Re: Combinometrics
David Heiser [EMAIL PROTECTED] wrote in message [EMAIL PROTECTED]">news:[EMAIL PROTECTED]... We seem to have a lot of recent questions involving combinations, and probabilities of combinations. I am puzzled. Are these concepts no longer taught as a fundamental starting point in stat? I remember all the urn problems and combinations of n taken m times, with and without replacements, the lot sampling problems, gaming problems, etc. These were all preliminary, early in the semester (fall). Now to see these questions popping up late in spring? Times may have changed, since the 1940's, and perhaps there is more important stuff to teach. Even if times hadn't changed, perhaps some of the posters aren't studying in the US, so their timetable may not match yours. (Right now it's late autumn where I am sitting.) Here in Australia, for example, the school year is the same as the calendar year - high schools will start in early February, universities will mostly start in early March (though it varies some from institution to institution). And not all posters are necessarily at university. However, I'd guess that many stats courses no longer do much combinatorial probability. Glen = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Combinometrics
Puzzle from last week: That said, there IS at least one natural application of such a sampling technique [random selection from equiprobable multisets], used in a major industry, where it saves millions of dollars a year. Answer: The casino/gaming industry... The wheel of fortune version of Crown and Anchor, which uses the [six multichoose 3 = 8 choose 3 = 56] multiset triples of six symbols on a wheel pays the casino far more handsomely than the chuckaluck version with three dice, which is in fact one of the more punter-friendly games. The player who wonders about the wheel game will notice that every possible combination is there (on the big wheels; there are smaller ones with some multisets omitted); but because it's every possible multiset, not every possible list, a higher proportion of outcomes are the doubles and triples, which (paying off at 2:1 for a double and 3:1 for a triple) at once look generous and actually lower the payout overall. If each number is covered equally, on a 1-1-1 outcome the house takes in $6 and pays out $6 ($3 returned bets + $3 winnings). On a 2-1, the house pays out $2 in returned bets and $3 winnings; and on a triple, only $4 in total. The wheel-of-fortune version keeps $0.125 for every $1 bet; the chuckaluck cage only $0.0787. (There are also smaller wheels which omit some of the 1-1-1 patterns (well, it wouldn't be fair to leave off the ones with _bigger_ prizes!) and do even better. Imagine the following scam, based on that psychology. The midway wheel operator has a couple accomplices in the crowd who do not hide the fact that they know him, but rather suggest that as friends they'd like a special game. Operator pretends that he's afraid of catching hell from the boss, but eventually gives in and explains to the other players that this means that all bets ride until there's a double or triple, and that he's not really meant to do this. Now, ladies and gentlemen, it's the same rules for everybody, so if you don't want to play keep your dollars in your pockets for this one game. When a shill loses he pleads for one more chance under the good rules unless one of the suckers is already doing it for him. And my, how the money rolls in... -Robert Dawson = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: cross-over study with baseline measurements
On 6 May 2001 21:00:26 -0700, [EMAIL PROTECTED] (Donald Burrill)
wrote:
Just to make sure I'm following the details: you have 36
observations, and the analysis above uses 11 df for subjects and 2 df
each for period drug, total 15; hence 20 df for error.
I think I would have modelled (at least initially) the PxD interaction
as well (4 df), as the only interaction involving only fixed effects,
and as costing relatively little (in terms of df) to ask about.
Yes, you are following correctly and yes, it is useful to include
PeriodxDrug interaction in the model. Because I have replicated the
3x3 Latin square four times there are enough df to do so. In my study
this PxD interaction is not significant BTW. In the literature on
cross-over trials PxD interaction is criticized as having insufficient
power to detect period ('learning') effects.
( A single square would not have had enough df to estimate
interaction. That's why standard textbooks do not include this in the
analysis)
However: I have also done baseline measurements before each treatment
and I wish to include these in the analysis for two reasons:
1. to compare treatments with baseline
2. to compare baseline measurements to check for carry-over.
snip, details of design layout
I'm not certain what, precisely, you want to find out. It would be
possible to carry out several slightly different analyses, but not all of
them may be of interest to you. For example, you could carry out the
analysis you've already done (with the treatment variable) on the
baseline variable; you could construct the (treatment minus baseline)
difference, and carry out the same analysis on that; you could carry out
your original analysis, adding the baseline variable as a covariate.
1.Yes, to ensure that there is no drift in baseline measurements (no
difference between baseline measurements), these can simply be
compared amongst each other using to the model Period x Subject
2.Yes, baseline measurements can be incorporated in the analysis by
subtracting them from treatment values or by adding them as
covariates. This would offer a degree of stabilization against period
('learning') effects. In my data this seems unneccessary.
