How many intervals before discrete becomes continuous

[Rick Bello]

Many statistical tests we use have an a priori assumption of normality.
This not only implies the shape for the distributions but also it's
continuous nature(ie all values of the variable under question are
possible). Despite the fact that many variables we measure are
theoretically continuous, the limitations of the instruments we use to
measure them quantize our measurements to some degree. Imagine a scale
that reads body weight in kg but not fractions thereof.  Yet we would not
think twice about comparing the mean weight of 2 groups measured with such
a scale using a t-test (provided the other assumptions are met).

Now imagine a scale that measures body weight but only to the nearest 10
kg so that weights of 0, 10, 20, etc are reported by the scale.  Clearly
the distribution of weights in this situation would be discontinuous but
the peaks at the 10kg multiples would trace out the envelope of the normal
distribution.  In any case, this would violate the normality assumption of
a t-test.  Would we still use the t-test to compare the mean weights of 2
groups measured with such a scale? What if the scale measured to the
nearest multiple of 25kg instead?

So my question is at what point is the assumption of normality violated?
Is there a general rule of thumb one can use to determine when to resort
to a non-parametric test?  For example, is there a minimum number of
measurable intervals within the expected range of measurements before it
can be considered "continuous"?


Rick Bello, MD, PhD
Albert Einstein College of Medicine

[peter link]

Hi Rick -

Pretty interesting question.  My comments might not directly answer your
question, but hopefully sheds some light on the issue at hand.

This to me is a 'precision of the instrument' type of issue rather than a
normality issue.  I would be much more concerned about how imprecise the
instrument is, if it is measuring bodyweight to the nearest 10kg, for
example.  How confident could you be in any statistical test (t-test, as in
your example) performed on data gathered using such an imprecise instrument?
To me at some point the data becomes pretty meaningless.  You will only
detect very large differences if you're using imprecise instruments.  And
even then how reliable that is, is very questionable.  You really need to
first address the issue of whether or not the instrument you are using is
precise enough for your needs, and then figure out if you can detect
meaningful differences using this instrument.

Just a quick comment about normality before I close.  Using your example of
bodyweight: this is clearly a continuous variable.  Just because your
instrument can not measure as precisely as other instruments doesn't really
make it discrete.  All instruments have a certain precision.  Better put,
all weights could be realized, you just might not be able to accurately
determine them.

Anyways, just a few things to consider.

Peter Link
VA San Diego Healthcare System

-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]]On Behalf Of
Rick Bello
Sent: Thursday, December 21, 2006 3:38 PM
To: [hidden email]
Subject: How many intervals before discrete becomes continuous


Many statistical tests we use have an a priori assumption of normality.
This not only implies the shape for the distributions but also it's
continuous nature(ie all values of the variable under question are
possible). Despite the fact that many variables we measure are
theoretically continuous, the limitations of the instruments we use to
measure them quantize our measurements to some degree. Imagine a scale
that reads body weight in kg but not fractions thereof.  Yet we would not
think twice about comparing the mean weight of 2 groups measured with such
a scale using a t-test (provided the other assumptions are met).

Now imagine a scale that measures body weight but only to the nearest 10
kg so that weights of 0, 10, 20, etc are reported by the scale.  Clearly
the distribution of weights in this situation would be discontinuous but
the peaks at the 10kg multiples would trace out the envelope of the normal
distribution.  In any case, this would violate the normality assumption of
a t-test.  Would we still use the t-test to compare the mean weights of 2
groups measured with such a scale? What if the scale measured to the
nearest multiple of 25kg instead?

So my question is at what point is the assumption of normality violated?
Is there a general rule of thumb one can use to determine when to resort
to a non-parametric test?  For example, is there a minimum number of
measurable intervals within the expected range of measurements before it
can be considered "continuous"?


Rick Bello, MD, PhD
Albert Einstein College of Medicine

[Arthur Kramer]

This is an interesting question.  I think consideration has to be given to the
scale and ramifications of the decision to be made based on the results of the
test.

Hopefully, the scale is at least a true interval measurement.  We know the
t-test and ANOVA are robust to departures from normality and unequal sample
sizes, but the synergy of the two creates problems.

