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<title>Survival function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Survival function</span></span>

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<p>The <b>survival function</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that gives the <a href="Probability" title="Probability">probability</a> that a patient, device, or other object of interest will <a href="Survival_analysis" title="Survival analysis">survive</a> past a certain time.<sup id="cite_ref-KleinbaumKlein2012_1-0" class="reference"><a href="#cite_note-KleinbaumKlein2012-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

The survival function is also known as the <b>survivor function</b><sup id="cite_ref-TablemanKim2003_2-0" class="reference"><a href="#cite_note-TablemanKim2003-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> or <b>reliability function</b>.<sup id="cite_ref-Ebeling2010_3-0" class="reference"><a href="#cite_note-Ebeling2010-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

The term <i>reliability function</i> is common in <a href="Engineering" title="Engineering">engineering</a> while the term <i>survival function</i> is used in a broader range of applications, including human mortality. The survival function is the <a href="Complementary_cumulative_distribution_function" class="mw-redirect" title="Complementary cumulative distribution function">complementary cumulative distribution function</a> of the lifetime. Sometimes complementary cumulative distribution functions are called survival functions in general.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Let the lifetime <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> be a continuous random variable describing the time to failure. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> has <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t)}">

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<annotation encoding="application/x-tex">{\displaystyle F(t)}</annotation>

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</math></span><img src="./5b57ed3bbf501fb3c7f4bc5c4eafa96bb9e32165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.39ex; height:2.843ex;" alt="{\displaystyle F(t)}" loading="lazy"></span> and <a href="Probability_density_function" title="Probability density function">probability density function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

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<annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation>

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</math></span><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"></span> on the interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}">

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<annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation>

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</math></span><img src="./8dc2d914c2df66bc0f7893bfb8da36766650fe47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}" loading="lazy"></span>, then the <i>survival function</i> or <i>reliability function</i> is:

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t)=P(T>t)=1-F(t)=1-\int _{0}^{t}f(u)\,du}">

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<annotation encoding="application/x-tex">{\displaystyle S(t)=P(T>t)=1-F(t)=1-\int _{0}^{t}f(u)\,du}</annotation>

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</math></span><img src="./38ffcf89f82aaab81f79ded1bdaa8a5dc84cb524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.995ex; height:6.176ex;" alt="{\displaystyle S(t)=P(T>t)=1-F(t)=1-\int _{0}^{t}f(u)\,du}" loading="lazy"></span>

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<div class="mw-heading mw-heading2"><h2 id="Examples_of_survival_functions">Examples of survival functions</h2></div>

<p>The graphs below show examples of hypothetical survival functions. The <span class="texhtml"><i>x</i></span>-axis is time. The <span class="texhtml"><i>y</i></span>-axis is the proportion of subjects surviving. The graphs show the probability that a subject will survive beyond time <span class="texhtml"><i>t</i></span>.

</p>

<p>For example, for survival function 1, the probability of surviving longer than <span class="texhtml"><i>t</i> = 2</span> months is <span class="texhtml">0.37</span>. That is, 37% of subjects survive more than 2 months.

</p>

<p>For survival function 2, the probability of surviving longer than <span class="texhtml"><i>t</i> = 2</span> months is <span class="texhtml">0.97</span>. That is, 97% of subjects survive more than 2 months.

</p>

<p>Median survival may be determined from the survival function: The median survival is the point where the survival function intersects the value <span class="texhtml">0.5</span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For example, for survival function 2, 50% of the subjects survive 3.72 months. Median survival is thus <span class="texhtml">3.72</span> months.

</p>

<p>Median survival cannot always be determined from the graph alone. For example, in survival function 4, more than 50% of the subjects survive longer than the observation period of 10 months.

</p>

<p>The survival function is one of several ways to describe and display survival data. Another useful way to display data is a graph showing the distribution of survival times of subjects. Olkin,<sup id="cite_ref-OlkinGleserDerman1994_5-0" class="reference"><a href="#cite_note-OlkinGleserDerman1994-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> page 426, gives the following example of survival data. The number of hours between successive failures of an air-conditioning (AC) system were recorded. The time in hours, <span class="texhtml"><i>t</i></span>, between successive failures are 1, 3, 5, 7, 11, 11, 11, 12, 14, 14, 14, 16, 16, 20, 21, 23, 42, 47, 52, 62, 71, 71, 87, 90, 95, 120, 120, 225, 246 and 261. The mean time between failures is 59.6. The figure below shows the distribution of the time between failures. The blue tick marks beneath the graph are the actual hours between successive AC failures.

</p>

<p>In this example, a curve representing the <a href="Exponential_distribution" title="Exponential distribution">exponential distribution</a> overlays the distribution of AC failure times; the exponential distribution approximates the distribution of AC failure times. This particular exponential curve is specified by the parameter lambda, <span class="texhtml"><i>λ</i></span>:

</p><p><span class="texhtml"><i>λ</i> = 1/(mean time between failures) = 1/59.6 = 0.0168</span>.

</p><p>The distribution of failure times is the <a href="Probability_density_function" title="Probability density function">probability density function</a> (PDF), since time can take any positive value. In equations, the PDF is specified as <span class="texhtml"><i>f<sub>T</sub></i></span>. If time can only take discrete values (such as 1 day, 2 days, and so on), the distribution of failure times is called the <a href="Probability_mass_function" title="Probability mass function">probability mass function</a>. Most survival analysis methods assume that time can take any positive value, and <span class="texhtml"><i>f<sub>T</sub></i></span> is the PDF. If the time between observed AC failures is approximated using the exponential function, then the exponential curve gives the probability density function, <span class="texhtml"><i>f<sub>T</sub></i></span>, for AC failure times.

</p><p>Another useful way to display the survival data is a graph showing the cumulative failures up to each time point. These data may be displayed as either the cumulative number or the cumulative proportion of failures up to each time. The graph below shows the cumulative probability (or proportion) of failures at each time for the air conditioning system. The stairstep line in black shows the cumulative proportion of failures. For each step there is a blue tick at the bottom of the graph indicating an observed failure time. The smooth red line represents the exponential curve fitted to the observed data.

