Although Michael's answer is simple and almost correct, there is an aspect I consider relevant because that's a reason for some of the confusion: a linear model can describe a linear as well as a non-linear relationship between a predictor and a response, even in the "transformed space" (by the link function).

The term "model" refers to a mathematical construct, which is a linear combination of terms, each term being the product of a coefficient and some (function of) one ore more predictor/s. This is why the models are called linear models. Since the predictors may be transformed by any arbitrary function, such models can describe arbitrary non-linear relationships between the response and the predictor/s.

Here is the bit lengthy explanation:

In a logistic regression, the expected logit of the response is predicted or modelled by a so-called "linear predictor". This linear predictor is a simple function representing a weighted sum of the values of the predictor variables. The weight factors are simple coefficients in this function (often denoted as "betas": ß). These coefficients are the "effects" that are detemined so that the model fits the data (as good as possible). Since these coefficients are not transformed or inside some other function (exp, sqrt, sin, ...), they are said to be in their "linear form", and the sum is a linear combination of these coefficients, hence the name "linear predictor" and "linear model".

The most simple linear predictor is simply ß0. There is no predictor in the model, so this model predicts a constant expectation (for a logistic model this is the logit of "success").

If the expectation depends on a predictor X, the model can be written as ß0 + ß1X. Here, ß0 is the expected response (e.g. the logit od "success") when X=0, and ß1 is a difference in that expectation (the difference between logits is a log odds ratio!) for a one unit change in X. This is called the "effect" of X (on the logit).

The model can include any number of different predictors, like ß0 + ß1X1 + ß2X2 + ß3X3 + ... The model is linear as long as the betas are simple, untransformed factors. Note that the predictor variables may be transformed. For instance, this is also a linear model: ß0 + ß1X1 + ß2sin(X1) + ß2exp(X2) + ß3sqrt(X1X2), whereas this is not a linear model: ß0 + sin(ß1X1)

Yes, it’s a linear model but in a transformed space. It’s fit with the general linear model. yes, its link function is nonlinear, but the machinery is still linear. Hope that makes sense! The Wikipedia page on nonlinear regression touches on this topic.

Thanks Michael

It is a linear model, look here:

https://en.wikipedia.org/wiki/Logistic_regression

precisely:

https://wikimedia.org/api/rest_v1/media/math/render/svg/21135f8ddca09553a884ea00e7502d9c3f624385

Don't forget that the relevant inflection point can be found by using the methods of

Article On the Efficient Identification of an Inflection Point

after implementing them in:

https://cran.r-project.org/web/packages/inflection/index.html

Thanks Demetris

Although Michael's answer is simple and almost correct, there is an aspect I consider relevant because that's a reason for some of the confusion: a linear model can describe a linear as well as a non-linear relationship between a predictor and a response, even in the "transformed space" (by the link function).

The term "model" refers to a mathematical construct, which is a linear combination of terms, each term being the product of a coefficient and some (function of) one ore more predictor/s. This is why the models are called linear models. Since the predictors may be transformed by any arbitrary function, such models can describe arbitrary non-linear relationships between the response and the predictor/s.

Here is the bit lengthy explanation:

In a logistic regression, the expected logit of the response is predicted or modelled by a so-called "linear predictor". This linear predictor is a simple function representing a weighted sum of the values of the predictor variables. The weight factors are simple coefficients in this function (often denoted as "betas": ß). These coefficients are the "effects" that are detemined so that the model fits the data (as good as possible). Since these coefficients are not transformed or inside some other function (exp, sqrt, sin, ...), they are said to be in their "linear form", and the sum is a linear combination of these coefficients, hence the name "linear predictor" and "linear model".

The most simple linear predictor is simply ß0. There is no predictor in the model, so this model predicts a constant expectation (for a logistic model this is the logit of "success").

If the expectation depends on a predictor X, the model can be written as ß0 + ß1X. Here, ß0 is the expected response (e.g. the logit od "success") when X=0, and ß1 is a difference in that expectation (the difference between logits is a log odds ratio!) for a one unit change in X. This is called the "effect" of X (on the logit).

The model can include any number of different predictors, like ß0 + ß1X1 + ß2X2 + ß3X3 + ... The model is linear as long as the betas are simple, untransformed factors. Note that the predictor variables may be transformed. For instance, this is also a linear model: ß0 + ß1X1 + ß2sin(X1) + ß2exp(X2) + ß3sqrt(X1X2), whereas this is not a linear model: ß0 + sin(ß1X1)

Thanks Jochen. I wish I could spit out so much knowledge like you ;)

When we say that a model is linear, we mean that its predictions are a linear function of its parameters.Logistic regression definitely qualifies on that definition. Logistic regression is considered as a linear model because the decision boundary it generates is linear, which can be used for classification purposes.

See more here: https://www.quora.com/Why-is-logistic-regression-considered-a-linear-model

Thanks Abdullah