3.What I am really after is to compare treatment values with baseline
values.
Obviously, after inclusion of these baseline measurements the design
is no longer fully factorial and has become unbalanced.
I don't quite follow this. I don't perceive the design as having
been _fully_ factorial to start with (Latin squares being inherently
incomplete, although balanced, designs). And it appears to me that
including the baseline data is merely adding another factor of 2 levels
(baseline measure, experimental measure -- call it R for repetitions or
replications) crossed with the other factors, producing 72 observations
all together. Am I misunderstanding something about the baseline
information?
You haven't said what design you specified to SPSS or MINITAB,
but an analysis of main effects only would expend only 16 df (1 for the
two-level factor, 11 for subjects, 2 for period, 2 for drug), leaving 55
for error. You could even afford the 4 df for PxD, and 2 df each for RxP
and RxD, and 4 df for RxPxD, and still have 43 df for error.
What I mean is that mathematically, a Latin square ia analyzed as a
factorial design without interaction. This is how it is described in
the statistics textbooks.
The way I have included the baseline measurements is as follows:
I have added 3 more levels to the factor 'period', so that there are 6
levels (eg baseline1, drug1, baseline2, drug2, baseline3, drug3).
I have defined the baseline as an additional level of the treatment
factor, so that there now are 4 levels (eg none, druga, drugb, drugc)
The model is still period x treatment x subject
Thank you for your suggestion to include baseline measurements by
defining a fourth factor R with 2 levels and include the above
interactions.
I have now tried this and this does make it possible to compare
treatment values with baseline values. The difference with my method
is that each treatment is now compared to the baseline preceding it,
rather than with pooled measurements on a control treatment (=no
treatment). In my data one of the treatments no longer differs
significantly from baseline with your analysis, whereas it did with
mine.
(also:I am not sure wether this fourth factor should be nested in the
treatment factor or not.)
If it is technically correct I would probably prefer adding levels to
existing factors, rather than adding factors.
(But I
don't understand the term carryover effect, which may imply something
else about the effects you seek to analyze than I have so far perceived.)
Carry-over effect aka residual effect is persistence of some of the
treatment effect in subsequent periods. This and the period or
learning effect may invalidate the results of cross-over trials.
.
My question is:
Re: Arithmetic, Harmonic, Geometric, etc., Means
Simon, Steve, PhD wrote: Stan Alekman writes: What is the physical significance or meaning regarding a manufacturing process whose output over an extended period of time has the same value for the arithmetic, geometric and harmonic mean of a property, its purity, for example? Exactly the same value? I suspect that the only way this could happen would be if the data were constant. Almost the same value? Probably the data is very close to constant (i.e., the coefficient of variation is very small). The geometric and harmonic means represent averages on the log and inverse scale, respectively, that are back-transformed to the original units of measurement. You might want to review a bit on transformations, especially the stuff on Exploratory Data Analysis by Tukey et al. One rule of thumb I seem to remember is that transformations do not have much of an impact on the data analysis until there is a good amount of relative spread in the data, Yes. such as the maximum value being at least three times larger than the minimum value. This assumes of course that all your data is positive. Note that the ratio of the maximum to minimum values could be considered a measure of relative spread, just like the coefficient of variation. You might want to rethink your approach, however. Usually there are good physical reasons for preferring one measure of central tendency over another. Just blindly computing all possible measures of central tendency is an indication, perhaps, that you are not spending enough time thinking about the physical process that creates your data. You mention elsewhere, for example, that this data represents purity levels. Perhaps it might make more sense to look at impurity levels, since small relative changes in purity levels might be associated with large relative changes in impurity levels. Perhaps certain factors might influence impurity levels in a multiplicative fashion. when impurities get down to low levels, all kinds of interesting things can happen. Steve's is good advice. Steve Simon, [EMAIL PROTECTED], Standard Disclaimer. STATS: STeve's Attempt to Teach Statistics. http://www.cmh.edu/stats Watch for a change in servers. On or around June 2001, this page will move to http://www.childrens-mercy.org/stats Jay -- Jay Warner Principal Scientist Warner Consulting, Inc. North Green Bay Road Racine, WI 53404-1216 USA Ph: (262) 634-9100 FAX: (262) 681-1133 email: [EMAIL PROTECTED] web: http://www.a2q.com The A2Q Method (tm) -- What do you want to improve today? = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
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