I am analyzing behavioral data (i.e., decision-making) with a quite large N (> 10.000), but cannot access outcome variables in absolute form (5.000 choosing option A, 5.000 choosing option B), but only in percentages (50% choosing option A, 50% choosing option B).
How can I find out, whether along a series of >100 decision, participants favor, e.g., A over B?
As Bayesian statistics innately uses percentages/probabilities this might be a good fit, right?
However, I am a Bayesian noob an not quite sure about my approach:
In a first step I would use a 50/50 prior and integrate the probabilities from one decision.
The posterior distribution then serves a the new prior for integrating the next decision percentages.
In sum, I would perform Bayesian hierarchical modelling along all decision reporting the last posterior distribution.
Any help or feedback would be very, very welcome :)
I don't quite follow your discussion but why not an appropriate form of logistic regression if your problem fits an appropriate form. D. Booth
Hi Eugene, thanks for your comment.
How would you perform a logistic regression with percentage data?
Isn't it a problem that they have boundaries at 0 and 100?
Or would you try to "artifically" transform the percentages into absolute numbers?
If you need any further information on the original project, feel free to ask :).
There is a "natural" transformation: the logit transformation: logit(p) = log(p/(1-p)).
You could use the transformed values in a standard regression model (assuming that the transformed variable has an approximate normal distribution [conditional on the predictors]).
It ist usually better not to transform the observations but to model a transformation of the expectation. This can be achieved by generalized linear models (GLMs). There, the transformation is done via the link function.
If you don't need real likelihoods you could use a quasi-binomial model (with logit-link). This is said to be used for "real" binomial data where the variance is not np (i.e. where the data is over- or underdispersed). However, it works also perfectly fine with proportions given as real values between 0 and 1. I don't know if there might be issues depending on the implementation, but at least in R it works neatly.
The "most correct" but less stable way is a beta-model (again, with logit-link).
https://stats.stackexchange.com/questions/91724/what-is-quasi-binomial-distribution-in-the-context-of-glm
https://www.tandfonline.com/doi/abs/10.1080/03610929008830214?journalCode=lsta20
Article Beta Regression or Modeling Rates and Proportions
---
Using Bayesian vs. "standard" statistics has nothing to do with the kind of data and the kind of model you use. A Bayesian analysis adresses different questions. You can make a Bayesian analysis with any underlying model: the normal regression, the logistic regression, or the beta-regression. The standard analysis goeas up to the point of determining the likelihood function, and you can make likelihood-ratio tests and construct confidence intervals. This is about our expectation about hypothetical observations, conditional on given parameter values. In a Bayesian analysis, this likelihood is used to modify a prior distribution about all model parameters to get a posterior distribution. This posterior distribution is then used to express our (reasonable) believes about the parameter values, conditional on the observed data.
- Badges
- Science topic