## Re: [SPSS] Multivariable logistic regression - how to transform to achieve linearity?

In linear regression models and in logistic linear regression, the “linearity” is that the model is linear in the parameters, often called “betas”.

Let LP be a linear predictor:

LP= beta0 + beta1*x1+beta2*x2

It doesn't matter if the “x:es” are nonlinear like x1= log(x01) and x2= (x02)^2

Substitute betas for z:s and the x:es for k:s if it becomes more clear.
LP = z0 + z1*k1 + z2*k2

(When one is searching for least squares or maximum likelihood, then the x:es are as observed constants (“k:s”) and the betas (“z:s”) are varied to try to find the minimum or maximum.)

In linear regression: E(Y) = mu = LP

In linear logistic regression (Y= 1 or 0):

E(Y) =p and

log(p/(1-p)) = LP

which can be solved to the non-linear link:
p = exp(LP)/(1+exp(LP))

Which is an S-shaped function.

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Of course the model must fit! Maybe the original LP does not fit. Maybe it is needed to include a squared term:
LP2 = beta0 + beta1*x1+beta2*x2 + beta3*(x2)^2
But it will still be a linear model since it is linear in its parameters – the betas.
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When you run this in a software you just declare that you want to run logit (or logistic regression) and tell which variable is the 0/1 variables. So you absolutely do not transform the dependent variable. Then you also declare which are your independent variables (x-variables)

To transform a continuous variable by classifying it in “high” and “low” is to throw away information.