# Summation of terms in survival function giving negative probability

The following expression represents a survival function for a group size (n), where survival is a percentage on a scale of 0 to 100.

$$p(x) = 100 - \beta(x \cdot x) = 100 - \beta\sum_{i=1}^{n}x_i^2$$

According to this paper, plotting this expression looks as follows:

But isn't this only true if the summation leads to something equal or less than 1? Anything more than 1 in the summation will lead to negative probability (impossible) of survival.

For example, using a group size of 5, and an "action" value of 0.75, leads to a sum of:

(0.75^2) + (0.75^2) + (0.75^2) + (0.75^2) + (0.75^2) = 2.8125


...which gives a p(x) of (if using a beta of 100):

100 - (100*2.8125) = 100 - 281.25 = -181.25


Based on the plot by the authors, it seems as though the only way to get their results is if the group size (n) is 1. But this wouldn't be a group.

What am I missing? Am I doing the summation wrong?

No, you are not misunderstanding anything. As the authors say immediately before this figure:

Note in (2) that it is possible for some actions x to result in p(x) < 0, which is interpreted to mean that the group has no chance of survival.

You are also correct that Figure 1 is a plot of the survival probability for a single individual. Note that there is no summation in the image title. In general the different individuals in a group will have different $$x$$ values, and so an accurate plot of the survival probability for a group of 2 would be a 3-dimensional image. For larger groups an accurate plot would be impossible to visualize. A plot of the survival function for 1 individual might not be what you hoped to see, but it does illustrate the proposed mechanism. In particular, the smooth quadratic relationship between $$x$$ and the survival probability.

Finding negative estimated probabilities is a general problem with "linear probability models". This problem was recognized very early on, so models using non-linear link functions like logit or probit links are much more common. But sometimes linear models are still useful.

• Thank you for the great answer. I did see the “some actions…” comment in the paper related to negative survival probabilities, but since they said “for SOME actions…” I didn’t expect it be for all group numbers except 1. But I guess this was always under the assumption the group size was 1. As for removing the summation in order to show the “quadratic relationship”, this seems like an extreme simplification. Can we really say the “line” (high dimensional “surface”) would be quadratic in a group (e.g. n = 10). Jun 12 at 16:14
• "can we really say the 'line' would be quadratic in a group?" From the individual's perspective their actions have a quadratic effect on group survival. Eg, if the other 9 members have made actions resulting in p(x) = 0.8, then the 10th action reduces the survival probability by $\beta x_{10}^2$. Ie, if you view $x$ as a vector of group actions then survival probability is quadratic in each component of $x$. Proportionally changing the behavior of the whole group is also quadratically related to survival probability. Jun 12 at 16:51
• I think to show this ("changing the behavior of the whole group is also quadratically related to survival probability" one would have to prove that a sum of quadratic components is itself a quadratic result. Of course this includes the drastic, and quite unrealistic, approximation that a group would operate as a sum. Jun 12 at 16:58
• I don't think we understand each other? $\beta((px_1)^2+...+(px_n)^2)=\beta p^2(x_1^2+...+x_n^2)$. Perhaps the assumed model is unrealistic, but if the model is true then proportionally changing the actions of the group has a quadratic effect on group survival. Maybe that terminology is inaccurate, but there's clearly some sort of relation. Of course, this is not the point of the article. Jun 12 at 17:24