What Is Survival Evaluation? Examples by Hand and in R | by Antoine Soetewey | Dec, 2022

Study extra about survival evaluation (additionally known as time-to-event evaluation) and the way it’s used, and methods to apply it by hand and in R

For the final publish of the 12 months, I wish to current a somewhat unknown (but vital) statistical methodology: survival evaluation.

Though survival evaluation is a department of statistics, it’s normally not coated in introductory statistics programs and it’s somewhat unknown to most of the people. It’s largely taught in biostatistics programs or superior statistics research applications.

On this article, I’ll clarify what survival evaluation is, and the way and through which context it’s used. I’ll clarify the principle instruments and strategies utilized by biostatisticians to research survival information and methods to estimate and interpret survival curves.

I’ll present intimately methods to apply these methods in R with concrete examples. In follow, survival evaluation is sort of at all times carried out by way of a statistical program and by no means carried out by hand. Nevertheless, as for any statistical idea, I imagine that doing it by hand permits to essentially perceive the ideas and what these applications really do. For that reason, I will even present a quick instance on methods to carry out a fundamental survival evaluation by hand.

Word that this text is impressed from:

• The lecture notes of Prof. Van Keilegom and my private notes as educating assistant for her course entitled “Evaluation of Survival and Period Knowledge” given at UCLouvain
• The lecture notes of Prof. Legrand for her course entitled “Statistics in scientific trials” given at UCLouvain

Survival evaluation (additionally known as time-to-event evaluation or length evaluation) is a department of statistics geared toward analyzing the anticipated length of time till a number of occasions occur, known as survival instances or length instances.

In survival evaluation, we’re curious about a sure occasion and wish to analyze the time till the occasion occurs.

Whereas the occasion of curiosity is usually demise (on this case we research the time to demise for sufferers having a particular illness) or recurrence (on this case we research the time to relapse of a sure illness), it’s not restricted to the fields of medication and epidemiology.

In reality, it may be utilized in many domains. For instance, we can also analyze the time till…

• getting cured from a sure illness
• discovering a brand new job after a interval of unemployment
• being arrested once more after being launched from jail
• the primary being pregnant
• the failure of a mechanical system or a machine
• a financial institution or an organization goes bankrupt
• a buyer buys a brand new product or stops its present subscription
• a letter is delivered
• a taxi picks you up after having known as the taxi firm
• an worker leaves the corporate
• and many others.

As you possibly can see, the occasion of curiosity doesn’t essentially need to be the demise or a illness, however in all conditions we have an interest within the time till a particular occasion happens.

Survival information, additionally referred as time-to-event information, requires a particular set of statistical strategies for 3 principal causes:

1. Period instances are at all times optimistic: the time till an occasion of curiosity takes place can’t be lower than 0.

2. Completely different measures are of curiosity relying on the analysis query, context, and many others. For example, we could possibly be curious about:

• The chance {that a} most cancers affected person survives longer than 5 years?
• The standard ready time for a cab to reach after having known as the taxi firm?
• What number of, out of 100 unemployed individuals, are anticipated to have a job once more after 2 months?

3. Censoring is sort of at all times a problem:

• More often than not, the occasion occurred earlier than the top of the research and the survival length is thought.
• Nevertheless, generally, the occasion is just not but noticed on the finish of the research. Suppose that we research the time till demise of sufferers with breast most cancers. Fortunately, some sufferers is not going to die earlier than the top of the research.
• Different instances, one other occasion takes place earlier than the occasion of curiosity. For instance, a most cancers affected person might die from a automotive accident.
• And even generally the affected person withdraws from the research or strikes to a different nation so we can not observe her survival time (this is called misplaced to comply with up or drop out).
• Censoring will be seen, in some sense, as a kind of lacking information.
• For these causes, many “commonplace” descriptive statistics, speculation assessments and regression fashions aren’t applicable for this type of information. Particular statistical strategies are required to take note of the truth that we miss the actual survival length for some sufferers. We all know that they survived a sure period of time (till the top of the research or till the time of withdrawal), however we have no idea their actual survival time.

To your data, when the occasion is just not but noticed on the finish of the research (i.e., the survival length is bigger than the noticed length), that is referred as right-censoring. Then again, left-censoring happens if a participant is entered into the research when the occasion of curiosity occurred prior to review entry however we have no idea precisely when. In fact, we wish to analyze all obtainable information, together with details about censored sufferers.

