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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!!

Thanks

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  • $\begingroup$ Closely related: stats.stackexchange.com/q/9913. $\endgroup$
    – whuber ♦
    Commented Nov 1, 2011 at 17:17
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    $\begingroup$ 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%. $\endgroup$
    – whuber ♦
    Commented Nov 1, 2011 at 17:18
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    $\begingroup$ 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. $\endgroup$
    – user6666
    Commented Nov 1, 2011 at 17:28
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    $\begingroup$ Someone who stumbles across this question because of the title may be interested in reading the following thread: Interpretation of log transformed predictor. $\endgroup$
    – gung - Reinstate Monica
    Commented Nov 2, 2012 at 15:07

1 Answer 1

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(Adding answer from comments so that this question is no longer officially 'unanswered'.)

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%.

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