On 3 May 2000, Richard M. Barton wrote:
> Suppose Y does not appear to be normally distributed, but Z=ln(Y) does.
>
> I do a linear regression of Z on X, which is dichotomous (0,1).
>
> 1) In simple terms, what does the unstandardized regression
> coefficient b tell me about the relationship between X and Y?
You'd get a more interesting answer if X were continuous, (and
especially, perhaps, if X were a measure of time).
If you had regressed Y on dichotomous X, you have two predicted values
for Y: a, and a + b. Then b is the difference in means of Y between
the two groups identified by X. (You have in effect performed a t-test.)
Having regressed Z (= ln(Y)) on X, you have two predicted values for Z:
a, and a + b. The corresponding values of Y are e^a and e^a times
e^b. Then e^b is the ratio between the geometric means of Y for the
two groups aforementioned.
> 2) Is {e raised to the b} interpretable in terms of X and Y?
If Y is a time series and X is the time at which Y is observed, you are
estimating an exponential growth curve (when b is positive) or an
exponential decay curve (when b is negative). A simple function of b
gives the "doubling time" (the constant time, or difference in X, it
takes for Y to become twice as large); or the half-life, if B is
negative. If I recall correctly, the doubling time is b/ln(2), but I';m
too lazy to work it out at this hour. (Use Y1 = exp(a + bX1) and Y2 =
exp(a + bX2), set Y2/Y1 = 2, and solve for (X2-X1).)
> 3) Is there some other transformation of b that makes the relationship
> between X and Y easily interpretable?
With dichotomous X, how difficult is the interpretation that one is
carrying out a t-test on Z, and therefore finding an average ratio of
Y|X=1 to Y|X=0 ?
-- Don.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
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