</p><p><span typeof="mw:File"></span>

</p><p>A graph of the cumulative probability of failures up to each time point is called the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF). In survival analysis, the cumulative distribution function gives the probability that the survival time is less than or equal to a specific time, <span class="texhtml"><i>t</i></span>.

</p><p>Let <span class="texhtml"><i>T</i></span> be survival time, which is any positive number. A particular time is designated by the lower case letter <span class="texhtml"><i>t</i></span>. The cumulative distribution function of <span class="texhtml"><i>T</i></span> is the function

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<annotation encoding="application/x-tex">{\displaystyle F(t)=\operatorname {P} (T\leq t),}</annotation>

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</math></span><img src="./c62adf2be40b645672919812106af09b092e48cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.101ex; height:2.843ex;" alt="{\displaystyle F(t)=\operatorname {P} (T\leq t),}" loading="lazy"></span></dd></dl>

<p>where the right-hand side represents the probability that the random variable <span class="texhtml"><i>T</i></span> is less than or equal to <span class="texhtml"><i>t</i></span>. If time can take on any positive value, then the cumulative distribution function <span class="texhtml"><i>F(t)</i></span> is the integral of the probability density function <span class="texhtml"><i>f(t)</i></span>.

</p><p>For the air-conditioning example, the graph of the CDF below illustrates that the probability that the time to failure is less than or equal to 100 hours is <span class="texhtml">0.81</span>, as estimated using the exponential curve fit to the data.

</p><p><span typeof="mw:File"></span>

</p><p>An alternative to graphing the probability that the failure time is <i>less</i> than or equal to 100 hours is to graph the probability that the failure time is <i>greater</i> than 100 hours. The probability that the failure time is greater than 100 hours must be 1 minus the probability that the failure time is less than or equal to 100 hours, because total probability must sum to 1.

</p><p>This gives:

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<mtext>failure times</mtext>

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<mtext> hours</mtext>

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<mtext>failure times</mtext>

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<mtext> hours</mtext>

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<annotation encoding="application/x-tex">{\displaystyle P({\text{failure times}}>100{\text{ hours}})=1-P({\text{failure times}}\leq 100{\text{ hours}})=1-0.81=0.19}</annotation>

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</math></span><img src="./f9ac26e47f759db3fbd5daacbda2b87fa81388e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:83.005ex; height:2.843ex;" alt="{\displaystyle P({\text{failure times}}>100{\text{ hours}})=1-P({\text{failure times}}\leq 100{\text{ hours}})=1-0.81=0.19}" loading="lazy"></span></dd></dl>

<p>This relationship generalizes to all failure times:

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<mtext> cumulative distribution function.</mtext>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle P(T>t)=1-P(T\leq t)={\text{ cumulative distribution function.}}}</annotation>

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</math></span><img src="./ddd4f0e44ea03f6af9cfb12b99db489c9765c82e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.267ex; height:2.843ex;" alt="{\displaystyle P(T>t)=1-P(T\leq t)={\text{ cumulative distribution function.}}}" loading="lazy"></span></dd></dl>

<p>This relationship is shown on the graphs below. The graph on the left is the cumulative distribution function, which is P(T ≤ t). The graph on the right is P(T > t) = 1 - P(T ≤ t). The graph on the right is the survival function, S(t). The fact that the S(t) = 1 – CDF is the reason that another name for the survival function is the complementary cumulative distribution function.

</p><p><span typeof="mw:File"></span>

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<div class="mw-heading mw-heading2"><h2 id="Parametric_survival_functions">Parametric survival functions</h2></div>

<p>In some cases, such as the air conditioner example, the distribution of survival times may be approximated well by a function such as the exponential distribution. Several distributions are commonly used in survival analysis, including the exponential, Weibull, gamma, normal, log-normal, and log-logistic.<sup id="cite_ref-Ebeling2010_3-1" class="reference"><a href="#cite_note-Ebeling2010-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Klein2005_6-0" class="reference"><a href="#cite_note-Klein2005-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> These distributions are defined by parameters. The normal (Gaussian) distribution, for example, is defined by the two parameters mean and standard deviation. Survival functions that are defined by parameters are said to be parametric.

</p><p>In the four survival function graphs shown above, the shape of the survival function is defined by a particular probability distribution: survival function 1 is defined by an exponential distribution, 2 is defined by a Weibull distribution, 3 is defined by a log-logistic distribution, and 4 is defined by another Weibull distribution.

</p>

<div class="mw-heading mw-heading3"><h3 id="Exponential_survival_function">Exponential survival function</h3></div>

<p>For an exponential survival distribution, the probability of failure is the same in every time interval, no matter the age of the individual or device. This fact leads to the "memoryless" property of the exponential survival distribution: the age of a subject has no effect on the probability of failure in the next time interval. The exponential may be a good model for the lifetime of a system where parts are replaced as they fail.<sup id="cite_ref-Mendenhall2007_7-0" class="reference"><a href="#cite_note-Mendenhall2007-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> It may also be useful for modeling survival of living organisms over short intervals. It is not likely to be a good model of the complete lifespan of a living organism.<sup id="cite_ref-Brostrom2012_8-0" class="reference"><a href="#cite_note-Brostrom2012-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> As Efron and Hastie <sup id="cite_ref-EfronHastie2016_9-0" class="reference"><a href="#cite_note-EfronHastie2016-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

(p. 134) note, "If human lifetimes were exponential there wouldn't be old or young people, just lucky or unlucky ones".

</p>

<div class="mw-heading mw-heading3"><h3 id="Weibull_survival_function">Weibull survival function</h3></div>

<p>A key assumption of the exponential survival function is that the hazard rate is constant. In an example given above, the proportion of men dying each year was constant at 10%, meaning that the hazard rate was constant. The assumption of constant hazard may not be appropriate. For example, among most living organisms, the risk of death is greater in old age than in middle age – that is, the hazard rate increases with time. For some diseases, such as breast cancer, the risk of recurrence is lower after 5 years – that is, the hazard rate decreases with time. The <a href="Weibull_distribution" title="Weibull distribution">Weibull distribution</a> extends the exponential distribution to allow constant, increasing, or decreasing hazard rates.

</p>

<div class="mw-heading mw-heading3"><h3 id="Other_parametric_survival_functions">Other parametric survival functions</h3></div>

<p>There are several other parametric survival functions that may provide a better fit to a particular data set, including normal, lognormal, log-logistic, and gamma. The choice of parametric distribution for a particular application can be made using graphical methods or using formal tests of fit.

