Normalized regression coefficients - interpretation

I have data containing several variables. I ran a regression model. Prior to running the model I have normalized the dependent variable Y and the independent variables X1 and X2.

After receiving the output I want to interpret the results. For example, if the coefficient of X1 is 0.15, I know that it means that for addition of one standard deviation of X1, there is an increase of 0.15 standard deviations in Y, but this is not clear.

I want to go back to the original units of Y and X for interpretation. How do I do that ? Can I simply take the "normalized coefficients", multiply by the standard deviation of Y and add the mean of Y?

Something about it doesn't make me comfortable.

• Would you please post a link to the raw data? I would like to surface fit "Y = f(X1, X2)"and see what I find with an equation search. Jan 23, 2019 at 16:02

Firstly, why did you normalized Y? It will make your output harder to interpret, and it is often not necessary to standardize the dependent variable.

I presume you have centered and scaled you X's, you can backtransform them to interpret,

        # run your model with the X's standardized

mean <- mean(x1)
sd <- sd(x1)

b1*(x1-mean)/sd

#you can also plot

plot(y~x1)

Standardising both the dependent and independent variables can be useful for presentation and coefficient interpretation, normally in simple linear regression, whenever the Pearson correlation coefficient $$r_{xy}$$ is of interest. The OLS estimation $$\hat{\beta} = r_{xy} \times \frac{s_y}{s_x}$$. Clearly, if a variable $$z$$ has been standardised to $${z\prime}$$ then it's standard deviation must equal unity, since $$\sqrt{\mathbb{V}[{z\prime}]} = \sqrt{\mathbb{V}\left[\frac{z-\bar{z}}{\sqrt{\mathbb{V}[z]}} \right]} = \sqrt{\frac{1}{\mathbb{V}[z]}{\mathbb{V}[z-\bar{z}]}} = 1$$ Therefore, when both $$y$$ and $$x$$ have been standardised $$s_y=s_x=1$$ and the OLS estimator for $$\hat\beta$$ is the correlation coefficient $$r_{xy}$$. Note also that the OLS estimator of $$\alpha$$ is zero for standardised variables since $$\bar{x}=\bar{y}=0$$.
In your multiple regression context, if your $$X_i$$ are independent then the $$\beta_i$$ will be equal to the correlation coefficients if you had performed separate simple regressions.
If the $$X_i$$ are not independent, each $$\beta_k$$, a partial correlation coefficient, shows the relationship between $$Y$$ and $$X_k$$ with the other $$X_i$$ fixed. This can have a different value to the simple correlation coefficient between $$Y$$ and $$X_k$$ in a simple linear regression ignoring the other $$X_i$$.