Should the regression coefficient of a log transformed dependent variable be interpreted as a percentage change?
logv.pdf (278.2 KB)
No. You are predicting log Y, and difference in two predicted values is the predicted difference in logs, which equals the predicted log ratio of the two conditions. Anti-logging that gives you the predicted fold-change (ratio).
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If the model is \ln(y)=\beta_0 +\beta_1 x, and you let y_{new} denote the value of y after increasing x with k units then \ln(y_{new})=\beta_0 +\beta_1 (x+k). Subtracting the two equations from each other and using the properties of the logarithm shows that \frac{y_{new}}{y}=\exp(k\beta_1).
If you use the example in example 3.1.1. of the document you link to then \ln(\text{weight})= 2.14 + 0.00055 \cdot \text{height}. Thus, if we take to subjects who differ with 1 unit (e.g. cm) in height we would expect that the taller of the two has a weight that is \exp(1\cdot\beta_1)=\exp(0.00055)=1.0006. times (or 0.06%) higher than the other.
If we take two subjects who differ with 10 units in height then we expect the taller of the two has a weight that is \exp(10\cdot\beta_1)=\exp(0.0055)=1.006. times (or 0.6%) higher than the other.
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