These distributions and tests are described in textbooks on survival analysis.<sup id="cite_ref-KleinbaumKlein2012_1-1" class="reference"><a href="#cite_note-KleinbaumKlein2012-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ebeling2010_3-2" class="reference"><a href="#cite_note-Ebeling2010-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Lawless<sup id="cite_ref-Lawless2002_10-0" class="reference"><a href="#cite_note-Lawless2002-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

has extensive coverage of parametric models.

</p><p>Parametric survival functions are commonly used in manufacturing applications, in part because they enable estimation of the survival function beyond the observation period. However, appropriate use of parametric functions requires that data are well modeled by the chosen distribution. If an appropriate distribution is not available, or cannot be specified before a clinical trial or experiment, then non-parametric survival functions offer a useful alternative.

</p>

<div class="mw-heading mw-heading2"><h2 id="Non-parametric_survival_functions">Non-parametric survival functions</h2></div>

<p>A parametric model of survival may not be possible or desirable. In these situations, the most common method to model the survival function is the non-parametric <a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator</a>. This estimator requires lifetime data. Periodic case (cohort) and death (and recovery) counts are statistically sufficient to make non-parametric maximum likelihood and least squares estimates of survival functions, without lifetime data.

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<ul><li>Every survival function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S(t)}</annotation>

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</math></span><img src="./ed5c98226fcae6540afa928ccb8c2245844ac0a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.148ex; height:2.843ex;" alt="{\displaystyle S(t)}" loading="lazy"></span> is <a href="Monotonic_function" title="Monotonic function">monotonically decreasing</a>, i.e. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(u)\leq S(t)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S(u)\leq S(t)}</annotation>

</semantics>

</math></span><img src="./58dba8522a1c294afe02be3bd290a1dda2365a3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.885ex; height:2.843ex;" alt="{\displaystyle S(u)\leq S(t)}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u>t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u>t}</annotation>

</semantics>

</math></span><img src="./456550cab70359cd9d81424f7319f82199401d22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.009ex;" alt="{\displaystyle u>t}" loading="lazy"></span>.

<ul><li>It is a property of a <a href="Random_variable" title="Random variable">random variable</a> that maps a set of events, usually associated with mortality or failure of some system, onto <a href="Time" title="Time">time</a>.</li></ul></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=0}</annotation>

</semantics>

</math></span><img src="./43469ec032d858feae5aa87029e22eaaf0109e9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}" loading="lazy"></span>, represents some origin, typically the beginning of a study or the start of operation of some system. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(0)}</annotation>

</semantics>

</math></span><img src="./f113163faf55307ea92fcdbbbf83372c4be3cd3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.471ex; height:2.843ex;" alt="{\displaystyle S(0)}" loading="lazy"></span> is commonly unity but can be less to represent the <a href="Probability" title="Probability">probability</a> that the system fails immediately upon operation.</li>

<li>Since the CDF is a <a href="Right-continuous" class="mw-redirect" title="Right-continuous">right-continuous</a> function, the survival function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(t)=1-F(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(t)=1-F(t)}</annotation>

</semantics>

</math></span><img src="./3cb7d0b64ac71de275fbf7af0038b6cf85abbf8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.639ex; height:2.843ex;" alt="{\displaystyle S(t)=1-F(t)}" loading="lazy"></span> is also right-continuous.</li>

<li>The survival function can be related to the <a href="Probability_density_function" title="Probability density function">probability density function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation>

</semantics>

</math></span><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"></span> and the <a href="Failure_rate#Failure_rate_in_the_continuous_sense" title="Failure rate">hazard function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (t)}</annotation>

</semantics>

</math></span><img src="./e0e1cc2c2f9e6d8b22c0da0bb5fe8f0f47791e4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \lambda (t)}" loading="lazy"></span>

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle f(t)=-S'(t)}</annotation>

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</math></span><img src="./1a5a71d604ab07577fb67a45e38a0f6f4819ec85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.689ex; height:3.009ex;" alt="{\displaystyle f(t)=-S'(t)}" loading="lazy"></span></li>

<mfrac>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (t)=-{d \over {dt}}\log S(t)}</annotation>

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</math></span><img src="./7063f340712a2d3b8382735e453c27e656b8bfb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.696ex; height:5.509ex;" alt="{\displaystyle \lambda (t)=-{d \over {dt}}\log S(t)}" loading="lazy"></span></li></ul></li></ul>

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S(t)=\exp[-\int _{0}^{t}\lambda (t')dt']}</annotation>

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</math></span><img src="./fdfa9a2850a79370ca4d1295dc45b48f0344e37d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.394ex; height:6.176ex;" alt="{\displaystyle S(t)=\exp[-\int _{0}^{t}\lambda (t')dt']}" loading="lazy"></span>

</p>

<ul><li>The expected survival time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} (T)=\int _{0}^{\infty }S(t)dt}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} (T)=\int _{0}^{\infty }S(t)dt}</annotation>

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</math></span><img src="./115564a457de8112b0d31bc9b9ad114551d18a70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.025ex; height:5.843ex;" alt="{\displaystyle \mathbb {E} (T)=\int _{0}^{\infty }S(t)dt}" loading="lazy"></span></li></ul>

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<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof of expected survival time formula</div>

</th></tr>

<tr>

<td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);">The <a href="Expected_value" title="Expected value">expected value</a> of a random variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in [0,\infty )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle T\in [0,\infty )}</annotation>

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</math></span><img src="./f342471bd952a00b46cfbd558949ab800cbfd13e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.549ex; height:2.843ex;" alt="{\displaystyle T\in [0,\infty )}" loading="lazy"></span> is defined as:

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} (T)=\int _{0}^{\infty }tf(t)dt}</annotation>

</semantics>

</math></span><img src="./bd4b5a5db9f62a733da55e458287971b9027f578.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.644ex; height:5.843ex;" alt="{\displaystyle \mathbb {E} (T)=\int _{0}^{\infty }tf(t)dt}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation>

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</math></span><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"></span> is the <a href="Probability_density_function" title="Probability density function">probability density function</a>. Using the relation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=-S'(t)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t)=-S'(t)}</annotation>

