Re: Rates and proportions
p HAVE DISTIBUTION BAYESIAN COMO FUNCTION BETA NO NECESARY NORMAL === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
RE: Rates and proportions
Here's what I got for the confidence interval: Let n = sample size, K = number of successes, p = sample proportion (=K/n), pi = true proportion. If n = 1250 and K = 1 (p = 1/1250), we can be 95% sure that pi > about 0.41 (small-sample one-sided 95% confidence interval using the binomial distribution). In particular, pi = 0 is rejected. Here are a couple of hypothesis tests to verify this: H0: pi = 0 H1: pi > 0 p-value = P(K >= 1) assuming H0 is true = 0 So pi = 0 is rejected (and it always will be if there are a positive number of successes.) H0: pi = 0.41 H1: pi > 0.41 p-value = P(K >= 1) assuming H0 is true = 0.04996 pi = 0.41 is the limit of the confidence interval. (using BINOMDIST from Excel) Rodney ~~ Rodney Carr School of Management Information Systems Deakin University PO Box 423 Warrnambool VIC 3280 Australia email: [EMAIL PROTECTED] phone: + 61 3 5563 3458 mobile: 0417 307 692 fax: + 61 3 5563 3320 www: http://www.man.deakin.edu.au/rodneyc -Original Message- From: Donald Burrill [SMTP:[EMAIL PROTECTED]] Sent: Wednesday, June 21, 2000 5:28 PM To: Dale Berger Cc: [EMAIL PROTECTED]; [EMAIL PROTECTED] Subject: Re: Rates and proportions On Tue, 20 Jun 2000, Dale Berger wrote: > If we observe one escape out of 1250 inmates, why can't we reliably > rule out zero as the population escape rate? Because k = 1 (for n = 1250) is not significantly different from k = 0. > The normal approximation to the binomial may not be appropriate here. No, I don't expect it is. So use the binomial distribution. That's supposing that one wants a statistical argument. If a purely logical argument suffices, it is indeed the case that a counterexample demonstrates the falsity of a proposition. But it may still be not unreasonable to ask, with what probability may one observe one (or more) escapes outof n=1250 (or whatever n actually applies), if the true probability of an escape is ? (I specify non-zero only because it's difficult to carry out some computations when p=0 exactly.) And it is certainly reasonable to ask what confidence interval on p is associated with k = 1. -- Don. Donald F. Burrill [EMAIL PROTECTED] 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] MSC #29, Plymouth, NH 03264 603-535-2597 184 Nashua Road, Bedford, NH 03110 603-471-7128 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ === === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
- Forwarded message from Robert Dawson - Again, a confidence interval may be useful (if not optimal) while including values that are obviously absurd. Examples are: the Z interval for proportion, in cases where the confidence level is greater than 98% and the np>=5 criterion is only just met. Because the critical value is greater than sqrt(5), the interval contains 0 and some negative values for p. - End of forwarded message from Robert Dawson - Which just shows that np>5 is a poor criterion for 98% CI. (I think it is the least stringent standard commonly offered for 95% CIs.) There are a variety of rules of thumb for when the normal approximation is reasonable. My own favorite is that it's fairly good whenever the CI is completely within the interval (0,1). I don't know if this has any great theoretical properties but it does make intuitive sense, is easy to understand and implement, encourages people to look at their results and check them for plausibility, and it does avoid the problems under discussion here. _ | |Robert W. Hayden | | Work: Department of Mathematics / |Plymouth State College MSC#29 | |Plymouth, New Hampshire 03264 USA | * |fax (603) 535-2943 /| Home: 82 River Street (use this in the summer) | )Ashland, NH 03217 L_/(603) 968-9914 (use this year-round) Map of New[EMAIL PROTECTED] (works year-round) Hampshire http://mathpc04.plymouth.edu (works year-round) === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
