# Re: [R] Statistical question: one-sample binomial test for clustered data

From: Thomas Lumley <tlumley_at_u.washington.edu>
Date: Tue, 25 Nov 2008 12:58:25 -0800 (PST)

The bootstrap that Greg Snow suggested is probably the best approach, but it is possible to estimate the variance of the proportion.

The total T number of yes reponses is the sum of twenty totals for blocks, and these are independent, so the variance of Y is 20 times the variance of these twenty numbers.

The variance of the proportion is the variance of Y divided by 100^2

-thomas

On Tue, 25 Nov 2008, Matthias Gondan wrote:

> Dear list,
>
> I hope the topic is of sufficient interest, because it is not
> R-related. I have N=100 yes/no-responses from a psychophysics
> paradigm (say Y Yes and 100-Y No-Responses). I want to see
> whether these yes-no-responses are in line with a model
> predicting a certain amount p of yes-responses. Standard
> procedure would be a one-sample binomial test for the observed
> proportion,
>
> chi²(1 df) = (Y-Np)²/(Np) + [(100-Y)-N(1-p)]²/[N(1-p)]
>
> Actually, this is the approximate chi²-test, but the sample
> size seems to be reasonably high for an asymptotic test.
>
> The problem is that the experiment took quite a while, and
> the 100 responses are grouped into 20 blocks of 5 responses
> each. The responses within the blocks are clustered, ICC is
>
> Can anyone point me to some literature explaining a one-sample
> binomial test / or chi² test for correlated data? Most of the
> literature I found starts with more advanced stuff, e.g.
> 2x2 cross-tabulated data.
>
> Best wishes,
>
> Matthias
>
> ______________________________________________
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> and provide commented, minimal, self-contained, reproducible code.
>

```Thomas Lumley			Assoc. Professor, Biostatistics
tlumley_at_u.washington.edu	University of Washington, Seattle

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```
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