</semantics>

</p>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} (T)=-\int _{0}^{\infty }tS'(t)dt}</annotation>

</semantics>

</math></span><img src="./89891ccd4e3bb1cde7538926393459dc1d76e7e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.767ex; height:5.843ex;" alt="{\displaystyle \mathbb {E} (T)=-\int _{0}^{\infty }tS'(t)dt}" loading="lazy"></span></dd></dl>

<p>This may be further simplified by employing <a href="Integration_by_parts" title="Integration by parts">integration by parts</a>:

</p>

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

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</msubsup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\int _{0}^{\infty }tS'(t)dt=-tS(t){\bigg |}_{0}^{\infty }+\int _{0}^{\infty }S(t)dt}</annotation>

</semantics>

</math></span><img src="./49d8b9919430dfd1b7ec4018c7f7b27d841cef91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.86ex; height:5.843ex;" alt="{\displaystyle -\int _{0}^{\infty }tS'(t)dt=-tS(t){\bigg |}_{0}^{\infty }+\int _{0}^{\infty }S(t)dt}" loading="lazy"></span></dd></dl>

<p>By definition, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\infty )=0}">

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<annotation encoding="application/x-tex">{\displaystyle S(\infty )=0}</annotation>

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</math></span><img src="./b89df006f18ec0017acf448d1930adb3bdad1139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.893ex; height:2.843ex;" alt="{\displaystyle S(\infty )=0}" loading="lazy"></span>, meaning that the boundary terms are identically equal to zero. Therefore, we may conclude that the expected value is simply the integral of the survival function:

</p>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} (T)=\int _{0}^{\infty }S(t)dt}</annotation>

</semantics>

</td></tr></tbody></table></div>

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<ul><li><a href="Failure_rate" title="Failure rate">Failure rate</a></li>

<li><a href="Frequency_of_exceedance" title="Frequency of exceedance">Frequency of exceedance</a></li>

<li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator</a></li>

<li><a href="Mean_time_to_failure" class="mw-redirect" title="Mean time to failure">Mean time to failure</a></li>

<li><a href="Residence_time_(statistics)" title="Residence time (statistics)">Residence time (statistics)</a></li>

<li><a href="Survivorship_curve" title="Survivorship curve">Survivorship curve</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFKleinbaumKlein2012" class="citation cs2">Kleinbaum, David G.; Klein, Mitchel (2012), <i>Survival analysis: A Self-learning text</i> (Third ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1441966452</bdi></cite></span>

</li>

<li id="cite_note-TablemanKim2003-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-TablemanKim2003_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFTablemanKim2003" class="citation cs2">Tableman, Mara; Kim, Jong Sung (2003), <i>Survival Analysis Using S</i> (First ed.), Chapman and Hall/CRC, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1584884088</bdi></cite></span>

</li>

<li id="cite_note-Ebeling2010-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ebeling2010_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ebeling2010_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Ebeling2010_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFEbeling2010" class="citation cs2">Ebeling, Charles (2010), <i>An Introduction to Reliability and Maintainability Engineering</i> (Second ed.), Waveland Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1577666257</bdi></cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Machin, D., Cheung, Y. B., Parmar, M. (2006). Survival Analysis: A Practical Approach. Deutschland: Wiley. Page 36 and following <a rel="nofollow" class="external text" href="https://books.google.com/books?id=z6_Hr9NGjr0C&pg=PA36">Google Books</a></span>

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<li id="cite_note-OlkinGleserDerman1994-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-OlkinGleserDerman1994_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFOlkinGleserDerman1994" class="citation cs2">Olkin, Ingram; Gleser, Leon; Derman, Cyrus (1994), <i>Probability Models and Applications</i> (Second ed.), Macmillan, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-02-389220-X</bdi></cite></span>

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<li id="cite_note-Klein2005-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Klein2005_6-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKleinMoeschberger2005" class="citation cs2">Klein, John; <a href="Melvin_L._Moeschberger" title="Melvin L. Moeschberger">Moeschberger, Melvin</a> (2005), <i>Survival Analysis: Techniques for Censored and Truncated Data</i> (Second ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0387953991</bdi></cite></span>

</li>

<li id="cite_note-Mendenhall2007-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mendenhall2007_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFMendenhallTerry2007" class="citation cs2">Mendenhall, William; Terry, Sincich (2007), <i>Statistics for Engineering and the Sciences</i> (Fifth ed.), Pearson / Prentice Hall, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0131877061</bdi></cite></span>

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<li id="cite_note-Brostrom2012-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brostrom2012_8-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFBrostrom2012" class="citation cs2">Brostrom, Göran (2012), <i>Event History Analysis with R</i> (First ed.), Chapman & Hall/CRC, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1439831649</bdi></cite></span>

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<li id="cite_note-EfronHastie2016-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-EfronHastie2016_9-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFEfronHastie2016" class="citation cs2">Efron, Bradley; Hastie, Trevor (2016), <i>Computer Age Statistical Inference: Algorithms, Evidence, and Data Science</i> (First ed.), Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1107149892</bdi></cite></span>

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<li id="cite_note-Lawless2002-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lawless2002_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFLawless2002" class="citation cs2">Lawless, Jerald (2002), <i>Statistical Models and Methods for Lifetime Data</i> (Second ed.), Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0471372158</bdi></cite></span>

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</style><div id="Statistics654" style="font-size:114%;margin:0 4em"><a href="Statistics" title="Statistics">Statistics</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Outline_of_statistics" title="Outline of statistics">Outline</a></li>

<li><a href="List_of_statistics_articles" title="List of statistics articles">Index</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Descriptive_statistics654" style="font-size:114%;margin:0 4em"><a href="Descriptive_statistics" title="Descriptive statistics">Descriptive statistics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">Continuous data</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Central_tendency" title="Central tendency">Center</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_mean" title="Arithmetic mean">Arithmetic</a></li>

<li><a href="Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">Arithmetic-Geometric</a></li>

<li><a href="Contraharmonic_mean" title="Contraharmonic mean">Contraharmonic</a></li>

<li><a href="Cubic_mean" title="Cubic mean">Cubic</a></li>

<li><a href="Generalized_mean" title="Generalized mean">Generalized/power</a></li>

