Discussion: View Thread

Hi

In my primary efficacy analysis I have a signfiicant center effect (with a non-significant center-by-treatment interaction effect p-value 0.23).  I have a total of 8 centers.  

There are 2 centers that have a visibly stronger treatment effect that the others.  Is it possible to pool these two centers together, and also pool the remianing 6 centers together (thus creating only two composite centers) and re-run the analysis? 

I am thinking this would lead to a significant interaction term, and thus two estimates of the treatment effect.

Does anyone have any experience in pooling centers based on the efficacy results?

Thanks
Shelley

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Shelley-Ann Walters
3M
-------------------------------------------

While  it is not uncommon for centers to be pooled based on some criteria specified in advance of breaking the blind (with some potential issues associated with that, as well), one really shouldn't do this kind of thing, after the fact.  This is especially true if it is based on the an assessment of relative treatment effects among centers.

I wouldn't try to justify it.......
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Jeffrey Finman
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But isn't that analogous to pooling non-significant effects in a designed experiment to increase the degrees of freedom used to estimate the error variance - something done quite frequently (and taught by the experts)?  

And if this research is to determine which centers are the "same" in the treatment effect, and which are different, shouldn't we act on the findings...especially if we're looking for clues as to new hypotheses? 

-------------------------------------------
Wayne Fischer
Statistician
University of Texas Medical Branch
-------------------------------------------







Original Message:
Sent: 04-30-2013 18:02
From: Jeffrey Finman
Subject: Justifying pooling of centers in multicenter study

While  it is not uncommon for centers to be pooled based on some criteria specified in advance of breaking the blind (with some potential issues associated with that, as well), one really shouldn't do this kind of thing, after the fact.  This is especially true if it is based on the an assessment of relative treatment effects among centers.

I wouldn't try to justify it.......
-------------------------------------------
Jeffrey Finman
-------------------------------------------

(This is not a reply to Wayne Fischer; I just entered the string here.)
There has been a lot of talk over whether pooling Centers should be permitted or prohibited. It sounds a little like a court of law. What statistics should be about is learning from data. If I saw two Centers quite different from the others, I would ask what variables make these Centers different and if that tells me a way I can improve the operations of my Centers.
--Bob

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Robert Riffenburgh
Naval Medical Center
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Original Message:
Sent: 05-01-2013 13:19
From: Wayne Fischer
Subject: Justifying pooling of centers in multicenter study

But isn't that analogous to pooling non-significant effects in a designed experiment to increase the degrees of freedom used to estimate the error variance - something done quite frequently (and taught by the experts)?  

And if this research is to determine which centers are the "same" in the treatment effect, and which are different, shouldn't we act on the findings...especially if we're looking for clues as to new hypotheses? 

-------------------------------------------
Wayne Fischer
Statistician
University of Texas Medical Branch
-------------------------------------------

I think what some respondents to this post are hinting at is that you don't want to be accused of starting a statistical "fishing expedition" with your data. Trying to get a signficant interaction result when your designed experiment shows that there isn't one as your experiment stands. The right way to do this is to finish this experiment and show non-significance but note the potential for a significant result as "of interest for future research". Then construct another experiment that explores exactly what you think is there using your pooled variance concept. This preserves the ethical integrity of your initial study and sets up another study to appropriately look at what may or may not be there using pooled centers.

If you go fishing, you may not like what you catch.

-------------------------------------------
Daniel Butorovich
Research Analyst
Cochise College
-------------------------------------------







Original Message:
Sent: 05-02-2013 11:55
From: Robert Riffenburgh
Subject: Justifying pooling of centers in multicenter study

(This is not a reply to Wayne Fischer; I just entered the string here.)
There has been a lot of talk over whether pooling Centers should be permitted or prohibited. It sounds a little like a court of law. What statistics should be about is learning from data. If I saw two Centers quite different from the others, I would ask what variables make these Centers different and if that tells me a way I can improve the operations of my Centers.
--Bob

-------------------------------------------
Robert Riffenburgh
Naval Medical Center
-------------------------------------------







Original Message:
Sent: 05-01-2013 13:19
From: Wayne Fischer
Subject: Justifying pooling of centers in multicenter study

But isn't that analogous to pooling non-significant effects in a designed experiment to increase the degrees of freedom used to estimate the error variance - something done quite frequently (and taught by the experts)?  

