Alternatives to Logistic Regression I have age, gender, height, weight and some other similar parameters of 15000 subjects. I also have one column showing if they had a medical condition (present in about 20% subjects). I now want to analyze the factors which are most contributing to the medical condition. The usual test for this is Logistic regression analysis. However, that assumes that the relationships are linear. What other machine learning tests can I apply for this purpose with this kind of data? Thanks for your insight.
Any classifier that naturally provides feature importance can be applied here. For example, take a look at Decision Trees and Random Forrest.

For other algorithms that do not provide such information (eg KNN and Neural Nets), you may take a (forward/backward/bidirectional) feature selection. For example, chose a subset of features, train a model and evaluate it with any preferred metrics (better to use bootstrapping or cross-validation). Choose another subset if features and try again. After several trials, you can find the subset with the best prediction power. This is the set of the mist contributing features. 

Note that different techniques/measures may give you different results.