<li><a href="Geometric_mean" title="Geometric mean">Geometric</a></li>

<li><a href="Harmonic_mean" title="Harmonic mean">Harmonic</a></li>

<li><a href="Heronian_mean" title="Heronian mean">Heronian</a></li>

<li><a href="Heinz_mean" title="Heinz mean">Heinz</a></li>

<li><a href="Lehmer_mean" title="Lehmer mean">Lehmer</a></li></ul></li>

<li><a href="Median" title="Median">Median</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_dispersion" title="Statistical dispersion">Dispersion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Average_absolute_deviation" title="Average absolute deviation">Average absolute deviation</a></li>

<li><a href="Coefficient_of_variation" title="Coefficient of variation">Coefficient of variation</a></li>

<li><a href="Interquartile_range" title="Interquartile range">Interquartile range</a></li>

<li><a href="Percentile" title="Percentile">Percentile</a></li>

<li><a href="Range_(statistics)" title="Range (statistics)">Range</a></li>

<li><a href="Standard_deviation" title="Standard deviation">Standard deviation</a></li>

<li><a href="Variance#Sample_variance" title="Variance">Variance</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Shape_of_the_distribution" class="mw-redirect" title="Shape of the distribution">Shape</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Moment_(mathematics)" title="Moment (mathematics)">Moments</a>

<ul><li><a href="Kurtosis" title="Kurtosis">Kurtosis</a></li>

<li><a href="L-moment" title="L-moment">L-moments</a></li>

<li><a href="Skewness" title="Skewness">Skewness</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Count_data" title="Count data">Count data</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Index_of_dispersion" title="Index of dispersion">Index of dispersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Summary tables</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Frequency_distribution" class="mw-redirect" title="Frequency distribution">Frequency distribution</a></li>

<li><a href="Grouped_data" title="Grouped data">Grouped data</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Dependence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Pearson_correlation_coefficient" title="Pearson correlation coefficient">Pearson product-moment correlation</a></li>

<li><a href="Rank_correlation" title="Rank correlation">Rank correlation</a>

<ul><li><a href="Kendall_rank_correlation_coefficient" title="Kendall rank correlation coefficient">Kendall's τ</a></li>

<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's ρ</a></li></ul></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_graphics" title="Statistical graphics">Graphics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bar_chart" title="Bar chart">Bar chart</a></li>

<li><a href="Biplot" title="Biplot">Biplot</a></li>

<li><a href="Control_chart" title="Control chart">Control chart</a></li>

<li><a href="Correlogram" title="Correlogram">Correlogram</a></li>

<li><a href="Fan_chart_(statistics)" title="Fan chart (statistics)">Fan chart</a></li>

<li><a href="Forest_plot" title="Forest plot">Forest plot</a></li>

<li><a href="Histogram" title="Histogram">Histogram</a></li>

<li><a href="Pie_chart" title="Pie chart">Pie chart</a></li>

<li><a href="Radar_chart" title="Radar chart">Radar chart</a></li>

<li><a href="Run_chart" title="Run chart">Run chart</a></li>

<li><a href="Scatter_plot" title="Scatter plot">Scatter plot</a></li>

<li><a href="Stem-and-leaf_display" title="Stem-and-leaf display">Stem-and-leaf display</a></li>

<li><a href="Violin_plot" title="Violin plot">Violin plot</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_collection654" style="font-size:114%;margin:0 4em"><a href="Data_collection" title="Data collection">Data collection</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Design_of_experiments" title="Design of experiments">Study design</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Effect_size" title="Effect size">Effect size</a></li>

<li><a href="Missing_data" title="Missing data">Missing data</a></li>

<li><a href="Optimal_design" class="mw-redirect" title="Optimal design">Optimal design</a></li>

<li><a href="Statistical_population" title="Statistical population">Population</a></li>

<li><a href="Replication_(statistics)" title="Replication (statistics)">Replication</a></li>

<li><a href="Sample_size_determination" title="Sample size determination">Sample size determination</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Statistical_power" class="mw-redirect" title="Statistical power">Statistical power</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survey_methodology" title="Survey methodology">Survey methodology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sampling_(statistics)" title="Sampling (statistics)">Sampling</a>

<ul><li><a href="Cluster_sampling" title="Cluster sampling">Cluster</a></li>

<li><a href="Stratified_sampling" title="Stratified sampling">Stratified</a></li></ul></li>

<li><a href="Opinion_poll" title="Opinion poll">Opinion poll</a></li>

<li><a href="Questionnaire" title="Questionnaire">Questionnaire</a></li>

<li><a href="Standard_error" title="Standard error">Standard error</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Experiment" title="Experiment">Controlled experiments</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Blocking_(statistics)" title="Blocking (statistics)">Blocking</a></li>

<li><a href="Factorial_experiment" title="Factorial experiment">Factorial experiment</a></li>

<li><a href="Interaction_(statistics)" title="Interaction (statistics)">Interaction</a></li>

<li><a href="Random_assignment" title="Random assignment">Random assignment</a></li>

<li><a href="Randomized_controlled_trial" title="Randomized controlled trial">Randomized controlled trial</a></li>

<li><a href="Randomized_experiment" title="Randomized experiment">Randomized experiment</a></li>

<li><a href="Scientific_control" title="Scientific control">Scientific control</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Adaptive designs</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_clinical_trial" class="mw-redirect" title="Adaptive clinical trial">Adaptive clinical trial</a></li>

<li><a href="Stochastic_approximation" title="Stochastic approximation">Stochastic approximation</a></li>

<li><a href="Up-and-Down_Designs" class="mw-redirect" title="Up-and-Down Designs">Up-and-down designs</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Observational_study" title="Observational study">Observational studies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohort_study" title="Cohort study">Cohort study</a></li>

<li><a href="Cross-sectional_study" title="Cross-sectional study">Cross-sectional study</a></li>

<li><a href="Natural_experiment" title="Natural experiment">Natural experiment</a></li>

<li><a href="Quasi-experiment" title="Quasi-experiment">Quasi-experiment</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Statistical_inference654" style="font-size:114%;margin:0 4em"><a href="Statistical_inference" title="Statistical inference">Statistical inference</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Statistical_theory" title="Statistical theory">Statistical theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Population_(statistics)" class="mw-redirect" title="Population (statistics)">Population</a></li>