And if this research is to determine which centers are the "same" in the treatment effect, and which are different, shouldn't we act on the findings...especially if we're looking for clues as to new hypotheses? 

-------------------------------------------
Wayne Fischer
Statistician
University of Texas Medical Branch
-------------------------------------------

The natural solution (to me) is to avoid pooling and model centers as random effects.
Centers naturally vary (if for no other reason than luck of the draw) and as a random effect one gets a sense of the variability between centers.  (If some centers seem to be outliers, a random effects model gives a sense of how far out they are)
-------------------------------------------
Dennis Sweitzer
Principal Biostatistician
Medidata Solutions
-------------------------------------------







Original Message:
Sent: 05-02-2013 12:40
From: Daniel Butorovich
Subject: Justifying pooling of centers in multicenter study

I think what some respondents to this post are hinting at is that you don't want to be accused of starting a statistical "fishing expedition" with your data. Trying to get a signficant interaction result when your designed experiment shows that there isn't one as your experiment stands. The right way to do this is to finish this experiment and show non-significance but note the potential for a significant result as "of interest for future research". Then construct another experiment that explores exactly what you think is there using your pooled variance concept. This preserves the ethical integrity of your initial study and sets up another study to appropriately look at what may or may not be there using pooled centers.

If you go fishing, you may not like what you catch.

-------------------------------------------
Daniel Butorovich
Research Analyst
Cochise College
-------------------------------------------







Original Message:
Sent: 05-02-2013 11:55
From: Robert Riffenburgh
Subject: Justifying pooling of centers in multicenter study

(This is not a reply to Wayne Fischer; I just entered the string here.)
There has been a lot of talk over whether pooling Centers should be permitted or prohibited. It sounds a little like a court of law. What statistics should be about is learning from data. If I saw two Centers quite different from the others, I would ask what variables make these Centers different and if that tells me a way I can improve the operations of my Centers.
--Bob

-------------------------------------------
Robert Riffenburgh
Naval Medical Center
-------------------------------------------

Hi Dennis:

In studies with a large number of site, and often smaller number of subjects per site, this often makes sense - even to FDA.  In my experience they are the ones who will push for pooling; my preference is as per your suggestion - random.  Gary Koch equated this issue to survey sampling many years ago - subjects within many centres are like a survey sample within some unit like county.
-------------------------------------------
Janet McDougall
President
McDougall Scientific Ltd
-------------------------------------------







Original Message:
Sent: 05-02-2013 12:58
From: Dennis Sweitzer
Subject: Justifying pooling of centers in multicenter study

The natural solution (to me) is to avoid pooling and model centers as random effects.
Centers naturally vary (if for no other reason than luck of the draw) and as a random effect one gets a sense of the variability between centers.  (If some centers seem to be outliers, a random effects model gives a sense of how far out they are)
-------------------------------------------
Dennis Sweitzer
Principal Biostatistician
Medidata Solutions
-------------------------------------------







Original Message:
Sent: 05-02-2013 12:40
From: Daniel Butorovich
Subject: Justifying pooling of centers in multicenter study

I think what some respondents to this post are hinting at is that you don't want to be accused of starting a statistical "fishing expedition" with your data. Trying to get a signficant interaction result when your designed experiment shows that there isn't one as your experiment stands. The right way to do this is to finish this experiment and show non-significance but note the potential for a significant result as "of interest for future research". Then construct another experiment that explores exactly what you think is there using your pooled variance concept. This preserves the ethical integrity of your initial study and sets up another study to appropriately look at what may or may not be there using pooled centers.

If you go fishing, you may not like what you catch.

-------------------------------------------
Daniel Butorovich
Research Analyst
Cochise College
-------------------------------------------







Original Message:
Sent: 05-02-2013 11:55
From: Robert Riffenburgh
Subject: Justifying pooling of centers in multicenter study

(This is not a reply to Wayne Fischer; I just entered the string here.)
There has been a lot of talk over whether pooling Centers should be permitted or prohibited. It sounds a little like a court of law. What statistics should be about is learning from data. If I saw two Centers quite different from the others, I would ask what variables make these Centers different and if that tells me a way I can improve the operations of my Centers.
--Bob

-------------------------------------------
Robert Riffenburgh
Naval Medical Center
-------------------------------------------

Thank you for your responses to my original post.  I found the responses helpful and insightful.  Thank you all for your quick replies, they were very much appreciated.  In the end, I did not pool the centers. Other exploratory work was done to see if there were something unique about the two sites, and nothing came up.  So Scott Berry's statement comes to mind..that this is probably good 'ol variation at work.

