Best practices for extrapolating data

Question

I have a set of variables that parameterize a logistic equation bacterial growth model. The parameters change based on temperature (e.g., growth speeds up at higher temperatures) and so it is necessary to fit some secondary models in order to be able to estimate growth under a variety of temperatures. Only five temperatures (16, 20, 25, 30 and 35 C) have been measured under laboratory conditions, and so the secondary equations are fit from these five sets of parameters.

I have noticed that the growth model is very sensitive to slight changes in the parameters - even the small difference between the observed and predicted parameters resulted in a large end difference in growth. So it is very desirable to include the real observed set of parameters in the model. I can do this using a higher order polynomial and I am comfortable with the values that are interpolated in between (because at the very least they are probably close, and the model then includes the exact values at temp=16, 20, etc). However, to make a long story long, once I extrapolate below the lowest observed values (with this higher order polynomial) things get crazy.

I know extrapolation is always kind of dicey in terms of uncertainty, but I do need to have values for parameters down to 10 C. Are there some methods that tend to work better than others? Or what could I even search for? (This is a new subject to me and wikipedia's entry on extrapolation mostly suggests "don't do it".) If all else fails I can hold the parameters constant from 10 - 16 C (which is conservative) but it would be nice to use something better.

Edits: In order to elaborate, the logistic equation I am using is: $y(t)= \frac{\alpha }{1+e^{\beta-rt}}$

$\alpha ,\beta$ and a $r$ are my parameters that change with temperature. Here are the parameters that were fit to growth curves measured in a laboratory:

Temperature  α        β         r
16          3.2366    3.4698    0.0297
20          3.5557    3.3770    0.1256
25          3.6941    3.9876    0.3300
30          3.9081    4.0331    0.5422
35        3.9868      3.8333    0.8405

And I would like to fit curves to these to know (for instance) what these three will be at 18 C. It is standard in the microbiological literature to use certain types of relationships but I find that they're incredibly inaccurate and I trust the laboratory observed values the most so I'm willing to use something ad hoc. Here is an image of what it looks like for $\beta$ using a polynomial

And, here is a truncated image from the paper where I got the original data just to give you an idea (Yang et al. 2009. Food microbiology 26: 606-614).

Answer 1

To answer your general question, in order to extrapolate you need to be confident that the form of your model is rooted in reality and applies over the extrapolated region. For instance, puppychart.com can extrapolate a dog's weight from very little data because it knows that dogs grow according to a logistic curve. An anti-ballistic missile works because it can extrapolate a ballistic missile's position from a few observations using a physical model of a ballistic flight path (quadratic effect of gravity among other factors).

Your extrapolation problem (thanks for revising the question with details) is perhaps an extra step removed from reality since you are using the parameters of another model as the response in your model. But at least those parameters have physical meanings (e.g., $\alpha$ is the asymptotic growth limit). Surely, there's no basis for expecting $\beta$ to behave as a fourth order polynomial.

Best would be to fit the original data to a new model containing both time and temperature as factors. If you must extrapolate (for instance, to get starting points for another experiment), use models that are simple and smooth and don't extrapolate too far. Then you'll have some safety under the principle that all processes are locally smooth.

You might model $\alpha$ and $\beta$ as linear and $r$ as logistic. All of the parameters need to be positive for this to be a growth model, but only $r$ is in danger of going negative from a local linear extrapolation. Plus $r$ appears to be exponentially approaching 0 as temperature decreases. Given the exponential decay of $r$, even going down to 14 C may be a stretch.

In case a real statistician wants to explore further, here is my screen-scraped version of the data from the original four plots.

Answer 2

On your plot of beta, there's really no way to tell whether the curve should turn up/down or remain flat below 16. I tried making some plots using different smoothing splines, and a small tweak in a tuning parameter can completely change the shape of the curve outside the range of the data.

If you have good reason to believe the curve should be flat below 16, I think you can just make it be so. It seems to me that, in this case, the subject matter knowledge you have will take you much further than any automatic/quantitative methods can.