To address Rick's question, though, too wide an interval may mask the point at
which a true difference exists in the resultant means, even though the means
will probably be expressed in terms of a fraction of the interval.  Confidence
intervals may be useful in analyzing the results.

Arthur Kramer

>===== Original Message From peter link <[hidden email]> =====
>Hi Rick -
>
>Pretty interesting question.  My comments might not directly answer your
>question, but hopefully sheds some light on the issue at hand.
>
>This to me is a 'precision of the instrument' type of issue rather than a
>normality issue.  I would be much more concerned about how imprecise the
>instrument is, if it is measuring bodyweight to the nearest 10kg, for
>example.  How confident could you be in any statistical test (t-test, as in
>your example) performed on data gathered using such an imprecise instrument?
>To me at some point the data becomes pretty meaningless.  You will only
>detect very large differences if you're using imprecise instruments.  And
>even then how reliable that is, is very questionable.  You really need to
>first address the issue of whether or not the instrument you are using is
>precise enough for your needs, and then figure out if you can detect
>meaningful differences using this instrument.
>
>Just a quick comment about normality before I close.  Using your example of
>bodyweight: this is clearly a continuous variable.  Just because your
>instrument can not measure as precisely as other instruments doesn't really
>make it discrete.  All instruments have a certain precision.  Better put,
>all weights could be realized, you just might not be able to accurately
>determine them.
>
>Anyways, just a few things to consider.
>
>Peter Link
>VA San Diego Healthcare System
>
>-----Original Message-----
>From: SPSSX(r) Discussion [mailto:[hidden email]]On Behalf Of
>Rick Bello
>Sent: Thursday, December 21, 2006 3:38 PM
>To: [hidden email]
>Subject: How many intervals before discrete becomes continuous
>
>
>Many statistical tests we use have an a priori assumption of normality.
>This not only implies the shape for the distributions but also it's
>continuous nature(ie all values of the variable under question are
>possible). Despite the fact that many variables we measure are
>theoretically continuous, the limitations of the instruments we use to
>measure them quantize our measurements to some degree. Imagine a scale
>that reads body weight in kg but not fractions thereof.  Yet we would not
>think twice about comparing the mean weight of 2 groups measured with such
>a scale using a t-test (provided the other assumptions are met).
>
>Now imagine a scale that measures body weight but only to the nearest 10
>kg so that weights of 0, 10, 20, etc are reported by the scale.  Clearly
>the distribution of weights in this situation would be discontinuous but
>the peaks at the 10kg multiples would trace out the envelope of the normal
>distribution.  In any case, this would violate the normality assumption of
>a t-test.  Would we still use the t-test to compare the mean weights of 2
>groups measured with such a scale? What if the scale measured to the
>nearest multiple of 25kg instead?
>
>So my question is at what point is the assumption of normality violated?
>Is there a general rule of thumb one can use to determine when to resort
>to a non-parametric test?  For example, is there a minimum number of
>measurable intervals within the expected range of measurements before it
>can be considered "continuous"?
>
>
>Rick Bello, MD, PhD
>Albert Einstein College of Medicine

[John White]

akramer wrote:
> Hopefully, the scale is at least a true interval measurement.  We know the
> t-test and ANOVA are robust to departures from normality and unequal sample
> sizes, but the synergy of the two creates problems.

This is more about the underlying distribution, rather than the scale of
measurement, correct?  Unless the scale were somehow adjusted to reflect
a non-normal distribution.

The original question seems to ask how thick or thin will we slice the
baloney? It is still baloney, either way.  So long as the slices are
equal in size, the math works.  The number of intervals in the scale
don't matter.  The assumption of equal intervals would seem to trump this.

Ask the question differently.  Is there any point, at which unequal
intervals, of any amount (say 30 categories) become a scale?  The point
of this is to ask whether a 10-point Likert scale is preferable to a
4-point scale.

I know this gets away from Rick's original question about distributions,
but it is more along something I'm curious about.  At least this
morning.  :)

JW

--
======================================================================
John B. White, Ph.D.                |               [hidden email]
Dept. of Public Health - WKU        |             Office: 270.745.5867
1 Big Red Way                       |                Fax: [phone]
Bowling Green, KY 42101-3576        |   http://www.wku.edu/~john.white
======================================================================