Re: Presenting results of categorical data?
Dennis Roberts wrote: > sorry ... i can't agree with this ... > > it could be that in the "serious" cases ... there is a unidentifiable gene > factor that INTERACTS with the treatment ... that is not available in the > "mild" cases group (that's why you have serious and mild cases) ... so, it > is not the treatment that is doing this ... it is the presence or lack of > presence of the gene factor > > in the above ... you are trying to identify ... IF there is an effect, WHAT > it is due to and, the design tendered above will not do that On reflection, my example was very poorly chosen as random allocation to groups here is important as a control. My example conflated random sampling with random allocation to groups. I still think that there might be cases where a pattern observed in a convenience sample might be more informative than one from a random sample. I think its a moot point because I'm not convinced that random sampling is possible in most cases. Even if you know what population you want to sample, I'm not sure an adequate sample strategy could be devised to sample it. Thom = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Presenting results of categorical data?
At 12:39 PM 8/16/01 +0100, Thom Baguley wrote: > For example, if a new drug is administered to a >treatment group made up of serious cases and compared to a control >group of mild cases obtaining more "cures" for the treatment group >might be considered better evidence than a random sample. > >Thom sorry ... i can't agree with this ... it could be that in the "serious" cases ... there is a unidentifiable gene factor that INTERACTS with the treatment ... that is not available in the "mild" cases group (that's why you have serious and mild cases) ... so, it is not the treatment that is doing this ... it is the presence or lack of presence of the gene factor in the above ... you are trying to identify ... IF there is an effect, WHAT it is due to and, the design tendered above will not do that >= >Instructions for joining and leaving this list and remarks about >the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ >= _ dennis roberts, educational psychology, penn state university 208 cedar, AC 8148632401, mailto:[EMAIL PROTECTED] http://roberts.ed.psu.edu/users/droberts/drober~1.htm = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Presenting results of categorical data?
Thom Baguley wrote:
however, I think the
> defence of convenience samples can be stronger than this. Unless we
> have reason to believe that a sample is biased in such a way as to
> generate our pattern of results a convenience sample is just as good
> evidence as a (hypothetical) random sample.
This gets onto tricky philosophical ground. On the one hand, the
"argument from ignorance" ("if we don't have reason to believe
otherwise...") can be abused dreadfully. On the other hand it
underlies everything we do; it would seem to be perfectly rational to
start your car in the morning without looking under it on the grounds
that you don't have reason to believe that anybody wired a bomb to the
ignition.
Ideally I'd like positive reasons to believe that the sample WILL be
like the population to which inference is extended; but I'm not sure of
the extent to which this process can be made either formal or foolproof,
while still practical.
Random sampling manages "foolproof" but not "practical" in many
circumstances. I'm not sure to what extent it is formal, either; random
numbers work fine as abstract mathematical constructs but the
definitions don't tell you where in the real world to look for these
things.
Most so-called random numbers are pseudorandom and work just fine.
Whether mechanical methods like coin-tossing are truly random seems to
depend on your interpretation of quantum mechanics; again, if somebody
should come up with a souped-up hidden-variable theory that gets around
Bell's inequality next week, does statistical practice become invalid?
I think not.
Ultimately scientists still seem to need common sense.
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Re: Presenting results of categorical data?
"Robert J. MacG. Dawson" wrote: > Oh, it never is (strictly), outside of a few industrial > applications. Nobody ever took a random equal-probability sample from > all turnips, all cancer patients, all batches of stainless steel, all > white mice, or all squirrels. However, there are good common-sense > reasons to believe that *some* convenience samples will act enough like > true random samples to be useful. Otherwise we could stop teaching > statistics to biology and psychology students. I agree with pretty much everything you said, however, I think the defence of convenience samples can be stronger than this. Unless we have reason to believe that a sample is biased in such a way as to generate our pattern of results a convenience sample is just as good evidence as a (hypothetical) random sample. In some cases (e.g., a convenience sample we think, a priori, should be biased against the observed pattern) it might constitute stronger evidence than a random sample. For example, if a new drug is administered to a treatment group made up of serious cases and compared to a control group of mild cases obtaining more "cures" for the treatment group might be considered better evidence than a random sample. Thom = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
RE: Presenting results of categorical data?
