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Survival analysis is a branch of statistics which deals with analysis of time duration to until one or more events happen, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, and duration analysis or duration modeling in economics or event history analysis in sociology. Survival analysis attempts to answer questions such as: what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival?

To answer such questions, it is necessary to define "lifetime". In the case of biological survival, death is unambiguous, but for mechanical reliability, failure may not be well-defined, for there may well be mechanical systems in which failure is partial, a matter of degree, or not otherwise localized in time. Even in biological problems, some events (for example, heart attack or other organ failure) may have the same ambiguity. The theory outlined below assumes well-defined events at specific times; other cases may be better treated by models which explicitly account for ambiguous events.

More generally, survival analysis involves the modeling of time to event data; in this context, death or failure is considered an "event" in the survival analysis literature – traditionally only a single event occurs for each subject, after which the organism or mechanism is dead or broken. Recurring event or repeated event models relax that assumption. The study of recurring events is relevant in systems reliability, and in many areas of social sciences and medical research.

General formulation

Survival function

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The object of primary interest is the survival function, conventionally denoted S, which is defined as

S (t) = \Pr (T > t)

where t is some time, T is a random variable denoting the time of death, and "Pr" stands for probability. That is, the survival function is the probability that the time of death is later than some specified time t. The survival function is also called the survivor function or survivorship function in problems of biological survival, and the reliability function in mechanical survival problems. In the latter case, the reliability function is denoted R(t).

Usually one assumes S(0) = 1, although it could be less than 1 if there is the possibility of immediate death or failure.

The survival function must be non-increasing: S(u) ≤ S(t) if u ≥ t. This property follows directly because T>u implies T>t. This reflects the notion that survival to a later age is only possible if all younger ages are attained. Given this property, the lifetime distribution function and event density (F and f below) are well-defined.

The survival function is usually assumed to approach zero as age increases without bound, i.e., S(t) → 0 as t → ∞, although the limit could be greater than zero if eternal life is possible. For instance, we could apply survival analysis to a mixture of stable and unstable carbon isotopes; unstable isotopes would decay sooner or later, but the stable isotopes would last indefinitely.

Lifetime distribution function and event density

Related quantities are defined in terms of the survival function.

The lifetime distribution function, conventionally denoted F, is defined as the complement of the survival function,

F (t) = \Pr (T \leq t) = 1 - S (t) .

If F is differentiable then the derivative, which is the density function of the lifetime distribution, is conventionally denoted f,

f (t) = F^{'} (t) = \frac{d}{d t} F (t) .

The function f is sometimes called the event density; it is the rate of death or failure events per unit time.

The survival function can be expressed in terms of probability distribution and probability density functions

S (t) = \Pr (T > t) = \int_{t}^{\infty} f (u) d u = 1 - F (t) .

Similarly, a survival event density function can be defined as

s (t) = S^{'} (t) = \frac{d}{d t} S (t) = \frac{d}{d t} \int_{t}^{\infty} f (u) d u = \frac{d}{d t} [1 - F (t)] = - f (t) .

Hazard function and cumulative hazard function

The hazard function, conventionally denoted $λ$ , is defined as the event rate at time t conditional on survival until time t or later (that is, T ≥ t),

λ (t) = \lim_{d t \to 0} \frac{\Pr (t \leq T < t + d t | T \geq t)}{d t} = \frac{f (t)}{S (t)} = - \frac{S^{'} (t)}{S (t)} .

Force of mortality is a synonym of hazard function which is used particularly in demography and actuarial science, where it is denoted by $μ$ . The term hazard rate is another synonym.

The hazard function must be non-negative, λ(t) ≥ 0, and its integral over $[0, \infty]$ must be infinite, but is not otherwise constrained; it may be increasing or decreasing, non-monotonic, or discontinuous. An example is the bathtub curve hazard function, which is large for small values of t, decreasing to some minimum, and thereafter increasing again; this can model the property of some mechanical systems to either fail soon after operation, or much later, as the system ages.

The hazard function can alternatively be represented in terms of the cumulative hazard function, conventionally denoted $Λ$ :

Λ (t) = - \log S (t)

so transposing signs and exponentiating

S (t) = \exp (- Λ (t))

or differentiating (with the chain rule)

\frac{d}{d t} Λ (t) = - \frac{S^{'} (t)}{S (t)} = λ (t) .

The name "cumulative hazard function" is derived from the fact that

Λ (t) = \int_{0}^{t} λ (u) d u

which is the "accumulation" of the hazard over time.

