Converting linear regression estimates to percentages?

joshAU

New Member
Hi all.
Is there a method to convert two estimates that have been calculated using linear regression, to a percentage?
eg, I have two X scores, two corresponding estimated Y scores, and a standard error of estimate.
Is it valid to say a given X will score higher than another given X, a specified percentage of the time?

I guess to rephrase the question...
Is there a way to calculate the percentage likelihood that the true value of the estimate for a given X score will be larger than that for another given X score?
i.e %chance that Y¹ > Y² for given X¹ and X²

example data:
X¹ is predicted as Y¹= 50
X² is predicted as Y² = 60
Standard error of estimate = 10
so, 95% of the time the true value of Y¹ will be between 30 and 70.
and 95% of the time the true value of Y² will be between 40-80.
What would be the percentage chance that Y¹ will be higher than Y², or vice versa?
(Assuming we ignore the other 5%, or so, of the time).

I've tried working it out manually, and its doing my head in..
Eg, I can work out using only 1 std deviation, ie, 65% of the time, that, for the give figures above, Y¹ will be higher than Y² 25% of the time, and therefore Y² will be higher 75% of the time... but taking it to 2 std devs gets more complicated, and I'm sure there's some sort of confidence interval equation I could be using.

I do realise that the 65% of the time is for one Y value only, however, for brevity, I'm ignoring that that figure applies to all X's..
Hope this makes sense, and any help or suggestions appreciated.

obh

Active Member
Ai Josh,

You started in the right direction.

The regression will predict the mean of Y for the relevant X.
So you may convert the question to a new question:
you should check the distribution and Sigma.
Under an assumption of Normal distribution and that the standard deviation is not dependent on X
(those assumptions are almost ..the linear regression assumption. Homoscedasticity but the normality is only for the residuals)

Y1~N(m1,Sigma) Y2~N(m2,Sigma)

D=Y2-Y1

You want to find the probability that Y2>Y2 => the probability that D>0

From here I assume you can continue.

The example was for a normal distribution, but you should check the distribution.
If for example, you don't know the standard deviation and the population distribute normally, you should use T distribution.

Thanks obh.