Nolan Madson writes: >I have a data set of answers to questions on employee performance. >The answers available are: > >Exceeded Expectations >Met Expectations >Did Not Meet Expectations > >The answers can be assigned weights of 3,2,1 (Exceeded, Met, Did Not >Meet). > >One of my colleagues says that it is not valid to average categorical >data such as this. This is one of those debates that almost takes on a religious overtone. I would not be terribly offended to see a mean reported for this data, though the gap between "Met" and "Did Not Meet" is typically a larger gap than between "Met" and "Exceeded". The "Did Not Meet" rating is typically reserved for those people that a supervisor would like to fire. Other people get quite upset about this, though. You could get some buy-in to an average by first asking any critic what their grade point average was. Unless they respond with a statement like "73% of my grades were B or higher" you have them in a contradictory response. The grade point average is an average of data that is clearly ordinal and where (in my humble opinion) the gap between a "D" and an "F" is much larger than the gap between an "A" and a "B". If a grade point average is considered valid then you might ask how would it differ from the above employee rating scale. Of course, you should read what Deming and others say about employee rating systems, but that is a topic for a different email. If I were doing this project, I would try to get a sense of the folks who will be reading your report. Are they nit pickers? If so, avoid an issue like averaging ordinal data that would give them an extra nit to pick at. Or are they people who despise details and yearn for simplicity. Then use an average which provides a single numeric summary that has a simple (perhaps overly simple) interpretation. And I wouldn't put a lot of time into this problem, since that would tend to add legitimacy to a system (numerical rating of employees) that deserves no legitimacy. Steve Simon, 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 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Presenting results of categorical data?
On 14 Aug 2001 15:45:04 -0700, [EMAIL PROTECTED] (Nolan Madson) wrote: > I have a data set of answers to questions on employee performance. > The answers available are: > > Exceeded Expectations > Met Expectations > Did Not Meet Expectations > > The answers can be assigned weights of 3,2,1 (Exceeded, Met, Did Not > Meet). > > Our client wants to see the results averaged, so, for example, we see > that all employees in all Ohio offices for the year 2001 have an > average performance rating of 1.75 while all employees in all Illinois > offices have an average performance rating of 2.28. > > One of my colleagues says that it is not valid to average categorical > data such as this. His contention is that the only valid form of > representation is to say that 75% of all respondents ranked Ohio > employees as having "Met Expectations" or "Exceeded Expectations." Here is a perspective not emphasized so far, in the half dozen responses I have read. At the start, information and sampling is rooted in *respondents* and not employees. Is there one value used, as the average for each respondent, which is then averaged? Unless each respondent rated the same number of employees, you have two obvious choices for the weighting. But a) one of those doesn't give you "independence of ratings"; b) the other one doesn't give you scores that are equally precise and reliable. (X rated 1 employee, and his counts the same as Y who rated 500?) Now, if there are hundreds of respondents, who were all socialized to the same ideal, *that* is useful information to work with. However, I imagine that these numbers could easily be collected from one guy in one state, and three others in the other. If that was so, any comparison you imagine is based on faith, until you add more information to show that the raters mean the same thing. And that, I think, is a much, MUCH bigger problem, than whether the raters conceive of unequal "intervals". > > Can anyone comment on the validity of using averages to report on > categorical data? Or point me to reference sources which would help > clarify the issue? For my points: any book on experimental design. -- Rich Ulrich, [EMAIL PROTECTED] http://www.pitt.edu/~wpilib/index.html = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Presenting results of categorical data?
Jon Cryer wrote:
>
> I do not see how (probabilistic) inference is appropriate here at all.
Oh, it never is (strictly), outside of a few industrial
applications. Nobody ever took a random equal-probability sample from
all turnips, all cancer patients, all batches of stainless steel, all
white mice, or all squirrels. However, there are good common-sense
reasons to believe that *some* convenience samples will act enough like
true random samples to be useful. Otherwise we could stop teaching
statistics to biology and psychology students.