> On Wed, 21 Jun 2000, Dale Berger wrote: > > > Yet, p=0 is a special case where an outcome is impossible. A > > reasonable confidence interval for p should not include zero if the > > outcome has been observed in a sample. Not so? and Donald Burrill replied: > I am unable to reconcile this assertion with the fact that the only > values one can observe, in the vicinity of (some small) p, are 0/n, > 1/n, 2/n, ... ; and that if 1/n is observed, 0/n is possible to have > observed, in which case one's estimate of p would, presumably, have > been 0, at least to the precision available in the data. I do not see the conflict between these statements. A "reasonable" confidence interval should be computed from the data that _were_ observed, not from what they might have been. If the outcome has been observed, not only is 0 a value that is flatly contradicted by the data, but moreover values very close to 0 have very low likelihood. A "reasonable" confidence interval would thus omit such values. Now, not all confidence intervals are reasonable. In particular, as has been said before, a specified confidence level gives no guarantee that the interval estimator retains any of the information about the parameter that was present in the data. It may achieve its confidence level merely by mixing intervals that are far too large with a few that are far too small in a random or arbitrary fashion. Again, a confidence interval may be useful (if not optimal) while including values that are obviously absurd. Examples are: the Z interval for proportion, in cases where the confidence level is greater than 98% and the np>=5 criterion is only just met. Because the critical value is greater than sqrt(5), the interval contains 0 and some negative values for p. Another example: one could, motivated by the method of moments, derive a confidence interval for the parameter A based on a sample of data from the uniform distribution on [0,A], of the form [c x-bar, d x-bar]. This would of course be nonoptimal, but it would not be positively stupid! For appropriate combinations of sample size and confidence level, however, it would have a nonzero probability of yielding an interval containing _no_ values consistent with the data. Finally (and this time we _are_ being silly) if we drop the (itself irrelevant) condition of connectedness, we could create a 95% confidence region for the mean of the form (-infinity, xbar - t_0.475,n-1 s/sqrt(n)) union (xbar + t_0.525,n-1 s/sqrt(n), infinity) ---) ( containing precisely the _least_ likely values for mu. As somebody once said, the main reason that confidence intervals, as usually constructed, work is that they often resemble likelihood intervals. -Robert Dawson === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
On Wed, 21 Jun 2000, Dale Berger wrote: > Yet, p=0 is a special case where an outcome is impossible. A > reasonable confidence interval for p should not include zero if the > outcome has been observed in a sample. Not so? I am unable to reconcile this assertion with the fact that the only values one can observe, in the vicinity of (some small) p, are 0/n, 1/n, 2/n, ... ; and that if 1/n is observed, 0/n is possible to have observed, in which case one's estimate of p would, presumably, have been 0, at least to the precision available in the data. Possibly the dissonance arises from a (as it were) theological interpretation of "impossible", and the fact that p cannot be observed directly (one can only observe k instances, and relate that to the n potential instances); possibly it turns on something rather more mundane, such as the precision with which one is estimating p, which is necessarily finite. (One thinks of those computer programs that cheerfully report p = 0. in connection with a statistical test.) -- Don. Donald F. Burrill [EMAIL PROTECTED] 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] MSC #29, Plymouth, NH 03264 603-535-2597 184 Nashua Road, Bedford, NH 03110 603-471-7128 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
Yet, p=0 is a special case where an outcome is impossible. A reasonable confidence interval for p should not include zero if the outcome has been observed in a sample. Not so? -Dale - Original Message - From: Donald Burrill <[EMAIL PROTECTED]> To: Dale Berger <[EMAIL PROTECTED]> Cc: <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]> Sent: Wednesday, June 21, 2000 12:27 AM Subject: Re: Rates and proportions > On Tue, 20 Jun 2000, Dale Berger wrote: > > > If we observe one escape out of 1250 inmates, why can't we reliably > > rule out zero as the population escape rate? > > Because k = 1 (for n = 1250) is not significantly different from k = 0. > > > The normal approximation to the binomial may not be appropriate here. > > No, I don't expect it is. So use the binomial distribution. > > That's supposing that one wants a statistical argument. If a purely > logical argument suffices, it is indeed the case that a counterexample > demonstrates the falsity of a proposition. But it may still be not > unreasonable to ask, with what probability may one observe one (or more) > escapes outof n=1250 (or whatever n actually applies), if the true > probability of an escape is ? > (I specify non-zero only because it's difficult to carry out some > computations when p=0 