Functional Relationship of Survival Parameters – One Group and Two Groups Tabs

The parameter conversions in this tool assume the event times follow an exponential survival distribution. Using the hazard rate equations below, any of the four survival parameters can be obtained from any of the other parameters.

Exponential Distribution

The density function of the exponential is defined as

f(t)=he−ht

The probability of surviving the first t years is

S(t)=e−ht

The mortality (probability of an adverse event during the first t years) is

M(t)=1−e−ht

For an exponential distribution, the mean survival is 1/h, and the median is ln(2)/h.

Based on these formulas it is straightforward to translate between the hazard rate, the proportion surviving, the mortality, and the median survival time.

Hazard Rate from Median Survival Time

The median survival time is transformed to a hazard rate using the relationship h = ln(2) / MST.

Hazard Rate from Proportion Surviving

In this case, the proportion surviving until a given time T0 is specified. The proportion surviving is transformed to a hazard rate using the relationship h = –ln(S(T0)) / T0.

Hazard Rate from Mortality

Here, the mortality until a given time T0 is specified. The mortality is transformed to a hazard rate using the relationship h = –ln(1 – M(T0)) / T0.

Hazard Ratio

The hazard ratio is the simple ratio of two hazard rates: HR = h1 / h2.

Mortality Ratio

The mortality ratio is the simple ratio of two mortalities: MR = M2 / M1.

Reference

  1. NCSS PASS Sample Size Softwar Chapter 903 Survival Parameter Conversion Tool
    https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/PASS/Survival_Parameter_Conversion_Tool.pdf