Indeed, it could be argued that pure science *only* makes inferences
about populations for which no sampling frame can be constructed. Yes,
you may be as random as you like about choosing a control group of 50
from your 100 white mice. But unless you title your paper
"The effect of resublimated thiotimoline on the reaction times of one
particular population of 100 white mice (now deceased)"
you are making inferences about a *different* population - that of all
white mice - and the only reason to extend the inference to that
population (rather to that of termites, or light bulbs) is because your
original 100 are a convenience sample from the population of all white
mice.
> I assume that _all_ employees are rated. There is no sampling, random
> or otherwise.
True. However, it seems reasonable to consider such a data set as a
pseudo-random sample from the "Platonic" set of "all potential employees
under this system" and interpret the inference in terms of the existence
of a pattern.
For instance: if I determine the mean height of all named peaks in the
Rockies whose names start with A,B,...,M, and compare that with the mean
height of all named peaks whose names start with N,...,Z, presumably one
mean will be greater. However, this is presumably a "fact without a
reason"; and many of us would place moderate wagers, if the odds were
right, on the difference between the means behaving much like the
difference between two random samples from the same population.
Moreover, should it turn out that this was *not* the case, no doubt you
(like most of us) would ask "why?" (perhaps the most prominent peaks had
been named for saints ("S")? Perhaps a couple super-high peaks
dominated and invalidated distributional assumptions?)
Now, if the same comparison were done between the Rockies and the
Himalayas, we would *not* be surprised to see a difference bigger than
the t distribution might predict. (Again - these are not random
samples.) And why?
"Because the Himalayas are higher than the Rockies." (Duh!)
Ah, but (let's say) the A-M Rockies are bigger than the N-Z Rockies.
"Ah, but that's just chance, there's no real pattern there."
Precisely.
To summarize: In most disciplines, there are no true random samples.
There are better and worse convenience samples
In most disciplines, there are no sampling frames. There are
populations, usually to some extent abstract or at least unlistable.
Inference is ultimately not about populations, but about patterns. A
property of an entire well-defined population is one example of a
pattern; there are others.
It behooves us to accept this and work with it, rather than to delude
ourselves that we sometimes have a true random sample.
-Robert Dawson
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Re: Presenting results of categorical data?
I do not see how (probabilistic) inference is appropriate here at all. I assume that _all_ employees are rated. There is no sampling, random or otherwise. Jon Cryer At 11:14 AM 8/15/01 -0300, you wrote: > > >"Silvert, Henry" wrote: >> >> I would like to add that with this kind of data [three-level ordinal] >> we use the median instead of the average. > > Might I suggest that *neither* is appropriate for most purposes? In >many ways, three-level ordinal data is like dichotomous data - though >there are a couple critical differences. > > Nobody would use the median (which essentially coincides with the >mode) for dichotomous data unless thay had a very specific reason for >wanting that specific bit of information (and I use the word "bit" in >its technical sense.) By contrast, the mean (=proportion) is a lossless >summary of the data up to permutation (and hence a sufficient statistic >for any inference that assumes an IID model) - about as good as you can >get. > > With three levels, the mean is of course hopelessly uninterpretable >without some way to establish the relative distances between the levels. >However, the median is still almost information-free (total calorie >content per 100-gram serving <= log_2(3) < 2 bits). I would suggest >that unless there is an extremely good reason to summarize the data as >ONE number, three-level ordinal data should be presented as a frequency >table. Technically one row could be omitted but there is no strong >reason to do so. > > "What about inference?" Well, one could create various nice >modifications on a confidence interval; most informative might be a >confidence (or likelihood) region within a homogeneous triangle plot, >but a double confidence interval for the two cutoff points would be >easier. As for testing - first decide what your question is. If it *is* >really "are the employees in state X better than those in state Y?" you >must then decide what you mean by "better". *Do* you give any weight to >the number of "exceeded expectations" responses? Do you find 30-40-30 >to be better than 20-60-20, equal, or worse? What about 20-50-30? If >you can answer all questions of this type, by the way, you may be ready >to establish a scale to convert your data to ratio. If you can't, you >will have to forego your hopes of One Big Hypothesis Test. > > I do realize that we have a cultural belief in total ordering and >single parameters, and we tend to take things like stock-market and >cost-of-living indices, championships and MVP awards, and quality- of- >living indices, more seriously than we should. We tend to prefer