# Linear regression of dependent variable squared & retransformation

I have performed linear regression of a dependent variable squared, & my statistics package produced least squares means for each level of categorical variables that I would like in original units. There is much research on the retransformation bias when the transformation power is a positive fractional one (i.e. <0 AND <1), though what about when the power is >1 - is there bias & if so how may it be corrected?

Best Regards,

Mick.

• Was the purpose of squaring the data solely for the purpose of using linear regression rather than non-linear regression? – James Phillips Nov 16 '18 at 13:35
• where you said "<0 AND <1" did you mean > 0 AND <1? – Glen_b Nov 17 '18 at 3:30
• James Phillips - yes, the purpose of squaring was to use linear regression, as I am not that familiar with non-linear... though my statitics package (JMP 14) does have that platform. – Mick Nov 19 '18 at 1:09
• Glen_b - apologies - yes I mean between 0 AND 1 as you described. – Mick Nov 19 '18 at 1:10

## 1 Answer

In statistics, if $$\hat \theta$$ is unbiased estimate of $$\theta$$, $$f(\hat\theta)$$ is biased estimate of $$f(\theta)$$, given function $$f(\theta)$$ is not linear on $$\theta$$.

In your situation, suppose least squares means ($$\hat\theta$$) are your unbiased estimate of $$\theta$$, and you want the estimate of $$\sqrt{\theta}$$. Because it is not linear function, so $$\sqrt{\hat\theta}$$ is biased estimate of $$\sqrt{\theta}$$.

Let look at the Taylor series:

$$\sqrt\theta = \sqrt{\hat\theta} + \frac 12\hat\theta^{-1/2}(\theta-\hat\theta)-\frac18\hat\theta^{-3/2}(\theta-\hat\theta)^2 +...$$

Take the expectation on right hand side, you will know the unbiased estimate of $$\sqrt\theta$$. But we have no knowledge on the high order moments of the estimate $$\hat\theta$$, the approximation is stooped at the second order. So you can use following ti adjust the biased:

$$\sqrt{\hat\theta} -\frac18\hat\theta^{-3/2}var(\hat\theta)$$

From this estimate, you can say, if your sample size is large such that $$var(\hat\theta)$$ is very small, and/or $$\hat\theta$$ is large (far from zero), the effect of second order adjustion is so small such that it is ignorable.