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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!!