What is the difference between logistic and logit regression?

Question

What is the difference between logistic and logit regression? I understand that they are similar (or even the same thing) but could someone explain the difference(s) between these two? Is one about odds?

Answer 1

The logit is a link function / a transformation of a parameter. It is the logarithm of the odds. If we call the parameter $\pi$, it is defined as follows:
$$ {\rm logit}(\pi) = \log\bigg(\frac{\pi}{1-\pi}\bigg) $$ The logistic function is the inverse of the logit. If we have a value, $x$, the logistic is:
$$ {\rm logistic}(x) = \frac{e^x}{1+e^x} $$ Thus (using matrix notation where $\boldsymbol X$ is an $N\times p$ matrix and $\boldsymbol\beta$ is a $p\times 1$ vector), logit regression is:
$$ \log\bigg(\frac{\pi}{1-\pi}\bigg) = \boldsymbol{X\beta} $$ and logistic regression is:
$$ \pi = \frac{e^\boldsymbol{X\beta}}{1+e^\boldsymbol{X\beta}} $$ For more information about these topics, it may help you to read my answer here: Difference between logit and probit models.

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The odds of an event is the probability of the event divided by the probability of the event not occurring. Exponentiating the logit will give the odds. Likewise, you can get the odds by taking the output of the logistic and dividing it by 1 minus the logistic. That is:
$$ {\rm odds} = \exp({\rm logit}(\pi)) = \frac{{\rm logistic}(x)}{1-{\rm logistic}(x)} $$ For more on probabilities and odds, and how logistic regression is related to them, it may help you to read my answer here: Interpretation of simple predictions to odds ratios in logistic regression.

Answer 2

This answer applies for scikit-learn in python.

Both logit from statsmodels and LogisticRegression from scikit-learn can be used to fit logistic regression models. However, there are some differences between the two methods.

Logit from statsmodels provides more detailed statistical output, including p-values, confidence intervals, and goodness-of-fit measures such as the deviance and the likelihood ratio test. It also allows for more advanced modeling options, such as specifying offset terms, incorporating robust standard errors, and modeling hierarchical data structures.

LogisticRegression from scikit-learn, on the other hand, provides a more user-friendly interface and is better suited for large-scale machine learning applications. It allows for easy cross-validation, regularization, and feature selection, and is generally faster and more scalable than logit from statsmodels.

In this case, either logit or LogisticRegression could be used to fit the logistic regression model with the two indicator variables. The choice between the two methods may depend on the specific needs of the analysis, such as the desired level of statistical inference or the computational resources available.