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

There are two quantities you should be careful not to mix up.

  • One is the number of individuals who will die during a given interval: $d_x = N_x - N_{x+1}$.
  • One is the fraction, out of those alive at the beginning of a given interval, who will die during the interval: $q_x = \frac{d_x}{N_x}$

($N_x$ being the number of survivors at age $x$.)

On a regular plot, if $d_x$ is constant then survivorship will decrease linearly; if $q_x$ is constant then survivorship will decrease geometrically. On a semilog plot, when $q_x$ is constant then this geometric decrease will look like a straight line. This is, as far as I know, the main reason to even use semilog plots in the first place; it makes geometric decrease easy to recognize. $d_x$ is sometimes called the death rate and $q_x$ is sometimes called the mortality rate.

(In my opinion this usage is careless; they aren't rates, but just numbers. Compare: there is a distinction between travelling 10 miles in an hour, and travelling at 10 miles per hour.)

$q_x$ can also be thought of as estimating the probability that someone who is age $x$ will die before age $(x+1)$. It is therefore sometimes also called the "Age-Specific Probability of Death". Notice that it is bounded between 0 and 1.

(Probabilities can't be lower than 0 or higher than 1; and the fraction dying has to be somewhere in between "none of them" and "all of them"!)

Here's an example survivorship curve, on a regular plot, I made for constant $q_x = 0.1$.

enter image description here

This much should be enough to answer your question. But, for completeness...

For Continuous Time

I complained about calling $d_x$ the death rate. That's because rates have a unit of $time^{-1}$ ("...per second", "...per hour" etc.) Of course, dividing $d_x$ by the length of the interval, does give you the average death rate over the interval. And,

  • The slope of the survivorship curve (multiplied by -1; we want a positive rate of decrease, instead of a negative rate of increase) gives you the instantaneous death rate at age $x$: $-N^{'}_{x}$.

(For instantaneous rates to be useful, we have to "smooth-out" the survivorship curve instead of letting it be a stepped curve. If it was stepped, then the slope would just always be horizontal or vertical.)

This brings us to another quantity: the "Force of Mortality":

  • The instantaneous, age-specific, per-capita death rate; or, the death rate at age $x$, divided by the cohort size at age $x$: $\mu_x = -\frac{N^{'}_{x}}{N_x}$

To illustrate, if the death rate at age $x$ is "60 deaths per minute", and if at that instant there are 60 people, then the per-capita death rate is "1 death per person per minute". Because choice of time unit is arbitrary, this is the same as saying "60 deaths per person per hour" or "525600 deaths per person per year".

The Force of Mortality is unintuitive ("per-capita" death rate? - but everybody only dies once!) but the formula is coherent. The point of having a "per-capita rate" is to enable comparison between cohorts of different sizes, or the same cohort at different ages when it was bigger vs. smaller, and so on; in general the idea is that death rate has something to do with the number of members, but also something to do with the condition of each member, and $\mu_x$ is trying to get at the latter. It can be visualized as the slope of the survivorship curve (multiplied by -1 to make it positive), divided by the height of the survivorship curve.

enter image description here

There is a similarity between these two quantities, and the two which were defined for discrete time. If $-N^{'}_{x}$ is constant, then so is $d_x$; and if $\mu_x$ is constant, then so is $q_x$. And, again: on a regular plot, if $-N^{'}_{x}$ is constant then survivorship will decrease linearly; and if $\mu_x$ is constant then survivorship will decrease exponentially.

But they are distinct. It is the $time^{-1}$ dimension, and the fact that time units are interchangeable, which makes the difference.

If there are only 60 people, then it's obviously not possible for more than 60 people to die in the next year $(0 \le d_x \le 60)$; and it's not possible for the fraction dying to be higher than "all of them" $(0 \le q_x \le 1)$.

But suppose the rate of death at this instant is "60 deaths per hour". That's the same thing as saying it's "1 death per minute" or "525600 deaths per year". There's no implication that that many people will in fact die, because there's no implication that the rate will be kept up for the whole year. If this particular death rate were held constant as long as possible, then the cohort would all be dead in exactly an hour and then the rate would then hit 0. The instantaneous death rate can be as high as you like; it just can't stay high forever.

(If a death rate of "60 deaths per hour" is measured at an instant when there are 60 people, then the per-capita death rate is "1 death per person per hour" - or, equivalently, "$\frac{1}{60}^{th}$ of a death per person per hour", or "8760 deaths per person per year".)

Although the absolute death rate can't stay high forever, the per-capita death rate ($\mu_x$) can. That's what happens during constant $\mu_x$ (exponentially decreasing $N_x$), for example. The reason why this is not paradoxical is that the slope ($-N^{'}_{x}$; the numerator) is constantly decreasing; it's just that so is the height ($N_x$; the denominator).

The Force of Mortality $\mu_x$ has also sometimes been referred to as the "mortality rate". The equivocal use of language is unfortunate. It is also, especially in reliability engineering, known as the "hazard rate".

By the way

$q_x$ and $\mu_x$ have often been conflated, at least in gerontology (with which I am most familiar). But they are not the same. The first estimates a probability, and is bounded between 0 and 1. The second is not a probability, and has no upper bound.

One way they have been conflated is with the "Gompertz equation".

A Type II Survivorship Curve is one which is decreasing exponentially. When this is the case, both $q_x$ and $\mu_x$ are constant. We would call such a species "non-aging" or "non-senescing": your age makes no difference to your vulnerability.

But many species do age.

Benjamin Gompertz proposed that for many species including humans $\mu_x$ grows exponentially throughout adulthood:

  • Gompertz' Law of Mortality: $\mu_x = \mu_0 \cdot e^{Gx}$
  • Or, in its logarithmic form: $ln(\mu_x) = ln(\mu_0) + Gx$

(Where $\mu_0$ is the "initial mortality" and $G$ is the exponential "Gompertz parameter". In the logarithmic form, $ln(\mu_0)$ is the intercept, and $G$ is the slope.)

Whether or not $\mu_x$ follows such a law indefinitely is an empirical matter. (It's actually debated whether it decelerates in late life, and if so why.) But notice that $q_x$ couldn't possibly grow exponentially forever, since $q_x$ couldn't possibly ever exceed 1. And yet, due to the conflation, in recent times it has often been in terms of $q_x$ that the Gompertz equation has been presented and discussed.

Further Reading

Unfortunately, as far as I know the literature (both professional and pedagogical) is sorely lacking in treatments which are both intuitive and accurate!

Peter Medawar's "The Definition and Measurement of Senescence" (Chapter 1 in The CIBA Foundation Colloquia on Ageing, Vol. 1) is probably still the best introduction to the rationale behind why we care about survivorship curves and mortality, from the perspective of biogerontology. Medawar gives the correct definition and formula for $\mu_x$ (though he doesn't stress its distinction from $q_x$; indeed, he doesn't mention $q_x$ at all).

The distinction between $q_x$ and $\mu_x$ is discussed in Gavrilov and Gavrilova's The Biology of Life Span: A Quantitative Approach, as well as in some of their papers. They present the correct forumula for $\mu_x$, but do not really try to explain it.

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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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