We all know that logistic regression is used to calculate probabilities through the logistic function. For a dependent categorical random variable $y$ and a set of $n$ predictors $\textbf{X} = [X_1 \quad X_2 \quad \dots \quad X_n]$ the probability $p$ is

$$p = P(y=1|\textbf{X}) = \frac{1}{1 + e^{-(\alpha + \boldsymbol{\beta}\textbf{X})}}$$

The cdf of the logistic distribution is parameterized by its scale $s$ and location $\mu$

$$F(x) = \frac{1}{1 - e^{-\frac{x - \mu}{s}}}$$

So, for $\textbf{X} = X_1$ it is easy to see that

$$s = \frac{1}{\beta}, \quad \mu = -\alpha s$$

and this way we map the two fashions of the sigmoid curve. However, how does this mapping works when $\textbf{X}$ has more than one predictor? Say $\textbf{X} = [X_1 \quad X_2]$, what I see from a tri-dimensional perspective is depicted in the figure below.

So, $\textbf{s} = [s_1 \quad s_2]$ and $\boldsymbol{\mu} = [\mu_1 \quad \mu_2]$ would become

$$\textbf{s} = \boldsymbol{\beta}^{-1}, \quad \boldsymbol{\mu} = -\alpha\textbf{s}$$

and $p$ would derive from the linear combination of the parameters and the predictors in $\textbf{X}$. The way the unknown parameters of the logistic regression function relate to the cdf of the logistic distribution is what I am trying to understand here. I would be glad if someone could provide with insights on this matter.


2 Answers 2


One way of defining logistic regression is just introducing it as $$ \DeclareMathOperator{\P}{\mathbb{P}} \P(Y=1 \mid X=x) = \frac{1}{1+e^{-\eta(x)}} $$ where $\eta(x)=\beta^T x$ is a linear predictor. This is just stating the model without saying where it comes from.

Alternatively we can try to develop the model from some underlying principle. Say there is maybe, a certain underlying, latent (not directly measurable) stress or antistress, we denote it by $\theta$, which determines the probability of a certain outcome. Maybe death (as in dose-response studies) or default, as in credit risk modeling. $\theta$ have some distribution that depends on $x$, say given by a cdf (cumulative distribution function) $F(\theta;x)$. Say the outcome of interest ($Y=1$) occurs when $\theta \le C$ for some threshold $C$. Then $$ \P(Y=1 \mid X=x)=\P(\theta \le C\mid X=x) =F(C;x) $$ and now the logistic distribution wiki have cdf $\frac1{1+e^{-\frac{x-\mu}{\sigma}}}$ and so if we assume the latent variable $\theta$ has a logistic distribution we finally arrive at, assuming the linear predictor $\eta(x)$ represent the mean $\mu$ via $\mu=\beta^T x$: $$ \P(Y=1\mid x)= \frac1{1+e^{-\frac{C-\beta^T x}{\sigma}}} $$ so in the case of a simple regression we get the intercept $C/\sigma$ and slope $\beta/\sigma$.

If the latent variable has some other distribution we get an alternative to the logit model. A normal distribution for the latent variable results in probit, for instance. A post related to this is Logistic Regression - Error Term and its Distribution.

  • $\begingroup$ How do we, in this case, calculate the latent variable value corresponding to origin x? Is it just Wx? $\endgroup$ Commented Feb 15 at 14:59

One way to think of it is to consider the latent variable interpretation of logistic regression. In this interpretation, we consider a linear model for $Y^*$, a latent (i.e., unobserved) variable that represents the "propensity" for $Y=1$.

So, we have $Y^*=X\beta + \epsilon$. We get the observed values of $Y$ as $Y=I(Y^*>0)$, where $I(.)$ is the indicator function.

When $\epsilon$ is distributed as the logistic distribution with mean 0 and variance $\frac{\pi^2}{3}$, a logistic regression model correctly describes $Y$. That is, $P(Y=1)=\frac{1}{1+e^{-X \beta}}$ is the correct model for $Y$. When $\epsilon$ is distributed as the normal distribution with mean 0 and variance 1, a probit regression model correctly describes $Y$. The polychoric correlation between two variables $Y_1$ and $Y_2$ is the implied correlation of $Y^*_1$ and $Y^*_2$ assuming a probit model.

A benefit of the latent variable interpretation is that the model coefficients can be interpreted as the linear change in $Y^*$ corresponding to a 1-unit change in a predictor holding others constant, in contrast to the log odds ratio interpretation often used for logistic regression (and it seems almost impossible to interpret a probit regression coefficient). The modeled implied mean and standard deviation of $Y^*$ can be computed to see how much in standardized units of $Y^*$ a 1-unit change in a predictor is associated with, just as you would with a continuous outcome of arbitrary scale. In addition, this interpretation works regardless of whether logistic, probit, or some other type of regression model or error distribution is used.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.