< Probability and statistics definitions < *inverse survival function*

The **inverse survival function**, S^{-1}(p), is the inverse of the survival function. It is time at which a certain probability of survival is achieved. While “survival” generally refers to surviving death, in the context of survival analysis the word refers to the length of time until an event happens.

As with the percent point function, the horizontal axis represents a probability. Therefore the horizontal axis ranges from 0 to 1 regardless of the particular probability distribution you’re working with.

## Example: the normal distribution inverse survival function

**Example**: Given that you are searching for a subject who is larger than 95% of all other subjects, what size does the subject need to be?

The inverse survival function of the normal distribution can be used to calculate this. It is defined as the mean (μ), plus the standard deviation (σ) multiplied by the inverse cumulative distribution function (Φ^{-1}) of the normal distribution times the probability *p*:

*S*^{-1} (p) = μ + σ · Φ^{-1}**(p)**

If you are trying to find a subject who is larger than 95% of all other subjects, then the subject must be larger than the 95th percentile of subject’s heights — the value that is greater than 95% of other values — a probability is 0.95.

Let’s say we know that distribution of heights has a mean of 175 cm and a standard deviation of 8 cm. The value of Φ^{-1}(0.95) is approximately 1.645. Therefore, the inverse survival function at p = 0.95 is:

*S*^{-1} (.05) = 175 + 8 · 1.645

Consequently, the inverse survival function at p = 0.95 is 188.16 cm. This implies that a subject with a height of 188.16 cm is greater 95% of all other subjects.

Thus, if you are seeking a subject larger than 95% of all other subjects, they must be at least 188.16 cm tall.

## Inverse survival function vs percent point function

While the percent point function (PPF) is the inverse of the cumulative distribution function (CDF), the inverse survival function is the inverse of the survival function (ISF).

The ISF can be defined in terms of the PPF [2]

**Z(α) = G(1 − α)**,

Where

- Z = the PPF
- α = the probability
- G = the CDF.

## Inverse survival function vs. quantile function

The inverse survival function and the quantile function are closely related. The quantile function, Q(p), is the time at which a certain quantile of the distribution is achieved. The ISF at *q* is the same as the *(*1-*q)* quantile of a distribution [1]. The ISF is the inverse of the survival function, while the quantile function takes a probability as input and returns the corresponding quantile

Another way to look at this, is that the (1 – q)^{th} quantile of a distribution is the time where 1 – q of the population has survived. This means that the ISF at q is equivalent to the (1 – q)^{th} quantile. For example, if theISF at q = 0.6 is 60 years, then this means that 60% of the population will survive to at least 60 years.

## References

- InverseSurvivalFunction. Retrieved July 23, 2023 from: https://reference.wolfram.com/language/ref/InverseSurvivalFunction.html
- NIST. 1.3.6.2. Related Distributions. https://www.itl.nist.gov/div898/handbook/eda/section3/eda362.htm