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,


  • 1
    $\begingroup$ Was the purpose of squaring the data solely for the purpose of using linear regression rather than non-linear regression? $\endgroup$ – James Phillips Nov 16 '18 at 13:35
  • $\begingroup$ where you said "<0 AND <1" did you mean > 0 AND <1? $\endgroup$ – Glen_b Nov 17 '18 at 3:30
  • $\begingroup$ 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. $\endgroup$ – Mick Nov 19 '18 at 1:09
  • $\begingroup$ Glen_b - apologies - yes I mean between 0 AND 1 as you described. $\endgroup$ – Mick Nov 19 '18 at 1:10

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.


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