exactly.) > And it is certainly reasonable to ask what confidence interval on p is > associated with k = 1. > -- Don. > > Donald F. Burrill [EMAIL PROTECTED] > 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] > MSC #29, Plymouth, NH 03264 603-535-2597 > 184 Nashua Road, Bedford, NH 03110 603-471-7128 > > > > === > This list is open to everyone. Occasionally, less thoughtful > people send inappropriate messages. Please DO NOT COMPLAIN TO > THE POSTMASTER about these messages because the postmaster has no > way of controlling them, and excessive complaints will result in > termination of the list. > > For information about this list, including information about the > problem of inappropriate messages and information about how to > unsubscribe, please see the web page at > http://jse.stat.ncsu.edu/ > === > === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
On Tue, 20 Jun 2000, Dale Berger wrote: > If we observe one escape out of 1250 inmates, why can't we reliably > rule out zero as the population escape rate? Because k = 1 (for n = 1250) is not significantly different from k = 0. > The normal approximation to the binomial may not be appropriate here. No, I don't expect it is. So use the binomial distribution. That's supposing that one wants a statistical argument. If a purely logical argument suffices, it is indeed the case that a counterexample demonstrates the falsity of a proposition. But it may still be not unreasonable to ask, with what probability may one observe one (or more) escapes outof n=1250 (or whatever n actually applies), if the true probability of an escape is ? (I specify non-zero only because it's difficult to carry out some computations when p=0 exactly.) And it is certainly reasonable to ask what confidence interval on p is associated with k = 1. -- Don. Donald F. Burrill [EMAIL PROTECTED] 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] MSC #29, Plymouth, NH 03264 603-535-2597 184 Nashua Road, Bedford, NH 03110 603-471-7128 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
One might also ask what is meant by the 'population escape rate' in this context. Is the data not population data? Alan Dale Berger wrote: > > Hi Don et al., > > If we observe one escape out of 1250 inmates, why can't we reliably rule out > zero as the population escape rate? The normal approximation to the > binomial may not be appropriate here. > > Dale Berger > > > "Unreliable" or "useless"? Well, the basic graininess in a rate > > is one escapee more (or less) than was reported. A rate of .08 per 100 > > is about 1 out of 1250. If the data on which the rate was based were 1 > > escapee out of 1250 inmates, one cannot _reliably_ tell the rate from > > zero. If the data were 13 escapees out of 16,200 inmates, one would have > > more faith in the rate, at least insofar as representing a small value > > different from (not equal to!) zero. Unfortunately, the rate itself > > does not tell one how grainy the data were. > > >- Alan McLean ([EMAIL PROTECTED]) Department of Econometrics and Business Statistics Monash University, Caulfield Campus, Melbourne Tel: +61 03 9903 2102Fax: +61 03 9903 2007 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
Hi Don et al., If we observe one escape out of 1250 inmates, why can't we reliably rule out zero as the population escape rate? The normal approximation to the binomial may not be appropriate here. Dale Berger Professor and Dean, Psychology Claremont Graduate University 123 East Eighth Street Claremont, CA 91711 FAX: 909-621-8905 Phone: 909-621-8084 http://www.cgu.edu/faculty/bergerd.html - Original Message - From: Donald Burrill <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Cc: <[EMAIL PROTECTED]> Sent: Tuesday, June 20, 2000 2:49 PM Subject: Re: Rates and proportions > On Tue, 20 Jun 2000 [EMAIL PROTECTED] wrote: > > > Hello, I "inherited" the reporting system for our escapes and have some > > questions about how data has been reported in the past. ; ; > "Unreliable" or "useless"? Well, the basic graininess in a rate > is one escapee more (or less) than was reported. A rate of .08 per 100 > is about 1 out of 1250. If the data on which the rate was based were 1 > escapee out of 1250 inmates, one cannot _reliably_ tell the rate from > zero. If the data were 13 escapees out of 16,200 inmates, one would have > more faith in the rate, at least insofar as representing a small value > different from (not equal to!) zero. Unfortunately, the rate itself > does not tell one how grainy the data were. > === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
