Dear all,

I was not surprised by the discussion on linearity or non-linearity. I am going to say, logistic regression (binomial or multinomial) is a linear model and the fitted curves are straight-lines. All these issues critically depend on the definition of the sample space of the response variables.

In the case of a binomial logistic regression, the response variable is a set of two complementary probabilities. Denote them as p0 and p1, p0,p1>0, p0+p1=1. The value of such a vector (p0,p1) is a point in the simplex of two parts. Probabilities have the property of "scale invariance": the information they provide is expressed as the ratio p0/p1, or even better as the log-ratio log(p0/p1), does not matter if they are expressed within the interval (0,1) or are given in percentages; the ratio cancels the units. With these assumptions in mind, (p0,p1) can be considered as a "composition" represented in the two part simplex. In 2001 the so called Aitchison geometry of the simplex was introduced (Pawlowsky-Glahn & Egozcue, SERRA, 2001), which is a Euclidean structure. In this sample space the sets of probabilities are represented by Cartesian coordinates (ilr transformation). In the particular case of two components the only coordinate is, up to a constant, logit(p0) = log(p0 /p1) and exhibits an absolute scale. Now, on this coordinate the logistic regression is the linear model z=log(p0/p1) = b0 + b1 x1 + ... + residual. It is clear that the model is linear and the line fitted model is a straight line (when there are only one explanatory variable x1). If one try to express this model in terms of (p0,p1), a couple of logistic curves are obtained:

p0 = exp( z ) / (1 + exp(z)) ; p1 = 1/ (1+ exp(z))

These equations are those considered as non-linear because we are thinking on p0 and p1 as real variables in an absolute scale.

However, when we think on (p0,p1) as an element of the simplex endowed with the Aitchison geometry, the model is written as

(p0,p1) = c0 circle+ x1 circle-dot c1 circle+ residual

where

circle+ is called perturbation and is the sum in the simplex

circle-dot is called powering and plays the role of multiplication by scalars in the simplex

c0 and c1 are 2-part compositions to be determined (possibly by maximum likelihood)

residuals are now 2-part compositions.

Using these operations circle+ and circle-dot the model (only one explanatory variable) is a straight line in the Aitchison geometry of the simplex.

Excuse me for this long and probably not clear explanation. A detailed information can be found in

V. Pawlowsky-Glahn, J. J. Egozcue and R. Tolosana-Delgado (2015), Modeling and analysis of compositional data, J Wiley & Sons, Chichester, UK.

I also attach figures representing a "straight-line" in a 3-part simplex: probabilities in time, probabilities in the 3-part simplex and the same probabilities in ilr-coordinates. They could be the model fitted in a multinomial logistic regression.

Hi Luke,

Thanks for your answer.

But that's exactly my point that there is no "true" linear relationship if the scale is infinite, because eventually it will reach the point where the linear increase slows down, and becomes nonlinear. And that's why I used the word paradox (which might not be apt): if we assume a genuine linear relationship, we must also assume a nonlinear relationship, which then refutes linearity (of course "nonlinear" is actually linear in terms of the linear regression context, but you get what I mean).

The logistic function, like the cumulative normal, is non-linear. It is an ogive. However, if the fit of the logistic function to your data is very good, then plotting the actual versus predicted values in a scatterplot will show a linear correlation.

Dear all,

I was not surprised by the discussion on linearity or non-linearity. I am going to say, logistic regression (binomial or multinomial) is a linear model and the fitted curves are straight-lines. All these issues critically depend on the definition of the sample space of the response variables.

In the case of a binomial logistic regression, the response variable is a set of two complementary probabilities. Denote them as p0 and p1, p0,p1>0, p0+p1=1. The value of such a vector (p0,p1) is a point in the simplex of two parts. Probabilities have the property of "scale invariance": the information they provide is expressed as the ratio p0/p1, or even better as the log-ratio log(p0/p1), does not matter if they are expressed within the interval (0,1) or are given in percentages; the ratio cancels the units. With these assumptions in mind, (p0,p1) can be considered as a "composition" represented in the two part simplex. In 2001 the so called Aitchison geometry of the simplex was introduced (Pawlowsky-Glahn & Egozcue, SERRA, 2001), which is a Euclidean structure. In this sample space the sets of probabilities are represented by Cartesian coordinates (ilr transformation). In the particular case of two components the only coordinate is, up to a constant, logit(p0) = log(p0 /p1) and exhibits an absolute scale. Now, on this coordinate the logistic regression is the linear model z=log(p0/p1) = b0 + b1 x1 + ... + residual. It is clear that the model is linear and the line fitted model is a straight line (when there are only one explanatory variable x1). If one try to express this model in terms of (p0,p1), a couple of logistic curves are obtained:

p0 = exp( z ) / (1 + exp(z)) ; p1 = 1/ (1+ exp(z))

These equations are those considered as non-linear because we are thinking on p0 and p1 as real variables in an absolute scale.

However, when we think on (p0,p1) as an element of the simplex endowed with the Aitchison geometry, the model is written as

(p0,p1) = c0 circle+ x1 circle-dot c1 circle+ residual

where

circle+ is called perturbation and is the sum in the simplex

circle-dot is called powering and plays the role of multiplication by scalars in the simplex

c0 and c1 are 2-part compositions to be determined (possibly by maximum likelihood)

residuals are now 2-part compositions.

Using these operations circle+ and circle-dot the model (only one explanatory variable) is a straight line in the Aitchison geometry of the simplex.

Excuse me for this long and probably not clear explanation. A detailed information can be found in

V. Pawlowsky-Glahn, J. J. Egozcue and R. Tolosana-Delgado (2015), Modeling and analysis of compositional data, J Wiley & Sons, Chichester, UK.

I also attach figures representing a "straight-line" in a 3-part simplex: probabilities in time, probabilities in the 3-part simplex and the same probabilities in ilr-coordinates. They could be the model fitted in a multinomial logistic regression.