The objective of survival evaluation is thus to mannequin and describe time-to-event information in an applicable method, taking the particularities of any such information into consideration.

We aren’t going to go to a lot into the small print, however it is very important lay the inspiration with the commonest features in survival evaluation.

Let T be a non-negative steady random variable, representing the time till the occasion of curiosity. We think about the next features:

1. Survival operate
2. Cumulative hazard operate
3. Hazard operate

The most typical one is the survival operate. For every tt:

S(t)=P(T>t)=1−F(t)

S(t) represents, for every time tt, the chance that the time till the occasion is bigger than this time tt. In different phrases, it fashions the chance that the occasion of curiosity occurs after tt.

Within the context of our examples talked about above, it offers the chance that:

• a randomly chosen affected person will survive past time tt,
• the cab takes greater than tt minutes to reach, or
• an unemployed individual take greater than tt months to discover a new job.

The survival operate S(t) is:

• a lowering operate,
• taking values in [0,1] (since it’s a chance), and
• equal to 1 at t=0 and 0 at t=∞.

Visually now we have:

The curve reveals the proportion of people (or experimental models) who, as time goes on, haven’t skilled an occasion. As time progresses, occasions happen, so the proportion who haven’t skilled the occasion decreases.

The cumulative hazard operate is outlined as:

H(t)=−logS(t)

and has the next properties:

• rising operate,
• taking worth in [0,+∞], and
• S(t)=exp(−H(t))

The cumulative hazard is the whole hazard skilled as much as time tt.

The hazard operate, or hazard fee, is outlined as:

and has the next properties:

• optimistic operate (not essentially rising or lowering)
• the hazard operate h(t) can have many alternative shapes and is due to this fact a great tool to summarize survival information

Within the instance of most cancers sufferers, h(t) measures the instantaneous danger of dying proper after time tt given the person is alive at time tt.

To hyperlink the hazard fee with the survival operate; the survival curve represents the hazard charges. A steeper slope signifies a better hazard fee as a result of occasions occur extra regularly, lowering the proportion of people who haven’t skilled the occasion at a quicker fee. Quite the opposite, a gradual and flatter slope signifies a decrease hazard fee as a result of occasions happen much less regularly, lowering the proportion of people who haven’t experiences the occasion at a slower fee.

Word that, in distinction to the survival operate which focuses on not having an occasion, the hazard operate focuses on the occasion occurring.

To estimate the survival operate, we have to use an estimator which is ready to cope with censoring. The most typical one is the nonparametric Kaplan and Meier (1958) estimator (additionally generally referred because the product-limit estimator, or extra merely, the Ok-M estimator).

Some great benefits of the Kaplan-Meier estimator are that:

• it’s easy and easy to make use of and interpret
• it’s a nonparametric estimator, so it constructs a survival curve from the info and no assumptions is made in regards to the form of the underlying distribution
• it offers a graphical illustration of the survival operate(s), helpful for illustrative functions

Word that an vital assumption for the estimation to carry is that censoring is unbiased of the incidence of occasions. We are saying that censoring is non-informative, that’s, censored topics have the identical survival prospects as topics that aren’t censored and proceed to be adopted.

To know the way it works, let’s first estimate it by hand on the next dataset:1

##    topic time occasion
## 1        1    3     0
## 2        2    5     1
## 3        3    7     1
## 4        4    2     1
## 5        5   18     0
## 6        6   16     1
## 7        7    2     1
## 8        8    9     1
## 9        9   16     1
## 10      10    5     0

the place:

• topic is the person’s identifier
• time is the time to occasion (in years)2
• occasion is the occasion standing (0 = censored, 1 = occasion occurred)

Do not forget that for every topic, we have to know a minimum of 2 items of data:

1. the time till the occasion of curiosity or the time till the censoring, and
2. whether or not now we have noticed the occasion of curiosity or if now we have noticed censoring.