From the definition of $Λ (t)$ , we see that it increases without bound as t tends to infinity (assuming that S(t) tends to zero). This implies that $λ (t)$ must not decrease too quickly, since, by definition, the cumulative hazard has to diverge. For example, $\exp (- t)$ is not the hazard function of any survival distribution, because its integral converges to 1.

Quantities derived from the survival distribution

Future lifetime at a given time $t_{0}$ is the time remaining until death, given survival to age $t_{0}$ . Thus, it is $T - t_{0}$ in the present notation. The expected future lifetime is the expected value of future lifetime. The probability of death at or before age $t + t_{0}$ , given survival until age $t_{0}$ , is just

P (T \leq t_{0} + t | T > t_{0}) = \frac{P (t_{0} < T \leq t_{0} + t)}{P (T > t_{0})} = \frac{F (t_{0} + t) - F (t_{0})}{S (t_{0})} .

Therefore the probability density of future lifetime is

\frac{d}{d t} \frac{F (t_{0} + t) - F (t_{0})}{S (t_{0})} = \frac{f (t_{0} + t)}{S (t_{0})}

and the expected future lifetime is

\frac{1}{S (t_{0})} \int_{0}^{\infty} t f (t + t_{0}) d t = \frac{1}{S (t_{0})} \int_{t_{0}}^{\infty} S (t) d t,

where the second expression is obtained using integration by parts.

For $t_{0} = 0$ , that is, at birth, this reduces to the expected lifetime.

In reliability problems, the expected lifetime is called the mean time to failure, and the expected future lifetime is called the mean residual lifetime.

As the probability of an individual surviving until age t or later is S(t), by definition, the expected number of survivors at age t out of an initial population of n newborns is n × S(t), assuming the same survival function for all individuals. Thus the expected proportion of survivors is S(t). If the survival of different individuals is independent, the number of survivors at age t has a binomial distribution with parameters n and S(t), and the variance of the proportion of survivors is S(t) × (1-S(t))/n.

The age at which a specified proportion of survivors remain can be found by solving the equation S(t) = q for t, where q is the quantile in question. Typically one is interested in the median lifetime, for which q = 1/2, or other quantiles such as q = 0.90 or q = 0.99.

One can also make more complex inferences from the survival distribution. In mechanical reliability problems, one can bring cost (or, more generally, utility) into consideration, and thus solve problems concerning repair or replacement. This leads to the study of renewal theory and reliability theory of aging and longevity.

Censoring

Censoring is a form of missing data problem which is common in survival analysis. Ideally, both the birth and death dates of a subject are known, in which case the lifetime is known.

If it is known only that the date of death is after some date, this is called right censoring. Right censoring will occur for those subjects whose birth date is known but who are still alive when they are lost to follow-up or when the study ends.

If a subject's lifetime is known to be less than a certain duration, the lifetime is said to be left-censored.

It may also happen that subjects with a lifetime less than some threshold may not be observed at all: this is called truncation. Note that truncation is different from left censoring, since for a left censored datum, we know the subject exists, but for a truncated datum, we may be completely unaware of the subject. Truncation is also common. In a so-called delayed entry study, subjects are not observed at all until they have reached a certain age. For example, people may not be observed until they have reached the age to enter school. Any deceased subjects in the pre-school age group would be unknown. Left-truncated data are common in actuarial work for life insurance and pensions.^[1]

We generally encounter right-censored data. Left-censored data can occur when a person's survival time becomes incomplete on the left side of the follow-up period for the person. As an example, we may follow up a patient for any infectious disorder from the time of his or her being tested positive for the infection. We may never know the exact time of exposure to the infectious agent.^[2]

Fitting parameters to data

Survival models can be usefully viewed as ordinary regression models in which the response variable is time. However, computing the likelihood function (needed for fitting parameters or making other kinds of inferences) is complicated by the censoring. The likelihood function for a survival model, in the presence of censored data, is formulated as follows. By definition the likelihood function is the conditional probability of the data given the parameters of the model. It is customary to assume that the data are independent given the parameters. Then the likelihood function is the product of the likelihood of each datum. It is convenient to partition the data into four categories: uncensored, left censored, right censored, and interval censored. These are denoted "unc.", "l.c.", "r.c.", and "i.c." in the equation below.

L (θ) = \prod_{T_{i} \in u n c .} \Pr (T = T_{i} | θ) \prod_{i \in l . c .} \Pr (T < T_{i} | θ) \prod_{i \in r . c .} \Pr (T > T_{i} | θ) \prod_{i \in i . c .} \Pr (T_{i, l} < T < T_{i, r} | θ) .

