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Let's say I have a study in which I send a 100 people a questionnaire in which the dependent variable is binary, such as "does x, y and z correlate with whether a person is obese or not obese".

Would a logistic regression be a good way to answer this question? Furthermore, would a multiple logistic regression be a good model to say "does x affect whether a person is obese or not obese, controlling for y?"

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    $\begingroup$ Your questionnaire seems to be about attitudes and perceptions. That is a long way from whatever causes obesity. $\endgroup$
    – Nick Cox
    Oct 2, 2019 at 8:57

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You can fit a multiple logistic regression. But your larger goal is to make a claim about causality, which is rarely easy to do with observational data. In surveys/questionnaires, for example, you need to worry about nonresponse bias, reverse causation, and human biases associated with people's perceptions of themselves (I'm sure there are others too, this is not an exhaustive list). Logistic regression is a tool that will not fix any of these problems for you - it will merely identify patterns within the data that you have.

You may be interested in some of the threads in our tag, especially Under what conditions does correlation imply causation?

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You have not stated, whether the "independent" data from the questionaire are binary as well or metric or pseudo-metric.

Assuming metric predictors (independent variables) and binary outcome (dependent variable) then logistic regression is a natural first choice. That is, if you believe in an additive and more or less monotonic relations. More complex, non-linear relationships might be better described by other regression models like regression trees or random forests or neural networks but a logistic regression is a reasonable first attempt.

For the bivariate case you might also consider simpler/more intuitive attempts such as an Receiver Operator Curve ROC with AUC analysis or even a t-test comparing obese and non-obese participants, if your audience is better acquainted with that then with logistic regression.

If there is data available concerning the weight of the participant, their body mass index or their hip-to-waist ratio it is very often advisable to examine these continuous measures instead of dichotomous values like obesity. Definitions of when exactly one is obese are to some point arbitrary.

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You might best be served by using Item Response Theory, which is like factor analysis with categorical DVs. In this case, there is a latent factor, obesity, that causes responses in the questionnaire. Then you can use the observed obesity outcome as a convergent validity factor.

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