censored data results from a particular choice of sampling scheme Usually such a sampling scheme is necessary due to logistical constraints There is nothing inherent in the mere presence of censored data that need alter the question which is deemed scientifically most important
52	Ultimate Goal My task, therefore, is to discuss the ways that we can answer our scientific questions in the presence of censored data How do we make inference about the summary measures of greatest interest? Sometimes, the presence of censored data does make us modify the summary measure used. Almost always, the presence of censored data requires that we estimate those summary measures with different computation formulas
53	(Right) Censored Data The Setting
54	Censored Variables A special type of missing data The exact value is not always known Right censoring: For some observations it is only known that the true value exceeds some threshold Left censoring: For some observations it is only known that the true value is below some threshold Interval censoring: For some observations it is only known that the true value is between some thresholds
55	Example A clinical trial is conducted to examine aspirin in prevention of cardiovascular mortality 10,000 subjects are randomized equally to receive either aspirin or placebo Subjects are randomized over a three year period Subjects are followed for fatal events for an additional three year period following accrual of the last subject
56	The Problem At the end of the clinical trial Some subjects have been observed to die True time to death is known for these subjects Most subjects are likely to be still alive Death times of these subjects are only known to be longer than the observation time “(Right) Censored observations”
57	What Should We Do? Cannot ignore These are our treatment successes Cannot just treat as binary (live/die) data: Potential time of follow-up (censoring time) differs across subjects due to time of study entry Confounding vs loss of precision (Censored data may also arise due to loss to follow-up, e.g., moved away) (Could figure out whether alive/dead at earliest censored observation, but this is inefficient and may not answer the question of interest)
58	Right Censored Data Notation:
59	(Right) Censored Data Motivating Example
60	Where Am I Going? I try to build an intuitive feel for The information present in the censored data, and How that information can be used to estimate the distribution of response
61	Motivating Example Hypothetical study of subject survival Subjects accrued to study and followed until time of analysis Study done at three centers, which started the studies in three successive years Censoring time thus differs across centers
62	Data (Real Time) Staggered study entry by site Accrual Group Year A B C 1990 On study 100 -- -- Died 43 Surviving 57 1991 On study 57 100 -- Died 27 53 Surviving 30 47 1992 On study 30 47 100 Died 13 22 55 Surviving 17 25 45
63	Data (Study Time) Realign data according to time on study Accrual Group Year A B C 1 On study 100 100 100 Died 43 53 55 Surviving 57 47 45 2 On study 57 47 -- Died 27 22 Surviving 30 25 3 On study 30 -- -- Died 13 Surviving 17
64	Combined Data Accrual Group Year A B C Combined 1 On study 100 100 100 300 Died 43 53 55 151 Surviving 57 47 45 149 2 On study 57 47 -- 104 Died 27 22 49 Surviving 30 25 55 3 On study 30 -- -- 30 Died 13 13 Surviving 17 17
65	Problem Posed by Missing Data Sampling scheme causes (informative) missing data Potentially, we might want to estimate three year survival probabilities Different centers contribute information for varying amounts of time One year survival can be estimated at A, B, C Two year survival can be estimated at A, B Three year survival can be estimated at A
66	Possible Remedies WRONG: Ignore missing E.g., 17 of 300 subjects alive at three years RIGHT BUT WRONG QUESTION: Use data only up to earliest censoring time E.g., 149 of 300 subjects alive at one year RIGHT BUT INEFFICIENT: Use only center A E.g., 17 of 100 subjects alive at three years
67	Best Remedy RIGHT AND EFFICIENT Use all available data to estimate that portion of survival for which it is informative Use Centers A, B, and C to estimate one year survival Use Centers A and B to estimate proportion of one-year survivors who survive to two years Use Center A to estimate proportion of two-year survivors who survive to three years
68	Theoretical Basis for Approach Properties of probabilities Probability of event A and B occurring is product of Probability that A occurs when B has occurred Probability that B has occurred
