A log-linear model indeed serves to analyze contingency tables, whereby the mean is modeled through a log link, i.e. as product of factors. These factors can be your margin totals, but also or your regressors such as ethnicity indeed.

However in order to fit a log-linear model you need to specify an error distribution, usually a Poisson. Log-linear analysis is thus simply Poisson regression applied to contingency tables.

Regarding your analysis, I would prefer a logistic regression model that predicts for every questionnaire the probability of being returned or not. You can use your regressors as intended (as fixed effects), and add a random effect for participant. The random effect accounts for heterogeneity of your population, and models dependency between questionnaire collection on the same individual.

A log-linear model indeed serves to analyze contingency tables, whereby the mean is modeled through a log link, i.e. as product of factors. These factors can be your margin totals, but also or your regressors such as ethnicity indeed.

However in order to fit a log-linear model you need to specify an error distribution, usually a Poisson. Log-linear analysis is thus simply Poisson regression applied to contingency tables.

Regarding your analysis, I would prefer a logistic regression model that predicts for every questionnaire the probability of being returned or not. You can use your regressors as intended (as fixed effects), and add a random effect for participant. The random effect accounts for heterogeneity of your population, and models dependency between questionnaire collection on the same individual. This approach allows you to take time-varying variables such as change of protocol into account. It zooms in on every single questionnaire instead of their sum. If this approach is impossible because you only have the totals than you can go ahead with Poisson regression as well.