What I've seen very often, is that people are saying, that logistic and probit regression are giving very similar results, however, logistic regression is more interpretabe. And in this question, I want to understand why.

In the logistic regression, interpretability lies on the log of the odds:

$$\log\left(\frac{p}{1 - p}\right) = \beta_0 + \sum_{i = 1}^n \beta_iX_i,$$

because if we take the derivative with respect to $j-$ th variable, we exactly obtain the $j-$th coefficient:

$$\frac{\partial\log\left(\frac{p}{1 - p}\right)}{\partial X_j} = \beta_j.$$

Because this happens, people are saying, that increase $X_j$ of one unit, increase the odds (i.e. $\frac{p}{1-p}$) of $\exp(\beta_j)$ units.

In the probit regression we can come up with the exactly same thing. We have the following relationship:

$$p = \Phi\left(\beta_0 + \sum_{i = 1}^n \beta_i X_i\right),$$

where $\Phi$ is the PDF of standard normal distribution. From this equation we have that:

$$Q(p) = \beta_0 + \sum_{i = 1}^n \beta_i X_i,$$

where $Q$ is a quantile function of standard normal distribution.

Now if we take the derivative :

$$\frac{\partial Q(p)}{\partial X_j} = \beta_j$$

So can we say the same? What I would like to say, is that increase $X_j$ by one unit, increases quantile function by one unit, and therefore, increase $X_j$ by one unit, increases $p$ by $\Phi(\beta_j)$ units. If that's true, why people are saying, that probit regression is less interpretable?

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    $\begingroup$ Perhaps because the ratio of two odds --- the "odd ratio" --- is a concept well known beyond logistic regression? I've never seen anyone using the ratio of two quantiles, much less heard a name for it. $\endgroup$
    – Igor F.
    Commented Sep 7, 2022 at 11:21
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    $\begingroup$ Having to take a derivative to make something interpretable is not what I'd recommend. $\endgroup$ Commented Sep 7, 2022 at 11:55
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    $\begingroup$ "increases $p$ by $\Phi(β_j)$ units" looks peculiar - if $\beta_j=1$, are you saying an increase of $X_j$ by one unit increases $p$ by $0.841$? You would not want to do that twice. Or are you saying it increases $p$ by $\Phi(\hat Y +\beta_j)-\Phi(\hat Y)$ perhaps approximately $\beta_j \phi(\hat Y)$ - neither of which seem easy to understand $\endgroup$
    – Henry
    Commented Sep 7, 2022 at 11:58
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    $\begingroup$ Count me as one of them, in terms of coefficient interpretation. $\endgroup$ Commented Sep 7, 2022 at 13:06
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    $\begingroup$ But the function $\Phi(x)$ is very interpretable not? A change of $x$ is a change of one standard variation on a normal bell curve. It is maybe not as easy to compute the exact probability as with log-odds, but it is not difficult to interpret. There are even fields where they describe probabilities in terms of the number of $\sigma$. $\endgroup$ Commented Sep 7, 2022 at 13:09

2 Answers 2


Logistic regression is a model for probabilities of binary events. Another concept that is closely related to probabilities is odds, i.e. ratios of probabilities. If the probability of observing a binary event is $p$, the odds of observing it is $\tfrac{p}{1-p}$. It's a fairly simple and commonly understood concept. Logistic regression predicts log-odds. They are "simpler" to interpret because odds are already related to probabilities of binary events, while normal quantiles do not directly translate to them in a meaningful way. If the predicted quantile is $Q(p)$, how "likely" is this to happen? To answer the question, you need to translate the value to probability, while in the case of odds it just re-phrases the question to probability relative to the probability of the opposite event.


Here's another way to think about this:

Let $\theta = β_0 +\sum_{i=1}^n \beta_i X_i $ be your linear predictor. If $X_j$ increases by one unit, the linear predictor increases by $\beta_j$. What can we say about the corresponding change in probability, $|p(\theta+ \beta_j)-p(\theta)|$?

In a logistic regression model, the probability is given by $p(\theta)=e^\theta/(1+e^\theta)$, the inverse of the logit (log-odds) function. It is straightforward to show that the derivative of this function is maximised at $\theta=0$, and that $p'(0)=1/4$. We deduce that $|p(\theta+ \beta_j)-p(\theta)| < \frac{|\beta_j|}{4}$. In other words, the maximum effect a unit change in the $j$th variable can have on the probability is equal to its coefficient divided by 4.

Can you work out the equivalent for the probit model?


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