69	Application of Theory to Survival For times T₁ < T₂ , probability of surviving beyond time T₂ is the product of Probability of surviving beyond time T₂ given survival beyond time T₁, and Probability of surviving beyond time T₁
70	Estimation of Conditional Survival Estimate conditional probability of survival within each time interval Condition on surviving up until the start of the time interval Denominator is number of subjects at start of interval Numerator is deaths during the interval
71	Requirements for Valid Estimates Consistent estimates of survival probabilities depend on The subjects available at the start of each time interval must be a random sample of the population suriviving to that time “Noninformative censoring” cf: Nonignorable missing, but noninformative censoring
72	Estimation of Survival Probability Estimate probability of survival at the endpoint of each time interval Multiply the conditional probabilities for all intervals prior to the time point of interest
73	Obtaining Estimates Within interval conditional probabilities Use A, B, C to estimate Pr(T > 1) Use A, B to estimate Pr(T > 2 \| T > 1) Use A to estimate Pr(T > 3 \| T > 2) Multiply to obtain unconditional cumulative survival Pr(T > 1) Pr(T > 2) = Pr(T > 2 \| T > 1) Pr(T > 1) Pr(T > 3) = Pr(T > 3 \| T > 2) Pr(T > 2)
74	Combined Data Accrual Group Year A B C Combined 1 On study 100 100 100 300 Died 43 53 55 151 Surviving 57 47 45 149 2 On study 57 47 -- 104 Died 27 22 49 Surviving 30 25 55 3 On study 30 -- -- 30 Died 13 13 Surviving 17 17
75	Survival Probability Estimates Survival Probabilities Yr Combined Each Year Cumulative 1 On study 300 Died 151 Surviving 149 149/300 = 49.67% 49.67% 2 On study 104 Died 49 Surviving 55 55/104 = 52.88% .4967.5288 = 26.27% 3 On study 30 Died 13 Surviving 17 17/ 30 = 56.67% .2627.5667 = 14.88%
76	Improved Precision Intuitively, these estimates would provide greater precision, because they are based on more data than using Center A alone We can show this exactly using confidence intervals
77	Number At Risk and Number Failed For notational convenience
78	Survival Probability Notation For notational convenience
79	Survival Probability Estimates Maximum likelihood estimates for Conditional survival probability within intervals Unconditional survival probability
80	Logarithmic Transformation Sums are easier to work with than products The log transformed unconditional survival probability is the sum of log transformed conditional survival probabilities
81	Basic Approach We will find the standard error of the log transformed survival probabilities by Estimating each conditional survival probability and finding the variance of the log transformed estimates Invoking noninformative censoring to argue that the sum of our log transformed estimates must have the same distribution as the sum of log transformed independent estimates
82	Standard Error of Proportions From the laws of expectation, for the jth interval
83	Large Sample Approximation From the central limit theorem
84	Logarithmic Transformation From the delta method
85	Noninformative Censoring In the presence of noninformative censoring, the risk set in any interval should look like a random sample of the population at risk Estimates of the conditional probability of survival for the intervals should be uncorrelated
86	Confidence Intervals Using the large sample approximation with plug-in estimates for standard errors
87	Survival Probability Estimates Note the improved precision (and accuracy) Narrower CI even for the third year estimates Survival Probabilities (95% CI) Yr Site A Only Combined 1 0.570 (0.473, 0.667) 0.497 (0.443, 0.557) 2 0.300 (0.210, 0.390) 0.263 (0.212, 0.325) 3 0.170 (0.096, 0.244) 0.149 (0.102, 0.217)
88	Aside: Greenwood’s Formula SE for the survival probabilities by a second application of the delta method
89	Aside: Alternatives for CI Three common methods for CI Based on log ( S(t) ) Based on Greenwood’s formula Based on log ( - log ( S(t) ) ) These intervals will always be between 0 and 1
90	(Right) Censored Data Product Limit (Kaplan-Meier) Estimates
91	Where Am I Going? I introduce the nonparametric estimate of survivor functions, by making analogy with the previous example I also provide an alternative derivation that provides intuition about the assumption of noninformative censoring
92	Life Table Methods In the actuarial (e.g., insurance) setting The time intervals are often chosen by years, decades, etc. The data are presented for each year as N_j: Number of subjects at risk at start of interval C_j: Number censored during interval (these will contribute half a person-year) D_j: Number of events in interval