On Tue, 20 Jun 2000 [EMAIL PROTECTED] wrote: > Hello, I "inherited" the reporting system for our escapes and have some > questions about how data has been reported in the past. > > First, I have a question about the formula used to calculate escape > rates which is (escapes)/(average daily population - escapes). Then > this is reported as a rate per 100 inmates. Isn't this actually a ratio > of escapees to non-escapees. Right. AKA an odds ratio (before multiplying by 100). One might take the reciprocal (e.g., 1/0.0008 = 1,250, from your .08 per 100 below) as representing the odds AGAINST escaping (1250:1), rather than the odds IN FAVOR OF escaping (0.08 chance in 100, or 1:1250). > Maybe I'm just picking at semantics, let me know. I thought that the > formula for rates was (a/(a+b)) * k where the numerator is included in > the denominator. Right again. > Then I also have a rule of thumb question. At what point is a rate > considered unreliable or a useless piece of information? My example > again and remember that it uses the "formula" I first presented above. > The previous reports show rates of .44 per 100 or .08 per 100, etc. > Of course I find this comical because I imagine that .44 means an > escapee with only a torso, legs and head and .08 as an escapee with > only the torso! Mmm. Only the left shin, I would have thought... But this is no more comical than expressions like 0.44% (do you remember the old Ivory Soap ads, claiming that Ivory was 99.44% pure? Only they wrote it as a fraction, 44/100.) "Unreliable" or "useless"? Well, the basic graininess in a rate is one escapee more (or less) than was reported. A rate of .08 per 100 is about 1 out of 1250. If the data on which the rate was based were 1 escapee out of 1250 inmates, one cannot _reliably_ tell the rate from zero. If the data were 13 escapees out of 16,200 inmates, one would have more faith in the rate, at least insofar as representing a small value different from (not equal to!) zero. Unfortunately, the rate itself does not tell one how grainy the data were. > But, many folks around here take those numbers to indicate that the > escape "rate" has decreased substantially! Well, in fairness, .08 is only 20% -- that is, 1/5 -- of 0.44. Dividing one's number of escapees by 5 might well reflect substantial success, in some terms. But part of the point, as Dennis has mentioned, is whether the comparison is between the same institution at two different times (then one could suppose the "average daily population" to be, if not essentially constant, at least comparable), or between two different institutions with very different sizes of population. > I have seen CDC tables with a caveat regarding > small rates and will pull those as evidence for my argument. > > So here's a real life problem for my colleagues out there. I am going > through all the statistics books in my office and have started to > search for references to present my case. I'm not kidding because I > was told that this is the way it has always been calculated so don't > mess with tradition. Sounds depressingly realistic. > If anyone has any references, suggestions, openings for positions, > cites [ Sites? -- dfb ] to search, etc. I would really appreciate it. > Many thanks in advance, Fran Donald F. Burrill [EMAIL PROTECTED] 348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED] MSC #29, Plymouth, NH 03264 603-535-2597 184 Nashua Road, Bedford, NH 03110 603-471-7128 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
>Reword these as per 10,000? That way you have "whole people" while >preserving the differences among the rates. this might ease the problem but, does not eliminate it (though makes more sense than a base of 100) ... for, what if the value comes out to be ... .04423? i guess it depends on how many tend to be IN each prison ... perhaps ... as that should be the yardstick to use ... if you are comparing escapee rates across institutions ... but, maybe the base should more be a function of the type of comparison being done ... across institutions might (logically) call for a smaller base ... across states might call for a larger base ... for example ... in small towns in a state ... to talk about the escapee rate in local jails as out of 10,000 seems totally unrealistic ... it might give you a nice "whole" number but, it seems rather meaningless as it might take 40 years to accumulate that many prisoners there will always be some "roundoff" ... of course, the bigger the base ... 