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Normally logistic regression uses the natural logarithm, as opposed to a $\log_{10}$ base. If we have a regression for binary $Y$ where $p = \Pr(Y=1\mid X)$,

$$ \log_{10}\left(\dfrac{p}{1-p}\right) = \beta_0 + \beta_1X_1 $$

and we found $\widehat{\beta_0} = -2$ and $\widehat{\beta_1} = 3$, then we can interpret $\widehat{\beta_0}$ as saying the log odds of $Y=1$ are 100 to 1 when $X_1= 0$. Similarly, $\widehat{\beta_1}$ can be interpreted by saying that if $X_1$ increasing by 1, it implies that the odds of $Y=1$ increases by $10^3$.

Why isn't this interpretation preferred to what you would get from the natural log $\ln$?

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I can't be certain, but I think this is more to do with how GLM is motivated theoretically.

As you may know, logistic regression is fit via maximum likelihood. We often instead maximize the log likelihood because it is easier and prevents underflow when using a computer. We use the natural log for the log likelihood because we have to take derivatives of the log likelihood for maximization. The natural log is nicer because we don't need to book keep the extra factor of $1/\ln(a)$ in the derivative when using a different base.

The use of the natural log in the log-likelihood then finds its way into the canonical link function in this case. So it isn't a preference per se, and if it is a preference then the preference is not to do with logistic regression but with book keeping in the equations for fitting these models.

You could also very easily use some algebra to take whatever base you prefer for the logit function. So long as you're able to book keep the factor when calculating confidence intervals etc, that sounds fine (if not a bit tedious) to me.

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    $\begingroup$ You're right about the annoying factors that will litter the results when using other bases of logs. However, we maximize the log likelihood not because it's easier but because that produces statistics needed to compute confidence intervals and other measures of uncertainty. How differentiation is carried out doesn't matter, either. The issues of computation are irrelevant to the question. $\endgroup$
    – whuber
    Apr 29, 2021 at 13:28

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