Isn't it just the number of standard deviations from the equivalent line, that indicates how linear it is?

It would be, but the equivalent line is uncertain due the the uncertainty of the data points. The only thing I have is that the regression model should be a linear fit, but due the introduced uncertainty, the best-of-fit curve would have this as well. I have though found two methods to include this uncertainty. One method is using GraphPad Prism to perform nonlinear regression analysis with an intial linear curve. Prism accounts for the existing uncertainty and can subsequently tell you the significance of an inadequate model by a replicates test for lack of fit. Another (manual) method would be scatter plotting the replicates that form the uncertain mean and applying a linear best of fit curve to it. One can subsequently derive the significance of the deviation from best-fit by using the same calculation as the previous method. Additionally/alternatively a residuals plot can be created to show this deviation visually.

Ok, I am interested in this myself because I am trying to model the uncertainty in the human brain.

The problem as I understand it, is that statistics tends to smooth the uncertainty away, So most of your tools like regression analysis are going to give you a best curve fit. I am not sure what the chaos theory adherents are using for mathematics, but lets think about the nature of the curve.

Essentially we can see the curve as an expression of uncertainty, or we can see the uncertainty as a deviation from the curve. As near as I can tell, statistics does the latter. As I understand it, chaos theory, sees the curve as an expression of strange attractors. In your scatter plot, this might be seen as dark spots where there is more deviant data. Dynamism is a problem for statistics it is not a problem for chaos theory, which would assume that the centroid of the set of deviant data, is probably an attractor.

As near as I can tell, once you find the attractors, the next step is to find the standard deviation of the data from the attractor, to find the range of dynamism. Now here is where it gets interesting. How do you model the curve from the attractors over the full range of dynamic options indicated by the standard deviation from the attractor? It's a different way of building your curve fitting algorithm isn't it.

I'm not too versed in chaos theory (or so much in statistics for that matter), but to suggest an answer to your question:

I reckon, as you have a dynamic system, you will have multiple curves which converge increasingly the closer you get to the attractors. Perhaps not a best-of-fit curve will exist, rather a range in which these curves can exist? Just speculating

One thing that I think is happening in the brain, is a sort of modeling step, where the range of results are compared to the norm to get a feel for how dynamic the results are likely to be. We want to reduce uncertainty when it is moving in a negative trend, and enhance uncertainty when it is moving in a positive trend, without getting stuck in a positive feedback loop.

Sebastian, given that the x-axis is on a log scale, these are *both* clearly non-linear. The general reason for transforming the data like this is to linearize it, or make it approximately linear to simplify further analysis. That said, given that the data are distributed about each point, I think you will have a hard time making any argument that one of these transformed groups is linear and the other is non-linear. First, you should strengthen your visual argument by showing boxplots to show the distribution about each point, and by limiting the graph to the part with data (e.g. don't show the y-axis below 15).

Aside from improving the visual argument, I think your best bet is to compute the Pearson correlation for each transformed group, and then test for statistical significance for the difference between the two correlation coefficients. The Pearson correlation coefficient will be higher for relationships that are more linear, so by showing statistical significance in the difference of coefficients, you can make an argument that the graph on the right is more linear. Unless your data are very tightly distributed, I am guessing from visual inspection of your graphs that you won't find a significant difference.

I'm not sure that you will get anywhere with this. However, as you have some expectation that the transformed graph on the left should be non-linear, maybe it would be more productive to try formalizing that expectation into a model, and then try fitting the data to that model.

@Vincent

Thanks for the tips. I'll indeed explore ways to visualise it better. The problem with the boxplot or error bars for this data is that the error is so small for each data point, that this becomes hard to visualise, even if I don't show y-axis below 15.

As for the Pearsons correlation, I've thought about this, however my goal is to treat each graph separately and test them against linearity, rather than asking whether one is more linear than the other. In addition, the resulting correlation coefficient is open to interpretation, which results in the question being of "how linear is the transformed data?", rather than "does the transformed not significantly deviate from a linear model?"

This is why, as you mentioned, I've tried formalising the expectation of linearity and tested the data against that. The results matched the visual data well and I'm glad to now have a mathematical argument in addition to the visual one.

Lastly, I could have performed logarithmic regression, but I found it easier to visualise this data as linear by transforming it.

>The problem with the boxplot or error bars for this data is that the error is so small for each data point, that this becomes hard to visualise

Well good, that is encouraging for your results. It might look weird to have what looks like a single hash mark through each point, but you can explain the small error in a caption. Some kind of visual indicator of the confidence in the data is required to make a persuasive argument. Otherwise people will look at the graph and have the first impression that both are pretty linear and they are unlikely different.

>which results in the question being of "how linear is the transformed data?", rather than "does the transformed not significantly deviate from a linear model?"

I think that these questions are philosophically equivalent here. I don't think you can pose the question in absolutist terms, because I am confident that even the graph on the right deviates from a true line to some small degree. If you posit that the one on the right is a line, then if its r is statistically different from the first graph, then I think you have a reasonable linear/not linear argument. If you are doing regression, recall that r^2 is the same for regression as it is when computed from correlation.

>This is why, as you mentioned, I've tried formalising the expectation of linearity and tested the data against that.

I was trying to suggest the converse of this, which would be to have some model for the first graph (i.e. the nonlinear one). That way you can propose what is happening in the first graph as an alternative to the linear model.

>The results matched the visual data well and I'm glad to now have a mathematical argument in addition to the visual one.

I'm not sure what this means in terms of how you came to a resolution to this. I am curious about this problem, so could you clarify what mathematical argument you are going with?

By the way, this may not come as welcome advice as you already have some tools that you are familiar with, but it is really worth it in the long run to learn some R. It usually makes very nice graphs on the first go without having to tweak a lot of parameters. Also, it's free.

I imagine you mean GNU R (also known as GNU S)? the Statistical Language?

Yes. It can be found here: http://www.r-project.org/

If you don't have experience in programming or in using a command line interface, it might be a little daunting at first. However, there is documentation on the site for getting started, and the built-in documentation for using different functions is generally very good. R is the basic statistical computing language, but there are hundreds of packages (collections of R code) written by different groups for a wide variety of applications beyond stats. One very large collection of packages geared towards biological applications is called BioConductor. If learning R is of interest to you, you will probably want to install the basic packages in the BioConductor suite.

When I start using R, it also took me sometime. However the documentation is pretty well organized. There are lots of code examples, which you can use for getting deeper into the structure and use of R. Good luck.

I've tried R in the past. Indeed, it's a bit daunting at first but started using it for some specific tasks. SPSS and Prism suit my needs at the moment, but might try R again in the future.