10,000 versus 1,000 or 100 ... the less importance it will have > >Disclaimer: I am in NO WAY an expert. > > >Jill Binker >Fathom Dynamic Statistics Software >KCP Technologies, an affiliate of Key College Publishing and >Key Curriculum Press >1150 65th St >Emeryville, CA 94608 >1-800-995-MATH (6284) >[EMAIL PROTECTED] >http://www.keypress.com >__ > > >=== >This list is open to everyone. Occasionally, less thoughtful >people send inappropriate messages. Please DO NOT COMPLAIN TO >THE POSTMASTER about these messages because the postmaster has no >way of controlling them, and excessive complaints will result in >termination of the list. > >For information about this list, including information about the >problem of inappropriate messages and information about how to >unsubscribe, please see the web page at >http://jse.stat.ncsu.edu/ >=== == dennis roberts, penn state university educational psychology, 8148632401 http://roberts.ed.psu.edu/users/droberts/droberts.htm === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
At 12:55 PM -0400 6/20/00, dennis roberts wrote: >At 11:10 AM 6/20/00 -0500, [EMAIL PROTECTED] wrote: >>Then I also have a rule of thumb question. At what point is a rate >>considered unreliable or a useless piece of information? My example again >>and remember that it uses the "formula" I first presented above. The >>previous reports show rates of .44 per 100 or .08 per 100, etc. Of course I >>find this comical because I imagine that .44 means an escapee with only a >>torso, legs and head and .08 as an escapee with only the torso! But, many >>folks around here take those numbers to indicate that the escape "rate" has >>decreased substantially! I have seen CDC tables with a caveat regarding >>small rates and will pull those as evidence for my argument. > > >well, just like the mean on a 50 item test might be 29.84 ... which no >person could actually obtain AS a score ... you have to take summary values >like these with a grain of salt ... for reporting purposes ... it would >seem to me to make more sense to say ... 30 items ... > >for escapee rates ... in either the case of .44 per 100 or .08 per 100 ... >you don't want to round to 0 ... and report that since ... it suggest NO >escapees ... but, saying about 1 per 100 seems not correct either ... >though, i would prefer saying "about 1" to saying .44 or .08 ... 1 gives a >more UNDERstandable idea of what is happening ... Reword these as per 10,000? That way you have "whole people" while preserving the differences among the rates. Disclaimer: I am in NO WAY an expert. Jill Binker Fathom Dynamic Statistics Software KCP Technologies, an affiliate of Key College Publishing and Key Curriculum Press 1150 65th St Emeryville, CA 94608 1-800-995-MATH (6284) [EMAIL PROTECTED] http://www.keypress.com __ === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: Rates and proportions
At 11:10 AM 6/20/00 -0500, [EMAIL PROTECTED] wrote: >Then I also have a rule of thumb question. At what point is a rate >considered unreliable or a useless piece of information? My example again >and remember that it uses the "formula" I first presented above. The >previous reports show rates of .44 per 100 or .08 per 100, etc. Of course I >find this comical because I imagine that .44 means an escapee with only a >torso, legs and head and .08 as an escapee with only the torso! But, many >folks around here take those numbers to indicate that the escape "rate" has >decreased substantially! I have seen CDC tables with a caveat regarding >small rates and will pull those as evidence for my argument. well, just like the mean on a 50 item test might be 29.84 ... which no person could actually obtain AS a score ... you have to take summary values like these with a grain of salt ... for reporting purposes ... it would seem to me to make more sense to say ... 30 items ... for escapee rates ... in either the case of .44 per 100 or .08 per 100 ... you don't want to round to 0 ... and report that since ... it suggest NO escapees ... but, saying about 1 per 100 seems not correct either ... though, i would prefer saying "about 1" to saying .44 or .08 ... 1 gives a more UNDERstandable idea of what is happening ... of course, what if you are comparing rates across different prisons ... where one is .44 ... another is .08 ... and another is .99 ... to call all about 1 seems not quite fair either the best rule of a thumb variety or not is ... take them all with a grain of salt using statistics ... and understanding what each might provide (ie, what IS the mean anyway) .. are not the same ... Dennis Roberts, EdPsy, Penn State University 208 Cedar Bldg., University Park PA 16802 Email: [EMAIL PROTECTED], AC 814-863-2401, FAX 814-863-1002 WWW: http://roberts.ed.psu.edu/users/droberts/drober~1.htm FRAMES: http://roberts.ed.psu.edu/users/droberts/drframe.htm === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
