# Solved – Confidence interval of a log-linear regression

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

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:

$$[0.10345-2*0.00592, 0.10345+2*0.00592]$$
$$[0.09161, 0.1153]$$
$$[exp(0.09161)-1,exp(0.1153)-1]$$
$$[0.096,0.122]$$

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?

Contents

$$ln(Capital) =beta_0 +beta_1Education+varepsilon,$$
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.