Thanks again!

-------------------------------------------
Shelley-Ann Walters
3M
-------------------------------------------







Original Message:
Sent: 05-02-2013 12:58
From: Dennis Sweitzer
Subject: Justifying pooling of centers in multicenter study

The natural solution (to me) is to avoid pooling and model centers as random effects.
Centers naturally vary (if for no other reason than luck of the draw) and as a random effect one gets a sense of the variability between centers.  (If some centers seem to be outliers, a random effects model gives a sense of how far out they are)
-------------------------------------------
Dennis Sweitzer
Principal Biostatistician
Medidata Solutions
-------------------------------------------







Original Message:
Sent: 05-02-2013 12:40
From: Daniel Butorovich
Subject: Justifying pooling of centers in multicenter study

I think what some respondents to this post are hinting at is that you don't want to be accused of starting a statistical "fishing expedition" with your data. Trying to get a signficant interaction result when your designed experiment shows that there isn't one as your experiment stands. The right way to do this is to finish this experiment and show non-significance but note the potential for a significant result as "of interest for future research". Then construct another experiment that explores exactly what you think is there using your pooled variance concept. This preserves the ethical integrity of your initial study and sets up another study to appropriately look at what may or may not be there using pooled centers.

If you go fishing, you may not like what you catch.

-------------------------------------------
Daniel Butorovich
Research Analyst
Cochise College
-------------------------------------------







Original Message:
Sent: 05-02-2013 11:55
From: Robert Riffenburgh
Subject: Justifying pooling of centers in multicenter study

(This is not a reply to Wayne Fischer; I just entered the string here.)
There has been a lot of talk over whether pooling Centers should be permitted or prohibited. It sounds a little like a court of law. What statistics should be about is learning from data. If I saw two Centers quite different from the others, I would ask what variables make these Centers different and if that tells me a way I can improve the operations of my Centers.
--Bob

-------------------------------------------
Robert Riffenburgh
Naval Medical Center
-------------------------------------------

Good old variation is the usual cause.  Certainly I have seen many investigators get fooled by subgroup analysis, and sites are a way to form subgroups.  I saw one investigator who had a lot of sites (16-20 or so) get quite excited about the single site that showed significance.  Three statisticians telling him that it was chance alone was not tempering him enthusiasm for finding out what "that site was doing right."  Certainly pooling sites for the purpose of getting a mall p-value is fishing.

However, when considering sites, it does matter what you are doing and how you are using the information.  I came in at the tail end of one study where half the investigators had all their subjects being protocol deviations.  The analyzing statistician and the clinical monitor disagreed on which ones the violations and which ones were OK.  Each brought me the protocol to prove their point.  I read the passage carefully.  It turned out that they, like the investigators, were not reading the poorly-worded protocol the same way.  At one point the protocol said the next step was do (essentially) "A or  B and C".  Now is that A or ( B and C)?  Or is that (A or B) and C?   The logical options were AB, AC, then AC, BC.  It seemed that most docs liked B, so most had done either AB or BC.  Different investigators read the protocol differently, and now there were 2 different substudies. Looking at sites mattered.

Another example when beta-testing instruments in development.  Some sites had very high variability in their results.  Were they not using the machine properly (not changing reagents/electrodes properly, etc), or was the instrument under development not stable?  It can be worth figuring this out!

So sometimes looking at sites matters.  But mostly, site difference is variability doing its thing.  As someone said, variability is job security for statisticians!  

-------------------------------------------
Katherine Monti
Rho, Inc.
-------------------------------------------







Original Message:
Sent: 05-03-2013 12:58
From: Shelley-Ann Walters
Subject: Justifying pooling of centers in multicenter study

Thank you for your responses to my original post.  I found the responses helpful and insightful.  Thank you all for your quick replies, they were very much appreciated.  In the end, I did not pool the centers. Other exploratory work was done to see if there were something unique about the two sites, and nothing came up.  So Scott Berry's statement comes to mind..that this is probably good 'ol variation at work.

Thanks again!