For an uncensored datum, with $T_{i}$ equal to the age at death, we have

\Pr (T = T_{i} | θ) = f (T_{i} | θ) .

For a left censored datum, such that the age at death is known to be less than $T_{i}$ , we have

\Pr (T < T_{i} | θ) = F (T_{i} | θ) = 1 - S (T_{i} | θ) .

For a right censored datum, such that the age at death is known to be greater than $T_{i}$ , we have

\Pr (T > T_{i} | θ) = 1 - F (T_{i} | θ) = S (T_{i} | θ) .

For an interval censored datum, such that the age at death is known to be less than $T_{i, r}$ and greater than $T_{i, l}$ , we have

\Pr (T_{i, l} < T < T_{i, r} | θ) = S (T_{i, l} | θ) - S (T_{i, r} | θ) .

An important application where interval censored data arises is current status data, where the actual occurrence of an event $T_{i}$ is only known to the extent that it known not to occurred before observation time and to have occurred before the next.

Non-parametric estimation

The Nelson–Aalen estimator can be used to provide a non-parametric estimate of the cumulative hazard rate function.

Distributions used in survival analysis

References

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External links

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[2] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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+{{More footnotes|date=November 2011}}
+'''Survival analysis''' is a branch of [[statistics]] which deals with analysis of time duration to until one or more events happen, such as death in biological organisms and failure in mechanical systems. This topic is called ''[[reliability theory]]'' or ''reliability analysis'' in [[engineering]], and ''duration analysis'' or ''duration modeling'' in [[economics]] or ''event history analysis'' in [[sociology]]. Survival analysis attempts to answer questions such as: what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival?
+To answer such questions, it is necessary to define "lifetime". In the case of biological survival, [[death]] is unambiguous, but for mechanical reliability, [[failure]] may not be well-defined, for there may well be mechanical systems in which failure is partial, a matter of degree, or not otherwise localized in [[time]]. Even in biological problems, some events (for example, [[myocardial infarction|heart attack]] or other organ failure) may have the same ambiguity. The [[theory]] outlined below assumes well-defined events at specific times; other cases may be better treated by models which explicitly account for ambiguous events.
+More generally, survival analysis involves the modeling of time to event data; in this context, death or failure is considered an "event" in the survival analysis literature – traditionally only a single event occurs for each subject, after which the organism or mechanism is dead or broken.  ''Recurring event'' or ''repeated event'' models relax that assumption. The study of recurring events is relevant in [[systems reliability]], and in many areas of social sciences and medical research.
+==General formulation==
+===Survival function===
+{{main|survival function}}
+The object of primary interest is the '''survival function''', conventionally denoted ''S'', which is defined as
+:<math>S(t) = \Pr(T > t)</math>
+where ''t'' is some time, ''T'' is a [[random variable]] denoting the time of death, and "Pr" stands for [[probability]]. That is, the survival function is the probability that the time of death is later than some specified time ''t''.
+The survival function is also called the ''survivor function'' or ''survivorship function'' in problems of biological survival, and the ''reliability function'' in mechanical survival problems. In the latter case, the reliability function is denoted ''R''(''t'').
+Usually one assumes ''S''(0) = 1, although it could be less than 1 if there is the possibility of immediate death or failure.
+The survival function must be non-increasing: ''S''(''u'') ≤ ''S''(''t'') if ''u'' ≥ ''t''. This property follows directly because ''T''>''u'' implies ''T''>''t''. This reflects the notion that survival to a later age is only possible if all younger ages are attained. Given this property, the lifetime distribution function and event density (''F'' and ''f'' below) are well-defined.
+The survival function is usually assumed to approach zero as age increases without bound, i.e., ''S''(''t'') → 0 as ''t'' → ∞, although the limit could be greater than zero if eternal life is possible. For instance, we could apply survival analysis to a mixture of stable and unstable [[Carbon#Isotopes|carbon isotopes]]; unstable isotopes would decay sooner or later, but the stable isotopes would last indefinitely.
