# When the regression line is not linear

I performed a regression analysis with two datasets, each of which has size 50. One dataset is called Spatial % and the other Min values, and I wanted to check whether the two are correlated. I did the analysis in SPSS and the resulting scatterplot is as follows:

I am not that much experienced but it seems to me that a line is not the perfect fit for this scatterplot. Would a power line fit better? Or what else do you suggest?

EDIT

If I figure out that there is some sort of linear relationship between my x and the square root of y, how does that translate into the situation where y is not transformed?

Your suggestion is correct, a power line would be appropriate but be aware that the choice of the power is crucial.

Have a look at Box-Cox regressions where you can obtain the optimal power to fit your regression according to $\tilde{y}=(y^\lambda-1)/\lambda$, a Box-Cox transformation. You will obtain an optimal power that is between -1 and 1.

Easily interpretable coefficients are:

• theta=-1: the variable is transformed as 1/x
• theta=0: the variable is transformed in logarithm
• theat=1: the variable is kept as it is.

You can test the statistical significance of the optimal value with respect to each of these three values and decide on the most appropriate transformation.

• A few thoughts regarding lambda and theta. The former is clearly the parameter of the Box-Cox model, while the latter is the power itself. 1) Are they related? 2) How do you compute lambda? – FaCoffee Nov 12 '15 at 10:09
• @FrancescoCastellani, I am sorry, I mistook theta for lambda. My understanding of the command in Stata is that they are the same. – user89073 Nov 12 '15 at 10:31
• Ok. If they are the same, then we have a big problem when theta=lambda=0 (second bullet point). I guess we should have a simple y_transf=ln(y). – FaCoffee Nov 12 '15 at 10:42
• Yes, that is exactly that. When theta=0, we have the logarithm. – user89073 Nov 12 '15 at 12:12