John,
Let me try to explain the rounding rule that I was taught. Suppose
that X falls within the range (XL, XU). If XXL<XUX, then round off
to XL. If XXL>XUX, then round off to XU. Now, what about the
situation where XXL=XUX? If we always round X to XU for all
intervals (XLi,XUi), then Y=round(X, <roundoffunit>) has the
property that E(Y)>E(X).
So, what is the rule that I suggest should be employed when XXL=XUX?
First, let's standardize the problem so that we are rounding off to
the nearest integer value. This roundoff problem is obtained if we
divide X by the roundoffunit. Note that now, XL and XU are integer
valued, and one of XL or XU will be even and the other odd. Choose
whichever of XL, XU is even as the value to round off to.
Examples:
round(10.5,1) = 10 XL=10, XU=11
round(11.5,1) = 12 XL=11, XU=12
round(7+1/128, 1/64) XL=7+0/128, XU=7+2/128
= round(64*7 + 1/2, 1)/64 XL=7+0/64, XU=7+1/64
= ((64*7) + round( 1/2, 1) )/64 XL=0, XU=1
= (64*7 + 0)/64
= 7
round(7+3/128, 1/64) XL=7+2/128, XU=7+4/128
= round(64*7 + 3/2, 1)/64 XL=7+1/64, XU=7+2/64
= ((64*7) + round( 3/2, 1) )/64 XL=1, XU=2
= (64*7 + 2)/64
= 7.03125
Is this clear as mud?
Dale
>Date: Thu, 14 Jun 2001 22:05:28 +0100
>To: Dale McLerran <dmclerra@mydeja.com>, SASL@LISTSERV.UGA.EDU
>From: John Whittington <John.W@mediscience.co.uk>
>SUBJECT
>At 11:24 14/06/01 0700, Dale McLerran wrote:
>
>>You and I apparently have different rounding rules. Consider x=i+.5
>>for i an integer value. Also, let y=round(x). I was always taught
>>that we should round to the nearest even value so that in expectation,
>>the mean of y is equal to the mean of x (assuming that the probability
>>of an odd value for i is 1/2).
>
>Hmmmm  yes, we do have different rounding rules  and the one I (and I
>think most other people, as well as, it seems, SAS) were taught is far less
>sophisticated than yours. In essence I was never taught anything more
>sophisticated that the 'elementary school' rule (for 'conventional
>rounding' to whole numbers of significant digits) that one 'rounded up' if
>the the first 'insignificant' digit was EQUAL TO OR greater than 5,
>otherwise one 'rounded down'.
>
>I'm not sure that I completely understand the rule you have described. I
>can't really believe that you are suggesting that, say, 0.49 'rounded to
>one decimal place' should be 0.4 (clearly the nearest even most significant
>digit); I think I must be undertsanding you.
>
>>I note, though, that SAS does not implement my rule, but rather
>>implements your rule as demonstrated in the following program.
>
>Indeed  and if one really wants to see where the transition (between
>rounding up and rounding down) occurs, it can be demonstrated with a
>program such as:
>
>data test;
> do i=0.499999 to 0.5000003 by 0.0000001;
> x = round(i);
> put i= x= ;
> end;
>run;
>
>>Personally, I do not like this behavior. Always rounding up for the
>>value that is halfway through the interval will result in a mean
>>value that is always larger than the true mean in any set of data.
>>Arrrrggggghhhhh!!!!
>
>Well, not ANY set of data  only sets that contain values exactly halfway
>through rounding intervals  but, yes, in such sets of data I agree with
>what you are saying, and also agree that the ideal method of rounding would
>be such that E[round(x)] = E[x]. I'm surprised that neither I nor anyone
>who has ever taught me has even raised this issue before, let alone
>proposed a better method of rounding! That better method (and I'm sure
>that 'your' method is an example of that) is one which results in equal
>probabilities of a 'halfway value' being rounded up or down. Random would
>be one (probably the best) way, but your method (when I understand it
>properly) presumably does it in a more structured, and therefore more
>readily applied, (albeit datadependent) fashion.
>
>Perhaps you could explain 'your' rule in a little more detail, because I
>clearly have not understood it.
>
>Kind Regards
>
>
>John
>
>
>Dr John Whittington, Voice: +44 (0) 1296 730225
>Mediscience Services Fax: +44 (0) 1296 738893
>Twyford Manor, Twyford, Email: John.W@mediscience.co.uk
>Buckingham MK18 4EL, UK mediscience@compuserve.com
>

Dale McLerran
Fred Hutchinson Cancer Research Center
mailto: dmclerra@fhcrc.org
Ph: (206) 6672926
Fax: (206) 6675977


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