We first must depend the variety of distinct occasion instances. Ignoring censored observations, now we have 5 distinct occasion instances:

2, 5, 7, 9 and 16

The best approach to do the calculation by hand is by filling the next desk (a desk with 5 rows since there are 5 distinct occasion instances):

We fill columns one after the other:

• y(j) = the ordered distinct occasion instances:

2, 5, 7, 9 and 16

So the desk turns into:

• d(j) = the variety of observations for every distinct occasion time. For this, the frequency for every distinct occasion time is helpful:

## time
##  2  5  7  9 16 
##  2  1  1  1  2

The desk turns into:

• R(j) = the remaining variety of people in danger. For this, the distribution of time (censored and never censored) is helpful:

## time
##  2  3  5  7  9 16 18 
##  2  1  2  1  1  2  1

We see that:

• At first there are 10 topics
• Simply earlier than time t=5, there are 7 topics left (10 topics — 2 who had the occasion — 1 who’s censored)
• Simply earlier than time t=7, there are 5 topics left (= 10–2–1–2)
• Simply earlier than time t=9, there are 4 topics left (= 10–2–1–2–1)
• Simply earlier than time t=16, there are 3 topics left (= 10–2–1–2–1–1)

The desk turns into:

• 1−(d(j)/R(j)) is simple, so the desk turns into:

The Kaplan-Meier estimator is:

For every j, we thus take the cumulative product:

• j1=0.8
• j2=0.8⋅0.857=0.6856
• j3=0.6856⋅0.8=0.54848
• j4=0.54848⋅0.75=0.41136
• j5=0.41136⋅0.333=0.1369829

So lastly, now we have the survival chances (rounded to three digits):

We will now characterize graphically the Kaplan-Meier estimator:

To attract this survival curve, keep in mind that:

• the x-axis corresponds to the time variable within the preliminary dataset, and
• the y-axis corresponds to the survival chances discovered above.

We now evaluate our outcomes with the outcomes present in R.

We first create the dataset with the time and occasion variables:

# create dataset
dat <- information.body(
time = c(3, 5, 7, 2, 18, 16, 2, 9, 16, 5),
occasion = c(0, 1, 1, 1, 0, 1, 1, 1, 1, 0)
)

We then run the Kaplan-Meier estimator with the survfit() and Surv() features:

# KM
library(survival)

km <- survfit(Surv(time, occasion) ~ 1,
information = dat
)

Discover that the Surv() operate accepts two arguments:

1. the time variable, and
2. the occasion variable.

The ~ 1 within the survfit() operate signifies that we estimate the Kaplan-Meier with none grouping. See extra on this later within the publish.

Lastly, we show the outcomes and draw the Kaplan-Meier plot:

# outcomes
abstract(km)

## Name: survfit(method = Surv(time, occasion) ~ 1, information = dat)
## 
##  time n.danger n.occasion survival std.err decrease 95% CI higher 95% CI
##     2     10       2    0.800   0.126       0.5868        1.000
##     5      7       1    0.686   0.151       0.4447        1.000
##     7      5       1    0.549   0.172       0.2963        1.000
##     9      4       1    0.411   0.176       0.1782        0.950
##    16      3       2    0.137   0.126       0.0225        0.834

# plot
plot(km,
xlab = "Time",
ylab = "Survival chance",
conf.int = FALSE
)

The survival chances will be discovered within the survival column. Comment that outcomes by hand and in R are related (any distinction with the outcomes by hand is because of rounding).

Alternatively, we are able to use the ggsurvplot() operate inside the {survminer} bundle:

library(survminer)

# plot
ggsurvplot(km,
conf.int = FALSE,
legend = "none"
)

Word that the crosses on the survival curve denote the censored observations.

The benefit with the ggsurvplot() operate is that it’s straightforward to attract the median straight on the plot:3

ggsurvplot(km,
conf.int = FALSE,
surv.median.line = "hv",
legend = "none"
)

To search out the median length:4

abstract(km)$desk["median"]

## median 
##      9

# or extra merely
km

## Name: survfit(method = Surv(time, occasion) ~ 1, information = dat)
## 
##       n occasions median 0.95LCL 0.95UCL
## [1,] 10      7      9       5      NA

Suppose that the occasion of curiosity is demise:

• At time zero, the survival chance is 1 (100% of the topics are alive).
• The median signifies that the median survival time is 9 years.5 That is the time at which the survival S(t) is 50%. In different phrases, is the time after which half of the topics are anticipated to have died.
• From the plot, we additionally see that S(5)=P(T>5 years)= Chance of survival of greater than 5 years for these topics = 75%. Because of this 75% of all topics survive longer than 5 years, and that 25% of all topics die inside the first 5 years.