+===Lifetime distribution function and event density===
+Related quantities are defined in terms of the survival function.
+The '''lifetime distribution function''', conventionally denoted ''F'', is defined as the complement of the survival function,
+:<math>F(t) = \Pr(T \le t) = 1 - S(t).</math>
+If ''F'' is [[differentiable]] then the derivative, which is the density function of the lifetime distribution, is conventionally denoted ''f'',
+:<math>f(t) = F'(t) = \frac{d}{dt} F(t).</math>
+The function ''f'' is sometimes called the '''event density'''; it is the rate of death or failure events per unit time.
+The survival function can be expressed in terms of [[probability distribution]] and [[probability density function]]s
+:<math>S(t) = \Pr(T > t) = \int_t^{\infty} f(u)\,du = 1-F(t).</math>
+Similarly, a survival event density function can be defined as
+:<math>s(t) = S'(t) = \frac{d}{dt} S(t) = \frac{d}{dt} \int_t^{\infty} f(u)\,du = \frac{d}{dt} [1-F(t)] = -f(t).</math>
+===Hazard function and cumulative hazard function===
+The '''[[hazard function]]''', conventionally denoted <math>\lambda</math>, is defined as the event rate at time ''t'' conditional on survival until time ''t'' or later (that is, ''T'' ≥ ''t''),
+:<math>\lambda(t) = \lim_{dt \rightarrow 0} \frac{\Pr(t \leq T < t+dt\,|\,T \geq t)}{dt} = \frac{f(t)}{S(t)} = -\frac{S'(t)}{S(t)}.</math>
+[[Force of mortality]] is a synonym of ''hazard function'' which is used particularly in [[demography]] and [[actuarial science]], where it is denoted by <math>\mu</math>. The term ''hazard rate'' is another synonym.
+The hazard function must be non-negative, λ(''t'') ≥ 0, and its integral over <math>[0, \infty]</math> must be infinite, but is not otherwise constrained; it may be increasing or decreasing, non-monotonic, or discontinuous.
+An example is the [[bathtub curve]] hazard function, which is large for small values of ''t'', decreasing to some minimum, and thereafter increasing again; this can model the property of some mechanical systems to either fail soon after operation, or much later, as the system ages.
+The hazard function can alternatively be represented in terms of the '''cumulative hazard function''', conventionally denoted <math>\Lambda</math>:
+:<math>\,\Lambda(t) = -\log S(t)</math>
+so transposing signs and exponentiating
+:<math>\,S(t) = \exp(-\Lambda(t))</math>
+or differentiating (with the chain rule)
+:<math>\frac{d}{dt} \Lambda(t) = -\frac{S'(t)}{S(t)} = \lambda(t).</math>
+The name "cumulative hazard function" is derived from the fact that
+:<math> \Lambda(t) = \int_0^{t} \lambda(u)\,du</math>
+which is the "accumulation" of the hazard over time.
+From the definition of <math>\Lambda(t)</math>, we see that it increases without bound as ''t'' tends to infinity (assuming that ''S''(''t'') tends to zero). This implies that <math>\lambda(t)</math> must not decrease too quickly, since, by definition, the cumulative hazard has to diverge. For example, <math>\exp(-t)</math> is not the hazard function of any survival distribution, because its integral converges to 1.
+===Quantities derived from the survival distribution===
+'''Future lifetime''' at a given time <math>t_0</math> is the time remaining until death, given survival to age <math>t_0</math>. Thus, it is <math>T - t_0</math> in the present notation. The '''expected future lifetime''' is the [[expected value]] of future lifetime. The probability of death at or before age <math>t + t_0</math>, given survival until age <math>t_0</math>, is just
+:<math>P(T \le t_0 + t | T > t_0) = \frac{P(t_0 < T \le t_0 + t)}{P(T > t_0)} = \frac{F(t_0 + t) - F(t_0)}{S(t_0)}.</math>
+Therefore the probability density of future lifetime is
+:<math>\frac{d}{dt}\frac{F(t_0 + t) - F(t_0)}{S(t_0)} = \frac{f(t_0 + t)}{S(t_0)}</math>
+and the expected future lifetime is
+:<math>\frac{1}{S(t_0)} \int_0^{\infty} t\,f(t+t_0)\,dt = \frac{1}{S(t_0)} \int_{t_0}^{\infty} S(t)\,dt,</math>
+where the second expression is obtained using [[integration by parts]].
+For <math>t_0 = 0</math>, that is, at birth, this reduces to the expected lifetime.
+In reliability problems, the expected lifetime is called the ''[[mean time to failure]]'', and the expected future lifetime is called the ''mean residual lifetime''.
+As the probability of an individual surviving until age ''t'' or later is ''S''(''t''), by definition, the expected number of survivors at age ''t'' out of an initial [[population]] of ''n'' newborns is ''n'' × ''S''(''t''), assuming the same survival function for all individuals. Thus the expected proportion of survivors is ''S''(''t'').
