I found this discussion very helpful for the analysis I need to conduct for my thesis:
https://stats.stackexchange.com/questions/259502/in-using-the-cbind-function-in-r-for-a-logistic-regression-on-a-2-times-2-t/615666#615666
I am not sure if I understand it right, that the multiple trials that are mentioned in the discussion are done within the same sample? (but I think so) In my case I have a control and a treatment group, and each respondent goes through 4 questions where he/she has to choose between train and plane. I now want to analyse the difference between control and treatment group over all four choices combined. So I think I could use the approach with a weighted logistic regression model and the dependent variable "proportion_train" (which represents how often the train was chosen out of the four choices).
I am also not sure about the interpretation of the coefficients of this model then. I know these are most likely the log-odds-ratios (or equivalently differences in log-odds). But do these log-odds-ratios show the probability combined over all trials (which I want to find out), or the per-trial probability?
Also, in some other forum someone used "family=quasibinomial" for a logistic regression model with proportion data. How do I find out if for my data I have to use "family=binomial" or "family=quasibinomial"? Or can you in general say that for a weighted logistic regression model with proportion data as dependent variable, the family is binomial?
I also read somewhere else that one needs to account for the correlation within the individuals by including random intercepts for the individual IDs (I guess as e. g. in my case the four questions were answered by the same individuals (but different individuals in control and treatment group of course). In the discussion I mentioned above, no one included such a random intercept for the individuals in the weighted logistic regression model (with proportion data as dependent variable), so I am wondering if it's necessary or not?
Thanks for your support!
If I understand correctly each participant makes four choices between plane and train coded as 0/1. If so you can't use a regular logistic regression because you have repeated measures. The solution is probably to use something like a multilevel generalised linear model (or maybe generalised estimating equations/GEEs which might be easier to run in some software). These can have convergence issues so I usually fit them using Bayesian approaches in R (e.g., using brms).
In general the models will produce coefficients on the log odds scale so e^b will give you them on the odds scale and you get probabilities out with some basic maths (or better still use prediction functions in your software to plot predicted probabilities).
What you get is a regression equation that predicts probabilities (with some calculation) overall or for different effects. Using marginal effects or predictions is a common way to plot these.
The quasi binomial allows for under- or over- dispersion. The easy way to check this is to see if changing the family alters the model. I'm guessing someone recommended the quasi binomial because your observations are clustered. I think its better to model the structure directly with a random intercept for the participant. Something like GEE treats this as a nuisance variable and handles it that way, but I find the multilevel approach more useful in handling a wide range of models.
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