For the sake of completeness, let’s do one other instance with a a lot bigger dataset; the tongue dataset inside the {KMsurv} bundle.6

# load information
library(KMsurv)
information(tongue)

# preview information
head(tongue)

##   sort time delta
## 1    1    1     1
## 2    1    3     1
## 3    1    3     1
## 4    1    4     1
## 5    1   10     1
## 6    1   13     1

• sort is the tumor DNA profile (1 = aneuploid tumor, 2 = diploid tumor)
• time is the time to demise or on-study time (in weeks)
• delta is the demise indicator (0 = alive, 1 = lifeless)

For this instance, we deal with the aneuploid sort:

anaploid <- subset(tongue, sort == 1)

We will now plot the estimated survival operate and estimate the median time to demise. Since it’s an estimator, we are able to additionally assemble a confidence interval for the estimated survival at every time tt and for the estimated median survival time.7

# outcomes
match <- survfit(Surv(time, delta) ~ 1,
information = anaploid,
conf.sort = "log-log"
)

match

## Name: survfit(method = Surv(time, delta) ~ 1, information = anaploid, conf.sort = "log-log")
## 
##       n occasions median 0.95LCL 0.95UCL
## [1,] 52     31     93      65     157

# plot
ggsurvplot(match,
surv.median.line = "hv",
legend = "none"
)

The median survival time is estimated to be 93 weeks, with a 95% confidence interval [65; 157].

Kaplan-Meier curves will be seen as descriptive statistics for survival information. We now deal with the second department of statistics, speculation testing which permits to attract conclusions on the inhabitants primarily based on a pattern (see a fast reminder in regards to the distinction between inhabitants and pattern in case you want).

Speculation testing within the discipline of survival evaluation largely considerations:

• The hazard operate of one inhabitants: on this case we take a look at whether or not a censored pattern comes from a inhabitants with a recognized hazard operate h0(t). For instance, we could also be to match survival in a pattern of sufferers to the survival within the general inhabitants (derived from the life tables).
• The comparability of the hazard operate of two or extra populations: on this case we’re curious about assessing whether or not there are variations in survival amongst totally different teams of topics. For instance:
• 2 teams: we’re curious about evaluating survival for feminine and male colon most cancers sufferers
• 3 teams or extra: we’re curious about evaluating survival for melanoma most cancers sufferers in accordance with their therapies (with therapies A, B and C for instance)8

On this article, we deal with evaluating survival between two teams utilizing the log-rank take a look at (often known as Mantel-Cox take a look at).

The instinct behind the take a look at is that if the 2 teams have totally different hazard charges, the 2 survival curves (so their slopes) will differ. Extra exactly, the log-rank take a look at compares the noticed variety of occasions in every group to what could be anticipated if the survival curves had been equivalent (i.e., if the null speculation had been true).

Word that, as for the Kaplan-Meier estimator, the log-rank take a look at is a nonparametric take a look at, which makes no assumptions in regards to the survival distributions.

For this instance, think about the next dataset:

##    affected person group time occasion
## 1        1     1  4.1     1
## 2        2     1  7.8     0
## 3        3     1 10.0     1
## 4        4     1 10.0     1
## 5        5     1 12.3     0
## 6        6     1 17.2     1
## 7        7     2  9.7     1
## 8        8     2 10.0     1
## 9        9     2 11.1     0
## 10      10     2 13.1     0
## 11      11     2 19.7     1
## 12      12     2 24.1     0

the place:

• affected person is the affected person’s identifier
• group is the group (group 1 or 2)
• time is the time to demise (in years)9
• occasion is the occasion standing (0 = censored, 1 = demise)

Suppose we’re curious about evaluating group 1 and a couple of when it comes to survival, that’s, we evaluate survival curves between the two teams:

• H0: S1(t)=S2(t) for all t
• H1: S1(t)≠S2(t) for some t

It’s a statistical take a look at, so if the p-value < α (normally 0.05), we reject the null speculation and we conclude that survival (or the time to occasion) is considerably totally different between the 2 teams thought of.

To carry out the log-rank take a look at, the next take a look at statistic might be helpful:

As for the Kaplan-Meier estimator by hand, it’s best to additionally fill in a desk for the log-rank take a look at by hand.