+If the survival of different individuals is independent, the number of survivors at age ''t'' has a [[binomial distribution]] with parameters ''n'' and ''S''(''t''), and the [[variance]] of the proportion of survivors is ''S''(''t'') ×  (1-''S''(''t''))/''n''.
+The age at which a specified proportion of survivors remain can be found by solving the equation ''S''(''t'') = ''q'' for ''t'', where ''q'' is the [[quantile]] in question. Typically one is interested in the '''[[median]] lifetime''', for which ''q'' = 1/2, or other quantiles such as ''q'' = 0.90 or ''q'' = 0.99.
+One can also make more complex inferences from the survival distribution. In mechanical reliability problems, one can bring cost (or, more generally, [[utility]]) into consideration, and thus solve problems concerning repair or replacement. This leads to the study of [[renewal theory]] and [[reliability theory of aging and longevity]].
+==Censoring==
+[[Censoring (statistics)|Censoring]] is a form of missing data problem which is common in survival analysis. Ideally, both the birth and death dates of a subject are known, in which case the lifetime is known.
+If it is known only that the date of death is after some date, this is called ''right censoring''. Right censoring  will occur for those subjects whose birth date is known but who are still alive when they are lost to follow-up or when the study ends.
+If a subject's lifetime is known to be less than a certain duration, the lifetime is said to be ''left-censored''.
+It may also happen that subjects with a lifetime less than some threshold may not be observed at all: this is called ''truncation''. Note that truncation is different from left censoring, since for a left censored datum, we know the subject exists, but for a truncated datum, we may be completely unaware of the subject. Truncation is also common. In a so-called ''delayed entry'' study, subjects are not observed at all until they have reached a certain age. For example, people may not be observed until they have reached the age to enter school. Any deceased subjects in the pre-school age group would be unknown.  Left-truncated data are common in actuarial work for life insurance and pensions.<ref>{{cite journal |last=Richards |first=S. J. |title=A handbook of parametric survival models for actuarial use |journal=Scandinavian Actuarial Journal |volume=2012 |year=2012 |issue=4 |pages=233–257 |doi=10.1080/03461238.2010.506688 }}</ref>
+We generally encounter right-censored data. Left-censored data can occur when a person's survival time becomes incomplete on the left side of the follow-up period for the person. As an example, we may follow up a patient for any infectious disorder from the time of his or her being tested positive for the infection. We may never know the exact time of exposure to the infectious agent.<ref>{{cite journal |last=Singh |first=R. |last2=Mukhopadhyay |first2=K. |title=Survival analysis in clinical trials: Basics and must know areas |journal=[[Perspectives in Clinical Research|Perspect Clin Res]] |year=2011 |volume=2 |issue=4 |pages=145–148 |doi=10.4103/2229-3485.86872 }}</ref>
+==Fitting parameters to data==
+Survival models can be usefully viewed as ordinary regression models in which the response variable is time.  However, computing the likelihood function (needed for fitting parameters or making other kinds of inferences) is complicated by the censoring. The [[likelihood function]] for a survival model, in the presence of censored data, is formulated as follows. By definition the likelihood function is the conditional probability of the data given the parameters of the model.
+It is customary to assume that the data are independent given the parameters. Then the likelihood function is the product of the likelihood of each datum. It is convenient to partition the data into four categories: uncensored, left censored, right censored, and interval censored. These are denoted "unc.", "l.c.", "r.c.", and "i.c." in the equation below.
+:<math> L(\theta) = \prod_{T_i\in unc.} \Pr(T = T_i|\theta)
+  \prod_{i\in l.c.} \Pr(T < T_i|\theta)
+  \prod_{i\in r.c.} \Pr(T > T_i|\theta)
+  \prod_{i\in i.c.} \Pr(T_{i,l} < T < T_{i,r}|\theta) .</math>
+For an uncensored datum, with <math>T_i</math> equal to the age at death, we have
+:<math> \Pr(T = T_i|\theta) = f(T_i|\theta) .</math>
+For a left censored datum, such that the age at death is known to be less than <math>T_i</math>, we have