93	Life Table Methods Computation of probability of survival
94	Life Table Methods Computation of probability of survival (cont.)
95	Kaplan-Meier Estimates With more precisely measured individual data The time intervals are defined by unique observation times The data are presented for each year as Nj: Number of subjects at risk at start of interval Dj: Number of events at end of interval (Note no censoring or events during interval by definition) (Note also that for ties, censoring occurs after deaths)
96	Kaplan-Meier Estimates Computation of probability of survival
97	Kaplan-Meier Estimates Product Limit Estimate
98	Kaplan-Meier Estimates Note that in the above definition An interval which ends in a censored observation with no observed events has conditional probability of surviving within the interval is 1. If the largest observation time is censored, the KM (PLE) survivor function never goes to zero We generally regard the KM (PLE) survivor function to be undefined for times beyond the largest observation time in this situation
99	Kaplan-Meier Estimates Properties The KM (PLE) survivor functions can be shown to be Consistent: As sample sizes go to infinity, they estimate the true value Nonparametric maximum likelihood estimates (but usual asymptotic theory for regular, parametric MLE’s does not necessarily hold)
100	Alternative Derivations The KM (PLE) survivor functions can also be derived as the Self-consistent estimator (see Miller, Survival Analysis) “Redistribute to the right” estimator Provides intuition regarding noninformative censoring
101	Redistribute to the Right Basic idea Recall the empirical cdf assigns probability 1/n to each observation Each subject in a sample is representative of 1/n of the population A censored observation should be equally likely to have event time like any of the remaining uncensored observations Recursively redistribute the mass of each censored observation among the subjects remaining at risk
102	Redistribute to the Right Example Data: 1, 3, 4, 5, 7, 9, 10 (asterisk means censored) Initially: each point has mass 1/7 Determine probability of events at earliest observed (uncensored) event times Pr (T = 1) = 1/7 Pr (T = 3) = 1/7
103	Redistribute to the Right Example Censored observation at 4 Divide the mass at 4 equally among the remaining subjects at risk Now mass of 1/7 + 1/28 = 5/28 for each of 5, 7, 9, 10 Determine probability of events at next observed (uncensored) event times Pr (T = 5) = 5/28
104	Redistribute to the Right Example Censored observation at 7 Divide the mass at 7 equally among the remaining subjects at risk Now mass of 5/28 + 5/56 = 15/56 for each of 9, 10 Determine probability of events at next observed (uncensored) event times Pr (T = 9) = 15/56 Pr (T = 10) = 15/56
105	General Analysis Models Risk Sets and Hazard Functions
106	Where Am I Going? I claim that Our ability to address scientific questions with censored data is heavily dependent upon the assumption of noninformative censoring Noninformative censoring guarantees that we can estimate the hazard function in a consistent fashion Hence, understanding the role of risk sets and estimation of the hazard function is crucial to interpreting the most commonly used survival analysis methods
107	Hazard Functions From the approach to nonparametric estimation of survival curves we see the importance of the hazard function Hazard = instantaneous risk of failure Conditional upon being still alive, what is the probability (rate) of failing in the next instant
108	General Notation
109	Relationship to Survivor Function The survivor function (and, hence, the cdf) is uniquely determined by the hazard and vice versa
110	Estimating the Hazard Function The intuitive estimator of the hazard function is thus the conditional probability of failure at each point in time
111	Risk Sets Survival analysis often focuses on the “risk set” at each time “Risk set at time t”= the set of subjects in the sample who are at risk for failure at t These subjects can be used to compute and compare hazard functions and, hence, survival probabilities
112	Risk Based Analysis Models Analyses based on hazard functions afford the opportunity to allow sampling schemes which sample the population at risk at each time Advantages: More efficient use of available data Time-varying covariates Disadvantages: Less intuitive summary measures Temptation to use time-varying covariates
113	General Analysis Models A Useful Analogy
114	Where Am I Going? I claim that The most commonly used methods for censored data have straightforward analogues in the urn model used in classical probability
115	Urn Model Balls in an urn of various colors and patterns Balls might represent people in a study At any given time, the balls that are in the urn are therefore the risk set Colors and patterns represent risk factors
116	Death Process Periodically, I come in and choose a ball from the urn and take it When a ball is chosen it fails My predilection for choosing certain colors or patterns identifies true risk factors Characteristics of the balls that I do not notice have no effect on survival probabilities
117	Evidence for Risk Factors A certain color/pattern must be my favorite if (Time based observations) I come in more often when that color/pattern is in the urn You need not consider what else is in the urn (Risk set based observations) I choose that color/pattern with a frequency disproportionate to its frequency in the urn If I am blind to a characteristic, my choices should look like random sampling You need not consider the times that I come in
