# How to test the linearity between two non normal distributed variables

I have two variables $(x_i,y_i), \; i=1, \dots 300$ and I would have liked to apply a linear regression on them, but as you can see in the scatterplot below I have a very bad linear trend.

As I need to motivate it in an essay, I would like to have a measure of the amount of linearity and I used the Pearson product-moment correlation coefficient, obtaining a value of -0.1559585. But after having tested the normality of the variables with a Shapiro-Wilk test, I have obtained that that the X values are not normally distributed, therefore I cannot use the Pearson coefficient to do that. I read that I could computed Spearman's rank correlation coefficient as the X values don't follow a normal distribution, but this coefficient gives an estimation of the monotonic association between X and Y, while I would like to have a quantification of the linearity between X and Y. Do you know how I can compute a quantity that express this, please?

Thank you very much.

Edit: The qqplot of X is the following

• Have you thought about using a Box-Cox transformation? – JohnK Dec 5 '15 at 16:44
• Thank you for your advice :) No, but I have just read about it and I tried a Box-Cox transformation on X and I have obtained a better qq plot. I draw an histogram of X and the data now seem gaussian. I have calculated the Pearson coefficient for Y and X transformed, and I have obtained a result of -0.1419915, slightly less than before the transformation. But I don't know how I should interpret this result. What does this tell about the relationship between the original data? Doesn't the transformation change the nature of the data? – Helm Dec 5 '15 at 18:47
• Two points. First, it seems that there is some hard limit with many points bunched up at about Y = 28, below which no points are found. If there is such a limit, then your investigation of a linear relation is fraught with difficulty. Second, why do you want "a measure of the amount of linearity" in the first place? Wouldn't a clear description of the relation between X and Y be better, even in transformed scales, regardless of linearity in the original scales? – EdM Dec 5 '15 at 19:53
• 1) Yes, the Y values are all of them measure of a quantity that has been recorded if and only if it has a value greater than 26, why does this affects the investigation of the linear relationship? 2) I need to do a lot of regressions (more than 75) on identical datasets aquired at different time scales and I would like to have something - a number - that quantifies if I can apply the regression or not for each dataset(and the only assumption that fails is the linearity), as I cannot show a scatter plot for each dataset. – Helm Dec 5 '15 at 20:17

Why are you even looking at the distribution of $X$? This has no effect on whether or not the relationship between $X$ and $Y$ is linear. But that aside, Pearson's correlation measures the strength of linear association, period. There are no distributional assumptions needed. Just look at a scatterplot of the points (which by the way you haven't shown, you've provided a Q-Q plot) to see if it's linear and report the correlation.

Also, goodness of fit tests will almost always result in a rejection of the null hypothesis with any reasonable sample size, so they shouldn't be relied on too much.

• Thank you for your answer. I am looking at the distribution of X because I hava read a lot of references(here an example statstutor.ac.uk/resources/uploaded/pearsons.pdf) that say that before computing Pearson's correlation coefficient, the normality of the variables should be checked. Aren't they correct? Thank you. P.S. Sorry, in my last question I have changed the scatterplot by chance, now it is fixed. – Helm Dec 5 '15 at 19:54
• The person in this article doesn't seem to give any reason for why data have to be bivariate normal in order for a correlation to be meaningful. People justifiably calculate correlations all the time without bothering to check for such an assumption. Just as a counterexample, let $X$ have any non-normal distribution and set $Y = X$. The correlation is one and this tells us quite a lot about the relationship without even mentioning the distributions involved. Statistics aside you can tell just by looking at the plot that $x$ and $y$ don't have much of a relationship here linear or otherwise – dsaxton Dec 5 '15 at 20:24

You might use a Q-Q plot to further assess normality. The X values do not appear too skewed. If the Q-Q plot does not suggest a drastic departure from normality I think $R^2$ (square of the Pearson) is a reasonable measure of linearity. I have never done a Shapiro-Wilk test on real data that did not turn out to be significant so I don't depend on it alone.

• Thank you very much for your help. I have posted in the question the QQ plot, it seems not a good normal distribution. Just to be sure, should I fit my points with a linear model and look at the obtained R2? As it is a measure of how much a straight line adapts to the data, can I consider it as a measure of linearity? Thank you very much. – Helm Dec 5 '15 at 16:43
• This is the QQ plot of X? It doesn't look that bad to me, maybe a bit skewed but it is not a U-shaped distribution or bi-modal. Try replacing the qqnorm command with qqline and see if the red line does not go straight through most of it. Yes, I would report $R^2$. Even if you transformed X to be normal, you can tell from the scatter plot that there is not much of a trend here so $R^2$ is still likely to be low. Try a box-cox transformation on X and see what happens. – Derrick Kaufman Dec 5 '15 at 16:48
• Thank you for your answer. I have attached to the question the qqplot with the qqline. I also tried a box-cox transformation and I reported the result in my answer above to JohnK, but I don't know how to interpret this result. Might you have any suggestion, please? – Helm Dec 5 '15 at 18:51
• I'd say after the transformation that makes X more normal and the Pearson coef is still low means there is no linear relationship and you have your answer. Lower by .01 is essentially the same. – Derrick Kaufman Dec 5 '15 at 19:08
• But doesn't the transformation change the nature of the data? Why I can rely on a coefficiente computed after a modification of the values of the data? Sorry, I have difficulties to understand this. – Helm Dec 5 '15 at 19:56