<li><a href="Statistic" title="Statistic">Statistic</a></li>

<li><a href="Probability_distribution" title="Probability distribution">Probability distribution</a></li>

<li><a href="Sampling_distribution" title="Sampling distribution">Sampling distribution</a>

<ul><li><a href="Order_statistic" title="Order statistic">Order statistic</a></li></ul></li>

<li><a href="Empirical_distribution_function" title="Empirical distribution function">Empirical distribution</a>

<ul><li><a href="Density_estimation" title="Density estimation">Density estimation</a></li></ul></li>

<li><a href="Statistical_model" title="Statistical model">Statistical model</a>

<ul><li><a href="Model_specification" class="mw-redirect" title="Model specification">Model specification</a></li>

<li><a href="Lp_space" title="Lp space">L<sup><i>p</i></sup> space</a></li></ul></li>

<li><a href="Statistical_parameter" title="Statistical parameter">Parameter</a>

<ul><li><a href="Location_parameter" title="Location parameter">location</a></li>

<li><a href="Scale_parameter" title="Scale parameter">scale</a></li>

<li><a href="Shape_parameter" title="Shape parameter">shape</a></li></ul></li>

<li><a href="Parametric_statistics" title="Parametric statistics">Parametric family</a>

<ul><li><a href="Likelihood_function" title="Likelihood function">Likelihood</a> <a href="Monotone_likelihood_ratio" title="Monotone likelihood ratio"><span style="font-size: 85%;">(monotone)</span></a></li>

<li><a href="Location%E2%80%93scale_family" title="Location–scale family">Location–scale family</a></li>

<li><a href="Exponential_family" title="Exponential family">Exponential family</a></li></ul></li>

<li><a href="Completeness_(statistics)" title="Completeness (statistics)">Completeness</a></li>

<li><a href="Sufficient_statistic" title="Sufficient statistic">Sufficiency</a></li>

<li><a href="Plug-in_principle" class="mw-redirect" title="Plug-in principle">Statistical functional</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Optimal_decision" title="Optimal decision">Optimal decision</a>

<ul><li><a href="Loss_function" title="Loss function">loss function</a></li></ul></li>

<li><a href="Efficiency_(statistics)" title="Efficiency (statistics)">Efficiency</a></li>

<li><a href="Statistical_distance" title="Statistical distance">Statistical distance</a>

<ul><li><a href="Divergence_(statistics)" title="Divergence (statistics)">divergence</a></li></ul></li>

<li><a href="Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">Asymptotics</a></li>

<li><a href="Robust_statistics" title="Robust statistics">Robustness</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Frequentist_inference" title="Frequentist inference">Frequentist inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Point_estimation" title="Point estimation">Point estimation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Estimating_equations" title="Estimating equations">Estimating equations</a>

<ul><li><a href="Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">Maximum likelihood</a></li>

<li><a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">Method of moments</a></li>

<li><a href="M-estimator" title="M-estimator">M-estimator</a></li>

<li><a href="Minimum_distance_estimation" class="mw-redirect" title="Minimum distance estimation">Minimum distance</a></li></ul></li>

<li><a href="Bias_of_an_estimator" title="Bias of an estimator">Unbiased estimators</a>

<ul><li><a href="Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">Mean-unbiased minimum-variance</a>

<ul><li><a href="Rao%E2%80%93Blackwell_theorem" title="Rao–Blackwell theorem">Rao–Blackwellization</a></li>

<li><a href="Lehmann%E2%80%93Scheff%C3%A9_theorem" title="Lehmann–Scheffé theorem">Lehmann–Scheffé theorem</a></li></ul></li>

<li><a href="Median-unbiased_estimator" class="mw-redirect" title="Median-unbiased estimator">Median unbiased</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Interval_estimation" title="Interval estimation">Interval estimation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Confidence_interval" title="Confidence interval">Confidence interval</a></li>

<li><a href="Pivotal_quantity" title="Pivotal quantity">Pivot</a></li>

<li><a href="Likelihood_interval" class="mw-redirect" title="Likelihood interval">Likelihood interval</a></li>

<li><a href="Prediction_interval" title="Prediction interval">Prediction interval</a></li>

<li><a href="Tolerance_interval" title="Tolerance interval">Tolerance interval</a></li>

<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a>

<ul><li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrap</a></li>

<li><a href="Jackknife_resampling" title="Jackknife resampling">Jackknife</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Statistical_hypothesis_testing" class="mw-redirect" title="Statistical hypothesis testing">Testing hypotheses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="One-_and_two-tailed_tests" title="One- and two-tailed tests">1- & 2-tails</a></li>

<li><a href="Power_(statistics)" title="Power (statistics)">Power</a>

<ul><li><a href="Uniformly_most_powerful_test" title="Uniformly most powerful test">Uniformly most powerful test</a></li></ul></li>

<li><a href="Permutation_test" title="Permutation test">Permutation test</a>

<ul><li><a href="Randomization_test" class="mw-redirect" title="Randomization test">Randomization test</a></li></ul></li>

<li><a href="Multiple_comparisons" class="mw-redirect" title="Multiple comparisons">Multiple comparisons</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Parametric_statistics" title="Parametric statistics">Parametric tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio</a></li>

<li><a href="Score_test" title="Score test">Score/Lagrange multiplier</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="List_of_statistical_tests" title="List of statistical tests">Specific tests</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Z-test" title="Z-test"><i>Z</i>-test <span style="font-size: 85%;">(normal)</span></a></li>

<li><a href="Student's_t-test" title="Student's t-test">Student's <i>t</i>-test</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Goodness_of_fit" title="Goodness of fit">Goodness of fit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chi-squared_test" title="Chi-squared test">Chi-squared</a></li>

<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov</a></li>

<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling</a></li>

<li><a href="Lilliefors_test" title="Lilliefors test">Lilliefors</a></li>

<li><a href="Jarque%E2%80%93Bera_test" title="Jarque–Bera test">Jarque–Bera</a></li>

<li><a href="Shapiro%E2%80%93Wilk_test" title="Shapiro–Wilk test">Normality <span style="font-size: 85%;">(Shapiro–Wilk)</span></a></li>

