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I want to standardise my data for use in a regression. Taking my variable h and following Gelman 2007, I subtract the mean and divide by two standard deviations, resulting in my new variable z.h.

However, this results in low values of h translating into negative, but relatively high magnitude, values of z.h.

I now wish to fit a linear regression of the form y = h + h^2. If I use my transformed variable z.h for this y = z.h + z.h^2, then those negative, high magnitude values for z.h which were originally corresponding to low values of h result in high positive values for z.h^2, which gives me qualitatively different results than for using the non-standardised h.

I feel like this problem could be resolved by first taking the square of h and then transforming it afterwards - however Gelman is clear that this is not the correct way to do it (see link):

We are rescaling the input variables, not the predictors. For example, age is rescaled to z.age, and the new model includes z.age and its square as predictors. The ‘age-squared’ predictor is not itself standardized. Similarly, we standardize income and ideology, and interact these standardized inputs; we do not directly standardize the income × ideology interaction.

The other solution seems to be to only divide by the two standard deviations and not subtract the mean at all as this avoids any values becoming negative. Gelman says that subtracting the mean is not strictly necessary:

Subtracting the mean of each input variable and dividing by its standard deviation. (Strictly speaking, subtracting the mean is not necessary, but this step allows main effects to be more easily interpreted in the presence of interactions.)

However, I find it very strange that choosing to take this step or not has such a large effect on and can completely change the outcome of the regression. Also, I want to go on to use multiple regression and include interactions and I also read that centring by subtracting the mean is essential for correct interpretation of this. Please can someone advise what the best course of action is? Thankyou.

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