Let’s current the ultimate desk and remark beneath on methods to fill it, column by column:

Column j is the variety of distinct occasion instances. We see that there are 5 (ignoring censored observations), so we write 1 to five within the desk.

Column y(j) is the ordered distinct occasion instances:

4.1, 9.7, 10, 17.2 and 19.7

Column d(j)1 is the variety of observations for every distinct occasion time, for group 1:

## time
##  4.1   10 17.2 
##    1    2    1

When there is no such thing as a occasion, we merely write 0 within the desk.

Column R(j)1 is the remaining variety of sufferers in danger, for group 1. For this, the distribution of time (censored and never censored, for group 1) is helpful:

## time
##  4.1  7.8   10 12.3 17.2 
##    1    1    2    1    1

We see that:

• At first, there are 6 sufferers
• Earlier than time 9.7, there are 4 sufferers left (6–1 who had the occasion at time 4.1–1 who was censored at time 7.8)
• Earlier than time 10, there are 4 sufferers left (6–2)
• Earlier than time 17.2, there are 1 affected person left (6–5)
• Earlier than time 19.7, there are 0 affected person left (6–6)

Columns d(j)2 and R(j)2 comply with the identical precept, however for group 2 this time. So now we have, respectively for d(j)2 and R(j)2:

## time
##  9.7   10 19.7 
##    1    1    1

## time
##  9.7   10 11.1 13.1 19.7 24.1 
##    1    1    1    1    1    1

Columns d(j) and R(j) additionally comply with the identical precept, however this time contemplating each teams. So now we have, respectively for d(j)d(j) and R(j)R(j):

## time
##  4.1  9.7   10 17.2 19.7 
##    1    1    3    1    1

## time
##  4.1  7.8  9.7   10 11.1 12.3 13.1 17.2 19.7 24.1 
##    1    1    1    3    1    1    1    1    1    1

Column Ej is the anticipated variety of occasions within the first group assuming that h1≡h2. It’s obtained as follows

Column Oj is the noticed variety of occasions within the first group, so it is the same as the d(j)1 column.

Column Oj−Ej is simple.

Column N(j) is outlined as follows

Column D(j) is R(j)−1.

Column N(j)/D(j) is simple.

We’ve got |Uobs|=1.275<z0.975=1.96. Therefore, on the 5% significance degree we don’t reject H0. Because of this, primarily based on the info, we aren’t capable of conclude that survival is totally different between the 2 teams (which is equal than saying that we don’t reject the speculation that survival is equal between the 2 teams).

If you’re curious about computing the p-value:

p-value =2×P(Z>1.275)=2×0.101=0.202>0.05.

We now evaluate our leads to R with the survdiff() operate:

dat <- information.body(
group = c(rep(1, 6), rep(2, 6)),
time = c(4.1, 7.8, 10, 10, 12.3, 17.2, 9.7, 10, 11.1, 13.1, 19.7, 24.1),
occasion = c(1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0)
)

dat

##    group time occasion
## 1      1  4.1     1
## 2      1  7.8     0
## 3      1 10.0     1
## 4      1 10.0     1
## 5      1 12.3     0
## 6      1 17.2     1
## 7      2  9.7     1
## 8      2 10.0     1
## 9      2 11.1     0
## 10     2 13.1     0
## 11     2 19.7     1
## 12     2 24.1     0

survdiff(Surv(time, occasion) ~ group,
information = dat
)

## Name:
## survdiff(method = Surv(time, occasion) ~ group, information = dat)
## 
##         N Noticed Anticipated (O-E)^2/E (O-E)^2/V
## group=1 6        4     2.57     0.800      1.62
## group=2 6        3     4.43     0.463      1.62
## 
##  Chisq= 1.6  on 1 levels of freedom, p= 0.2

Alternatively, we are able to use the ggsurvplot() operate to attract the survival curves and carry out the log-rank take a look at on the identical time:

match <- survfit(Surv(time, occasion) ~ group, information = dat)

ggsurvplot(match,
pval = TRUE,
pval.methodology = TRUE
)

As we are able to see, the p-values and the conclusions are the identical (any distinction with the outcomes by hand is because of rounding).

