Hello all,

I predict an u-shaped relation between two variables, y and x. If I perform a quadratic linear regression, i.e. use a model like

y = b1 + b2x + b3x^2 + error,

then the coefficients b2 and b3 are not significant. However, if I change the exponent to something less than 2, e.g. 1.5, I obtain significance. In other words a model like

y = b1 + b2x + b3x^1.5 + error,

yields significant estimates of b2 and b3. The curvature is still quite marked using the exponent of 1.5. I can even use an exponent of 1.1 and obtain significance and a nice shape. But of course I dont think I can simply choose the exponent based on what gives me significance. Or can I?

I have tried to run a non-linear regression where the exponent was a free parameter, but although it tend to yield an exponent around 1 to 2, everything became highly insignificant.

I have also tried to use a splines as well as a piecewise constant formulation, but again the results are less than ideal.

I am wondering if it would be considered bad form to use an exponent of e.g. 1.5 based on the fact that it yields significance? It would appear a little arbitrary in my eyes, since then I could just as well choose an exponent of 1.1.

The non-linearity is rather apparant in a scatterplot (although extremely noisy), and the problem mostly shows up when controlling for other covariates and a simple graphical/nonparametric approach is unfeasible.

I am very interested in suggestions as how to approach this problem. Needless to say, I have been searching high and low for an answer before posting here.

Thank you in advance!

I predict an u-shaped relation between two variables, y and x. If I perform a quadratic linear regression, i.e. use a model like

y = b1 + b2x + b3x^2 + error,

then the coefficients b2 and b3 are not significant. However, if I change the exponent to something less than 2, e.g. 1.5, I obtain significance. In other words a model like

y = b1 + b2x + b3x^1.5 + error,

yields significant estimates of b2 and b3. The curvature is still quite marked using the exponent of 1.5. I can even use an exponent of 1.1 and obtain significance and a nice shape. But of course I dont think I can simply choose the exponent based on what gives me significance. Or can I?

I have tried to run a non-linear regression where the exponent was a free parameter, but although it tend to yield an exponent around 1 to 2, everything became highly insignificant.

I have also tried to use a splines as well as a piecewise constant formulation, but again the results are less than ideal.

I am wondering if it would be considered bad form to use an exponent of e.g. 1.5 based on the fact that it yields significance? It would appear a little arbitrary in my eyes, since then I could just as well choose an exponent of 1.1.

The non-linearity is rather apparant in a scatterplot (although extremely noisy), and the problem mostly shows up when controlling for other covariates and a simple graphical/nonparametric approach is unfeasible.

I am very interested in suggestions as how to approach this problem. Needless to say, I have been searching high and low for an answer before posting here.

Thank you in advance!

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