The question I'm trying to answer is (fundamentally) this: I have a bag of coins that I suspect are weighted, some towards heads, some towards tails. I toss each coin 4 times and record the outcomes (e.g., 3H1T). As a group, do the coins tend to be unfair?
I can't figure out what an appropriate test would be, though it seems like there ought to be one. Here are some relevant thoughts and options I've considered.
(1) Binomial test - Appropriate way to test EACH COIN's fairness, but (a) 4 tosses isn't enough for statistical significance ($\alpha$ = .05) at the level of the individual coin and (b) since I suspect different coins may be weighted in opposite directions, lumping all the data together would make these coins cancel each other out. (See similar comments on this question)
(2) Chi-square goodness-of-fit or multinomial test over counts - This will tell me if my observed counts for each outcome (4H0T, 3H1T, 2H2T...) differ from the expected counts (they do), but not how. It will return a high test statistic whether my coins are all magically fair (all 2H2T results) or if they are all weighted (all either 4H0T, or 0H4T). It also ignores the underlying binomial nature of the data.
(3) Regression seems like overkill for this data, and linear regression/linear mixed models wouldn't answer the right question anyway: coins with opposite weighting would cancel each other out.
For reference, my actual counts are as follows, of a total of 56 "coins". Unfortunately, since they're not real coins, and the experiment is over, I can't just go flip each one a few more times!
16 4H, 10 3H, 9 2H, 8 1H, 13 0H