As for the Kaplan-Meier estimation, we do one other instance on a bigger dataset. Contemplate the info on the instances till staphylococcus an infection of burn sufferers, additionally obtainable within the {KMsurv}:12

# load information
information(burn)

# preview information
head(burn)

##   Obs Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 T1 D1 T2 D2 T3 D3
## 1   1  0  0  0 15  0  0  1  1  0   0   2 12  0 12  0 12  0
## 2   2  0  0  1 20  0  0  1  0  0   0   4  9  0  9  0  9  0
## 3   3  0  0  1 15  0  0  0  1  1   0   2 13  0 13  0  7  1
## 4   4  0  0  0 20  1  0  1  0  0   0   2 11  1 29  0 29  0
## 5   5  0  0  1 70  1  1  1  1  0   0   2 28  1 31  0  4  1
## 6   6  0  0  1 20  1  0  1  0  0   0   4 11  0 11  0  8  1

Utilizing the log-rank take a look at, we wish to take a look at the speculation of distinction within the time to staphylococcus an infection (T3 variable) between sufferers whose burns had been cared for with a routine bathing care methodology (Z1 = 0) versus these whose physique cleaning was initially carried out utilizing 4% chlorhexidine gluconate (Z1 = 1). The occasion indicator is in variable D3.

For this take a look at, we use a two-sided different and a 5% significance degree.

# match
match <- survfit(Surv(T3, D3) ~ Z1, information = burn)

# plot with log-rank take a look at
ggsurvplot(match,
pval = TRUE,
pval.methodology = TRUE
)

Within the pattern, plainly the time to an infection for sufferers with routine bathing (Z1 = 0) is smaller than for sufferers with physique cleaning (Z1 = 1). That is the case as a result of the share of sufferers who haven’t skilled the an infection decreases extra shortly, so the hazard fee is bigger.

Nevertheless, this conclusion can’t be generalized to the inhabitants with out performing a sound statistical take a look at. And primarily based on the results of the log-rank take a look at, we don’t reject the speculation that point to an infection is identical between the 2 teams of sufferers (p-value = 0.051).

On this article, now we have offered what’s survival evaluation, when, why and methods to use it. We mentioned about censoring and survival curves. We confirmed methods to estimate the survival operate by way of the Kaplan-Meier estimator and methods to take a look at survival between two teams by way of the log-rank take a look at. We illustrated these approaches each by hand and in R.

As you seen, we didn’t present methods to mannequin survival information. There are a number of regression fashions that may be utilized to survival information, the commonest one being the semiparametric Cox Proportional Hazards mannequin (1972). It originated from the medical space to analyze and assess the connection between the survival instances of sufferers and their corresponding predictor variables.

We’ve got seen that the Kaplan-Meier estimator is helpful to visualise survival between teams and the log-rank take a look at to check whether or not survival considerably differs between teams (so each approaches use a categorical variable as predictor). Nevertheless, it doesn’t work effectively for assessing the impact of quantitative predictor. The Cox mannequin has the benefit that it really works for each quantitative in addition to for categorical predictors, and for a number of danger components on the identical time (so it could possibly mannequin the impact of a number of variables without delay).

With the Cox mannequin, we mannequin the affect of various components X1,X2,…,Xq on survival by way of their affect on the hazard operate:

the place:

• h(t|X) is the instantaneous demise fee conditional on having survived as much as time tt.
• h0(t) is the population-level baseline hazard — the underlying hazard operate. It describes how the typical individual’s danger evolves over time.
• exp(β1X1+β2X2+⋯+βqXq) describes how covariates have an effect on the hazard. Particularly, a unit enhance in xixi results in a rise of the hazard by an element of exp(βi).

This publish geared toward presenting the introductory ideas in survival evaluation, so this mannequin might be developed in one other publish. Within the meantime, if you want to study extra about modeling survival information (due to the Cox mannequin and different fashions), see this publish from Joseph Rickert.

Thanks for studying.

As at all times, when you’ve got a query or a suggestion associated to the subject coated on this article, please add it as a remark so different readers can profit from the dialogue.

Cox, David R. 1972. “Regression Fashions and Life-Tables.” Journal of the Royal Statistical Society: Sequence B (Methodological) 34 (2): 187–202.

Kaplan, Edward L, and Paul Meier. 1958. “Nonparametric Estimation from Incomplete Observations.” Journal of the American Statistical Affiliation 53 (282): 457–81.