 in broadband use was a key factor in making the service popular and praised moves by AAPT to launch an unlimited ADSL 2+ service.
“When users move away from toll-type data services to unlimited all-you-can-eat services with broader connectivity, it offers the availability of film and TV,” he said. “AAPT are innovators and that’s a good thing.”
Record labels participating in the venture locally include EMI, Shock Records and INgrooves. Artists such available include Katy Perry, Lily Allen and Megadeath.
But music lovers expecting Lady GaGa or Cold Chisel will be disappointed, as other major labels yet to sign up or confirm involvement. This is despite payment being offered per downloaded track.
“They’re very complicated negotiations and often take years to complete. We don’t have opposition from major labels, it just takes time to do the deals,” Thompson said.
“Advertisers get the ability to provide an engagement service with the user, whereby they enter a branded mini-website,” he said. “They can present the messages to the members with no interruptions to the music and you don’t have to watch any pre-roll advertisements and there’s no spamming.”
Although personal details are required to get credits that ‘pay’ for each song, Thompson claimed no personal identifying information would be given to the advertisers.
While other companies had attempted similar offers in the past, he said Guvera would succeed thanks to its low-disruption advertising and lack of copyright protection.
“We have a completely different approach in the advertising model, we don’t make our users sit through 30-second clips to get a piece of content, other services had DRM wrapped around them and you could only play the music using their software,” Thompson said.
“The portability and the ability to play it on personal players is key to the success of this. Really it’s a matching process of delivering the best available content in the best format and making the process less painful.”Offsite Services – what are they?
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For further information about Leap Health please see our Hobart, BodyTech Salamanca, New Town, Kingston, Rosny Park, Huonville, services, staff, FAQ, blog or contact us pages.# Binomial GLM and different sample sizes

I have a data set which consists of binomial proportions, let's say the success rate of converting a customer depending on the advertisement, the customer age, and various other factors.

For some common combinations of covariates, I have a lot of data, and therefore the binomial proportion of successes has low variance. For rare combinations of covariates, however, I have very little data, and therefore the variance of the proportion is high.

The magnitude of differences is very large, for example I might have 1 million trials for some combinations of covariates, and only 50 for others. However, I want to include ALL data in my model and weight it appropriately to get the best model fit.

I've tried to use R to do binomial (logistic) regression using a generalized linear model:

lrfit <- glm ( cbind(converted,not_converted) ~ advertisement + age, family = binomial)


This is a good start because it automatically weights the observations by the number of trials.

However, it's not good enough. Here's why: Let's say you have some observations with 100,000 trials and others with 1,000,000 trials. If you weight by number of trials the latter group is going to receive 10 times the weight. This seems nonsensical, however, because both observations are easily precise enough to receive equal treatment in the model. Clearly you want to penalize groups with only 10 or 100 trials, but as the number of trials gets larger, the weight should stop increasing.

Since in weighted least squares the reciprocal of the error variance is used as the weight, my idea would be to use calculate the posterior variance of the proportion (using Jeffrey's prior), then add some constant term to it (this will make sure the variance stops increasing at a certain number of trials) and then use the reciprocal of the sum as the weight.

Is this approach reasonable? Am I missing something? Can someone give me more information about this method?

• I don't understand this line "both observations are easily precise enough to receive equal treatment in the model": one has 10 times the observations than the other, and so has a 1/10th of the variance. Granted these variances will be small, but one is still smaller than the other. – Simon Byrne Mar 15 '13 at 16:34
• Yes, that wasn't very clear. What I meant was that although the two observations have different variances, it intuitively doesn't seem right to give one of them 10 times the weight of the other, if you consider the fact that the model is not going to be accurate enough for such small differences to matter anyway. Adding a billion more trials when you're already very confident about a proportion simply shouldn't result in major parameter changes. – M. Cypher Mar 15 '13 at 16:40
• Does down-weighting actually result in major parameter changes? By the way if you want to re-weight, you can just use the weights argument in glm, which will automatically deal with the fact that different probabilities give different variances. – Simon Byrne Mar 16 '13 at 13:47
• You can use any weights you like, but you are wrong that the standard approach is "nonsensical." Even a huge ratio of weights will not entirely suppress the influence of the low-weighted cases, especially if they have unique or unusual combinations of independent variable values. If introducing more data creates "major parameter changes," then you have a more fundamental problem: either the model is a bad fit and/or it changes over time. Fiddling with weights won't address that problem; it will only give you false confidence in the results. – whuber Jun 13 '13 at 18:17
• You seem to be worried about the weights. Are you able to produce some data which confirms your theory? For example, choose a "true beta", choose some covariates, simulate binomial data using these probabilities (using n's ranging from $5$ up to $1000000$). Then fit the glm and show it is producing "non sensical" results. – probabilityislogic Jan 12 '14 at 11:44

## 3 Answers

Could you get the correct level of inferences from a Generalized Linear Mixed Model? Treating the advertisement as a practical random effect should introduce the correct level of penalization only on the smaller sampled levels. Those levels with any reasonable magnitude will get the appropriate fixed effect estimates.

Assuming the use of lme4 in R, I'm not sure if you'd need to replicate the rows or not. (I'm leaning towards yes). Then you would have something like:

glmm.fit <- glmer(success~ns(age,df=6)+(1|advertisement),family=binomial(),data=rep.df)

(Notice I splined the age using ns() from splines package since I can't imagine it would actually be linear.)

With so many observations I would be worried about extra-binomial variance, that is, overdispersion. Did you try with a quasibinomial family?

That is, change your R code to

lrfit.quasi <- glm ( cbind(converted,not_converted) ~ advertisement + age, family = quasibinomial)


Can you tell us what happens when you do that? What was the estimated overdispersion factor? I would guess it could be much larger than one, the binomial case. Also, with 1 million observations, I guess there could be some extra structure to the sampling, like different sampling sites, subgroups, differing structure with time, autocorrelation... That would introduce dependencies which would have to be modeled. You need to tell us much more about how you obtained this data.

The binomial model also assumes that the count is exact, with 1 million counts that could easily be false... That is another source of extrabinomial variance! An alternative to using a quasibinomial model is to use some mixed model (or a negative binomial model).

Say, with $n=1000000$ and binomial $p$ around 0.5, then the (binomial) standard error of $\hat{p}$ is