<li><a href="Likelihood-ratio_test" title="Likelihood-ratio test">Likelihood-ratio test</a></li>

<li><a href="Model_selection" title="Model selection">Model selection</a>

<ul><li><a href="Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross validation</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Rank_statistics" class="mw-redirect" title="Rank statistics">Rank statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Sample_median" class="mw-redirect" title="Sample median">Sample median</a></li></ul></li>

<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Signed rank <span style="font-size: 85%;">(Wilcoxon)</span></a>

<ul><li><a href="Hodges%E2%80%93Lehmann_estimator" title="Hodges–Lehmann estimator">Hodges–Lehmann estimator</a></li></ul></li>

<li><a href="Mann%E2%80%93Whitney_U_test" title="Mann–Whitney U test">Rank sum <span style="font-size: 85%;">(Mann–Whitney)</span></a></li>

<li><a href="Nonparametric_statistics" title="Nonparametric statistics">Nonparametric</a> <a href="Analysis_of_variance" title="Analysis of variance">anova</a>

<ul><li><a href="Kruskal%E2%80%93Wallis_test" title="Kruskal–Wallis test">1-way <span style="font-size: 85%;">(Kruskal–Wallis)</span></a></li>

<li><a href="Friedman_test" title="Friedman test">2-way <span style="font-size: 85%;">(Friedman)</span></a></li>

<li><a href="Jonckheere's_trend_test" title="Jonckheere's trend test">Ordered alternative <span style="font-size: 85%;">(Jonckheere–Terpstra)</span></a></li></ul></li>

<li><a href="Van_der_Waerden_test" title="Van der Waerden test">Van der Waerden test</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>

<ul><li><a href="Prior_probability" title="Prior probability">prior</a></li>

<li><a href="Posterior_probability" title="Posterior probability">posterior</a></li></ul></li>

<li><a href="Credible_interval" title="Credible interval">Credible interval</a></li>

<li><a href="Bayes_factor" title="Bayes factor">Bayes factor</a></li>

<li><a href="Bayes_estimator" title="Bayes estimator">Bayesian estimator</a>

<ul><li><a href="Maximum_a_posteriori_estimation" title="Maximum a posteriori estimation">Maximum posterior estimator</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="CorrelationRegression_analysis654" style="font-size:114%;margin:0 4em"><div class="hlist"><ul><li><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></li><li><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></li></ul></div></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Correlation_and_dependence" class="mw-redirect" title="Correlation and dependence">Correlation</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">Pearson product-moment</a></li>

<li><a href="Partial_correlation" title="Partial correlation">Partial correlation</a></li>

<li><a href="Confounding" title="Confounding">Confounding variable</a></li>

<li><a href="Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Regression_analysis" title="Regression analysis">Regression analysis</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Errors_and_residuals" title="Errors and residuals">Errors and residuals</a></li>

<li><a href="Regression_validation" title="Regression validation">Regression validation</a></li>

<li><a href="Mixed_model" title="Mixed model">Mixed effects models</a></li>

<li><a href="Simultaneous_equations_model" title="Simultaneous equations model">Simultaneous equations models</a></li>

<li><a href="Multivariate_adaptive_regression_splines" class="mw-redirect" title="Multivariate adaptive regression splines">Multivariate adaptive regression splines (MARS)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Linear_regression" title="Linear regression">Linear regression</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simple_linear_regression" title="Simple linear regression">Simple linear regression</a></li>

<li><a href="Ordinary_least_squares" title="Ordinary least squares">Ordinary least squares</a></li>

<li><a href="General_linear_model" title="General linear model">General linear model</a></li>

<li><a href="Bayesian_linear_regression" title="Bayesian linear regression">Bayesian regression</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em">Non-standard predictors</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_regression" title="Nonlinear regression">Nonlinear regression</a></li>

<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric</a></li>

<li><a href="Semiparametric_regression" title="Semiparametric regression">Semiparametric</a></li>

<li><a href="Isotonic_regression" title="Isotonic regression">Isotonic</a></li>

<li><a href="Robust_regression" title="Robust regression">Robust</a></li>

<li><a href="Homoscedasticity_and_heteroscedasticity" title="Homoscedasticity and heteroscedasticity">Homoscedasticity and Heteroscedasticity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Generalized_linear_model" title="Generalized linear model">Generalized linear model</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Exponential_family" title="Exponential family">Exponential families</a></li>

<li><a href="Logistic_regression" title="Logistic regression">Logistic <span style="font-size: 85%;">(Bernoulli)</span></a> / <a href="Binomial_regression" title="Binomial regression">Binomial</a> / <a href="Poisson_regression" title="Poisson regression">Poisson regressions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Partition_of_sums_of_squares" title="Partition of sums of squares">Partition of variance</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analysis_of_variance" title="Analysis of variance">Analysis of variance (ANOVA, anova)</a></li>

<li><a href="Analysis_of_covariance" title="Analysis of covariance">Analysis of covariance</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Multivariate ANOVA</a></li>

<li><a href="Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">Degrees of freedom</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible uncollapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Categorical_/_multivariate_/_time-series_/_survival_analysis654" style="font-size:114%;margin:0 4em"><a href="Categorical_variable" title="Categorical variable">Categorical</a> / <a href="Multivariate_statistics" title="Multivariate statistics">multivariate</a> / <a href="Time_series" title="Time series">time-series</a> / <a href="Survival_analysis" title="Survival analysis">survival analysis</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Categorical_variable" title="Categorical variable">Categorical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a></li>

<li><a href="Contingency_table" title="Contingency table">Contingency table</a></li>

<li><a href="Graphical_model" title="Graphical model">Graphical model</a></li>

<li><a href="Poisson_regression" title="Poisson regression">Log-linear model</a></li>

<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a></li>

<li><a href="Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics" title="Cochran–Mantel–Haenszel statistics">Cochran–Mantel–Haenszel statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Multivariate_statistics" title="Multivariate statistics">Multivariate</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="General_linear_model" title="General linear model">Regression</a></li>

<li><a href="Multivariate_analysis_of_variance" title="Multivariate analysis of variance">Manova</a></li>

<li><a href="Principal_component_analysis" title="Principal component analysis">Principal components</a></li>

<li><a href="Canonical_correlation" title="Canonical correlation">Canonical correlation</a></li>

