Solved – Confidence interval of a log-linear regression

AIM: Make a confidence interval statement on a log-linear regression

I have read posts like:

Interpreting Standard Deviation of Natural Log Transformed Data

Lognormal Regression?

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_capita is $exp(-9.92029)=0.00004917$
  • A one unit increase in the edu_index_percentage, in this case, one percentage point, is expected to increase n_capita by $exp(0.10345)-1*100=10.9$ 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:

$$[0.10345-2*0.00592, 0.10345+2*0.00592]$$
$$[0.09161, 0.1153]$$

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?

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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