# Determining where no correlation exists

I need to determine at what value of a variable mictrb there would be almost no correlation between two other variables: Equivalent width and Abundance (description and graphs below).

I have been told that when mictrb = 0.9, then there is no correlation between Equivalent Width and Abundance. The first chart below, I've been told, shows this. The next two charts show different values of 'mictrb' and the correlations seen between Equivalent width and Abundance.

The technique often used here is to adjust mictrb to remove any trend in abundance as a function of Equivalent width.

Could someone describe what I'm looking at here? What is it about the first chart which indicates there is no correlation between the two axes, whilst the other two charts don't show this?

mictrb is the micro-turbulence present in the atmosphere of a star.
Equivalent width is the width of an element's waveform, think sine-wave (x-axis)
Abundance is the amount of that element present in the atmosphere (y-axis).   • What is mictrb? And abundance and equivalent width of what? Please give some context, people here aren't soothsayers.. – naught101 Apr 27 '12 at 5:57
• Also, any chance you could invert the colours on those images? – naught101 Apr 27 '12 at 6:36
• Is there a reason to believe that the width and the abundance would be uncorrelated when mictrb = 0.9? Looking at the plots, that doesn't seem to be the case (although they seem uncorrelated when mictrb = 0). – MånsT Apr 27 '12 at 12:24
• (+1) This question may seem confusing because what the OP has been told is exactly wrong--but that's what makes it a great question! It invites us to explain how to interpret the statistics and plots shown here. – whuber Apr 27 '12 at 14:43
• I think my answer from a couple of days ago still addresses your question, even after the edit. The relation b/t EW & A is better described as 'almost none' when MT = 0, as in the 3rd plot. When MT = .9, the cor b/t EQ & A is moderate (-.43) & 'highly significant', all just as before. If you fit a multiple regression model w/ an interaction term, you can compute exactly where there would be zero correlation, as I described. If you need more info, can you describe what specifically you want to know that I didn't mention? – gung - Reinstate Monica Apr 30 '12 at 0:47

## 1 Answer

I think someone is confused here. Looking at the figures you provide, it is clear that there is a correlation between Equivalent-width and Abundance when mictrb=.9. The top figure plainly states that the correlation is -.43, and the probability of getting a value that far from 0, if 0 were the true value, is .01; looking at the scatterplot, that seems correct, the points tend to get lower as you move from left to right. On the other hand, as @MansT points out, the last figure reports that when mictrb=0, the correlation is -.2, and the p-value is .27. (I should note in passing that I'm not sure if the data look bivariate normal, and that the true relationship appears somewhat curvilinear in all three plots.)

In general, you can think of this as a multiple regression scenario in which you want to describe how Abundance is related to Equivalent-width, micro-turbulence, and the interaction between them:
$$\text{log}_{10}A=\beta_0+\beta_1\text{mictrb}+\beta_2w_\lambda\text{[A]}+\beta_3\text{mictrb*}w_\lambda\text{[A]}$$

(Sorry, I can't figure out the mathjax for Angstrom, maybe I'll edit it later.)

A first approximation of the level of mictrb where there is no correlation between Equivalent-width and Abundance would be to solve for the value of mictrb that causes the last two terms to cancel each other out. For instance, if $\beta_2=1$ and $\beta_3=2$, then there should be no relation between Abundance and Equivalent-width when mictrb = -.5 (I suspect it doesn't make any sense to have a negative Micro-turbulence, this is just an example, and that would imply that they are never truly uncorrelated.)

• Thanks gung. You're absolutely correct that there is never truly no correlation. I have clarified this and updated the question. The objective here is to find the point where there is almost no correlation between the two. I have also been told that the figures aren't always accurate as they can be skewed by outliers. Hence why this is also done visually. – Carl Apr 30 '12 at 0:41
• To inspect this visually, you want to look at plots of EW vs A at different levels of MT. You want to find the level of MT where the red regression line is as close as possible to perfectly horizontal. Based on the 3 plots shown, that appears to be the last plot where MT = 0. B/c the slope gets flatter as MT gets lower, I'm guessing that it would be perfectly flat if MT were negative (although I'm guessing that's not possible in reality). – gung - Reinstate Monica Apr 30 '12 at 0:55