I'm assuming that there is some kind of interesting relationship here as evidenced by the correlation numbers, but that the outliers are killing the covariance number. Does that sound reasonable?

Look at the scatterplot.

Look at the scatterplot again.

Describe what you see.

Is the line shown a good model to describe how A changes with B?

Can you imagine a different line (curve) to describe the relationship between A and B?

Forget about the line alltogether, look only at the points. Without using any description based on a line or a curve: how would you dewscripe what you see to someone who should reproduce a similar plot just based on your description? (this helps to focus on patterns you actually see, without overinterpreting trends you'd like to see).

If someone (without knowing the original plot) can reproduce the important featurs of this plot based only on your description, you have the relevant features. Describe them, interpret them.

And forget the correlation coefficients, determination coefficients, OLS estimates, standard errors, F- and p-values. They are not helpful.

I'd say they the linear relationship (straight line) describes the scatter fairly well for most of the states. The linear relationship is fairly strong (especially for a psychologist like me who is used to modest correlations). There are a number of states, however, that do not follow the linear trend (e.g., LA, GA, MD, and perhaps SC and AL; UT also seems to be a "special case"). You should think about how these 6 or so states may be different (substantively) with regard to the 2 variables and their relationship.

Without knowing the specific stats software that you used, my guess is that "non-robust covariance" simply means that in the OLS linear regression that you ran, the standard covariance was used as "input" ("standard" meaning the one that the Pearson correlation coefficient is based on) rather than some sort of adjusted or "robust" covariance measure. This is the default in most stats programs for linear regression.

Christian Geiser, that’s very helpful, thanks. I did everything in Python—Pandas and Plotly. Do you think I should run some other type of regression on the data here?

My guess is that there are various drivers creating a wide spread around the A axis but B starts driving the results more as A goes down.

thanks!

What is your research question and the unit in which you are interested? Strength of what relationship? Google "ecological fallacy" and see if this is applicable. Other than that, I wish Jochen Wilhelm had mentioned scatterplot a few more times. ;o)

@Jochen, this one is for you...

Thank you, Russ Mcbride , that's nice. However, it still seems that the fit strongly over-estimates almost all values for 50 < B < 100 and it still predicts a clearly negative B value for UT (what I suppose should be impossible).

I attached a picture showing a log-log-linear model, which seems more reasonable to me. The solid line indicates the fit, the dashed lines indicate the limits of the 95% prediction interval. Both plots are identical and show the same data and the samemodel, except that the axes in the right plot are logarithmically scaled. This makes the linear relationsship of the logs more clear.

But is still shows the problem with these data: the variance for B < 50 is huge, and if you only look in this range, there is no trend visisble. For B > 50, there seems to be a slight decreasing trend, but way less clear than from the whole set of data. I don't know the theory behind, but there might be a reason that values (countries?) with small B may not be simply comparable to values with larger B. The entire negative correlation you see may not reflect a direct relationship between A and B, and a different predictor (other than B) might be much better. In any case it seems that you cannot say much about how A will change if B increases "just a bit": for lower B, A will just wildly scatter between 0.05 and 0.4, and for larger B, A will scatter at a lower level between 0.02 and 0.1. To me - being completely ignorant of any possibly sensible theory behind these data - anything more than this seems to be an over-interpretation.

A few general points. As noted by others there a curvilinear relationship that is consistent with relationships which are super additive including log linear ones . For state level data as noted by Daniel Wright there may be clustering which can produce ecological correlations (in which the pattern at the state level might be very different from that at the individual level within state - this might or might not matter depending on your research question).

Here the Spearman correlation is (as one would expect) higher than the other two coefficients you used. That's because the Spearman correlation just looks at the ranks and in effect tests whether the relationship is monotonically increasing or decreasing (which would be the case if the relationship follows a roughly log linear pattern or any increasing/decreasing curve rather than say U-shaped).

Strength of relationship isn't a defined statistical term and there are numerous to define strength (including r and r^2 from the Pearson correlation).

This is all very helpful, thanks!

Jochen Wilhelm, yes, there is wide variance where B < 50. I neglected to mention that B is the dependent variable. My sense is that A does not "drive" B when B < 50. This makes sense given the events that B represents (which I will leave mysterious) but as A drops it does drive B. And your "log linear" model definitely looks better. I wish I knew how to generate one in python stats model + plotly....

https://itecnote.com/tecnote/python-log-log-plot-linear-regression/

https://stackoverflow.com/questions/32536226/log-log-plot-linear-regression

https://stats.stackexchange.com/questions/445038/fitting-a-regression-model-to-log-log-distributed-data

https://www.statology.org/logarithmic-regression-python/

https://www.codegrepper.com/code-examples/python/fitting+linear+regression+on+log+scale+in+python

etc.

https://www.datasciencecentral.com/data-science-simplified-part-7-log-log-regression-models/

Also, I look at this more B = f(A) so rotating the scatter plot 90o might help visualizing. Then of course substantive theory comes into play, and I don't know Russ Mcbride 's area as well as it seems the other people answering, but given his "my sense is that A does not 'drive' B when ..." and then I might say below some threshold for A rather than B (which would be confusing), a broken stick (two straight lines joined at a knot) might be a guess (but again, I don't know the theory about these data as well as others). I assume there is some reason he is not sharing his research question or why the unit is state rather than something else.

Daniel Wright , I had the same thought: switch the axes. And then look at a linear-plateau model or quadratic-plateau, or, use something that can add some wiggles, like local regression or a general additive model. A Cate-Nelson model is also useful for data shaped like this, though it's a bit different approach than these other models.

Jochen Wilhelm

Daniel Wright

Sal Mangiafico

It sounds like it's accepted practice to put the dependent variable on the Y axis instead of the X axis. Good to know! I will see if I can get some log-log-linear models drawn as suggested by Jochen Wilhelm and then see if a bivariate approach with something like a Cate-Nelson model suggested by Sal Mangiafico might make more sense. Thanks for all the help! This is part of a completed paper that I'm now convinced will require an additional author before I send it out.