# How do I test a hypothesis that proposes existence of a positive relationship between two variables?

For my empirical experiment, I have analyzed the behavior between two variables. My one-sided hypothesis is "Increase variable A, then will variable B also increase." Now, I want to do a statistical test to confirm my hypothesis.

I have plotted the two variables. Below are some plots of my results. The blue circles are the measured data, and the orange line is the plotted linear regression.

I do a regression over the data, but it seems to be non-linear. Which test do I have to do now to test my hypothesis?

• which type of regression did you implement ? And which graph is showing nonlinear relationship ? – Subhash C. Davar Jul 16 '19 at 4:34
• how do you say it seems non-linear ? – Subhash C. Davar Jul 17 '19 at 14:09
• I did a linear regression. The orange graph is the regression of my data. It seems to be non-linear, because of the residuen, they are huge. – joe Jul 18 '19 at 6:06
• which figure shows non-linear regression results ? – Subhash C. Davar Jul 19 '19 at 13:05

Spearman rank correlation is another option. It can "catch" any, not only linear, monotonic (increasing or decresing) relationship between variables.

You can translate your hypothesis "Increase variable A, then will variable B also increase" to "Spearman rank correlation coefficient is positive (larger than zero)" and perform significance test.

Major drawback is that, as every nonparametric procedure, it performs poorly with small sample sizes.

• OP is looking for answer based on regression ? what are the factors that prompt you to talk of Spearman rank correlation. – Subhash C. Davar Jul 17 '19 at 14:20
• OP tries to "Find the right significance test to confirm a hypothesis". OK, he's using regression but it is not the only option. – Łukasz Deryło Jul 18 '19 at 6:36
• Has anybody an offer for my problem? If there is a good test without regression, it will also be fine. I thougt to assess the hypothesis with regression will be good. – joe Jul 19 '19 at 10:01
• @SubhashC.Davar You can test significance of Pearson correlation. – Łukasz Deryło Jul 19 '19 at 10:35
• @joe I still think that Pearson correlation coefficient + test of it's significance is good option for you – Łukasz Deryło Jul 19 '19 at 10:37

Even though the relation is non-linear (or maybe it is but with a lot of noise!), you can use linear-regression and then check for significance of the coefficient corresponding to variable $$A$$ (test for $$\beta_A:=0$$ in a model $$B= \beta_0 + \beta_A A + \epsilon$$ where $$\epsilon$$ is the "error term")

The reasoning is that you just want to see whether ("B increases with A") rather than determining the underlying relationship between A and B. So, you can just if you get a "big enough positive $$\beta_A$$" with that test