[R] Weights in binomial glm

[Jan van der Laan]

Thomas, Thierry,

Thank your for your answers and my appogies for my late reply. Thomas,
from your reply it seems that dividing the weights by their average
would also make finding a suitable starting value more robust. This
indeed seems to be case from test I've ran.

However, your comments about me misusing glm() make me wonder at how
the weights are used in glm(). I was under the impression that weights
in glm are usually replicae/frequency weights. When I look at how
weights are usually taken into account in the likelihood, namely

L = Product_i L_i^w_i

each record i contributes w_i times a factor L_i to the total
likelihood, which  is the same as saying that record i represents w_i
observations. This is exactly what I want to express with my weights.
Writing out the likelihood for binomial regression also shows that in
that case it is possible to have Y's that are not only 0 or 1, but can
also have values between 0 and 1 as in R. However, looking at this I
do not see any problems with large weights and Y's equal to 0 or one.

Is it just the fact that R has problems generating the starting values
when the Y's are all between 0 and 1, or are there also some other
reasons for this?

Regards,
Jan





On Fri, Apr 16, 2010 at 6:28 PM, Thomas Lumley <tlumley at u.washington.edu> wrote:
>
>
> Jan,
>
> Thierry is correct in saying that you are misusing glm(), but there is also
> a numerical problem.
>
> You are misusing glm() because your model specification claims to have
> Binomial(n,p) observations with w in the vicinity of 100, where there is a
> single common p but the observed binomial proportion is either 1 or 0, never
> anything in between.  These data are a very poor fit to a binomial model.
>
> The correct specification if you have what you call replicate weights and I
> call frequency weights is to produce a single data record for each covariate
> pattern that has both the 1 and 0 observations. This can either be two
> columns for successes and failures, or one column of proportions and one
> column of weights.  As your quote from MASS says "weights are used to give
> the number of trials when the response is the proportion of successes." In
> your data the response is *not* the proportion of successes.
>
>
> However, the MLE should still be equal to the weighted mean even with this
> misuse.  The reason it is not is because of the starting values.  R has to
> find some starting values for the iterative maximization of the likelihood,
> and for binomial data with y successes out of n it uses  starting values for
> the fitted means of  (y+0.5)/(n+1).  Starting the iteration at the data in
> this way usually makes the Fisher scoring algorithm very reliable -- it is
> correctly scaled to the data, in some sense.   Unfortunately, if you
> separate out the successes and failures, you have some points starting with
> values very close to 0.  When I used your code the starting value for the
> point with the largest weight was 0.5/199.   At iteration 2, the estimated
> mean ends up very small for all observations, and then the iteration
> diverges.  However, if you provide a starting value then the fitting works,
> even if you start the iteration at, say beta=1, corresponding to a fitted
> mean of over 70%.
>
> So, the result is wrong in the sense that it is not the mle, because of a
> failure of convergence, which happens because specifying the weights the way
> you did rather than the documented way leads to bad default starting values
> for the iteration.  You need either to specify the data as recommended or
> supply starting values.
>
>    =thomas
>
>
> On Fri, 16 Apr 2010, Jan van der Laan wrote:
>
>> I have some questions about the use of weights in binomial glm as I am
>> not getting the results I would expect. In my case the weights I have
>> can be seen as 'replicate weights'; one respondent i in my dataset
>> corresponds to w[i] persons in the population. From the documentation
>> of the glm method, I understand that the weights can indeed be used
>> for this: "For a binomial GLM prior weights are used to give the
>> number of trials when the response is the proportion of successes."
>>>
>>> From "Modern applied statistics with S-Plus 3rd ed." I understand the
>>
>> same.
>>
>> However, I am getting some strange results. I generated an example:
>>
>> Generate some data which is simular to my dataset
>>>
>>> Z <- rbinom(1000, 1, 0.1)
>>> W <- round(rnorm(1000, 100, 40))
>>> W[W < 1] <- 1
>>
>> Probability of success can either be estimated using:
>>>
>>> sum(Z*W)/sum(W)
>>
>> [1] 0.09642109
>>
>> Or using glm:
>>>
>>> model <- glm(Z ~ 1, weights=W, family=binomial())
>>
>> Warning message:
>> In glm.fit(x = X, y = Y, weights = weights, start = start, etastart =
>> etastart,  :
>>  fitted probabilities numerically 0 or 1 occurred
>>>
>>> predict(model, type="response")[1]
>>
>>          1
>> 2.220446e-16
>>
>> These two results are obviously not the same. The strange thing is
>> that when I scale the weights, such that the total equals one, the
>> probability is correctly estimated:
>>
>>> model <- glm(Z ~ 1, weights=W/sum(W), family=binomial())
>>
>> Warning message:
>> In eval(expr, envir, enclos) : non-integer #successes in a binomial glm!
>>>
>>> predict(model, type="response")[1]
>>
>>        1
>> 0.09642109
>>
>>
>> However scaling of the weights should, as far as I am aware, not have
>> an effect on the estimated parameters. I also tried some other
>> scalings. And, for example scaling the weights by 20 also gives me the
>> correct result.
>>
>>> model <- glm(Z ~ 1, weights=W/20, family=binomial())
>>
>> Warning message:
>> In eval(expr, envir, enclos) : non-integer #successes in a binomial glm!
>>>
>>> predict(model, type="response")[1]
>>
>>        1
>> 0.09642109
>>
>>
>> Am I misinterpreting the weights? Could this be a numerical problem?
>>
>> Regards,
>>
>> Jan
>>
>> ______________________________________________
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>> PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html
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>>
>
> Thomas Lumley                   Assoc. Professor, Biostatistics
> tlumley at u.washington.edu        University of Washington, Seattle
>
>