# If my goal is to show very low correlation, how should I check for statistical significance?

I am working on a research project in social sciences. My null hypothesis is that there is a high correlation between two variables and my alternative hypothesis is that there is very low to no correlation. Why would I do it this way? I want to show that under certain conditions the mainstream theories that rely on high correlation between the two variables do not work and lead to poor policy choices.

When correlation drops to near zero, so does the p-value, the benchmark measure of statistical significance. What can I use to show that a near-zero correlation is statistically significant and is NOT due to a chance?

Edit: I am doing a time-series cross-country analysis of correlation between two macroeconomic indicators. I stratify my sample to control for other variables that are also measured on a country level. One stratum has, as expected, ~0 correlation, the other has positive correlation around 0.3. The stratum with positive correlation has p < 0.05, the one with ~0 correlation has p closer to 1, even though it has the same number of observations. How do I show that the ~0 correlation in that stratum is not due to a chance?

• @Tim I haven't finished the analysis for all samples yet, but I think given like >1000 observations I would consider even a 0.3 correlation "high" compare to ~0 correlation in the control group. I think I would refer to it as significantly positive, not high, because realistically it would be medium to low if taken out of the context. – Arthur Tarasov Jan 26 '16 at 12:22
• What does it prove if you show that some things are unrelated? Most things in the world are unrelated to each other... – Tim Jan 26 '16 at 12:41
• It matters if, say, you spend a lot of money on a policy to increase one variable hoping to affect the other variable. If I show that for certain cases there is no correlation between them, then keeping that policy would be a waste of money. – Arthur Tarasov Jan 26 '16 at 12:46
• The "taken out of context" phrase in an early comment suggests you are not actually looking at correlations--it sounds instead like you are performing some kind of regression analysis. Is that the case? If so, please disclose more information about what you're actually doing, for otherwise the answers you get may be appropriate for the question you are asking but wrong for the analysis you're actually attempting. – whuber Jan 26 '16 at 13:45
• This sounds like a job for Equivalence Testing™!! Arthur, you might try having a gander at TOST (which is one of the methodologically simpler forms of equivalence testing), and see if you want to edit your question further, which if I understand it, aims to provide evidence of equivalent correlation. – Alexis Jan 27 '16 at 3:04