Re: How to test a distribution for uniformity?

From: Ben (benjamin.kenward_at_zoology.ox.ac.uk)
Date: 07/19/04 

Date: Mon, 19 Jul 2004 07:28:33 +0000 (UTC)

Ben <benjamin.kenward@zoology.ox.ac.uk> wrote in
news:Xns952A838207006benjaminkenwardzoolo@163.1.2.7:

> Hi all,
>
> I have an experiment for which I make certain observations several
> times a day. The observations can occur in any 45 minute time slot
> from 7.30am to 7.30pm. I need to balance the observations so that,
> over a given number of days, the distribution of times at which
> observations occured is roughly uniform. The obvious way to achieve
> this would be to block it out perfectly - unfortunately I can't do
> this because I need to fit the observations in around other
> commitments. So, what I am looking for is a way to test if the
> distribution of observation times differs significantly from
> uniformity. That way I can block out a week in advance in an arbitrary
> fashion that suits me, and then check that it is still roughly
> uniform.
>
> I haven't got any good ideas about how to go about this so I'd really
> appreciate some help. First I thought perhaps I could use some non-
> parametric test to check the difference between my observation times
> and what they ideally would have been had I done it completely
> uniformly, but this has two big problems. First, it would tell me if I
> was biased towards observing too early or too late, but not if, for
> example, there was a hump in the middle. Secondly, the ideal number of
> observations at each time would be non-integral so I'm not sure how I
> could make the ideal distribution anyway.
>
> Thanks in advance for any advice!
>
> Cheers,
>
> Ben Kenward
> Department of Zoology
> Oxford University
>

Thanks for your advice guys. I am using bins (those are my 45 minute
slots), but the problem with a chi-square test is that there is no
contiguity assumed between adjacent bins, so it wouldn't distinguish
between say:

Time slot: 07.30 08.15 09:00 09:45 10:30 11:15
No. of observations: 2 0 2 0 2 0

which isn't so bad for me, and:

Time slot: 07.30 08.15 09:00 09:45 10:30 11:15
No. of observations: 2 2 2 0 0 0

which is much worse.

Using the distribution of intervals between observations is an
interesting approach which hadn't occured to me, because I was only
thinking along the lines of bins. Sorry to not make that more clear. I
would prefer to use bins if I could, as it will greatly simplify things.
(This is a fairly small detail of my design and ideally I would just
like to have a sentence or two in a methods section to briefly say how I
tested that observations were distributed uniformly).

I wonder is there such a thing as a chi-square test which is adjusted to
assume contiguous bins, or something similar?

To answer the general experimental design points: I realise that I could
have quite a 'patchy' distribution, which still wasn't significantly
different from uniform. However, a) additional complications which I
haven't mentioned (like there being several classes of observations,
varying hunger levels, etc) mean that once I depart from a perfectly
blocked design it all gets very messy, and b) pilot observations make me
think that time of day is not going to make any big differences, plus I
will be testing for its effect anyway.

And the other points:

Yes I am the Ben Kenward who works on New Caledonian crows! We just had
the first succesful captive rearing attempt of this species (well we are
actually the only ones to have any of them captive full stop, apart from
one zoo in NC) and my task is to observe how the tool use behaviour
develops in the youngsters.

And, I was not aware of the luxuriant flowing hair club for
scientists. Thankyou for bringing it to my attention! I'm not sure I
qualify as I almost always have a pony tail (that picture was taken on
the millenium eve and I was a little bit, as Private Eye would say,
tired and emotional)... But I may see about joining as soon as I have a
minute away from my darling animals...

Cheers,

Ben

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