# Unstandardize the slope of standardised variables in a linear regression

If I standardize my dependent and independent variable, and run a linear regression between them, the slope estimate which I have will be standardised. The variables were standardised by subtracting the mean and dividing by twice the standard deviation. I know the means and SD of the input variables

Is there any way, I can unstandardise the slope (perhaps the obvious answer is to run the regression between dependent and independent variable without standardizing them, but lets say, for the sake of argument, I cannot unstandardize my dependent and independent variables). What would be the best solution to this.

If the question is not clear, please let me know and I will try my best to restructure the question.

Thanks

• If you want to recover the full regression equation, do you know the means and standard deviations for your variables? – Silverfish Nov 22 '14 at 18:01
• yes, I know the means and standard deviations of the variable. Also the variables were standardised by subtracting the mean and dividing by 2SD – user53020 Nov 22 '14 at 18:40
• Are you sure you wanted to divide by 2 sd? I'm curious what the rationale is to do that rather than by one sd. – Silverfish Nov 22 '14 at 19:29
• Well the package that I am using in R arm does this internally. So it is actually not in my hand. – user53020 Nov 22 '14 at 19:34
• To quote Andrew Gelman it's "so that the generic comparison is with inputs equal to the mean ±1 standard deviation. The resulting coefficients are then directly comparable for untransformed binary predictors... We recommend our rescaling as a default option — an improvement upon the usual approach of including variables in whatever way they are coded in the data file — so that the magnitudes of coefficients can be directly compared as a matter of routine statistical practice" STATISTICS IN MEDICINE Statist. Med. 2008; 27 :2865–2873 – Silverfish Nov 22 '14 at 21:14

Assuming you know, or can calculate, the standard deviations of the values of $x$ and $y$, the standardized estimate $b'$ is given by:

$$b'=b \times \frac{s_x}{s_y}$$

So to unstandardize:

$$b=b' \times \frac{s_y}{s_x}$$

• Hi Sorry I forgot to mention, the variables were standardised by subtracting the mean and dividing by twice the standard deviation – user53020 Nov 22 '14 at 18:41
• Also can I use this to calculate the confidence interval of my slope estimate (the actual confidence interval which I have are also standardised) – user53020 Nov 22 '14 at 18:47
• First it doesn't matter what multiple of SD you divide by, as long as both $x$ and $y$ are divided by the same sd. For the bottom equation, you're just going to put $\times 0.5$ on the top and bottom, and they will cancel out. – Jeremy Miles Nov 22 '14 at 19:23
• I think you can. Do you know the sample size? Once you have the slope and the sample size, you can work out the standard error and from that the confidence intervals. (Assuming you only have one predictor. If you have more, you need to know the correlations too.) – Jeremy Miles Nov 22 '14 at 19:26
• Ya. I know the sample size. I have 33 observations of x and y. – user53020 Nov 22 '14 at 19:35