118	(Semi)parametric Models Two general (semi)parametric probability models used in survival analysis Accelerated failure time models Consider time of failure Proportional hazards models Consider relations among hazards (Additive hazards models also used, but less frequently)
119	Accelerated Failure Time Models Two groups that differ in some risk factor have survivor functions related by a parameter measuring acceleration or deceleration of time E.g., A smoker ages twice as fast as a nonsmoker Each human year is seven dog years
120	Proportional Hazards Models Two groups that differ in some risk factor have survivor functions related by a parameter measuring increased hazard E.g., At any given time, a smoker is ten times more likely to develop lung cancer as a nonsmoker
121	Scientific Studies As a scientist you may Observe When I come into the room and take a ball, The colors/patterns on all the balls in the urn, and The color/patterns on the ball that I take Experiment Change the compostion in the urn and see Whether I come in the room more or less often, and The lengths to which I might go to find balls with certain colors or patterns by restricting my choices
122	Altering the Risk Set Censoring and time-varying covariates are analogous to changes in the composition of the urn Censoring = removing balls from the urn Time-varying covariates = repainting the balls or adding different balls
123	Caveats: Informative Censoring Altering the risk set can be problematic Recall that in order for survival estimates to be consistent, the risk set in the sample must look like a random sample from the population You should not selectively remove or change balls that were (for their risk factors) particularly more likely or less likely to be chosen If you notice that I search the urn from top to bottom, Don’t just change the balls sitting at the top of the urn Make sure you stir the urn after each change
124	Caveats: Time-varying Covariates Time-varying covariates are far more easily implemented in the hazard based models Risk set approach makes this easy However, scientifically we run the risk of overfitting our data using variables we are less interested in A priest delivering last rites is highly predictive of death and that may obscure that it was a gunshot wound that led to the death
125	General Analysis Models Noninformative Censoring
126	Where Am I Going? I claim that Noninformative censoring is a crucial, but untestable, assumption Hence, it is important to think about situations where it might not be satisfied
127	Noninformative Censoring Censoring must not be informative about subjects who were either more or less likely to have an event in the immediate future The censored individuals must look like a random sample of those individuals at risk at the time of censoring (Later we shall say that they are a random sample from all subjects at risk having similar modeled covariates)
128	Examples of Informative Censoring Subjects in a clinical trial are withdrawn due to treatment failure (likely they would die sooner than those remaining) Subjects in a clinical trial in a fatal condition are lost to follow up when they go on vacation (likely they are healthier than those remaining)
129	Examples of Informative Censoring Leukemia patients in a clinical trial of bone marrow transplantation are censored if they die of infections rather than dying of cancer (the subjects who died of infections might have had a more effective regimen to wipe out existing cancer)
130	Detecting Informative Censoring As a general rule it is impossible to use the data to detect informative censoring The necessary data is almost certainly missing in the data set In some cases, it is impossible to ever observe the missing data Nonfelines can only die once We cannot observe whether subjects dying of one cause are more or less likely to die of another if we cure them of the first cause
131	Competing Risks This last situation is often referred to as “Competing Risks” Some “nuisance” event sometimes precludes your ability to ever observe the event of interest In the presence of competing risks, we must decide how best to address the scientific question of interest
132	Example: Censoring Mechanisms Consider a study of smoking as a risk factor for incidence of cancer Possible causes of censored observations Subject still alive at time of data analysis Subject lost to follow-up during study Subject died in airline accident Subject died in single car accident Subject died of MI Subject died of emphysema
133	Censoring Competing Risks Time to cancer, but competing risk of death Suppose we censor deaths If deaths represent noninformative censoring People who died of, say, MI neither more nor less likely to get cancer in the near term Estimates desired hazard rate If deaths represent informative censoring Estimates cause specific hazard in presence of unchanged risk of competing event Results are not generalizable to a population with an altered risk of death
134	Competing Risks: Alternatives Model informative censoring Model must be based on untestable assumptions Event free survival Like censoring deaths if competing risk hazard low Like censoring deaths if everyone gets cancer first Loss of power if truly noninformative censoring Wilcoxon like statistic Rank first on death times; break ties with cancer dx Like survival only if everyone dies Survival only Not really the question, especially if competing risk hazard is high
135	General Analysis Models Time-varying Covariates
136	Where Am I Going? I claim that The hazard based methods for the analysis of censored data have the attractive capability to model time-varying covariates The difficulties of choosing the appropriate model to address a scientific question is magnified manifold when considering time-varying covariates
