# Can these data be aggregated into a proportion for a binomial glm?

We asked 60 people to list as many restaurant franchises in Atlanta as they could. The overall list included over 70 restaurants, but we eliminated those that were mentioned by fewer than 10% of the people, leaving us with 45. For these 45, we calculated the proportion of informants who listed the franchise, and we're interested in modeling this proportion as a function of the franchises's (log-transformed) advertising budget and years since becoming a franchise.

So I wrote this code:

model <- glm ( cbind (listed, 55-listed) ~ log.budget + years, family = binomial, data = list.45)


As predicted, both variables exhibit strong, significant effects.

But even though I know that proportional data should never be modeled with OLS regression, I subsequently wrote this code:

model.lm <- lm ( proportion.55 ~ log.budget + years, data = list.45)


In this case, "budget" is still a significant predictor, but "years" is relatively weak and not significant.

It makes me worried that the confidence in the estimates is artificially inflated by the aggregation. Doesn't the binomial glm essentially vectorize the data such that the model is based on 45 * 55 = 2,475 rows? Is that appropriate given that there are really only 45 restaurants and 55 informants? Would this call for mixed-effects modeling?

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hint: see what happens with family=quasibinomial –  Ben Bolker Sep 10 '12 at 2:54
Interesting. The estimated coefficients are the same, but standard errors are more conservative (and years is not significant in the quasibinomial model). I'm looking up the help files for quasibinomial, but can you explain what's going on? My impression has been that quasibinomial is used primarily for overdispersion . . . –  Jeremy _ Sep 10 '12 at 3:11
Exactly. There are a variety of differences between lm and glm(...,family=binomial), but one of the important ones is that a binomial GLM makes strong assumptions about the variance. If the data are not overdispersed, then aggregating/disaggregating makes no difference. –  Ben Bolker Sep 10 '12 at 3:17
The R output shows that the dispersion paramater is taken to be 8.7. I'm trying to figure out what this says about overdispersion. Meanwhile, Ben, I see that you've got quite a lot of background with mixed models. Am I safe using a binomial glm without mixed effects for either informant or franchise (in which case I would presumably have to vectorize all of the data while adding a column for "Informant ID")? –  Jeremy _ Sep 10 '12 at 3:24