Did Model 1 or Model 2 generate these sequences? Why? Model 1: Pólya urn, starting with one black and one red ball. 1 (0) stands for black (red). Model 2: Empty probit model, random intercepts with mean 0, ICC = 1/2. All 100,000 sequences here: tinyurl.com/35pkyfxa - ThreadSky

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arnoutvanderijt.bsky.social • 15 days ago

Did Model 1 or Model 2 generate these sequences? Why?

Model 1: Pólya urn, starting with one black and one red ball. 1 (0) stands for black (red).

Model 2: Empty probit model, random intercepts with mean 0, ICC = 1/2.

All 100,000 sequences here: https://tinyurl.com/35pkyfxa

Comments

aksoyundan.bsky.social•15 days ago

I couldn't check the data, but if I did I'd look at: is there a secular upward or downward trend in P over time (if so Polya) or there are momentary but somwehat stable shifts in P which goes back and fort (if so multilevel)

bthalpin.bsky.social•15 days ago

Summing the sequences yields an almost perfectly even distribution, so I suspect (1).

alexgelas.bsky.social•15 days ago

Interesting. Wouldn't you expect model 2 to (also?) have this property?

bthalpin.bsky.social•15 days ago

My initial intuition was that it would be more like a binomial distribution, but I made a crude simulation later that had a similar distribution.

stephenvaisey.com•15 days ago

I did the same thing and found the same

alexgelas.bsky.social•14 days ago

My point is that the only way time enters the picture in Model 2 is through the error terms, which are i.i.d, so this pattern shouldn’t be surprising.

(Disclaimer: I know the answer to the original question.)

bthalpin.bsky.social•14 days ago

The urn model should therefore show more variability earlier and less later.

stephenvaisey.com•15 days ago

I'm working on it. I know the punchline but I'm still working on it!

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