137	Fixed Covariates In a typical study, we compare the distribution of some outcome across groups defined at the start of the study Example: Risk of hang gliding Identify two groups Hang gliders Cowards Follow survival experience over time
138	Problem What if a coward obtains courage? Misclassification will attenuate the true effect of hang gliding on survival Biased estimates Less precision
139	A Wrong Approach We cannot divide the sample into groups according to lifetime habits Suppose we consider Ever hang glided (hung glide?) vs Constant coward We might detect spurious associations due to “survivorship” If we started study at birth, we might find hang gliding is beneficial Most people don’t start hang gliding until teenaged We would detect the fact that hang gliders survived at least that long
140	A Correct Approach Let each subject contribute observation time to the appropriate group according to covariate at the relevant time Proportional hazards model Easily done, if noninformative censoring results Accelerated failure time model Difficult due to need to integrate hazards over disjoint intervals
141	Issues Issues related to the use of time-varying covariates are analogous to those when deciding to adjust for any variable Can regard measurements made at different times as different covariates Need to consider Causal pathway of interest Confounding (bias) Precision Time aspect does increase the dimensionality
142	Issues: Informative Censoring Possibility that impending event causes informative censoring (confounding?) Types of variables Extrinsic: Unaffected by individual decisions As a rule, time-varying extrinsic variables will not cause informative censoring E.g., Air pollution on a given day in an asthma study (providing it does not affect relocation) Intrinsic: Potentially affected by impending event E.g., Marijuana use
143	Causation versus Association Example: Scientific interest in causal pathways between marijuana use and heart attacks (MI) Pictorial representation of hypothetical causal effect of marijuana on MI that might be of scientific interest
144	Causation versus Association Statistical analysis can only detect associations reflecting causation in either direction Only experimental design and understanding of the variables allows us to infer cause and effect Statistical analysis will identify causation in either direction
145	Causation versus Association In an observational study, we cannot thus be sure which causative mechanism an association might represent Either of these mechanisms will result in an association between marijuana use and MI
146	Causation versus Association Thus, in using statistical associations to try to investigate causation, we must further consider the role other variables might play A statistical association can exist between two variables due to a network of causal pathways in either direction between the two variables
147	Causation versus Association Furthermore, an association between two variables exists if they are each caused by a third variable This is the classic case of a confounder that we would like to adjust for in order to avoid finding spurious associations when looking for cause and effect
148	Causation versus Association But not all such networks of causal pathways will produce an association Two variables are not associated just because they each are the cause of a third variable E.g., no association between marijuana use and MI if the following are the only pathways
149	Causation versus Association Adjustment for the third variable in this case can produce a spurious association in this example Missing days off work is informative about MI incidence among those who do not use marijuana Among people missing work, marijuana users will have lower incidence of MI The incidence of MI will likely be similar between marijuana users and nonusers who do not miss work The resulting interaction will seem to be an association in an adjusted analysis
150	Causation versus Association In the previous example, we might know not to adjust for Days Off Work, because that occurs after the response We regard that causes of events must be in the correct temporal sequence However, there are situations where this criterion can be hard to judge Furthermore, there are situations where similarly inappropriate adjustment of variables can occur with variables measured before the event
151	Causation versus Association Similar problems can arise from more complicated causal pathways Adjustment for Variable C would produce a spurious association Note that the association between C and marijuana and C and MI are not causal, but C can occur before an MI
152	Issues: Obscuring Effect of Interest With time-varying covariates, we have increased opportunity to measure short term effects This is good if that is our interest Immediate effects of blood pressure on hemorrhagic stroke This is bad if we wanted to assess long acting risk factors Chronic effect of asbestos on lung cancer A former asbestos worker is still at high risk
153	Issues: Causal Pathway of Interest Capability for modeling time-varying covariates also increases chances for modeling a variable in the causal pathway of interest
154	Causation versus Association Adjustment for covariates changes the question being answered by the statistical analysis Adjustment can be used to isolate associations that are of particular interest Adjustment should not be used if the variable represents a “causal pathway of interest”
155	Causation versus Association Scientific question: Marijuana bad in any way? Do not adjust for arrest Marijuana causes MI by cardiovascular effect? Do adjust for arrest
