How to interprete beta (b value) of an OLS regression where the dependent variable is square root transferred

[Iqbal Chowdhury]

Hello Clyde Schechter, Bruce Weaver, George Ford, lorenabarberia, and Noor Sethi,

Thank you so much for your continuous support to thrive in my educational endeavour. I am a bit confused about the correct way of interpreting the result of my regression models (OLS). In my OLS regression models I had to perform square root transformation to my dependent variable positive mental health (higher value is higher level of positive mental health). One of the results I have found with the three-category of immigrant variable (Canadian born, Recent immigrant, and Long residing immigrant) as follows:

Immigration Status (Ref Cat: Canadian Born	b
Recent Immigrant	0.25**
Long Residing immigrant	-0.67***
Constant	12.00***

I am a bit confused how to interpret the result. Can you please help me out with your knowledge and understanding of this types of results?

Thank you so so much again and again.

Iqbal

[Jeff Wooldridge]

Why are you using the square root? Does y sometimes take the value zero? What is it's range. My hunch is that the measure has only ordinal and not quantitative meaning. Is it roughly continuous or does it take on a handful of values?

If y > 0 and it has quantitative meaning, you can use log(y) to obtain a percentage change interpretation -- and that's more straightforward than if you use sqrt(y).

[Iqbal Chowdhury]

Why are you using the square root? Does y sometimes take the value zero? What is it's range. My hunch is that the measure has only ordinal and not quantitative meaning. Is it roughly continuous or does it take on a handful of values?

If y > 0 and it has quantitative meaning, you can use log(y) to obtain a percentage change interpretation -- and that's more straightforward than if you use sqrt(y).

Thank you so much, Jeff Wooldridge

I did sqrt transfer as the scale variable of Positive variable constructed from 14 items is moderately positively skewed as in the literature it says if the variable is moderately but positively skewed, the sqrt transfer is the ideal one. What do you think?

[Iqbal Chowdhury]

Hello Jeff Wooldridge,
the range is from 14 to 84. And Y is >0. Sorry for skepping this part in the last reply.

[George Ford]

Wages/income are skewed right, but never have I seen a sqrt transformation used. I'm with Jeff, use ln(y). The results will be nearly the same.

But, I suspect you're following normal procedure in your field and will get harassed if you do something else.

This may be of some help, and maybe Jeff will correct me if I'm wrong.

use https://www.stata-press.com/data/r18/wageed , clear

hist wage
g lwage = ln(wage)
g swage = sqrt(wage)

center lwage , g(lwagec)
center swage , g(swagec)

twoway kdensity lwagec || kdensity swagec

g lage = ln(age)
summ lage

reg lwage lage
margins, at(lage=(3.6 3.8))  expression(exp(predict(xb))*exp((`e(rmse)'^2)/2)) post
di e(b)[1,2] / e(b)[1,1] - 1

reg swage lage
margins, at(lage=(3.6 3.8)) expression(predict(xb)^2) post
di e(b)[1,2] / e(b)[1,1] - 1

[Bruce Weaver]

Re the -center- command George used in #5:

ssc describe center

[Iqbal Chowdhury]

Hello Jeff Wooldridge, George Ford, and Bruce Weaver, thank you so much for your valuable direction. Would it be possible for you to share with me one or two references to that I can have support to argue with my committee? Take care.

[Iqbal Chowdhury]

Wages/income are skewed right, but never have I seen a sqrt transformation used. I'm with Jeff, use ln(y). The results will be nearly the same.

But, I suspect you're following normal procedure in your field and will get harassed if you do something else.

This may be of some help, and maybe Jeff will correct me if I'm wrong.

use https://www.stata-press.com/data/r18/wageed , clear

hist wage
g lwage = ln(wage)
g swage = sqrt(wage)

center lwage , g(lwagec)
center swage , g(swagec)

twoway kdensity lwagec || kdensity swagec

g lage = ln(age)
summ lage

reg lwage lage
margins, at(lage=(3.6 3.8)) expression(exp(predict(xb))*exp((`e(rmse)'^2)/2)) post
di e(b)[1,2] / e(b)[1,1] - 1

reg swage lage
margins, at(lage=(3.6 3.8)) expression(predict(xb)^2) post
di e(b)[1,2] / e(b)[1,1] - 1

Hello @George Ford,
Thank you. If I want to calculate margins of interaction term of two variables, say logGDP and Immigration status, what should be the stata code? Thank you once again.

[Nick Cox]

I have used square root scales for visualization, usually for showing counted data where the detail that the root of zero is also zero is profoundly helpful.

For regression, I am less convinced that rooting the outcome variable is ever the best solution.

The marginal distribution of the outcome is itself not important and only pertinent if associated with massive outliers or nonlinearity that throw doubt on whether the particular y and X that appear in y = Xb imply a well-chosen functional form.

With a square root transformation, you can't rule out negative predictions somewhere -- and the fact that negative values can be squared is utterly unhelpful in this context.

As already pointed, a log scale for the outcome is consistent with thinking in terms of percentage change, and indeed with always predicting positive outcomes.

In practice, that means Poisson regression or equivalently a generalized linear model with log link, and robust errors. The glm framework for some reason appears unpopular or even unfamiliar to many economsts in particular.

A plot

twoway function log(x), ra(14 84)

shows that over the observed range, this is a mild transformation, so you can have it both ways, it is worth trying and it doesn't warp your data much.

[George Ford]

I doubt there is a citation to always you log and not sqrt. But most textbooks will propose log with skewed data.

[Iqbal Chowdhury]

George Ford and Nick Cox thank you so much.