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I haven't found much info on this by googling so i thought maybe someone where has some answers for me.

When it comes to binary logistic regression the model assumes that the log odds ratio has a linear relationship with the independent variables. I'm wondering, what would happen to the model if this assumption was not fulfilled and furthermore, how would one tackle this problem to solve it?

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If the functional relationship between the exposure and average response is not an S-shaped logistic curve, there are still reasons why we might consider an S-shaped logistic curve as a meaningful summary of those data.

As an example, we might have omitted a prognostic factor from a model, meaning that the true marginal relationship between the exposure and the outcome is not logistic, but a complicated semi-logistic function that averages up risks across several conditional logistic curves. This is the principal of non-collapsibility in logistic regression.

Basically, we can rarely ever be certain that the S-shaped logistic trend is, in fact, the "right" one... but it is a useful one! All models are wrong, some models are useful.

Kenji is right that when we try to approximate an S-shaped trend, and the data show strong distributional violations, there may be some sensitivity analyses to consider like testing for higher order polynomial effects. Another type of test to consider is breakpoints, adjusting for "knots" so that trends can change direction. These approaches are hybridized in splines and made even more general by using LOESS curves to explore general non-linear relationships between exposures and outcomes.

Nonetheless, you may revert to the original question: you may say "I want to summarize these data using a single logistic curve whose intercept represents log-odds of the outcome for exposure=0 and whose slope is the log-odds ratio as a measure of association between an exposure and an outcome." The desire then is to obtain a robust error estimate that is unbiased and consistent. The S-curve then is taken to summarize a first order trend in the data, which you can think of as a rule of thumb: does the risk tend to increase or decrease as the exposure goes up, and by how much? To do this, you need only to apply sandwich based standard errors. This can be done using Generalized Estimating Equations with working independence covariance structure, logistic link, and binomial variance structure.

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  • $\begingroup$ +1 for mentioning splines and LOESS. Robust errors deal with the SE's being messed up by the violation of functional form, but won't coefficients be biased as well? $\endgroup$ – Kenji May 19 '17 at 17:56
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    $\begingroup$ @Kenji yes and no. If the true trend relating an outcome to a covariate is exponential, the linear term has a true coefficient of 0. However, we can draw a straight line through that exponential trend over a domain and interpolate it somewhat. That linear approximation can be expressed analytically. The misspecified GEE actually estimates that approximation without bias and consistently. $\endgroup$ – AdamO May 19 '17 at 18:05
  • $\begingroup$ Oh. Makes sense. Would that work for all functional forms apart from exponential? $\endgroup$ – Kenji May 19 '17 at 18:13
  • $\begingroup$ My bad. I misread your answer. You are trying to approximate only the first order trend to know if risk increases or decreases. Now I get it. $\endgroup$ – Kenji May 19 '17 at 18:15
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You get biased and inconsistent coefficient estimates, and biased standard errors. Bias in standard errors can be in both directions and the probability of types I and II errors could increase.

You can tackle non-linearity by introducing different functional forms of the predictor that had a non-linear relationship with Y. Common functional forms are quadratic, logarithmic, cubic, square roots, among others. You can also think about including splines and possibly interactions between two or more predictors. A last possibility is to use a different link function for the binary relationship, as functions such as probit and clog-log have slightly different shapes, albeit all of them following a synodal shape.

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    $\begingroup$ +1. You can also use different link functions. $\endgroup$ – whuber May 19 '17 at 15:23
  • $\begingroup$ Indeed. I'll edit the answer to reflect that. $\endgroup$ – Kenji May 19 '17 at 17:54
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The assumption that your target probability can be modelled as a linear combination of log-odds ratios scaled by your inputs is equivalent to assuming that it is a combination of independent pieces of Bernoulli evidence. When that's not the case, you typically build a more complex model with cross terms.

Seeing the logistic function as some arbitrary sigmoid link function really hides the assumption you're making.

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