156	Issues: Summary Measure As illustrated previously, the interpretation of some of the statistics commonly used in survival analysis is heavily dependent upon the censoring distribution It is very difficult to explore how the changing size of risk sets might be altering the interpretation of the time-averaged hazard ratio in a proportional hazards model
157	Issues: Final Comments Time-varying covariates are definitely of scientific interest However, they should not be used casually Usually, my first choice is to try to address scientific questions with fixed covariates I will put up with some misclassification, to avoid making mistakes that are due to incorrect, untestable assumptions
158	General Analysis Models Choice of Summary Measures Used for Inference
159	Where Am I Going? I claim that The presence of censored data is a technical (rather than scientific) issue raised by the sampling scheme Every summary measure of interest in the absence of censored data can be estimated using censored data in some probability model Nonparametric estimation places some limitations on choice of summary measures
160	Summary Measures Marginal summary measures Means (arithmetic, geometric, harmonic, …) Medians (or other quantiles) Proportion exceeding some threshold Odds of exceeding some threshold Time averaged hazard function (instantaneous risk) …
161	Summarizing Effect Based on marginal distributions Difference / ratio of means (arithmetic, geometric, …) Difference / ratio of proportion exceeding some threshold Difference / ratio of medians (or other quantiles) Ratio of odds of exceeding some threshold Ratio of hazard (averaged across time?) … Based on joint distribution Median difference / ratio of paired observations Probability that a randomly chosen measurement from one population might exceed that from the other …
162	Statistical Models Options for inference Parametric models Weibull, lognormal, etc. Semiparametric models Proportional hazards, etc. Nonparametric Weighted rank tests: logrank, Wilcoxon, etc. Comparison of Kaplan-Meier estimates
163	(Semi)parametric vs Parametric Choice of statistical model can affect Computational methods for estimating the summary measure Precision of summary measure estimates Robustness of inference about the summary measure Ability to estimate the summary measure
164	General Analysis Models Probability Models
165	Right Censored Data Notation:
166	Probability Distributions
167	Parametric Models F is known up to some finite dimensional parameter vectors
168	Parametric Survival Models Commonly used parametric survival models are generally accelerated failure time models Exponential Weibull Gamma Lognormal Log logistic Families joining several of the above
169	Weibull Survival Models Weibull distribution Log hazard function is linear Special case: exponential is constant hazard Memorylessness Only distribution both AFT and PH Can be motivated as earliest failure of components “A chain is as strong as its weakest link”
170	Gamma Survival Models Gamma distribution Special case: exponential is constant hazard Memorylessness Can be motivated as time to failure of last component Parallel components with exponential lifetimes
171	Lognormal Survival Models Lognormal distribution log(T) is normal with mean Φ₁ and variance Φ₂
172	Parametric Inference Parametric inference generally proceeds through likelihood methods MLE found by Newton-Raphson iteration Asymptotic distributions from theory of regular problems
173	Parametric Summary Measures
174	Parametric Models: Issues Advantages Can estimate any of the summary measures Can handle sparse data Disadvantages Not robust to other distributions Parametric estimates do not generally have easy nonparametric interpretation E.g., lognormal model is not particularly robust Little reason to suggest particular distribution But motivation does exist for Weibull and Gamma
175	Semiparametric Models Exact form of within group distributions are unknown, but related to each other by some finite dimensional parameter vector Full inference only for comparing distributions One group’s distn can be found from another group’s and a finite dimensional parameter (Most often: Distributions equal under H₀) (My definition of semiparametric models is a little stronger than some statisticians’, but agrees with commonly used semiparametric survival models)
176	Semiparametric Models: Notation
177	Semiparametric Survival Models
178	Semiparametric Inference Semiparametric inference generally proceeds through estimating equations Estimates found by iterative search Asymptotic distributions from special theory
179	PH Partial Likelihood Proportional hazards regression based on hazard of observed failure relative to sum of hazards in the risk set
180	Semiparametric Summary Measures Estimation of summary measures is generally limited to the parameter fundamental to the semiparametric model Proportional hazards Can only make inference about hazard ratio Accelerated failure time Can only make inference about ratio of quantiles
181	Semiparametric Models: Issues Advantages Can handle sparse data More robust than any single parametric model Disadvantages Not easily interpreted when semiparametric model does not hold Little reason to suggest a given risk factor would affect distribution in only one way
182	A Logical Disconnect
183	Inflammatory Assertion (Semi)parametric models are not typically in keeping with the state of knowledge as an experiment is being conducted The assumptions are more detailed than the hypothesis being tested, e.g., Question: How does the intervention affect the first moment of the probability distribution? Assumption: We know how the intervention affects the 2nd, 3rd, …, ∞ central moments of the probability distribution.
