# Survivorship Curves: Type II curve mortality rate is not constant?

In graphs of survivorship curves, I'm seeing that the Type II curves are straight lines, and the supplementary text says that the mortality rate is constant (i.e. the slope of the line is constant). However, it's also clearly stated that the y-axis is a logarithmic scale, which means that the original Type II curve is exponential: $$\ln y=-rx+b$$ $$y=Ae^{-rx}$$

This implies that the real mortality rate is not constant, but changes as $$y'=-Ar(e^{-rx})$$ How are we defining the word rate, anyway?

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Presumably $r$ (or $-r$) is the rate, which would appear to stay constant. – Amory Sep 24 '13 at 2:38

## 2 Answers

$r$ is the individual mortality rate per time step. Survivorship curves (plotted on a log scale) show the proportion of individuals surviving with time, and with a Type II curve a constant proportion is dying at each time step (constant mortality with age, $r$). When the model is expressed as $y$ you are looking at the number of individuals surviving to a time step, which is an negative exponential for Type II, since a smaller and smaller proportion of the population remains at each time step. Therefore $y'$ is the change (slope) in the number of individuals surviving over time. This can be seen as a rate, but it is not the mortality rate of individuals, and it is not accurate to describe it as the "..real mortality rate..".

For an introduction to survivorship curves you can also look at "Survivorship Curves" from Nature Education

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This is a statistical property of the curve - in time to event analysis (which is what a survivorship curve is), a constant hazard (the instantaneous probability of an event occurring in time t given it has not occurred already) will yield an exponentially distributed survival function. When graphed on a log axis, this function looks like a straight line.

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