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When $a$ and $b$ are not given, just use the usual logistic model (or whatever is appropriate), because (if it uses a suitable link function) it is guaranteed to fit probabilities with a lower bound no smaller than $0$ and an upper bound no greater than $1.$ These bounds give interval estimates for $a$ and $b.$
The interesting question concerns when $a$ ...
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You could simply instead of using logistic function $1/(1 + e^{-x})$, use the one that forces the bounds, e.g. $a + (b-a)/(1 + e^{-x})$. In such case the optimization algorithm would need to find the probabilities within the bounds.
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Yes, that makes sense. For a post doing this, and showing some plots, see Make Nonlinear Smooth Interpretable in Logistic GAM Regression. You should not only compare the predictions numerically, but look at plots of the estimated nonlinear smooth.
For how to do formal tests, see Anova on logistic regressions linearity. Here is a list of other relevant Qs ...
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It sounds like you are checking each predictor separately against the binary outcome. That's not a good idea with logistic regression, as is has an inherent omitted-variable bias. Omitting from a logistic regression any predictor associated with outcome will bias the coefficients for the included predictors. Unlike in linear regression, the omitted predictor ...
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Propensity matching is designed to reduce the influence of variables you would normally adjust for. Propensity matching will "balance" variables exactly that were used for matching. Propensity matching will also approximately "balance" variables that were not matched for depending on how well they're predicted by the matching. Basically, there should be very ...
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Although you could use a common ID for each matched pair, as suggested in a comment on the question, I would recommend that you use (inverse) propensity-score weighting rather than propensity-score matching to deal any bias resulting from how the method was chosen in each case.
When you match you are necessarily throwing away all the information included in ...
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