events >not to end in draws; sports that can end in a draw tend to have >(sometimes rather silly) tiebreaking mechanisms added to them. Even in >sports (chess, boxing) in which the outcomes of (one-on-one) events are >known to be sometimes intransitive, we insist on "finding a champion". >But perhaps the statistical community ought to take the lead in opposing >this bad habit! > > To say that "75% of all respondents ranked Ohio employees as having >'Met Expectations' or 'Exceeded Expectations.' ", as a single measure, >is not a great deal better than taking the mean in terms of information >content *or* arbitrariness. Pooling two levels and taking the >proportion is just taking the mean with a 0-1-1 coding. It says, in >effect, that we will consider > > (Exceed - Meet)/(Meet - Fail) = 0 > >while taking the mean with a 0-1-2 coding says that we will consider > > (Exceed - Meet)/(Meet - Fail) = 1. > >One is no less arbitrary than the other. (An amusing analogy can be >drawn with regression, when users of OLS regression, implicitly assuming >all the variation to be in the dependent variable, sometimes criticise >the users of neutral regression for being "arbitrary" in assuming the >variance to be equally divided.) > > -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/ >= > ___ --- | \ Jon Cryer, Professor Emeritus ( ) Dept. of Statistics www.stat.uiowa.edu/~jcryer \\_University and Actuarial Science office 319-335-0819 \ * \of Iowa The University of Iowa home 319-351-4639 \/Hawkeyes Iowa City, IA 52242 FAX319-335-3017 |__ ) --- V "It ain't so much the things we don't know that get us into trouble. It's the things we do know that just ain't so." --Artemus Ward = Instructions for joining and leaving t
Re: Presenting results of categorical data?
"Silvert, Henry" wrote: > > I would like to add that with this kind of data [three-level ordinal] > we use the median instead of the average. Might I suggest that *neither* is appropriate for most purposes? In many ways, three-level ordinal data is like dichotomous data - though there are a couple critical differences. Nobody would use the median (which essentially coincides with the mode) for dichotomous data unless thay had a very specific reason for wanting that specific bit of information (and I use the word "bit" in its technical sense.) By contrast, the mean (=proportion) is a lossless summary of the data up to permutation (and hence a sufficient statistic for any inference that assumes an IID model) - about as good as you can get. With three levels, the mean is of course hopelessly uninterpretable without some way to establish the relative distances between the levels. However, the median is still almost information-free (total calorie content per 100-gram serving <= log_2(3) < 2 bits). I would suggest that unless there is an extremely good reason to summarize the data as ONE number, three-level ordinal data should be presented as a frequency table. Technically one row could be omitted but there is no strong reason to do so. "What about inference?" Well, one could create various nice modifications on a confidence interval; most informative might be a confidence (or likelihood) region within a homogeneous triangle plot, but a double confidence interval for the two cutoff points would be easier. As for testing - first decide what your question is. If it *is* really "are the employees in state X better than those in state Y?" you must then decide what you mean by "better". *Do* you give any weight to the number of "exceeded expectations" responses? Do you find 30-40-30 to be better than 20-60-20, equal, or worse? What about 20-50-30? If you can answer all questions of this type, by the way, you may be ready to establish a scale to convert your data to ratio. If you can't, you will have to forego your hopes of One Big Hypothesis Test. I do realize that we have a cultural belief in total ordering and single parameters, and we tend to take things like stock-market and cost-of-living indices, championships and MVP awards, and quality- of- living indices, more seriously than we should. We tend to prefer events not to end in draws; sports that can end in a draw tend to have (sometimes rather silly) tiebreaking mechanisms added to them. Even in sports (chess, boxing) in which the outcomes of (one-on-one) events are known to be sometimes intransitive, we insist on "finding a champion". But perhaps the statistical community ought to take the lead in opposing this bad habit! To say that "75% of all respondents ranked Ohio employees as having 'Met Expectations' or 'Exceeded Expectations.' ", as a single measure, is not a great deal better than taking the mean in terms of information content *or* arbitrariness. Pooling two levels and taking the proportion is just taking the mean with a 0-1-1 coding. It says, in effect, that we will consider (Exceed - Meet)/(Meet - Fail) = 0 while taking the mean with a 0-1-2 coding says that we will consider (Exceed - Meet)/(Meet - Fail) = 1. One is no less arbitrary than the other. (An amusing analogy can be drawn with regression, when users of OLS regression, implicitly assuming all the variation to be in the dependent variable, sometimes criticise the users of neutral regression for being "arbitrary" in assuming the variance to be equally divided.) -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: Presenting results of categorical data?