+:<math> \Pr(T < T_i|\theta) = F(T_i|\theta) = 1 - S(T_i|\theta) .</math>
+For a right censored datum, such that the age at death is known to be greater than <math>T_i</math>, we have
+:<math> \Pr(T > T_i|\theta) = 1 - F(T_i|\theta) = S(T_i|\theta) .</math>
+For an interval censored datum, such that the age at death is known to be less than <math>T_{i,r}</math> and greater than <math>T_{i,l}</math>, we have
+:<math> \Pr(T_{i,l} < T < T_{i,r}|\theta)
+ = S(T_{i,l}|\theta) - S(T_{i,r}|\theta) .</math>
+An important application where interval censored data arises is current status data, where the actual occurrence of an event <math> T_i</math> is only known to the extent that it known not to occurred before observation time and to have occurred before the next.
+==Non-parametric estimation==
+The [[Nelson–Aalen estimator]] can be used to provide a [[non-parametric statistics|non-parametric]] estimate of the cumulative hazard rate function.
+==Distributions used in survival analysis==
+* [[Exponential distribution]]
+* [[Weibull distribution]]
+* [[Log-logistic distribution]]
+* [[Gamma distribution]]
+* [[Exponential-logarithmic distribution]]
+==See also==
+{{Columns-list|2|
+* [[Kaplan–Meier estimator]]
+* [[Survival rate]]
+* [[Reliability theory]]
+* [[Proportional hazards models]]
+* [[Accelerated failure time model]]
+* [[Failure rate]]
+* [[Logrank test]]
+* [[Survival function]]
+* [[MTBF]]
+* [[Censoring (statistics)]]
+* [[Maximum likelihood]]
+* [[Cell survival curve]]
+}}
+==References==
+{{reflist}}
+==Further reading==
+*{{cite book |first=David |last=Collett |title=Modelling Survival Data in Medical Research |edition=Second |location=Boca Raton |publisher=Chapman & Hall/CRC |year=2003 |isbn=1584883251 }}
+*{{cite book |first=Regina |last=Elandt-Johnson |first2=Norman |last2=Johnson |title=Survival Models and Data Analysis |location=New York |publisher=John Wiley & Sons |year=1999 |isbn=0471349925 }}
+*{{cite book |first=J. D. |last=Kalbfleisch |first2=Ross L. |last2=Prentice |title=The statistical analysis of failure time data |location=New York |publisher=John Wiley & Sons |year=2002 |isbn=047136357X }}
+*{{cite book |first=Jerald F. |last=Lawless |title=Statistical Models and Methods for Lifetime Data |edition=2nd |publisher=John Wiley and Sons |location=Hoboken |year=2003 |isbn=0471372153 }}
+*{{cite book |last=Rausand |first=M. |last2=Hoyland |first2=A. |title=System Reliability Theory: Models, Statistical Methods, and Applications |publisher=John Wiley & Sons |location=Hoboken |year=2004 |isbn=047147133X }}
+==External links==
+*{{cite web |first=Terry |last=Therneau |title=A Package for Survival Analysis in S |url=http://www.mayo.edu/hsr/people/therneau/survival.ps }} At: http://mayoresearch.mayo.edu/mayo/research/biostat/therneau.cfm
+*{{cite web |title=Engineering Statistics Handbook |publisher=NIST/SEMATEK |url=http://www.itl.nist.gov/div898/handbook/ }}
+* [[SOCR]], [http://www.socr.ucla.edu/htmls/ana/Survival_Analysis.html Survival analysis applet] and [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_AnalysisActivities_Survival interactive learning activity].
+* [http://www.statsoft.com/textbook/stsurvan.html Survival/Failure Time Analysis] @ [[Statistics]]' [http://www.statsoft.com/textbook/ Textbook Page]
+* [http://www.netstorm.be/home/survival Survival Analysis in R]
+* [http://www.nag.co.uk/numeric/fl/nagdoc_fl24/html/G12/g12conts.html Survival Analysis in NAG Fortran Library]
+{{Statistics|analysis}}
+{{Portal bar|Statistics}}
+{{DEFAULTSORT:Survival Analysis}}
+[[Category:Survival analysis| ]]
+[[Category:Senescence]]

Parseval's identity: Difference between revisions

Revision as of 04:35, 10 January 2014

Contents

General formulation

Survival function

Lifetime distribution function and event density

Hazard function and cumulative hazard function

Quantities derived from the survival distribution

Censoring

Fitting parameters to data

Non-parametric estimation

Distributions used in survival analysis

See also

References

Further reading

External links

Navigation menu

Parseval's identity: Difference between revisions

Revision as of 04:35, 10 January 2014

General formulation

Survival function

Lifetime distribution function and event density

Hazard function and cumulative hazard function

Quantities derived from the survival distribution

Censoring

Fitting parameters to data

Non-parametric estimation

Distributions used in survival analysis

See also

References

Further reading

External links

Navigation menu

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