-------------------------------------------
Shelley-Ann Walters
3M
-------------------------------------------







Original Message:
Sent: 05-02-2013 12:58
From: Dennis Sweitzer
Subject: Justifying pooling of centers in multicenter study

The natural solution (to me) is to avoid pooling and model centers as random effects.
Centers naturally vary (if for no other reason than luck of the draw) and as a random effect one gets a sense of the variability between centers.  (If some centers seem to be outliers, a random effects model gives a sense of how far out they are)
-------------------------------------------
Dennis Sweitzer
Principal Biostatistician
Medidata Solutions
-------------------------------------------







Original Message:
Sent: 05-02-2013 12:40
From: Daniel Butorovich
Subject: Justifying pooling of centers in multicenter study

I think what some respondents to this post are hinting at is that you don't want to be accused of starting a statistical "fishing expedition" with your data. Trying to get a signficant interaction result when your designed experiment shows that there isn't one as your experiment stands. The right way to do this is to finish this experiment and show non-significance but note the potential for a significant result as "of interest for future research". Then construct another experiment that explores exactly what you think is there using your pooled variance concept. This preserves the ethical integrity of your initial study and sets up another study to appropriately look at what may or may not be there using pooled centers.

If you go fishing, you may not like what you catch.

-------------------------------------------
Daniel Butorovich
Research Analyst
Cochise College
-------------------------------------------

"variability is job security for statisticians!"  A long time ago, I was a stat teaching assistant.  The professor teaching the class asked me what the most important statistic was, I said "the mean".  He sagely shook his head no and said "it's variability".  He was so right.  Our job is basically the control of and reporting of results in the face of variability.

-------------------------------------------
Allen Fleishman, PhD, PStat®
President
Allen Fleishman Biostatistics Inc.


Original Message:
Sent: 05-03-2013 13:58
From: Katherine Monti
Subject: Justifying pooling of centers in multicenter study

Good old variation is the usual cause.  Certainly I have seen many investigators get fooled by subgroup analysis, and sites are a way to form subgroups.  I saw one investigator who had a lot of sites (16-20 or so) get quite excited about the single site that showed significance.  Three statisticians telling him that it was chance alone was not tempering him enthusiasm for finding out what "that site was doing right."  Certainly pooling sites for the purpose of getting a mall p-value is fishing.

However, when considering sites, it does matter what you are doing and how you are using the information.  I came in at the tail end of one study where half the investigators had all their subjects being protocol deviations.  The analyzing statistician and the clinical monitor disagreed on which ones the violations and which ones were OK.  Each brought me the protocol to prove their point.  I read the passage carefully.  It turned out that they, like the investigators, were not reading the poorly-worded protocol the same way.  At one point the protocol said the next step was do (essentially) "A or  B and C".  Now is that A or ( B and C)?  Or is that (A or B) and C?   The logical options were AB, AC, then AC, BC.  It seemed that most docs liked B, so most had done either AB or BC.  Different investigators read the protocol differently, and now there were 2 different substudies. Looking at sites mattered.

Another example when beta-testing instruments in development.  Some sites had very high variability in their results.  Were they not using the machine properly (not changing reagents/electrodes properly, etc), or was the instrument under development not stable?  It can be worth figuring this out!

So sometimes looking at sites matters.  But mostly, site difference is variability doing its thing.  As someone said, variability is job security for statisticians!  

-------------------------------------------
Katherine Monti
Rho, Inc.
-------------------------------------------

Hi, Shelley-Ann, I think your question can be answered without pooling centers.  If you leave the nonsignificant center-by-treatment term in the model, you can then write a contrast statement that is designed specifically to test whether the two centers of interest have a significantly stronger treatment effect than the other six centers.  And since it's a DF=1 contrast, you could even estimate how much stronger the treatment effect is in the first two centers versus the other six centers. 

-------------------------------------------
Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
-------------------------------------------

There may be some difference of opinion on this, but mine is that there is rarely, if ever, a need to pool centers. I might consider it if the total sample size within a center is small, eg less than the block size, for several centers. However, the results are not interpretable and it is a statistical "game". I would never pool based on a significant center effect. A significant interaction indicates that results for different centers go in different directions (or at least in different magnitudes if in the same direction); yours isn't extreme enough to be significant. Your suggested pooling of the centers does not solve any problem; you do not have a problem. Presenting the results by center will provide the information that you wish to convey - that the effect isn't as consistent across centers as one might like.

