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The study involved $20$ families, each of $\sim3$ people. For each patient continious parameter $X$ is measured. Each patient has diagnosis $Y=\{\text{ill},\text{healthy}\}=\{1,0\}$.

Problem is to investigate the association between $X$ and $Y$.

May you advice me what methods are used for this case, please?

How wrong it is to use statistical tests without looking at the family origin of the data?

P.S. The problem statement is similar to the family of ANOVA problems, but I can't find this particular case.

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Assuming you want to predict $Y$ from $X$ then you could consider a mixed effect logistic regression. You would use $X$ as the fixed effect and then specify family as the grouping variable which would have a random intercept. You can think of the random intercept as equivalent to fitting a separate intercept for each family. Such an analysis would take account of the fact that one would expect people to be more similar within families than between families. You can extend the model to have a random slope (for $X$) but with 20 clusters of average size 3 I think that might be overdoing it.

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