AIM: Make a confidence interval statement on a log-linear regression
I have read posts like:
Interpreting Standard Deviation of Natural Log Transformed Data
But they do not tackle the confidence interval of log-linear regressions.
I have the following log-linear regression:
lm( log(n_capita) ~ edu_index_percent, data = full_maps_edu)
With the summary table:
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -9.92029 0.38053 -26.1 <0.0000000000000002 *** edu_index_percent 0.10345 0.00592 17.5 <0.0000000000000002 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.33 on 167 degrees of freedom Multiple R-squared: 0.646, Adjusted R-squared: 0.644 F-statistic: 305 on 1 and 167 DF, p-value: <0.0000000000000002 So far, I think it is correct to say:
- The geometric mean of
n_capitais $exp(-9.92029)=0.00004917$ - A one unit increase in the
edu_index_percentage, in this case, one percentage point, is expected to increasen_capitaby $exp(0.10345)-1*100=10.9$ percent.
Now I would like to make a statement about the confidence interval, something like:
There is approximately a 95% change that the following interval contains the true value of the edu_index_percentage coefficient:
$$[0.10345-2*0.00592, 0.10345+2*0.00592]$$
$$[0.09161, 0.1153]$$
$$[exp(0.09161)-1,exp(0.1153)-1]$$
$$[0.096,0.122]$$
QUESTION: Since I have to exponentiate the Estimate to interpret, I know I also have to exponentiate the Std.Error. But when it comes to building the interval, I don't know at what point I should exponentiate.
In other words, is the interval $[0.096,0.122]$ correct?
Best Answer
I am afraid, the interpretation of the log-linear model is slightly different. In general terms, if one has the model
$$ln(Capital) =beta_0 +beta_1Education+varepsilon,$$
then a one unit increase in $Education$ increases $Capital$ by $beta_1*100$ percent. In your case that is always 10.345% independent of the exact value of $Education$. This generality is the beauty of a log-linear model.
The 95% confidence interval is also easy. It ranges in your case from an 100%*(0.10345-2*0.00592)=9.161% to (0.10345+2*0.00592)=11.529% increase of $Capital$ following an increase of $Education$ by one unit.
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