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David Bristol
Statistical Consulting Services
-------------------------------------------

Shelley-Ann, Strikes me as a very bad idea.  Lumping by empirical results then testing and saying -- hey there is an interaction effect has huge multiplicity issues that are being more than ignored, they are being abused.  I suspect in "almost" every 8 center trial you can find two lumped together compared to 6 that result in significant interaction-effects... 

You could look in to some kind of cluster modeling ... but Seems hard to go anywhere here with individual site interaction effects when there is little evidence for that from the overall p-value.... after all you have no evidence that the variations you are visually seeing are anything but good ol' variation...  

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Scott Berry
Berry Consultants
-------------------------------------------

First off, since the interaction is not even close to significant (0.23), this is really a non-problem.  No one should ever expect centers to be identical.  Some will always have greater means and some lower means.  This could be due to many factors: instrumentation, investigator biases, patient population differences (including demographic and baseline medical condition differences).  While we tend to analyze the data using a fixed center model, we typically assume that investigators are random, with a population mean and standard deviation.  So seeing a difference among centers is to be expected.  As others have said, there is NO reason to a posteri pool centers based on their differences.  I agree with the consensus that you shouldn't pool. 

However, the important question is whether you see a 'qualitative' or 'quantitative' interaction.  In your case, the interaction was n.s. 

There is often reason to expect some centers might have a larger treatment effect than others.  For example, some centers have severely ill patients.  In that case, they could have quite large improvement, while the centers with less severe patients have more modest improvements.  Small N and differential adherence to the protocol (greater error variance in some centers) also impact on the magnitude of individual center's treatment effect.  By this logic, some center's treatment are small and positive and others are large and positive.  As the treatment effect among centers is a random effect with a mean and standard deviation, with some centers it might even be numerically negative. 

Gail and Simon (Biometrics 1985) wrote the key paper on analyzing for qualitative interactions.  The gist of the approach is one looks at the treatment effects per center and examines the ones going in the 'wrong' direction.  If the effects going in the wrong way is statistically significant, one would say the interaction was qualitative.  Since the interactions are divided between the positive and negative effects, it is usually more difficult to have a qualitative interaction.  For example, I just did a simulation for a dichotomous d.v. for a client and observed that overall, the incidence of any interaction (at the 0.05 alpha level) was 5%.  However, the incidence of a qualitative interaction was around 0.7% (with an upper 95% CI of 0.9%).  I used proc freq's table option of 'GailSimon'.

In sum, ignore center effects (one expects them), ignore a quantitative interaction (one expects them), but test and comment on the qualitative interaction.  Since your overall interaction was n.s. tell your client that you are fully justified in reporting on the treatment effect and ignore the center main effect.

-------------------------------------------
Allen Fleishman, PhD, PStat®
President
Allen Fleishman Biostatistics Inc.
-------------------------------------------

Some dialects of statistics use the term "ordinal" for interactions when differences (changes) are in the same direction but of different magnitude. In a visualization the line segments from, e.g., pre to post will differ by more than chance occurrence from being parallel.  The line segments will not cross in the area of the that the data covers.  The line segments are ordered in the data area. 

Those dialects use the term "disordinal" for interactions there the differences (changes) are in opposite directions.  The line segments will cross in the  data area.
 In describing interactions ordinal : quantitative :: disordinal : qualitative .

-------------------------------------------
Arthur Kendall
Social Research Consultants
-------------------------------------------







Original Message:
Sent: 05-01-2013 12:41
From: Allen Fleishman
Subject: Justifying pooling of centers in multicenter study

First off, since the interaction is not even close to significant (0.23), this is really a non-problem.  No one should ever expect centers to be identical.  Some will always have greater means and some lower means.  This could be due to many factors: instrumentation, investigator biases, patient population differences (including demographic and baseline medical condition differences).  While we tend to analyze the data using a fixed center model, we typically assume that investigators are random, with a population mean and standard deviation.  So seeing a difference among centers is to be expected.  As others have said, there is NO reason to a posteri pool centers based on their differences.  I agree with the consensus that you shouldn't pool. 

However, the important question is whether you see a 'qualitative' or 'quantitative' interaction.  In your case, the interaction was n.s. 

