Under- or Over-dispersion of GROUPED binary data

[Daniel Smith]

Dear Frank,

Following up from our earlier conversation, unless I misunderstood you, you are saying that even GROUPED binary data cannot be under or over dispersed (unless the data are clustered / not independent)? For example, if we repeatedly throw batches of coins (grouped binary data) and record the PROPORTION coming up heads each time, the distribution will be binomial. This is because each toss of the group of coins is independent? The probability of success (i.e. heads) is 0.5 but if we change this by say weighting the coins then presumably the distribution will still be binomial (just with a different probability of success). So under- or over- dispersion can only occur when we have say a random factor in the model which results in each toss of the coins being non-independent or something?!

Thanks!

[Frank Harrell]

It's best not to make a problem more complex than it needs to be.  Independent Bernoulli trials with each Y=0 or 1 add up to a binomial distribution.  The Bernouilli distribution has only one parameter: p, the probability that Y=1.  The binomial distribution has only two parameters: n and p.  Neither of these require any adjustment if the sampling design is based on sampling independent units.  Over- or under-dispersion occurs only when there is non-independence among the events, e.g., when sampling families as clusters or when mice in a cage are competing for food, making their responses negatively correlated with each other.  The logistic model is built up from Bernoulli or binomial distributions.