Solved – interpret the intercept and coefficients of normalized input variables in regression model

normalizationregression coefficients

My target variable of regression model is a rate between 0 – 1. When using the original input variables, it is easy to interpret the intercept, say 0.8, and coefficients of each input variable, say 0.005 etc.

But when using the normalized input variables to fit same regression model, the intercept was changed to 1.2, and coefficients of each input variable was also changed.

So in this case, how to interpret the intercept and coefficients of the normalized input variables? If the rescaling is a right way to do?

Best Answer

The intercept is still the predicted value when all the input variables are 0, only now, that is 0 after normalization which would be (usually) the mean of the variables on the original scale.

The coefficients are still the predicted change in the target per unit change in the input, but now, they are per normalized unit (usually) the standard deviation.

As a side note, if your target is bounded, the assumptions of linear regression are likely to be violated. You might want to look at beta regression.

Related Solutions

Solved – How does one interpret regression coefficients when no dumthe variables nor intercept are dropped

It is certainly a valid way to run a regression. The interpretation of the coefficients in your example ridge regression is simple:

If you are a male, then your predicted value is $\widehat{y}=\beta_0+\beta_m$
If you are a female, then your predicted value is $\widehat{y}=\beta_0+\beta_f$
The predicted difference between males and females is $\beta_f-\beta_m$

Then the interpretation is completely analogous if you have more than one categorical value, or if your variable has more than two categories. For example, if gender had a "not given" category that you wanted to include in your model with $x_n$, then you would simply add that:

If gender is not given, then your predicted value is $\widehat{y}=\beta_0+\beta_n$
The predicted difference between males and "not given"'s is $\beta_m-\beta_n$
The predicted difference between females and "not given"'s is $\beta_f-\beta_n$

And you can keep adding similar examples. There's no limitation on the interpretation of the coefficients because of the intercept/dummy issue.

Solved – Normalized regression coefficients – interpretation

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)
curve(b0 + b1*(x1-mean)/sd, add=TRUE)

I also recommend to use Y at its original scale, since standardization does not change the distribution shape.

Best Answer

Related Solutions

Solved – How does one interpret regression coefficients when no dumthe variables nor intercept are dropped

Solved – Normalized regression coefficients – interpretation

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