# How to interpret regression coefficients in a log-log model

I'm using a linear model to analyse some data,

    y~N(mu, sigma) where
mu[y] <- Intercept + Beta1X + Beta2X1 + Beta3X2
and Beta2 = Beta1^2
Beta[n] ~ N(mu.b[n], sigma.b[n])


but I have had to log-transform both the predicted and all the predictor variables, because I'm using BUGS, just for efficiency. Gelman alludes to this being called "elasticity" and says the coefficients can be directly interpreted as " a Beta% increase in X is associated with a 1% increase in y".

However, my results :

              Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.1135924  0.1495142   0.760    0.448
B1           1.4934436  0.0580981  25.706   <2e-16 ***
I(B1^2)     -0.1477196  0.0062205 -23.747   <2e-16 ***
B2           0.0003612  0.0515368   0.007    0.994


suggest that there is a 150% increase in y, with each 1% increase in Beta1 which would be bonkers. Also, how do interpret the "non-linear" or self-interaction term? My suggestion is : A 15% decrease in the effect of B1 on y occurs with each 1% increase in Beta1.

However If I don't transform either the predicted or the predictors I get:

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  5.382e+01  3.410e+00  15.782  < 2e-16 ***
B1          -3.026e-02  8.775e-03  -3.449 0.000591 ***
I(B1^2)      8.654e-06  7.828e-06   1.106 0.269264
B2           2.363e+00  2.490e+00   0.949 0.342789


In which the effect of X1 seems to be reversed, and the effect sizes are miniscule ( certainly not in the order of 150% and 15% respectively)

Hoping for correction!!

Thanks

-
Closely related: stats.stackexchange.com/q/9913. –  whuber Nov 1 '11 at 17:17
Where do you get 150% from? The coefficient of 1.49 for B1 turns 1% into 1.49*1% = 1.49%, not 15% and certainly not 150%. –  whuber Nov 1 '11 at 17:18
Bingo - I'm an idiot - problem solved. Submit it as an answer and I'll accept it. I'm confusing it, apparently with the interpretation of a logistic regression coefficient, close to zero. –  rosser Nov 1 '11 at 17:28
Would be lovely if this question got an official answer @whuber. :) –  Roman Luštrik Jun 12 '12 at 10:13
Someone who stumbles across this question because of the title may be interested in reading the following thread: Interpretation of log transformed predictor. –  gung Nov 2 '12 at 15:07