There is often reason to expect some centers might have a larger treatment effect than others.  For example, some centers have severely ill patients.  In that case, they could have quite large improvement, while the centers with less severe patients have more modest improvements.  Small N and differential adherence to the protocol (greater error variance in some centers) also impact on the magnitude of individual center's treatment effect.  By this logic, some center's treatment are small and positive and others are large and positive.  As the treatment effect among centers is a random effect with a mean and standard deviation, with some centers it might even be numerically negative. 

Gail and Simon (Biometrics 1985) wrote the key paper on analyzing for qualitative interactions.  The gist of the approach is one looks at the treatment effects per center and examines the ones going in the 'wrong' direction.  If the effects going in the wrong way is statistically significant, one would say the interaction was qualitative.  Since the interactions are divided between the positive and negative effects, it is usually more difficult to have a qualitative interaction.  For example, I just did a simulation for a dichotomous d.v. for a client and observed that overall, the incidence of any interaction (at the 0.05 alpha level) was 5%.  However, the incidence of a qualitative interaction was around 0.7% (with an upper 95% CI of 0.9%).  I used proc freq's table option of 'GailSimon'.

In sum, ignore center effects (one expects them), ignore a quantitative interaction (one expects them), but test and comment on the qualitative interaction.  Since your overall interaction was n.s. tell your client that you are fully justified in reporting on the treatment effect and ignore the center main effect.

-------------------------------------------
Allen Fleishman, PhD, PStat®
President
Allen Fleishman Biostatistics Inc.
-------------------------------------------

Wow - such diversity of terminology!  In the engineering / physical sciences we try to keep it simple:

positive interaction = quantitative = ordinal  /  negative interaction = qualitative = disordinal

-------------------------------------------
Wayne Fischer
Statistician
University of Texas Medical Branch
-------------------------------------------







Original Message:
Sent: 05-01-2013 13:55
From: Arthur Kendall
Subject: Justifying pooling of centers in multicenter study

Some dialects of statistics use the term "ordinal" for interactions when differences (changes) are in the same direction but of different magnitude. In a visualization the line segments from, e.g., pre to post will differ by more than chance occurrence from being parallel.  The line segments will not cross in the area of the that the data covers.  The line segments are ordered in the data area. 

Those dialects use the term "disordinal" for interactions there the differences (changes) are in opposite directions.  The line segments will cross in the  data area.
 In describing interactions ordinal : quantitative :: disordinal : qualitative .

-------------------------------------------
Arthur Kendall
Social Research Consultants
-------------------------------------------

I believe an angry letter should have been sent to Drs. Gail and Simon (twenty eight years ago).  How dare they publish with an inappropriate dialect!  Imagine slang in statistical literature. 

I, a humble and obedient statistician, only reported a very frequently used approach to analyzing the verity of 'disordinal' interactions.  Gail and Simon's paper provided an objective methodology to determine if 'disordinality' actually occurred in a study.

In any case, the key issue here is the, ahem, 'disordinality' of the center effect.  Since it was not statistically significant (p=0.23), there is little evidence that it was, ahem, disordinal.  There should be little motivation to explore the differential effects of treatment at the center level and even less interest in exploring pure center differences. 

Is the client motivated in exploring why some centers have greater differences than others?  All of my clients regard such exercises contraindicated, center main and interaction effects are to them a nuisance effect.  If told that the interaction was n.s. then they would go on to report the treatment differences, the purpose of the study.

"I come to bury the interaction, not to praise it." 

Allen Fleishman, PhD, PStat®
President
Allen Fleishman Biostatistics Inc.
-------------------------------------------







Original Message:
Sent: 05-01-2013 13:55
From: Arthur Kendall
Subject: Justifying pooling of centers in multicenter study

Some dialects of statistics use the term "ordinal" for interactions when differences (changes) are in the same direction but of different magnitude. In a visualization the line segments from, e.g., pre to post will differ by more than chance occurrence from being parallel.  The line segments will not cross in the area of the that the data covers.  The line segments are ordered in the data area. 

Those dialects use the term "disordinal" for interactions there the differences (changes) are in opposite directions.  The line segments will cross in the  data area.
 In describing interactions ordinal : quantitative :: disordinal : qualitative .

-------------------------------------------
Arthur Kendall
Social Research Consultants
-------------------------------------------