On Wed, 15 Aug 2001, Silvert, Henry wrote:
> I would like to add that with this kind of data
[three ordered categories per item, scored {3,2,1}]
> we use the median instead of the average.
I cannot resist pointing out two things:
(1) At the level of individual items, the median of three ordered
categories is pretty imprecise UNLESS one makes some interesting
assumptions about the scale values that are tantamount to treating them
as interval data;
(2) If the items are summed into a total score, they are being treated
as interval (and the total is a fortiori interval).
In either case, it is not clear whether the additional effort involved
in calculating medians (and their standard errors?), rather than means
etc., is worth one's while...
Donald F. Burrill [EMAIL PROTECTED]
184 Nashua Road, Bedford, NH 03110 603-471-7128
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RE: Presenting results of categorical data?
I would like to add that with this kind of data we use the median instead of the average. Henry M. Silvert Ph.D. Research Statistician The Conference Board 845 3rd. Avenue New York, NY 10022 Tel. No.: (212) 339-0438 Fax No.: (212) 836-3825 > -Original Message- > From: Donald Burrill [SMTP:[EMAIL PROTECTED]] > Sent: Wednesday, August 15, 2001 3:34 AM > To: Nolan Madson > Cc: [EMAIL PROTECTED] > Subject: Re: Presenting results of categorical data? > > On 14 Aug 2001, Nolan Madson wrote: > > > I have a data set of answers to questions on employee performance. > > The answers available are: > > > > Exceeded Expectations > > Met Expectations > > Did Not Meet Expectations > > > > The answers can be assigned weights [that is, scores -- DFB] > > of 3,2,1 (Exceeded, Met, Did Not Meet). > > > > Our client wants to see the results averaged, so, for example, we see > > that all employees in all Ohio offices for the year 2001 have an > > average performance rating of 1.75 while all employees in all Illinois > > offices have an average performance rating of 2.28. > > > > One of my colleagues says that it is not valid to average categorical > > data such as this. His contention is that the only valid form of > > representation is to say that 75% of all respondents ranked Ohio > > employees as having "Met Expectations" or "Exceeded Expectations." > > Your colleague is correct about "categorical data". It is not clear > whether he be correct about "data such as this". Your responses are > clearly at least ordinal (in the order you gave them, from most effective > to least effective). The question is whether the differences between > adjacent values are both approximately equal: that is, whether > "Exceeded Expectations" is roughly the same "distance" (in some > conceptual sense) from "Met Expectations" as "Did Not Meet Expectations" > is. (And whether this be the case for all the variables in question.) > These are difficult questions to argue in the abstract, either on > theoretical or empirical grounds -- although for empirical data you > could always carry out a scaling analysis and see if the scale values > thus derived are approximately equidistant. > > Probably more important than arguing about whether your data are "only > nominal" (i.e., categorical), or "only ordinal" or of "interval" quality > is, what do your clients (and/or the publics to whom they report) > understand of various styles of reportage? I suspect that some folks > would be much happier with "75% of respondents in Ohio met or exceeded > expectations, while only 60% of respondents in Illinois did so", > together with a statement that the difference is significant (or not), > than with a statement like "all employees in all Ohio offices ... had an > average performance rating of 1.75 while all employees in all Illinois > offices had an average performance rating of 2.28", also with a statement > about the statistical value of the distinction. OTOH, some people prefer > the latter. No good reason not to report in both styles, in fact. > > > Can anyone comment on the validity of using averages to report on > > categorical data? > > Well, now, as the question is put, the answer is (of course!) that > averages are NOT valid for categorical data (unless the categories are > at least ordinal and more or less equally spaced). But that begs the > question of whether "categorical data" be an adequate description of YOUR > data. I'd judge it is not: it appears to be at least ordinal. The > question whether it be also interval, at least approximately, depends on > the internal representations your respondents made of the questions and > the possible responses, which is a little hard to find out at this point. > However, if (as is often the case) the response medium depicted the three > possible responses on a linear dimension and at equal intervals, it's a > reaosnably good bet that most of your respondents internalized that > dimension accordingly. > > > Or point me to reference sources which would help > > clarify the issue? -- Nolan Madson > > I doubt that references would help much in dealing with the facts of the > matter, although they might provide you some information and help you to > sound more erudite to your clients... This is essentially a measurement > issue, so appropriate places to look are in textbooks on educational or > psychological measurement. > > ---
Re: Presenting results of categorical data?
