# Increasing effect size in a linear regression

In a linear regression, if I have a right hand term that has a small significant effect, because the variable is measured in tiny units, is there anything I can do about that (by adjusting the regression output) after I run the regression, or do I need to go back and change the units of the variable and run it again?

For example, if the number of units a factory creates each week is related to the perception of the factory among the local populace, and it makes 5000000 unit each week, each one will have a tiny effect. Can I recalculate units made as units made per hour or whatever to produce a bigger effect size? And if so, do I have to go back to the data to do this, or can I simply adjust the regression output?

(I suspect the recommendation will be to go back and adjust the data, but if I was working with an equation about car acceleration say, I could easily adjust velocity from meters per second to km per second in the equation without going back to the race track with a stop watch, so why couldn't I adjust the effect size from units to units per hour?)

## 2 Answers

You can adjust the data if you want to, but you don't need to and it will only affect the size of the coefficient on that particular variable, not its meaning, significance or anything else. The only reason to do this is to make the output easier to read.

You can also (at least for linear regression) simply multiply the coefficient by something as long as you change the units in the same way.

So, to change from units per week to units per hour, you can just multiply the coefficient by 168. Of course, this will also affect the standard error. But the p value and the meaning of the result will not change.

Quite the opposite. You do not need to adjust the units. The strength of the relationship between $X$ and $Y$ is the same no matter which units you pick. On the other hand, the magnitude of the regression coefficient is sensitive to the choice of units. This only tells us that the magnitude of the regression coefficient does not deserve any "respect" when assessing the predictive performance. We should look at

1) the change in $R^2$ when adding $X$ to the model,

2) the standardized regression coefficient of $X$

as well as many other metrics.