<li><a href="Linear_discriminant_analysis" title="Linear discriminant analysis">Discriminant analysis</a></li>

<li><a href="Cluster_analysis" title="Cluster analysis">Cluster analysis</a></li>

<li><a href="Statistical_classification" title="Statistical classification">Classification</a></li>

<li><a href="Structural_equation_modeling" title="Structural equation modeling">Structural equation model</a>

<ul><li><a href="Factor_analysis" title="Factor analysis">Factor analysis</a></li></ul></li>

<li><a href="Multivariate_distribution" class="mw-redirect" title="Multivariate distribution">Multivariate distributions</a>

<ul><li><a href="Elliptical_distribution" title="Elliptical distribution">Elliptical distributions</a>

<ul><li><a href="Multivariate_normal_distribution" title="Multivariate normal distribution">Normal</a></li></ul></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Time_series" title="Time series">Time-series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Decomposition_of_time_series" title="Decomposition of time series">Decomposition</a></li>

<li><a href="Trend_estimation" class="mw-redirect" title="Trend estimation">Trend</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationarity</a></li>

<li><a href="Seasonal_adjustment" title="Seasonal adjustment">Seasonal adjustment</a></li>

<li><a href="Exponential_smoothing" title="Exponential smoothing">Exponential smoothing</a></li>

<li><a href="Cointegration" title="Cointegration">Cointegration</a></li>

<li><a href="Structural_break" title="Structural break">Structural break</a></li>

<li><a href="Granger_causality" title="Granger causality">Granger causality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Specific tests</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dickey%E2%80%93Fuller_test" title="Dickey–Fuller test">Dickey–Fuller</a></li>

<li><a href="Johansen_test" title="Johansen test">Johansen</a></li>

<li><a href="Ljung%E2%80%93Box_test" title="Ljung–Box test">Q-statistic <span style="font-size: 85%;">(Ljung–Box)</span></a></li>

<li><a href="Durbin%E2%80%93Watson_statistic" title="Durbin–Watson statistic">Durbin–Watson</a></li>

<li><a href="Breusch%E2%80%93Godfrey_test" title="Breusch–Godfrey test">Breusch–Godfrey</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Time_domain" title="Time domain">Time domain</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autocorrelation" title="Autocorrelation">Autocorrelation (ACF)</a>

<ul><li><a href="Partial_autocorrelation_function" title="Partial autocorrelation function">partial (PACF)</a></li></ul></li>

<li><a href="Cross-correlation" title="Cross-correlation">Cross-correlation (XCF)</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">ARMA model</a></li>

<li><a href="Box%E2%80%93Jenkins_method" title="Box–Jenkins method">ARIMA model <span style="font-size: 85%;">(Box–Jenkins)</span></a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH)</a></li>

<li><a href="Vector_autoregression" title="Vector autoregression">Vector autoregression (VAR)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Frequency_domain" title="Frequency domain">Frequency domain</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li>

<li><a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li>

<li><a href="Wavelet" title="Wavelet">Wavelet</a></li>

<li><a href="Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Survival_analysis" title="Survival analysis">Survival</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier estimator (product limit)</a></li>

<li><a href="Proportional_hazards_model" title="Proportional hazards model">Proportional hazards models</a></li>

<li><a href="Accelerated_failure_time_model" title="Accelerated failure time model">Accelerated failure time (AFT) model</a></li>

<li><a href="First-hitting-time_model" title="First-hitting-time model">First hitting time</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Failure_rate" title="Failure rate">Hazard function</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nelson%E2%80%93Aalen_estimator" title="Nelson–Aalen estimator">Nelson–Aalen estimator</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Test</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Applications654" style="font-size:114%;margin:0 4em"><a href="List_of_fields_of_application_of_statistics" title="List of fields of application of statistics">Applications</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Biostatistics" title="Biostatistics">Biostatistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bioinformatics" title="Bioinformatics">Bioinformatics</a></li>

<li><a href="Clinical_trial" title="Clinical trial">Clinical trials</a> / <a href="Clinical_study_design" title="Clinical study design">studies</a></li>

<li><a href="Epidemiology" title="Epidemiology">Epidemiology</a></li>

<li><a href="Medical_statistics" title="Medical statistics">Medical statistics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Engineering_statistics" title="Engineering statistics">Engineering statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chemometrics" title="Chemometrics">Chemometrics</a></li>

<li><a href="Methods_engineering" title="Methods engineering">Methods engineering</a></li>

<li><a href="Probabilistic_design" title="Probabilistic design">Probabilistic design</a></li>

<li><a href="Statistical_process_control" title="Statistical process control">Process</a> / <a href="Quality_control" title="Quality control">quality control</a></li>

<li><a href="Reliability_engineering" title="Reliability engineering">Reliability</a></li>

<li><a href="System_identification" title="System identification">System identification</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Social_statistics" title="Social statistics">Social statistics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_science" title="Actuarial science">Actuarial science</a></li>

<li><a href="Census" title="Census">Census</a></li>

<li><a href="Crime_statistics" title="Crime statistics">Crime statistics</a></li>

<li><a href="Demographic_statistics" title="Demographic statistics">Demography</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Jurimetrics" title="Jurimetrics">Jurimetrics</a></li>

<li><a href="National_accounts" title="National accounts">National accounts</a></li>

<li><a href="Official_statistics" title="Official statistics">Official statistics</a></li>

<li><a href="Population_statistics" class="mw-redirect" title="Population statistics">Population statistics</a></li>

<li><a href="Psychometrics" title="Psychometrics">Psychometrics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:12.5em"><a href="Spatial_analysis" title="Spatial analysis">Spatial statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cartography" title="Cartography">Cartography</a></li>

<li><a href="Environmental_statistics" title="Environmental statistics">Environmental statistics</a></li>

<li><a href="Geographic_information_system" title="Geographic information system">Geographic information system</a></li>

<li><a href="Geostatistics" title="Geostatistics">Geostatistics</a></li>

<li><a href="Kriging" title="Kriging">Kriging</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span><b>Category</b></li>

<li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Statistics" class="extiw external" title="commons:Category:Statistics">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-04-11" href="https://en.wikipedia.org/wiki/?title=Survival_function&oldid=1285010531">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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