Donald Burrill wrote: I agree on all of this. I'd add that at issue is whether people find the mean format useful, whether it is misleading. I'd use -1, 0 and +1, rather than 1-3. In this case the mean gives you at-a-glance summary of the extent to which the people who exceeded expectations outnumbered those who failed to reach them. It shouldn't be too misleading unless people are using the scale in a very odd way. (OTOH a graphical method is probably even better). Thom = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Presenting results of categorical data?
On 14 Aug 2001, Nolan Madson wrote: > I have a data set of answers to questions on employee performance. > The answers available are: > > Exceeded Expectations > Met Expectations > Did Not Meet Expectations > > The answers can be assigned weights [that is, scores -- DFB] > of 3,2,1 (Exceeded, Met, Did Not Meet). > > Our client wants to see the results averaged, so, for example, we see > that all employees in all Ohio offices for the year 2001 have an > average performance rating of 1.75 while all employees in all Illinois > offices have an average performance rating of 2.28. > > One of my colleagues says that it is not valid to average categorical > data such as this. His contention is that the only valid form of > representation is to say that 75% of all respondents ranked Ohio > employees as having "Met Expectations" or "Exceeded Expectations." Your colleague is correct about "categorical data". It is not clear whether he be correct about "data such as this". Your responses are clearly at least ordinal (in the order you gave them, from most effective to least effective). The question is whether the differences between adjacent values are both approximately equal: that is, whether "Exceeded Expectations" is roughly the same "distance" (in some conceptual sense) from "Met Expectations" as "Did Not Meet Expectations" is. (And whether this be the case for all the variables in question.) These are difficult questions to argue in the abstract, either on theoretical or empirical grounds -- although for empirical data you could always carry out a scaling analysis and see if the scale values thus derived are approximately equidistant. Probably more important than arguing about whether your data are "only nominal" (i.e., categorical), or "only ordinal" or of "interval" quality is, what do your clients (and/or the publics to whom they report) understand of various styles of reportage? I suspect that some folks would be much happier with "75% of respondents in Ohio met or exceeded expectations, while only 60% of respondents in Illinois did so", together with a statement that the difference is significant (or not), than with a statement like "all employees in all Ohio offices ... had an average performance rating of 1.75 while all employees in all Illinois offices had an average performance rating of 2.28", also with a statement about the statistical value of the distinction. OTOH, some people prefer the latter. No good reason not to report in both styles, in fact. > Can anyone comment on the validity of using averages to report on > categorical data? Well, now, as the question is put, the answer is (of course!) that averages are NOT valid for categorical data (unless the categories are at least ordinal and more or less equally spaced). But that begs the question of whether "categorical data" be an adequate description of YOUR data. I'd judge it is not: it appears to be at least ordinal. The question whether it be also interval, at least approximately, depends on the internal representations your respondents made of the questions and the possible responses, which is a little hard to find out at this point. However, if (as is often the case) the response medium depicted the three possible responses on a linear dimension and at equal intervals, it's a reaosnably good bet that most of your respondents internalized that dimension accordingly. > Or point me to reference sources which would help > clarify the issue?-- Nolan Madson I doubt that references would help much in dealing with the facts of the matter, although they might provide you some information and help you to sound more erudite to your clients... This is essentially a measurement issue, so appropriate places to look are in textbooks on educational or psychological measurement. Donald F. Burrill [EMAIL PROTECTED] 184 Nashua Road, Bedford, NH 03110 603-471-7128 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