184	The Problem Incorrect parametric assumptions can lead to incorrect statistical inference Precision of estimators can be over- or understated Hypothesis tests do not attain the nominal size Hypothesis tests can be inconsistent Even an infinite sample size may not detect the alternative Interpretation of estimators can be wrong
185	(Semi)parametric Example Survival cure model (Ibrahim, 1999, 2000) Probability model Proportion π_i is cured (survival probability 1 at ∞) in the i-th treatment group Noncured group has survival distribution modeled parametrically (e.g., Weibull) or semiparametrically (e.g., proportional hazards) Treatment effect is measured by θ = π₁ – π₀ The problem as I see it: Incorrect assumptions about the nuisance parameter can bias the estimation of the treatment effect
186	Foundational Issues: Null Which null hypothesis should we test? The intervention has no effect whatsoever The intervention has no effect on some summary measure of the distribution
187	Foundational Issues: Alternative What should the distribution of the data under the alternative represent? Counterfactual An imagined form for F(t), G(t) if something else were true Empirical The most likely distribution of the data if the alternative hypothesis about were true
188	My Views The null hypothesis of greatest interest is rarely that a treatment has no effect Bone marrow transplantation Women’s Health Initiative National Lung Screening Trial The empirical alternative is most in keeping with inference about a summary measure
189	An Aside The above views have important ramifications regarding the computation of standard errors for statistics under the null Permutation tests (or any test which presumes F=G under the null) will generally be inconsistent
190	Problem with (Semi)parametrics Many mechanisms would seem to make it likely that the problems in which a fully parametric model or even a semiparametric model is correct constitute a set of measure zero Treatments are often directed to outliers Treatments are often only effective in subsets Factors affect rates; outcomes measure cumulative effects
191	A Non-Solution: Model Checking Model checking is apparently used by many to allow them to believe that their models are correct. From a recent referee’s report: “I know of no sensible statistician (frequentist or Bayesian) who does not do model checking.” Apparently the referee believes the following unproven proposition: If we cannot tell the model is wrong, then statistical inference under the model will be correct
192	A Non-Solution: Model Checking Counter example: Exponential vs Lognormal medians Pretest with Kolmogorov-Smirnov test (n=40) Power to detect wrong model 20% (exp); 12% (lnorm) Coverage of 95% CI under wrong model 85% (exp); 88% (lnorm)
193	A Non-Solution: Model Checking Model checking particularly makes little sense in a regulatory setting Commonly used null hypotheses presume the model fits in the absence of a treatment effect Frequentists would be testing for a treatment effect as they do model checking Bayesians should model any uncertainty in the distribution Interestingly, if one does this, the estimate indicating parametric family will in general vary with the estimate of treatment effect
194	Nonparametric Models Form of F is completely arbitrary and unknown within groups The summary measure measuring factor effect is just some difference between distributions The summary measure is estimated nonparametrically (preferably within groups and then compared across groups)
195	Comparison of Summary Measures Typical approaches to compare response across two treatment arms Difference / ratio of means (arithmetic, geometric, …) Difference / ratio of medians (or other quantiles) Median difference of paired observations Difference / ratio of proportion exceeding some threshold Ratio of odds of exceeding some threshold Ratio of instantaneous risk of some event (averaged across time?) Probability that a randomly chosen measurement from one population might exceed that from the other …
196	Nonparametric Summary Measures Nonparametric: Estimate summary measures from nonparametric empirical distribution functions E.g., use sample median for inference about population medians In the presence of censoring, use estimates based on Kaplan-Meier estimates Often the nonparametric estimate agrees with a commonly used (semi)parametric estimate Interpretation may depend on sampling scheme In this case, the difference will come in the computation of the standard errors
197	Nonparametric Summary Measures
198	Nonparametric Summary Measures Depending on the censoring scheme, not all summary measures are estimable The support of the censoring distribution may preclude estimation of the mean and some quantiles Can instead use the mean of the truncated distribution “Average increase in days alive during first 5 years”
199	Inference In most cases, variance estimates can be obtained from the asymptotic theory of the Kaplan-Meier estimates There are still some issues to be solved Regression modeling needs to be worked out Software is not readily available (Why not?)