Normalization across columns in linear regression

Question

I have a data set I would like to normalize in two different ways before building the multiple linear regression model. My data set looks as follows:

$$ x_{1} y_{1,1} y_{1,2}...y_{1,n-1}y_{1,n}$$ $$x_{2} y_{2,1} y_{2,2}...y_{2,n-1}y_{2,n}$$ $$... $$ $$x_{m} y_{m,1} y_{m,2}...y_{m,n-1}y_{m,n} $$

...where each $x_{i}, y_{i,j}$ is a count, and each row $i$ represents a data set collected from a video with a variable length $k$.

To make it so that all the rows have values with equivalent meanings, I normalize each row by dividing all of the counts by $k$, the length of the video. Now, instead of counts, I have counts per minutes. I also want to normalize across each column (variable) to be from 0 to 1, with the idea that I can then compare the relative importance of each variables' coefficient to other variable coefficients.

I am wondering if this is even a valid normalization. Normalizing across each row is fine, but I'm having trouble figuring out whether normalizing across each column using a different normalization factor is valid. My instinct is that it isn't. If it is not valid, is there another way to achieve what I want with being able to relatively compare the importance of variables?

Answer 1

Generally, any linear transformations on columns do not have an influence on linear regression statistics. Any linear model can be treated as a collection of linear transformations over columns, such that the result is closest to the response. For example, let we have ordinal regression $y=a+b*x$. Normalizing of x results in $(x-min(x))/max(x)=1/max(x)*x-min(x)/max(x)$, and, substituting x in a regression with its normalized value gives new $y=a-min(x)/max(x)+b/max(x)*x$. The same is with multiple linear regression. So no any statistics is changing - only regression coefficients have another prespecified values. So you can perform normalizing without any cautions. The only point is to keep in mind the normalizing made when interpreting the model.