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I have a logit model that comes up with a number between 0 and 1 for many cases, but how can we interprete this?

Lets take a case with a logit of 0.20

Can we assert that there is 20% probability that a case belongs to group B vs group A?

is that the correct way of interpreting the logit value?

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    $\begingroup$ In addition to @SvenHohenstein's good answer below, it may help you to read my answer here: Interpretation of simple predictions to odds ratios in logistic regression, which contains additional basic information about probabilities & odds. Note that the logit can be understood more abstractly as a link function; you can read more about that here: difference-between-logit-and-probit-models (although this answer might be a bit more technical). $\endgroup$ Mar 20, 2013 at 23:08
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    $\begingroup$ I want to know why the pronunciation of logit is neither like logarithm nor like logistic $\endgroup$
    – Henry
    Nov 15, 2013 at 1:14
  • $\begingroup$ @Henry - According to Wiktionary, the U.S. pronunciation of 'logit' /ˈloʊdʒɪt/ (en.wiktionary.org/wiki/logit) is like 'logistic' (/loʊˈdʒɪs.tɪk/) (en.wiktionary.org/wiki/logistic). $\endgroup$
    – shaneb
    Feb 8, 2019 at 22:39
  • $\begingroup$ @shaneb - fair enough - thought that only shifts the question to the unlogical pronounciation of logistic $\endgroup$
    – Henry
    Feb 10, 2019 at 20:38

3 Answers 3

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The logit $L$ of a probability $p$ is defined as

$$L = \ln\frac{p}{1-p}$$

The term $\frac{p}{1-p}$ is called odds. The natural logarithm of the odds is known as log-odds or logit.

The inverse function is

$$p = \frac{1}{1+e^{-L}}$$

Probabilities range from zero to one, i.e., $p\in[0,1]$, whereas logits can be any real number ($\mathbb{R}$, from minus infinity to infinity; $L\in (-\infty,\infty)$).

A probability of $0.5$ corresponds to a logit of $0$. Negative logit values indicate probabilities smaller than $0.5$, positive logits indicate probabilities greater than $0.5$. The relationship is symmetrical: Logits of $-0.2$ and $0.2$ correspond to probabilities of $0.45$ and $0.55$, respectively. Note: The absolute distance to $0.5$ is identical for both probabilities.

This graph shows the non-linear relationship between logits and probabilities:

enter image description here

The answer to your question is: There is a probability of about $0.55$ that a case belongs to group B.

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  • $\begingroup$ " Logits of $-0.2$ and $0.2$ correspond to probabilities of $0.45$ and $0.55$ respectively." How does it imply that logit distribution is symmetric? $\endgroup$
    – user 31466
    Feb 12, 2015 at 3:50
  • $\begingroup$ There is a probability of about $0.55$ that a case belong to group B. When will it belong to group A ? $\endgroup$
    – user 31466
    Feb 12, 2015 at 3:55
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    $\begingroup$ @Leaf Since there are only two groups, A and B, the probability for group A is $1 - 0.55 = 0.45$. $\endgroup$ Feb 12, 2015 at 6:59
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    $\begingroup$ @Here, symmetry is related to the absolute difference to a probability of $0.5$ or a logit of $0$. If probability is $0.5 + x$, the logit is $0 + y$; If probability is $0.5 - x$, the logit is $0 - y$. Here, $\text{sign}(x) = \text{sign}(y).$ $\endgroup$ Feb 12, 2015 at 7:02
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To add a more modern (but not very deep) perspective, consider how it's used in deep learning (ha, pun intended...):

logit is referred to the output of a function (e.g. a Neural Net) just before it's normalization (which we usually use the softmax). This is also known as the code. So if for label $y$ we have score $f_y(x)$ then the logit is:

$$ logit = \log \left( \frac{ e^{f_y(x)} }{Z} \right) = score = f_y(x)$$

Where $Z$ is the standard partition function. By the way, this is all over the place in the pytorch and tensorflow documentation.

So you can interpret it as:

the (unnormalized) score for a label or (functional confidence) for a specific class/label.

One of the many references: https://stackoverflow.com/questions/41455101/what-is-the-meaning-of-the-word-logits-in-tensorflow

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Could you maybe specify your model and give a screenshot of the output, then I could give you an detailed answer, but as a first try.... you may want to check out also the following examples on these websites:

http://www.ats.ucla.edu/stat/stata/seminars/stata_logistic/default.htm

http://www.ats.ucla.edu/stat/stata/dae/logit.htm

http://www.ats.ucla.edu/stat/stata/faq/oratio.htm

http://www.ats.ucla.edu/stat/mult_pkg/faq/general/odds_ratio.htm

so if the coefficient is 0.2 it depends on the variable, I guess you have a dummy, which is e.g. 0 for group B and 1 for group A?

odds ratio is given by: $OR = e^b$

so in your case: $e^{70.20}$

This would be the odds ratio of your group variable corresponding to your reference group.

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    $\begingroup$ I believe the OP is asking about how to interpret logits, not how to perform logistic regression. $\endgroup$
    – whuber
    Mar 20, 2013 at 16:22

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