অর্ধ-দ্বিপদী বিতরণ (জিএলএম প্রসঙ্গে) কী?

Question

আমি আশা করছি যে কোনওটি কোয়াসিবিনোমিয়াল বিতরণ কী এবং এটি কী করে তার একটি স্বজ্ঞাত ওভারভিউ সরবরাহ করতে পারে। আমি এই বিষয়গুলিতে বিশেষভাবে আগ্রহী:

1. দ্বিবার্ষিক বিতরণের ক্ষেত্রে কীভাবে কুইসিবিনোমিয়াল পৃথক হয়।

2. যখন প্রতিক্রিয়ার পরিবর্তনশীল একটি অনুপাত হয় (উদাহরণস্বরূপ মানগুলি 0.23, 0.11, 0.78, 0.98 অন্তর্ভুক্ত), একটি কাসিবিনোমিয়াল মডেল আরে চালিত হবে তবে একটি দ্বিপদী মডেল চলবে না।

3. যখন সত্য / মিথ্যা প্রতিক্রিয়াশীল ভেরিয়েবলকে অতিরঞ্জিত করা হয় তখন কেন কোয়াসিবিনোমিয়াল মডেলগুলি ব্যবহার করা উচিত।

Answer 1

1. দ্বিপদী বিতরণ এবং আধা-দ্বিপদী মধ্যে পার্থক্য তাদের সম্ভাব্যতা ঘনত্ব ফাংশন (পিডিএফ) এ দেখা যায়, যা এই বিতরণগুলিকে চিহ্নিত করে।

   দ্বিপদী পিডিএফ:

   $$P(X=k)={n \choose k}p^{k}(1-p)^{n-k}$$

   অর্ধ-দ্বিপদী পিডিএফ:

   $$P(X=k)={n \choose k}p(p+k\phi)^{k-1}(1-p-k\phi)^{n-k}$$

   $\phi$$|\phi| \le \min\{p/n, (1-p)/n\}$

   $p \sum_{i=0}^n \frac{n!\phi^i}{(n-k)!}$ rather than $p$ itself.)

2. I am not sure on this one, perhaps the glm function in R adds weights in the quasibinomial mode in order to account for this?

3. The purpose of the extra parameter $\phi$ is to estimate extra variance in the data. Every generalized linear model (GLM) makes a distributional assumption for the outcome/response and maximizes the likelihood of the data based on this distribution. It is a choice the analyst makes, and if you feel you need to account for more variance in your data, then you can choose the quasi-binomial distirbution to model the response for your glm. A great way to test if we need to fit a quasi-binomial model instead of a binomial is to fit a quasi-binomial model, and test to see if the $\phi$ parameter is 0.

Comments

Excellent Alejandro, now how can I test if the ϕ parameter is 0?

— Juanchi

Note that in R with glm.fit, binomial and quasibinomial are exactly the same, except that quasibinomial (1) removes the integer check, and (2) returns an AIC of NA. See this answer for more details.

— miguelmorin

-1 This kind of "quasi-binomial" distribution seems to be completely unrelated to quasi-binomial likelihoods in the context of glms, so it is hard to see why it got so many upvotes.

— Jarle Tufto

Answer 2

The quasi-binomial isn't necessarily a particular distribution; it describes a model for the relationship between variance and mean in generalized linear models which is $\phi$ times the variance for a binomial in terms of the mean for a binomial.

There is a distribution that fits such a specification (the obvious one - a scaled binomial), but that's not necessarily the aim when a quasi-binomial model is fitted; if you're fitting to data that's still 0-1 it can't be scaled binomial.

So the quasi-binomial variance model, via the $\phi$ parameter, can better deal with data for which the variance is larger (or, perhaps, smaller) than you'd get with binomial data, while not necessarily being an actual distribution at all.

When the response variable is a proportion (example values include 0.23, 0.11, 078, 0.98), a quasibinomial model will run in R but a binomial model will not

To my recollection a binomial model can be run in R with proportions*, but you have to have it set up right.

* there are three separate ways to give binomial data to R that I'm aware of. I am pretty sure that's one.

Comments

How is this related to quasilikelihood estimation?

— tim.farkas

+1 (but I would love to see a more comprehensive answer!). The three ways to set up binomial GLM with proportions are probably these: stats.stackexchange.com/a/26779/28666 ? A link might be helpful. Also, how does what you said about "quasibinomial" not really being a distribution relates to the second answer in this thread?

— amoeba says Reinstate Monica

@amoeba you can write a distribution for it, as was stated in my answer (a scaled binomial) but that cannot be a distribution for count data (quasibinomial is not on all the integers unless the dispersion parameter is 1) nor for continuous data (it's discrete!). People generally use it for count data because of its variance-structure (but in which case there is no such distribution in the exponential family)

— Glen_b -Reinstate Monica